Sizing and Placement of Distributed Generation in ...
Transcript of Sizing and Placement of Distributed Generation in ...
Sizing and Placement of Distributed Generation in Electrical Distribution Systems using Conventional and Heuristic Optimization Methods
by
Mohamad AlHajri
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy
at
Dalhousie University Halifax Nova Scotia
June 2009
copy Copyright by Mohamad AlHajri 2009
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DEDICATION PAGE
To my beloved parents my brothers Falah and Abdullah my sisters my wife OmFahad
my daughter Najla and my sons Fahad Falah and Othman
TABLE OF CONTENTS LIST OF TABLES x
LIST OF FIGURES xiii
ABSTRACT xvi
LIST OF ABBREVIATIONS AND SYMBOLS USED xvii
ACKNOWLEDGEMENTS xxiv
Chapter 1 INTRODUCTION 1
11 Motivation 1
12 Distribution Generation - Historic Overview 2
13 Distribution Generation 2
14 Thesis Objectives and Contributions 5
15 Thesis Outline 7
Chapter 2 LITERATURE REVIEW 9
21 Introduction 9
22 Distribution Power Flow 9
23 DG Integration Problem 13
231 Solving the DG Integration Problem via Analytical and Deterministic Methods 14
232 Solving the DG Integration Problem via Metaheuristic Methods 17
24 Summary 20
Chapter 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION NETWORKS 21
31 Introduction 21
32 Flexibility and Simplicity of FFRPF in Numbering the RDS Buses and Sections 22
321 Bus Numbering Scheme for Balanced Three-phase RDS 22
322 Unbalanced Three-phase RDS Bus Numbering Scheme 24
33 The Building Block Matrix and its Role in FFRPF 26
v
331 Three-phase Radial Configuration Matrix (RCM) 26
3311 Assessment of the FFRPF Building Block RCM 28
332 Three-phase Section Bus Matrix (SBM) 29
333 Three-phase Bus Section Matrix (BSM) 31
34 FFRPF Approach and Solution Technique 31
341 Unbalanced Multi-phase Impedance Model Calculation 32
342 Load Representation 38
343 Three-phase FFRPF BackwardForward Sweep 40
3431 Three-phase Current Summation Backward Sweep 40
3432 Three-phase Bus Voltage Update Forward Sweep 42
3433 Convergence Criteria 43
3434 Steps of the FFRPF Algorithm 44
344 Modifying the RCM to Accommodate Changes in the RDS 47
35 FFRPF Solution Method for Meshed Three-phase DS 48
351 Meshed Distribution System Corresponding Matrices 50
352 Fundamental Loop Currents 54
353 Meshed Distribution System Section Currents 56
354 Meshed Distribution System BackwardForward Sweep 59
36 Test Results and Discussion 60
361 Three-phase Balanced RDS 60
3611 Case 1 31-Bus with Single Main Feeder RDS 61
3612 Case 2 90-bus RDS with Extreme Radial Topology 70
3613 Case 3 69-bus RDS with Four Main Feeders 71
3614 Case 4 15-bus RDS-Considering Charging Currents 73
362 Three-phase Balanced Meshed Distribution System 74
3621 Case 1 28-bus Weakly Meshed Distribution System 74
3622 Case 2 70-Bus Meshed Distribution System 78
vi
3623 Case 3 201-bus Looped Distribution System 79
363 Three-phase Unbalanced RDS 80
3631 Case 1 10-bus Three-phase Unbalanced RDS 81
3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS 85
3633 Case 3 26-bus Three-Phase Unbalanced RDS 86
37 Summary 87
Chapter 4 IMPROVED SEQUENTIAL QUADRATIC PROGRAMMING
APPROACH FOR OPTIMAL DG SIZING 89
41 Introduction 89
42 Problem Formulation Overview 89
43 DG Sizing Problem Architecture 90
431 Objective Function 90
432 Equality Constraints 92
433 Inequality Constraints 92
434 DG Modeling 93
44 The DG Sizing Problem A Nonlinear Constrained Optimization Problem 94
45 The Conventional SQP 96
451 Search Direction Determination by Solving the QP Subproblem 96
4511 Satisfying Karush-Khun-Tuker Conditions 98
4512 Newton-KKT Method 101
4513 Hessian Approximation 103
452 Step Size Determination via One-Dimensional Search Method 104
453 Conventional SQP Method Summary 105
46 Fast Sequential Quadratic Programming (FSQP) 108
47 Simulation Results and Discussion 113
471 Case 1 33-busRDS 113
4711 Loss Minimization by Locating Single DG 114
4712 Loss Minimization by Locating Multiple DGs 118
vii
472 Case 2 69-bus RDS 124
4721 Loss Minimization by Locating a Single DG 125
473 Loss Minimization by Locating Multiple DGs 129
474 Computational Time of FSQP vs SQP 134
475 Single DG versus Multiple DG Units Cost Consideration 136
48 Summary 136
Chapter 5 PSO BASED APPROACH FOR OPTIMAL PLANNING OF
MULTIPLE DGS IN DISTRIBUTION NETWORKS 138
51 Introduction 138
52 PSO - The Motivation 138
53 PSO - An Overview 139
531 PSO Applications in Electric Power Systems 141 532 PSO - Pros and Cons 143
54 PSO - Algorithm 144
541 The Velocity Update Formula in Detail 145
5411 The Velocity Update Formula - First Component 146
5412 The Velocity Update Formula - Second Component 148
5413 The Velocity Update Formula-Third Component 149
5414 Cognitive and Social Parameters 150
542 Particle Swarm Optimization-Pseudocode 152
55 PSO Approach for Optimal DG Planning 153
551 Proposed HPSO Constraints Handling Mechanism 155
5511 Inequality Constraints 155
5512 Equality Constraints 157
5513 DG bus Location Variables Treatment 157
56 Simulation Results and Discussion 160
561 Case 1 33-bus RDS 161
viii
5611 33-bus RDS Loss Minimization by Locating a Single DG 161
5612 33-bus RDS Loss Minimization by Locating Multiple
DGs 169
562 Case 2 69-Bus RDS 180
5621 69-bus RDS Loss Minimization by Locating a Single DG 180
5622 69-bus RDS Loss Minimization by Locating Multiple
DGs 187
563 Alternative bus Placements via HP SO 195
57 Summary 196
Chapter 6 CONCLUSION 198
61 Contributions and Conclusions 198
62 Future Work 201
REFERENCES 203
APPENDIX 220
IX
LIST OF TABLES
Table 31 cok rd and De Parameters for Different Operation Conditions 34
Table 32 FFRPF Iteration Results for the 31-Bus RDS 67
Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method 68
Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models 69
Table 35 Comparisons between 31-Bus RDS Exponential Model Results 70
Table 36 31-bus RDS FFRPF Results vs Other Methods 70
Table 37 90-bus RDS FFRPF Results vs Other Methods 71
Table 38 69-bus RDS FFRPF Results vs Other Methods 73
Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods 74
Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network 77 t
Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods 78
Table 312 70-bus Meshed DS FFRPF Results vs Other Methods 79
Table 313 201-bus Meshed DS FFRPF Results vs Other Methods 80
Table 314 10-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 85
Table 315 IEEE 13-bus 3(j) RDS FFRPF Results vs Ref [52] and Gauss Zbus
Methods 86
Table 316 26-bus 34gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods 87
Table 41 Single DG Optimal Profile at the 33-bus RDS 115
Table 42 Optimal DG Profiles at all 33 buses 116
Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power
Factor 119
Table 44 SQP Method Double-DG Cycled Combinations 121
Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor 123
Table 46 Loss Reduction Comparisons for all DG Cases 123
Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles 128
Table 48 Optimal Double DG Profiles in the 69-bus RDS 131
Table 49 Optimal Three DG Units Profiles in the 69-bus RDS 133
Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS 134
Table 411 33-bus RDS CPU Execution Time Comparison 135
Table 412 69-bus RDS CPU Execution Time Comparison 135
x
Table 51 HPSO Parameters for the Single DG Case 162
Table 52 33-bus RDS Single DG Fixedpf Case 20 HPSO Simulations 162
Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Case 163
Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedDG Case 163
Table 55 33-bus RDS Single DG Unspecified pf Case 20 HPSO Simulations 163
Table 56 Descriptive Statistics for HPSO Results for an Unspecified pf Case 164
Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase 164
Table 58 HPSO Parameters for Both Double DG Cases 170
Table 59 33-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 171
Table 510 Descriptive Statistics for HPSO Results for Fixed pf Double DG Case 171
Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedCase 172
Table 512 33-bus RDS Double DG Unspecified pf Case 20 HPSO Simulations 172
Table 513 Descriptive Statistics for HPSO Results for UnspecifiedDouble DG
Case 173
Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedCase 173
Table 515 HPSO Parameters for Both Three DG Cases 174
Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 174
Table 517 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 175
Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedCase 175
Table 519 33-bus RDS Three DGs Unspecified pf Case 20 HPSO Simulations 175
Table 520 Descriptive Statistics for HPSO Results for the Fixed Three DG Case 176
Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase 176
Table 522 HPSO Parameters for the Four DG Case 177
Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations 178
Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases 178
Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase 179
Table 526 33-bus RDS Four DG Unspecified pf Case 20 HPSO Simulations 179
Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG
Case 179
Table 528 HPSO vs FSQP 33-bus RDS-Four -Unspecified pf Case 180
Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases 181
Table 530 69-bus RDS Single DG Fixed pf Case 20 HPSO Simulations 182
xi
Table 531 Descriptive Statistics for HPSO Results for the Fixed Single DG Case 182
Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedpDG Case 182
Table 533 69-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations 183
Table 534 Descriptive Statistics for Unspecified pSingle DG Case 183
Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-UnspecifiedpDG
Case 184
Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases 188
Table 537 69-bus RDS Double DG Fixed pf Case 20 HPSO Simulations 189
Table 538 Descriptive Statistics for HPSO Results for Fixed pDouble DG Case 189
Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case 190
Table 540 69-bus RDS Double DG UnspecifiedpfCase 20 HPSO Simulations 190
Table 541 Descriptive Statistics for HPSO Results for Unspecified ^Double DG
Case 191
Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedpDG Case 191
Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases 192
Table 544 69-bus RDS Three DG Fixed pf Case 20 HPSO Simulations 192
Table 545 Descriptive Statistics for HPSO Results for Fixed pf Three DG Case 193
Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedCase 193
Table 547 69-bus RDS Three DG Unspecified pf Case 20 HPSO Simulations 194
Table 548 Descriptive Statistics for HPSO Results for Unspecified pf Three DG
Case 194
Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-UnspecifiedpCase 195
Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed pf Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles 196
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LIST OF FIGURES
Figure 31 10-bus RDS 23
Figure 32 Different ways of numbering the system in Fig 31 24
Figure 33 The ease of numbering a modified and augmented RDS 24
Figure 34 Three-phase unbalanced 6-bus RDS representation 25
Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections 26
Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs 28
Figure 37 SBM for three-phase unbalanced 6-bus RDS 30
Figure 38 Three-phase section model 32
Figure 39 The final three-phase section model after Kron s reduction 34
Figure 310 Nominal ^-representation for three-phase RDS section 36
Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling 40
Figure 312 The FFRPF solution method flow chart 46
Figure 313 10-bus meshed distribution network 50
Figure 314 Fundamental cut-sets for a meshed 10-bus DS 57
Figure 315 31-bus RDS 62
Figure 316 The RCM of the 31-bus RDS 63
Figure 317 The RCM-1 of the 31-bus RDS 64
Figure 318 The SBM of the 31-bus RDS 65
Figure 319 The BSM of the 31-bus RDS 66
Figure 3 20 90-Bus RDS 71
Figure 321 69-bus multi-feeder RDS 72
Figure 322 Komamoto 15-bus RDS 73
Figure 323 28-bus weakly meshed distribution network 75
Figure 324 mRCM for 28-bus weakly meshed distribution network 75
Figure 325 mSBM for 28-bus weakly meshed distribution network 76
Figure 326 C for 28-bus weakly meshed distribution network 76
Figure 327 70-bus meshed distribution system 78
Figure 328 201-bus hybrid augmented test distribution system 80
Figure 329 10-bus three-phase unbalanced RDS 81
Figure 330 The 10-bus three-phase unbalanced RDS RCM 82
xni
Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1 83
Figure 332 The 10-bus three-phase unbalanced RDS SBM 84
Figure 333 The 10-bus three-phase unbalanced RDS BSM 85
Figure 334 IEEE 13-bus 3^ unbalanced RDS 86
Figure 41 The Conventional SQP Algorithm 107
Figure 42 The FSQP Algorithm 112
Figure 43 Case 1 33-busRDS 114
Figure 44 Optimal real power losses for placement of optimal DG size at all the 32
buses using APC method 117
Figure 45 Optimal real power losses vs different DG power factors at bus 30 117
Figure 46 Bus voltages improvement before and after installing a single DG at
bus 30 118
Figure 47 Voltage profiles comparisons of 33-bus RDS cases 120
Figure 48 Voltage improvement of the 33-bus RDS due to three DG installation
compared to pre-DG single and double-DG cases 122
Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases 124
Figure 410 Case 2 69-bus RDS test case 125
Figure 411 Optimal power losses obtained using APC procedure 126
Figure 412 Real power losses vs DG power factor 69-bus RDS 128
Figure 413 Bus voltage improvements via single DG installation in the 69-bus
RDS 129
Figure 414 Variation in power losses as a function of the DG output at bus 61 129
Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG
and double DGs cases 131
Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases 133
Figure 51 Number of publications in IEEEIET and ScienceDirect Databases
since the year 2000 140
Figure 52 Interaction between particles to share the gbest information 150
Figure 53 Illustration of velocity and position updates mechanism for a single particle during iteration k 151
Figure 54 PSO particle i updates its velocity and position vectors during two consecutive iterations A and k+ 152
Figure 55 The proposed HPSO solution methodology 159
xiv
Figure 56 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO proposed number of iterations = 30 164
Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle
DG case HPSO extended number of iterations = 50 165
Figure 58 Swarm particles on the first HPSO iteration 165
Figure 59 Swarm particles on the fifth HPSO iteration 166
Figure 510 Swarm particles on the tenth HPSO iteration 166
Figure 511 Swarm particles on 15th HPSO iteration 167
Figure 512 Swarm particles on the 20 HPSO iteration 167
Figure 513 Swarm Particles on the 25th HPSO iteration 168
Figure 514 Swarm Particles on the last HPSO iteration 168
Figure 515 A close-up for the particles on the 30th HPSO iteration 169
Figure 516 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 15 184
Figure 517 Convergence characteristics of HPSO in the 69-bus fixed single DG
case HPSO proposed number of iterations = 50 185
Figure 518 Swarm particles distribution at the first HPSO iteration 185
Figure 519 Swarm particles distribution at the 5 HPSO iteration 186
Figure 520 Swarm particles distribution at the 10 HPSO iteration 186
Figure 521 Swarm particles distribution at the 15l HPSO iteration 187
Figure 522 Close up of the HSPO particles at iteration 15 187
xv
ABSTRACT
Distribution Generation (DG) has gained increasing popularity as a viable element of electric power systems DG as small scale generation sources located at or near load center is usually deployed within the Distribution System (DS) Deployment of DG has many positive impacts such as reducing transmission and distribution network congestion deferring costly upgrades and improving the overall system performance by reducing power losses and enhancing voltage profiles To achieve the most from DG installation the DG has to be optimally placed and sized In this thesis the DG integration problem for single and multiple installations is handled via deterministic and heuristic methods where the results of the former technique are used to validate and to be compared with the latters outcomes
The unique structure of the radial distribution system is exploited in developing a Fast and Flexible Radial Power Flow (FFRPF) method that accommodates the DS distinct features Only one building block bus-bus oriented data matrix is needed to perform the proposed FFRPF method Two direct descendent matrices are utilized in conducting the backwardforward sweep employed in the FFRPF technique The proposed method was tested using several DSs against other conventional and distribution power flow methods Furthermore the FFRPF method is incorporated within Sequential Quadratic Programming (SQP) method and Particle Swarm Optimization (PSO) metaheuristic method to satisfy the power flow equality constraints
In the deterministic solution method the sizing of the DG is formulated as a constrained nonlinear optimization problem with the distribution active power losses as the objective function to be minimized subject to nonlinear equality and inequality constraints Such a problem is handled by the developed Fast Sequential Quadratic Programming method (FSQP) The proposed deterministic method is an improved version of the conventional SQP that utilizes the FFRPF method in handling the power flow equality constraints Such hybridization resultes in a more robust solution method and drastically reduces the computational time In a subsequent step the placement portion of the DG integration problem is dealt with by using an All Possible Combinations (APC) search method Afterward the FSQP methods outcomes were compared to those of the developed metaheuristic optimization method
The difficult nature of the overall problem poses a serious challenge to most derivative based optimization methods due to the discrete nature associated with the bus location Moreover a major drawback of deterministic methods is that they are highly-dependent on the initial solution point As such a new application of the PSO metaheuristic method in the DG optimal planning area is presented in this thesis The PSO is improved in order to handle both real and integer variables of the DG mixed-integer nonlinear constrained optimization problem The algorithm is utilized to simultaneously search for both the optimal IDG size and bus location The proposed approach hybridizes PSO with the developed FFRPF algorithm to satisfy the equality constraints The inequality constraints handling mechanism is dealt with in the proposed Hybrid PSO (HPSO) by combining the rejecting infeasible solutions method with the preserving feasible solutions method Results signify the potential of the developed algorithms with regard to the addressed problems commonly encountered in DS
xvi
LIST OF ABBREVIATIONS AND SYMBOLS USED
ACO
BFGS
BSM
CHP
CIGRE
CN
DER
DG
DG
DGs
DS
EG
EP
EPAct
EPRI
FD
FFRPF
FSQP
GA
GRG
GS
GWEC
HPSO
IP
KCL
KKT
KVL
LP
Ant Colony Optimization
Quasi-Newton method for Approximating and Updating the Hessian Matrix
Bus Section Matrix
Combined-Heat and Power
The International Council on Large Electric Systems
Condition Number
Distribution Energy Resources
Dispersed Generation
Decentralized Generation
Distribution Generation sources
Distribution System
Embedded Generation
Evolutionary Programming
English Policy Act of 1992
Electric Power Research Institute
Fast Decoupled
Fast and Flexible Radial Power Flow
Fast Sequential Quadratic Programming
Genetic Algorithm
Generalized Reduced Gradient
Gauss-Seidel
Global Wind Energy Council
Hybrid PSO
Interior Point method
Kirchhoff s Current Law
Karush-Khun-Tuker conditions
Kirchhoff s Voltage Law
Linear Programming
xvii
wBSM
mNS
wRCM
mRCM
mSBM
mSBMp
NB
NB
HDG
riL
NR
NS
NS
ftwDG
Pf PSO
PUHCA
PURPA
QP
RCM
RDS
RIT
RPF
SA
SBM
SE Mean
SQP
StDev
TS
UnSpec pf
Meshed BSM
Number of segments in meshed DS
Meshed RCM
Modified mRCM
Meshed SBM
Submatrix of wSBM that correspond to the RDS tree sections
Number of Buses
Number of DS Buses
Total number of DGs
Number of Links or number of the fundamental loops
Newton-Raphson
Number of Sections
Number of Sections in RDS AND in meshed DS tree
Total number of the unspecified pf DGs
power factor
Particle Swarm Optimization
Public Utilities Holding Company Act of 1935
Public Utilities Regulatory Policy Act of 1978
Quadratic Programming
Radial Configuration Matrix
Radial Distribution System
The Reduction in CPU execution Time
Radial Power Flow
Simulated Annealing
Section Bus Matrix
Standard Error of the Mean
Sequential Quadratic Programming
Standard Deviation
Tabu search algorithm
Unspecified power factor DG
xviii
U S P B Unique Set of Phase Buses
USPS Unique Set of Phase Sections
xf Unique set of phase buses
iff Unique set of phase sections
Zsec Section primitive impedance matrix
Z^ (3 X 3) section symmetrical impedance matrix
R D S section length
zu Per unit length self-impedance of conductor i
h Per unit length mutual- impedance be tween conductors a n d
rt Resis tance of conductor i
rd Ear th return conductor resistance
k Inductance multiplying constant
De Dis tance between overhead and its earth return counterpart
GMRj Geometr ic mean radius of conductor i
Dy Dis tance between conductors a n d
Vgbc Three-phase sending end voltages
Vg deg Three-phase receiving end voltages
Ias c Three-phase sending end section currents
lfc Three-phase receiving end section currents
Fscc Three-phase shunt admittance of section k
[]3x3 (3 X 3) identity matrix
[^Lx3 (3x3) zero matrix
^Klc Vol tage drop across three-phase section k
ysect Section k three-phase currents
V0 Nomina l bus voltage
V Operat ing bus voltage
xix
P0 Real power consumed at nominal voltage
Q0 Reactive power consumed at nominal voltage
S Bus load apparent power at single-phase bus sect
YsKus Total three-phase shunt admittance at bus i
Ic Three-phase shunt currents at bus i
IlucSi Bus three-phase currents
jabc Three-phase load current
IltLP Current through single-phase section p and phase ltjgt
its Current at bus and phase ^
Vss Substation voltage magnitude
Vls Substation complex phase voltage
VLt Voltage drop across section k in phase (j)
A and symbol
IMI oo-norm vector II I loo
91 (bull) Real part of complex value
3 (bull) Imaginary part of complex value
C Fundamental loop matrix which is a submatrix of mSBM
Zioop (laquoLx nL) loop-impedance matrix
Csec Upper submatrix of the C matrix with ((NB-1) x (nL) dimension
Zoop Loop-impedance matrix
setrade (NSxNS) meshed DS section-impedance diagonal matrix
ZtradeS (NSxNS) RDS or tree section-impedance diagonal matrix
IL (NB-1 x 1) RDS bus load currents vector
fnlsec (mNS x 1) segments currents column vector of meshed DS network vector
mILL (mNSx 1) meshed DS bus loads and links currents vector
Itrade (NB-1) tree section currents column
xx
( n L x 1) fundamental loop current vector which is also the meshed DS link loop
currents column vector
B ( N B - 1 xmNS) fundamental cut-sets matrix
^ s7 e c f a ( N B - 1 x nL) co-tree cut-set matrix
^ymesh Voltage drops across the tree sections of the meshed DS vector
ymesh j k g messed DS bus voltage profiles vector
PRPL Real power losses
Pj Generated power delivered to DS bus i
PjL Load power supplied by DS bus i
Yjj Magnitude of the if1 element admittance bus matrix
rv Phase angle of Yy = YyZyy
Vi Magnitude of DS bus complex voltage
8 Phase angle of V = ViZSl
bull Transpose of vector or matrix
bull Complex conjugate of vector or matrix
V (1 xNB) DS bus Thevenin voltages
Y (NB xNB) DS admittance matrix
A^ Real power mismatch at bus i
AQt Reactive power mismatch at bus i
|L| Infinity norm where llxll = max (|x|) II llco J II llraquo =l2iVBVI ]gt
bull+ Max imum permissible value
bull Minimum permissible value
bull0 Nominal value
PDG D G operating power factor
S^G D G generated apparent power
SsS Main DS substation apparent power
1 Scalar related to the allowable D G size
xxi
Sy Apparent power flow transmitted from bus to bus j
Stradex Apparent power maximum rating for distribution section if
(x) The objective function
z(x) Equality constraints
g(x) Inequality constraints
(bull) Independent unknown variables lower bounds
(bull) Independent unknown variables upper bounds
x Independent unknown variables vector
RPL ( X ) Distribution system real power losses objective function
d Search direction vector
a Positive step size scalar
WRPL (x ) Gradient of the objective function at point xk)
pound Lagrange function
H^ (nxri) Hessian symmetric matrix at point xw
h^ First-order Taylors expansion of the equality constraints at point xw
Vh(x^) (nxm) Jacobian matrix of the equality constraints at point xw
g ^ First-order Taylors expansion of the inequality constraints at point xw
Vg(x^) (nxp) Jacobian matrix of the inequality constraints at point xw
Xi Individual equality Lagrange multiplier scalar
Pi Individual inequality Lagrange multiplier scalar
k w-dimensional equality Lagrange multiplier vector
P (-dimensional inequality Lagrange multiplier vector
s A predefined small tolerance number
A Active set
m Number of all equality constraints
p Number of all inequality constraints
a Number of the active set equations
xxii
v 2 j6k)
XX
nTgtG
nuDG
y
v Y FFRPFbl
deg FFRPF bl
llAP II II lloo
Vi
Xi
Cj C2
rXgtr2
w
pbestj
gbesti
nk
X
APT Losses
pHPSO Losses
pFSQP Losses
Hessian of the Lagrange function
Total number of DGs
Total number of the unspecified DGs
The change in the Lagrange functions between two successive iterations
Voltage magnitude of bus i obtained by the FFRPF technique
Voltage phase angle of bus obtained by the FFRPF technique
Voltage deviation infinity norm ie II AV = max ( I F - K I ) deg llcD
=UAlaquoVI deg ngt
Particle i velocity
Particle i position vector
Individual and social acceleration positive constants
Random values in the range [0 l] sampled from a uniform distribution
Weight inertia
Personal best position associated with particle own experience
Global best position associated with the whole neighborhood experience
Maximum number of iterations
Constriction factor
The deviation of losses calculated by HPSO method from that determined
by FSQP method
Mean value of HPSO simulation results of real power losses
FSQP deterministic method result of real power losses
xxiii
ACKNOWLEDGEMENTS
All Praise and Thanks be to reg (Allah) Almighty whose countless bounties enabled me to
accomplish this thesis successfully I would like to express my deepest gratitude to my
parents who taught me the value of education and hard work A special note of gratitude
to my brothers Falah and Abdullah deer sisters my wife my daughter Najla and my
sons Fahad Falah and Othamn They endured the long road along with me and
provided me with constant support motivation and encouragement during the course of
my study
I would like to express my sincere gratitude to my advisor Dr M E El-Hawary for
his professional guidance valuable advice continual support and encouragement I also
appreciate the constructive comments of my PhD External Examiner Dr M A Rahman
I am also grateful to my advisory committee members Dr T Little and Dr W Phillips
for spending their valuable time in reading evaluating and discussing my thesis
I would like to acknowledge the academic discussions and the constant
encouragement I received from my dear friend Dr Mohammed AlRashidi- Thank you
Abo Rsheed I wish also to thank a special friend of mine Dr AbdulRahman Al-
Othman for his friendship and for believing in me
I would like to manifest my gratitude to the Public Authority for Applied Education
and Training in Kuwait who sponsored me through my PhD at Dalhousie University
From the Embassy of Kuwait Cultural Attache Office special thanks are due to Shoghig
Sahakyan for her efforts and help to make this work possible
xxiv
CHAPTER 1 INTRODUCTION
11 MOTIVATION
Electric power system networks are composed typically of four major subsystems
generation transmission distribution and utilizations Distribution networks link the
generated power to the end user Transmission and distribution networks share similar
functionality both transfer electric energy at different levels from one point to another
however their network topologies and characteristics are quite different Distribution
networks are well-known for their low XR ratio and significant voltage drop that could
cause substantial power losses along the feeders It is estimated that as much as 13 of
the total power generation is lost in the distribution networks [1] Of the total electric
power system real power losses approximately 70 are associated with the distribution
level [23] In an effort towards manifesting the seriousness of such losses Azim et al
reported that 23 of the total generated power in the Republic of India is lost in the form
of losses in transmission and distribution [4]
Distribution systems usually encompass distribution feeders configured radially and
exclusively fed by a utility substation Incorporating Distribution Generation sources
(DGs) within the distribution level have an overall positive impact towards reducing the
losses as well as improving the network voltage profiles Due to advances in small
generation technologies electric utilities have begun to change their electric
infrastructure and have started adapting on-site multiple small and dispersed DGs In
order to maximize the benefits obtained by integrating DGs within the distribution
system careful attention has to be paid to their placement as well as to the appropriate
amount of power that is injected by the utilized DGs In other words to achieve the best
results of DG deployments the DGs are to be both optimally placed and sized in the
corresponding distribution network
The motivation of this thesis research is to investigate placing and sizing single and
multiple DGs in Radial Distribution Systems (RDSs) The problem investigated involves
two stages finding the optimal DG placements in the distribution network and the
optimal size or rating of such DGs The optimal DG placement and sizing are dealt with
by utilizing deterministic and heuristic optimization methods
12 DISTRIBUTION GENERATION - HISTORIC OVERVIEW
During the first third of the twentieth century there were no restrictions on how many
utility companies could be owned by financial corporations known as utility holding
companies By 1929 80 of US electricity was controlled by 16 holding companies
and three of those corporations controlled 36 of the nations electricity market [5]
During the Great Depression most of these utility holding companies went bankrupt As
a result the US Congress Public Utilities Holding Company Act (PUHCA) of 1935
regulated the gas and electric industries and restricted holding companies to the
ownership of a single integrated utility PUHCA indirectly discouraged wholesale
wheeling of power between different states provinces or even countries The Public
Utilities Regulatory Policy Act (PURPA) of 1978 allowed grid interconnection and
required electric utilities to buy electricity from non-utility-owned entities called
Qualifying Facilities (QF) at each utilitys avoided cost The term QF refers to non-
utility-owned (independent) power generators The term at each utilitys avoided cost
is interpreted to mean that the utility shall buy the generated electricity at a price
equivalent to what it would cost the utility itself if had generated the same amount of
power in its own facility or if it had purchased the power from an open electricity market
ie what the utility saves by not generating the same amount of power This act heralded
the dawn of the DG industry era which paved the way to generate electricity arguably at
a lower cost compared to that of traditional utility companies and consequently have it
delivered to the end user at lower rates The English Policy Act of 1992 (EPAct)
intensified competition in the wholesale electricity market by opening the transmission
system for access by utilities and non-utilities electricity producers [67] entity A could
sell its power to entity B through entity Cs transmission infrastructure
13 DISTRIBUTION GENERATION
DG involves small-scale generation sources scattered within the distribution system level
atnear the load center ie close to where the most energy is consumed [8] The DG
2
generate electricity locally and in a cogeneration case heat can also be generated and
may be utilized in applications such as industrial process heating or space heating DG
generally has better energy efficiency than large-scale power plants The traditional
power stations usually have an efficiency of around 35 whereas the efficiency of DG
such as a Combined Heat and Power (CHP) gas turbine would be in the vicinity of 45-
65 [5]
It seems that there is no universal agreement on the definition of DG size range The
Electric Power Research Institute (EPRI) for example defined the DG size to be up to 5
MW in 1998 [9] in 2001 EPRI redefined the DG capacity to be less than 10 MW [10]
and by 2003 they identified the DG to have a power output ranging from 1 kW to 20 MW
[11] The IEEE published its DG-standards IEEE Std 1547-2003 and IEEE Std 15473-
2007 and emphasized that they are applicable to DGs that have total capacity below 10
MVA [1213] In its 2006 report about the impact of DG Natural Resources Canada
estimated that the DG size starts from few 10s of kW to perhaps 5 MW [14] In 2000
the International Council on Large Electric Systems (CIGRE) referred to the DG as non-
centrally dispatched usually attached to distribution level and smaller than 50-100 MW
[1516]
Many terms referring to DG technology are used in the literature such as Dispersed
Generation (DG) Decentralized Generation (DG) Embedded Generation (EG)
Distribution Energy Resources (DER) and on-site Generation [17] In particular the
term dispersed generation customarily refers to stationary small-scale DG with power
outputs ranging from 1 kW to 500 kW [7]
Late developments and innovations in the DG technology industry liberalization of
the electricity market transmission line congestion and increasing interest in global
warming and environmental issues expedited publicizing their deployment and adoption
world-wide Recent studies suggest that DG will play a vital role in the electric power
system An EPRI study predicts that by the year 2010 25 of the newly installed
generation systems will be DG [18] and a similar study by the Natural Gas Foundation
projects that the share of DG in new generation will be 30 [15] By 2003 around 40
of Denmarks power demand was served by DG while Spain the Netherlands Portugal
and Germany integrated nearly 20) of DG into their distribution networks [19] Of the
3
643 GW generated by the European Union in 2005 approximately 122 GW (19) was
generated by hydro 96 GW (15) of this generated capacity was cogeneration (CHP)
and 53 GW (8) generated by other renewable energy systems Half of the CHP
generated capacity was owned by utility companies and the other half was generated by
independent producers [20]
Globally in 2005 the total installed wind power capacity was 591 GW and the
Global Wind Energy Council (GWEC) expected the wind capacity to reach 1348 GW by
the year 2010 [21] Worldwide wind energy capacity of 19696 MW was added in the
year 2007 [22] and approximately 1400 MW of wind energy capacity was added in the
US during the second quarter of 2008 [23] GWEC also predicted in its Global Wind
Energy Outlook 2008 report that by the year 2020 15 billion tons of CO2 will be saved
every year and by the middle of the 21st century 30 of the worlds electricity will be
supplied by wind energy [24] compared to a total of 13 of the global electricity being
generated by wind at the end of 2007 [22]
DG technologies include a variety of energy sources ie powered by renewable or
by fossil fuel-based prime movers Renewable technologies used in DG include wind
turbines photovoltaic cells small hydro power turbines and solar thermal technologies
while DG based on conventional technologies may involve gas turbines CHP gas
turbines diesel engines fuel cells and micro-turbine technologies Some DGs are
installed by the utility company on the supply side of the consumers meter while some
are installed by the customers themselves on their side of a bi-directional meter thus
enabling them to benefit from the net-metering program offered by utility companies
[25]
Optimal deployment of DG technology would have an overall positive impact
although some negative traits would remain The noise and shadow flicker caused by
large wind blades and the noise caused by the wind turbine gearbox or gas turbines
especially when placed close to residential or populated areas are examples of negative
impacts of widespread use of DG Another drawback from an environmentalist point of
view is that wind DG could disturb bird immigration patterns and cause death to both
birds and bats [26] Renewable-source DGs also could be an indirect source of pollution
by causing the fossil-fuel power plants to shut down and start up more frequently as they
4
attempt to accommodate variable DG power output [27] Some plants have an emission
rate which is inversely proportional to its delivered power Voltage rise as a result of bishy
directional power flow caused by the interconnection of the DG in RDS is another
example of a negative impact caused by DG [28]
The integration of DG into electric power networks has many benefits Some
examples of such benefits could be summarized as follows
bull Improve both the reliability and efficiency of the power supply
bull Release the available capacity of the distribution substation as well as reducing
thermal stresses caused by loaded substations transformers and feeders
bull Improve the system voltage profiles as well as the load factor
bull Decrease the overall system losses
bull Generally DG development and construction have shorter time intervals
bull Delay imminent upgrading of the present system or the need to build newer
infrastructure and subsequently avoid related problems such as right-of-way
concerns
bull Decrease transmission and distribution related costs
bull In general DG tends to be more environmentally friendly when compared to
traditional coal oil or gas fired power plants
The extent of the benefits depends on how the DG is placed and sized in the system In
addition to supplying the system with the power needed to meet certain demands as an
installation incentive the real power losses could be minimal if the DG is optimally sited
and sized
14 THESIS OBJECTIVES AND CONTRIBUTIONS
Optimal integration of single and multiple DG units in the distribution network with
specified and unspecified power factors is thoroughly investigated from a planning
perspective in this thesis The DG problem is handled via deterministic and heuristic
optimization methods where the results of the former method are used to validate and to
be compared with those of the latter
The unique radial distribution structure is exploited in developing a Fast and Flexible
Radial Power Flow (FFRPF) method to deal with a wide class of distribution systems
5
eg radial meshed small large balanced and unbalanced three-phase networks The
proposed FFRPF algorithm starts by developing a Radial Configuration Matrix (RCM)
for radial topology DS or meshed RCM (mRCM) for meshed DS Both matrices consist
of elements with values 1 0 and -1 The RCM (or mRCM) corresponding building
algorithm is simple fast and practical as illustrated in Chapter 3 The RCM is inverted
only once to produce the Section Bus Matrix (SBM) which is then transposed to obtain
the Bus Section Matrix (BSM) For the meshed topology the corresponding resultant
matrices for the mRCM are the meshed Bus Section Matrix (mSBM) and the meshed Bus
Section Matrix (mBSM) The FFRPF technique relies on the backwardforward sweep
that only utilizes the basic electric laws such as Ohms law and both Kirchhoff s voltage
and current laws The backward current sweep is performed via SBM (or mSBM) and
the forward voltage update sweep is executed via the BSM (or mBSM) By utilizing the
two obtained matrices all the bus complex voltages can be obtained and consequently
left to be compared with the immediate previous obtained bus voltages The proposed
approach quickened the iterative process and reduced the CPU time for convergence It
is worth mentioning that the building block matrix is the only input data required by the
FFRPF method besides the DS parameters to perform the proposed distribution power
flow The FFRPF technique is incorporated in both utilized deterministic and
metaheuristic optimization methods to satisfy the power flow equality constraints
requirements
In the deterministic solution method the DG sizing problem is formulated as a
nonlinear optimization problem with the distribution active power losses as the objective
function to be minimized subject to nonlinear equality and inequality constraints
Endeavoring to obtain the optimal DG size an improved version of the Sequential
Quadratic Programming (SQP) methodology is used to solve for the DG size problem
The conventional SQP uses a Newton-like method which consequently utilizes the
Jacobean in handling the nonlinear equality constraints The radial low XR ratio and the
tree-like topology of distribution systems make the system ill-conditioned
A Fast Sequential Quadratic Programming (FSQP) methodology is developed in
order to handle the DG sizing nonlinear optimization problem The FSQP hybrid
approach integrates the FFRPF within the conventional SQP in solving the highly
6
nonlinear equality constraints By utilizing the FFRPF in dealing with equality
constraints instead of the Newton method the burden of calculating the Jacobean and
consequently its inverse as well as the complications of the ill-conditioned Y-matrix of
the RDS is eliminated Another advantage of this hybridization is the drastic reduction
of computational time compared to that consumed by the conventional SQP method
In this thesis a new application of the Particle Swarm Optimization (PSO) method in
the optimal planning of single and multiple DGs in distribution networks is also
presented The algorithm is utilized to simultaneously search for both the optimal DG
size and its corresponding bus location in order to minimize the total network power
losses while satisfying the constraints imposed on the system The proposed approach
hybridizes PSO with the developed distribution radial power flow ie FFRPF to
simultaneously solve the optimal DG placement and sizing problem The difficult nature
of the overall problem poses a serious challenge to most derivative based optimization
methods due to the discrete flavor associated with the bus location in addition to the
subproblem of determining the most suitable DG size Moreover a major drawback of
the deterministic methods is that they are highly-dependent on the initial solution point
The developed PSO is improved in order to handle both real and integer variables of the
DG mixed-integer nonlinear constrained optimization problem Problem constraints are
handled within the proposed approach based on their category The equality constraints
ie power flows are satisfied through the FFRPF subroutine while the inequality bounds
and constraints are treated by exploiting the intrinsic and unique features associated with
each particle The proposed inequality constraint handling technique hybridizes the
rejection of infeasible solutions method in conjunction with the preservation of feasible
solutions method One advantage of this constraint handling mechanism is that it
expedites the solution method converging time of the Hybrid PSO (HPSO)
15 THESIS OUTL INE
This thesis is organized in six chapters The research motivation brief description of the
DG and the thesis objectives are addressed in the first chapter The second chapter deals
with a literature review of the distribution power flow methods and the DG optimal
planning problem In the third chapter development of the proposed FFRPF method
7
utilized in the FSQP and the HPSO methods to satisfy the DG problem of nonlinear
equality constraints is presented The fourth chapter deals with the DG sizing problem
formulation and its solution based on the two deterministic solution methods The
problem is solved via the conventional SQP and the proposed FSQP methods and a
performance comparison between them is presented Basic concepts of the PSO are
presented in chapter five A brief literature review regarding the use of the PSO in
solving the electric power system problems is presented in this chapter In addition it
also addresses the development of the proposed HPSO in solving the DG planning
problem The last chapter provides the thesis concluding remarks and the scope of future
work
8
CHAPTER 2 LITERATURE REVIEW
21 INTRODUCTION
Recent publications in the areas of work relative to this thesis are reviewed and
summarised in this chapter which is organized in two sections as follows
bull The first section reviews the literature on distribution power flow methods A
brief background of conventional power flow methods is presented followed
by a review and summary of the literature on recent developments of the
distribution power flow algorithms
bull The DG integration problem is reviewed in the second section Recent work
on the optimal DG placement and sizing via analytical deterministic and
metaheuristic methods are analyzed and reviewed
22 DISTRIBUTION POWER FLOW
Power flow programs play a vital role in analyzing power systems The problem deals
with calculating unspecified bus voltage angles and magnitudes active and reactive
powers as well as (as a by-product) line loadings and their associated real and reactive
losses for certain operating conditions These values are typically obtained through
iterative numerical methods to analyze the status of a given power system
Since the middle of last century many methods were proposed to solve this problem
Even though Dunstan [29] was the first to demonstrate a digital method for solving the
power flow problem in 1954 Ward and Hale [30] are often credited with the successful
digital formulation and solution of the power flow problem in 1956 Most of the earlier
solution methods were based on both the admittance matrix and the Gauss-Seidel (GS)
iterative method The poor convergence characteristics of GS when large networks
andor ill-conditioned situations are encountered led to the development of the Gaussian
iterative scheme (Zbus) [3132] and later the Newton-Raphson (NR) method [33] as well
as the Decoupled [34] and Fast Decoupled (FD) power flow approaches [35] Though
the NR method generally converges faster than other methods it takes longer
computational time per iteration When Tinney et al [36] introduced the optimally
9
ordered and sparsity-oriented programming techniques Newton-based methods became
the de facto industry standard However the Jacobian matrix for the RDS is
approximately four times the size of the corresponding admittance matrix and it needs to
be evaluated at each iteration
Although conventional power flow methods are well developed in dealing with the
transmission and sub-transmission sections of the power system networks they are
deemed to be inefficient in handling distribution networks This is because the
Distribution System (DS) is different in several ways from its transmission counterpart
DS has a strictly radial topology nature or weakly meshed networks in contrast with
transmission systems which are tightly meshed networks DS is a low voltage system
having low XR ratio sections and a wide range of reactance and resistance values DS
may consist of a tremendously large number of sections and buses spread throughout the
network Sections of the DS could have unbalanced load conditions due to the
unbalanced three-phase loading as well as single and double phase loads in spurred
lateral lines The mutual couplings between phases are not negligible due to rarely
transposed distribution lines [37] All of these characteristics strongly suggest that DS is
to be classified as an ill-conditioned power system
The practical DSs low XR ratio sections may cause both the NR and FD
conventional methods to diverge [38-41] The line impedance angles are small enough to
deteriorate the dominance of the NR Jacobian main diagonal making it prone to
singularity Such a low XR value would also prevent the Jacobian matrix from being
decoupled and simplified
In addition to performance considerations a practical power flow technique needs to
consider all the DS distinctive features and to accommodate the imbalance introduced by
multiphase networks along with the distribution-level loads In the literature a number
of Newton and non-Newton power flow methods designed for distribution systems were
proposed Zhang et al [42] solved the distribution power flow based on the Newton
method although the proposed Jacobian is computed just once the solution converged
with a number of additional iterations more so than the conventional approach
Moreover shunt capacitor banks were ignored in the modeling as well as the line shunt
admittance (JI model) and the constant impedance loads Baran and Wu [43] solved the
10
power flow problem by utilizing three fundamental quadratic equations representing the
real and reactive section powers and the bus voltages in an iterative scheme as a
subroutine during the process of optimizing the capacitor sizing However they
computed the Jacobian using the chain rule within the proposed NR method which is in
turn time consuming Mekhamer et al [44] utilized the three equations developed in [43]
using a different iterative technique without the need for the Jacobian or the NR method
However their process is based on applying a multi-level iterative process on the main
feeder and laterals which makes the speed and the efficiency of their proposed algorithm
a function of the RDS configuration and topology
In [4546] the quadratic equation was also utilized in determining the relation
between the sending and receiving end voltage magnitudes along with the section power
flow They proposed to include the system power losses within their calculation while
solving for the system power flow However the voltage phase angles were ignored
during the solution of the radial power flow in order to speed up the convergence The
latter reference developed work was based on the assumption of balanced RDS and
sophisticated numbering scheme
The radial power flow introduced by [47-49] used a non-Newton power flow techshy
nique based on the ladder network theory This method adds the section currents and
calculates the RDS bus voltages including the substations during a backward sweep If
the difference between the calculated substation voltage value and substation predetershy
mined assigned bus voltage value is acceptable the iterations are concluded If not the
substation bus voltage is reset and the RDS bus voltages are computed for the second
time in the same iteration in the forward sweep Both the ladder and the backshy
wardforward methods are derivative-free instead they employ simple circuit laws
However the ladder method uses many sub-iterations on the laterals and calculates the
system bus voltages twice during a single iteration compared to once in the backshy
wardforward method Thukaram [50] utilized the backwardforward sweep technique to
solve the RDS power flow However the bus numbering procedure was a sophisticated
parent node and child node arrangement which may add some computational overshy
head if the system topology is changed Teng [51] used the backwardforward approach
as the solution procedure through the development of two matrices and multiplied them
11
together in a later stage of the solution process In assembling those matrices all the
system buses and sections have to be inspected carefully In a practical large RDS data
preparation for these matrices will be cumbersome and prone to errors Under continshy
gency situations switching operations or the addition of another feeder to the existing
one are quite common practices in the DSs hence changes in system topology need to be
accommodated by restructuring the corresponding matrices which would add an overshy
head to track modifications The weakly meshed DS was dealt with by adding extra
nodes in the middle of the new links Two equal currents with opposite polarities were
injected into each added node Each injection operation is represented by a two column
matrix which was subsequently added to the first proposed matrix and then the develshy
oped matrices were extended and multiplied together The resultant is a full matrix and
its dimension is reduced by the Kron method in every single iteration That is the
developed full matrix was inverted in each iteration of the solution method and such
procedure is expensive lengthy cumbersome and time consuming
Shirmohammadi et al [39] Cheng and Shirmohammadi [52] and Hague [53]
proposed an iterative solution method for both radial and weakly meshed DSs This
approach necessitates a special numbering scheme in which they number the DS sections
in layers starting from the root node The numbering scheme is to be carried out
carefully by examining the whole system when a new layer is to be numbered The
numbering process is cumbersome and prone to errors For weakly meshed networks
breakpoints are selected opened and consequently the meshed system is converted to a
radial system The loops are broken by adding two fictitious buses In each pair of
dummy buses equal and opposite currents are injected and the new system is evaluated
to produce a reduced order impedance matrix Their proposed method requires that the
breakpoint impedance matrix should be computed cautiously Such a procedure is highly
dependent on the distribution networks topology That is the more links that exist in the
DS the larger the break point impedance matrix and the more time will be consumed in
its computation
Goswami and Basu [38] introduced a direct solution method to solve for radial and
weakly meshed DS They applied a breakpoints method into the meshed DS similar to
that of [39] in order to convert it into RDS In their proposed methodology a restriction
12
was imposed on each of the system buses not to have more than three sections attached to
it Such limitation is a drawback of the method and moreover a difficult node numbering
scheme is a disadvantage
In this thesis the unique structure of the RDS is exploited in order to build up a new
fast flexible power flow technique that deals with radial and looped DSs The numbering
scheme of the DS is simple and straightforward All load types can be accommodated by
the proposed distribution power flow eg spot and distributed loads Unlike
conventional power flow methods no trigonometric functions are used in the proposed
distribution power flow method For weakly meshed and looped DSs the system is dealt
with as it is there is no need for radialization cuts or building breakpoints impedance
matrix The topology of the tested DS whether strictly radial weakly meshed or looped
is represented by a building block matrix which is the only one needed to perform the
backwardforward sweep technique
23 DG INTEGRATION PROBLEM
DG is gaining increasing popularity as a viable element of electric power systems The
presence of DG in power systems may lead to several advantages such as supplying
sensitive loads in case of power outages reducing transmission and distribution networks
congestion and improving the overall system performance by reducing power losses and
enhancing voltage profiles Some of the negative impacts of DG installations are
potential harmonic injections the need to adopt more complex control schemes and the
possibility of encountering reverse power flows in power networks Even though the
concept of DG utilization in electric power grids is not new the importance of such
deployment is presently at its highest levels due to various reasons Recent awareness of
conventionaltraditional thermal power plants harmful impacts on the environment and
the urge to find more environmentally friendly substitutes for electrical power generation
rapid advances made in renewable energy technologies and the attractive and open
electric power market are a few major motives that led to the high penetration of DG in
most industrial nations power grids To achieve the most from DG installation special
attention must be made to DG placement and sizing
13
The problem of optimal DG placement and sizing is divided into two subproblems
where is the optimal location for DG placement and how to select the most suitable size
Many researchers proposed different methods such as analytic procedures as well as
deterministic and heuristic methods to solve the problem
231 Solving the DG Integration Problem via Analytical and Deterministic Methods
In the literature the optimal DG integration problem is solved by means of employing
any analytical or optimization technique that suits the problem formulation Methods and
procedures of optimally sizing and locating the DGs within the DS are varied according
to objectives and solution techniques
Willis [54] presented an application of the famous 23 rule originally developed
for optimal capacitor placement to find a suitable bus candidate for DG placement That
is to install a DG with a rating of 23 of the utilized load at 23 the radial feeder length
down-stream from the source substation However this rule assumes uniformly
distributed loads in a radial configuration and a fixed conductor size throughout the
distribution network In any event the 23 rule was developed for all-reactive load
These assumptions limit its applicability to radial distribution systems and the fact that it
is only suitable for single DG planning
Kashem et al [55] developed an analytical approach to determine the optimal DG
size based on power loss sensitivity analysis Their approach was based on minimizing
the DS power losses The proposed method was tested using a practical distribution
system in Tasmania Australia However it assumes uniformly distributed loads with all
the connected loads along the radial feeder having the same power factor and it also
assumes no external currents injected into the system buses eg capacitors which limits
its practicality
Wang and Nehrir [56] developed an analytical approach to address the optimal DG
placement problem in distribution networks with different continuous load topologies
Minimizing the real power losses was the objective of the proposed method In their
approach the DG units were assumed to have unity power factor and only the overhead
distribution lines with neglected shunt capacitance are considered The candidate bus
was selected based on elements of the admittance matrix power generations and load
14
distribution of the distribution network The issue of DG optimal size was not addressed
in their formulation
Griffin et al [57] analyzed the DG optimal location analytically for two continuous
load distributions types ie uniformly distributed and uniformly increasing loads The
goal of their study was to minimize line losses One of the conclusions of their research
was that the optimal location of DG is highly dependent on the load distribution along the
feeder ie significant loss reduction would take place when placing the DG toward the
end of a uniformly increasing load and in the middle of uniformly distributed load feeder
Acharya et al [58] used the incremental change of the system power losses with
respect to the change of injected real power sensitivity factor developed by Elgerd [59]
This factor was used to determine the bus that would cause the losses to be optimal when
hosting a DG By equating the aforementioned factor to zero the authors solved for the
optimal real value of DG output They proposed an exhaustive search by applying the
sensitivity factor on all the buses and ranked them accordingly The drawback of their
work is the lengthy process of finding the candidate locations and the fact that they
sought to optimize only the DG real power output Furthermore they only considered
planning of a single DG
Popovic et al [60] utilized sensitivity analysis based on the power flow equations to
solve the DG placement and sizing Two indices were used in ranking all the DS buses
for the suitability of hosting the DG The first one is a voltage sensitivity index which is
derived directly from the NR power flow Jacobian inverse the second one exploits the
relation of incremental real power losses with respect to the injected real and reactive
power as developed in [61] Their objective for sizing the DG was to maximize its
capacity subject to boundary constraints such as bus voltage penetration level line flows
and fault current limits To solve the sizing DG problem they gradually increased the
DG capacity at selected most sensitive buses until one of the constraints is violated and
the direct previous installed DG size becomes the one chosen as the optimal rating This
process is a lengthy and impractical procedure and the authors did not elaborate on how
they would deal with multiple DG cases using the proposed scheme
Keane and OMalley [62] solved for the optimal DG size in the Irish system by using
a constrained Linear Programming (LP) approach To cope with the EU regulation which
15
emphasizes that Ireland should provide 132 of its electricity from renewable sources
by 2010 the objective of their proposed method was to maximize the DG generation
The nonlinear constraints were linearized with the goal of utilizing them in the LP
method A DG unit was installed at all the system buses and the candidate buses were
ranked according to their optimal objective function value
Rosehart and Nowicki [63] dealt with only the optimal location portion of the DG
integration problem They developed two formulations to assess the best location for
hosting the DG sources The first is a market based constrained optimal power flow that
minimized the cost of the generated DG power and the second is voltage stability
constrained optimal power flow that maximized the loading factor distance to collapse
Both formulations were solved by utilizing the Interior Point (IP) method The outcomes
of the two formulations were used in ranking the buses for DG installations The optimal
DG size problem was not considered in their paper
Iyer et al [64] employed the primal-dual IP method to find the optimal DG
placement through combined voltage profile improvement and line loss reduction indices
However the proposed approach was based on initially placing DGs at all buses in order
to determine proper locations for DG installations This methodology may not be
realistic for large scale distribution networks
Rau and Wan [65] only treated the DG sizing problem by utilizing the Generalized
Reduced Gradient (GRG) method The DG bus locations were assumed to be provided
by the system planner for the DG units to be installed In their proposed method they
considered minimizing the system active power losses In their formulation only the
power flow equality constraints were considered whereas the boundary conditions and
the inequality constraints were not taken into account
Hedayati et al [66] employed continuous power flow methodology to locate the
buses most sensitive to voltage collapse The sensitive bus set is ranked based on their
severity which is used accordingly to indicate potential bus locations for placement of
single and multiple DG sources An iterative method was proposed for optimally sitting
the DG A certain DG capacity which is known and fixed a priori is added to the DS
and the conventional power flow method was employed to determine the resultant DS
real power losses voltage profiles and power transfer capacity In the subsequent
16
iteration another DG with the same capacity was added to the next sensitive bus and
results were obtained This iterative process would continue until the system outcomes
reached acceptable values The proposed iterative method did not optimize the DG size
232 Solving the DG Integration Problem via Metaheuristic Methods
Metaheuristic techniques have proven their effectiveness in solving optimization
problems with appreciable feasible search space They can be easily modified to cope
with the discrete nature associated with different elements commonly used in power
systems studies Optimization methods such as Genetic Algorithm (GA) [67-71] GA
hybrid methods such as GA and fuzzy set theory [72] and GA and Simulated Annealing
(SA) [73] Tabu search algorithm (TS) [7475] GA-TS hybrid method [76] Ant Colony
Optimization (ACO) [77] Particle Swarm Optimization (PSO) [78] and Evolutionary
Programming (EP) [79] were utilized in the literature to solve for the DG integration
problem
Teng et al [67] developed a value-based method for solving the DG problem The
GA method was utilized in maximizing a DG benefit to cost ratio index subject to only
boundary constraints such as ratio index voltage drop and feeder transfer capacity A
drawback of their procedure is that the candidate DG bus locations were assumed to be
provided by the utility and consequently all combinations of the provided bus locations
were tested for obtaining the optimal DG capacities via the GA method
The proposal set forth by Mithulananthan et al [68] made use of the DS real power
losses as the fitness function to be minimized through GA Their formulation of the DG
size optimization problem is of an unconstrained type Moreover the NR method which
is usually inadequate in dealing with the DS topology was used in calculating the total
power losses Candidate DG bus locations were obtained by placing a DG unit at all
buses of the tested DS which is impractical for large DSs Furthermore the multiple
DGs case was not addressed
Haesen et al [69] and Borges et al [70] solved the DG integration problem by
basically employing the GA method Both utilized the metaheuristic technique in solving
for single and multiple DG sizing and placements Haesen et al used the GA method to
minimize the DS active power flow while the objective for Borges et al was to
17
maximize a DG benefit to total cost ratio index Reference [69] incorporated penalty
factors within the objective function to penalize constraint violations thus adding another
set of variables to be tuned The authors of the latter reference used a PV model for
modeling the DG
Celli et al [71] formulated the DG integration problem as an s-constraint
multiobjective programming problem and solved it using the GA method Their
proposed algorithm divided the set of the objective functions into one master and the rest
are considered as slave objective functions The master is treated as the primary
objective function that is to be minimized while the slaves are regarded as new
inequality constraints that are bounded by a predetermined e value They utilized their
hybrid method to minimize the following objective functions cost of network upgrading
energy losses in the DS sections and purchased energy (from transmission and DG) The
number of the DG sources to be installed was randomly assigned and the units were
randomly located at the network buses Whenever the constraints are violated the
objective function solution is penalized A Pareto set was calculated from this
multiobjective optimization problem to aid the distribution planner in the decision
making process
Kim et al [72] and Gandomkar et al [73] hybridized two methods to solve the DG
sizing problem The former hybridized GA with fuzzy set theory to optimally size the
single DG unit while the latter combined the GA and SA metaheuristic methods to solve
for the optimal DG power output In both references the DG sizing problem was
formulated as a nonlinear optimization problem subject to boundary constraints only
Unlike Gandomkar et al [73] added the nonlinear power flow equality constraints to
their problem formulation The former researchers utilized their methodology to
investigate multiple DG case while the latter solved only the single DG case Both sited
the DG at all DS buses in order to determine the optimal DG location and size
Nara et al [74] assumed that the candidate bus locations for the DG unit to be
installed were pre-assigned by the distribution planner Then they used the TS method in
solving for the optimal DG size The objective of their formulation was to minimize the
system losses The DG size was treated as a discrete variable and the number of the
18
deployed units was considered to be fixed The DS loads were modeled as balanced
uniformly distributed constant current loads with a unity power factor
Golshan and Arefifar [75] applied the TS method to optimally size the DG as well
as the reactive sources (capacitors reactors or both) within the DS They formulated
their constrained nonlinear optimization problem by minimizing an objective function
that sums the total cost of active power losses line loading and the cost of the added
reactive sources The DG locations were not optimized instead a set of locations were
designated to host the proposed DGs and the reactive sources
A hybrid method that combined the GA with the TS technique in order to solve the
DG sizing optimization problem was developed by Gandomkar et al [80] They solved
the DG integration problem by minimizing the distribution real power losses subject to
boundary conditions The authors restricted the number of DGs as well as their gross
capacity to be revealed prior to executing the optimization procedure They augmented
the objective function with penalty terms in their formulation to handle the constraint
violations
Falaghi and Haghifam [77] proposed the ACO methodology as an optimization tool
for solving the DG sizing and placement problems The minimized objective function for
the utilized method was the global network cost ie the summation of the DGs cost their
corresponding operational and maintenance cost the cost of energy bought from the
transmission grid and the cost of the network losses The DG sizes were treated as
discrete values They used a penalty factor to handle the violated constraints ie
infeasible solutions In addition to modeling the DG sources as exclusive constant power
delivering units ie with unity power factor the network loads were all assumed to have
09 power factor Thus it can be stated that such modeling is impractical especially when
real large DSs are encountered
Raj et al [78] dealt with the DG integration in two different steps They employed
the PSO method to optimally determine the size of single and multiple DGs The optimal
location portion of the problem was performed utilizing the NR power flow method to
assign those buses with the lowest voltage profiles as the optimal candidate DG locations
The PSO was used to minimize the system real power losses the voltage profiles
boundary conditions were the only constraints required by the authors to be satisfied
19
Constraint violations were handled via a penalty factor that was augmented with the
objective function The DG units were randomly sited at one or more of the candidate
buses obtained through the NR method and subsequently the PSO was used to find the
optimal size(s)
Rahman et al [79] derived sensitivity indices to identify the most suitable bus for a
single DG installation Subsequently the DG sizing problem was dealt with by
employing an EP approach The objective function of the proposed approach was to
minimize the DS real power losses subject only to the system bus voltage boundary
constraints The formulation of DG sizing in their work was not realistic for a variety of
reasons For instance they ignored the line loading restrictions power flow equality
constraints and DG size limits
In most of the reviewed work on the DG deployment problem the problems of DG
optimal sizing and placement were not simultaneously addressed due to the difficult
nature of the problem as it combines discrete and continuous variables for potential bus
locations and DG sizing in a single optimization problem This combination creates a
major difficulty to most derivative-based optimization techniques and it increases the
feasible search space size considerably In this thesis the DG sizing subproblem is
solved using an improved SQP deterministic method while the two subproblems are
addressed simultaneously via an enhanced PSO metaheuristic algorithm
24 SUMMARY
In this chapter distribution power flow techniques were reviewed in Section 22 The
literature review of DG integration problem solution methods was presented in Section
23 The analytical and deterministic methods that were utilized to handle the DG
integration problem were presented in Subsection 231 Then recent publications that
handled the DG sizing and placement problems via wide-class of metaheuristic methods
were reviewed and summarized
20
CHAPTER 3 FAST AND FLEXIBLE RADIAL POWER FLOW FOR
BALANCED AND UNBALANCED THREE-PHASE DISTRIBUTION
NETWORKS
31 INTRODUCTION
As discussed in Chapter 2 several limitations exist in radial power flow techniques
presently reported in the literature such as complicated bus numbering schemes
convergence related problems and the inability to handle modifications to existing DS in
a straightforward manner This motivated the development of an enhanced distribution
power flow solution method In this thesis the unique structure of the RDS is exploited in
order to build up a Fast Flexible Radial Power Flow (FFRPF) technique The tree-like
RDS configuration is translated into a building block bus-bus oriented data matrix
known as a Radial Configuration Matrix (RCM) which consequently is utilized in the
solution process The developed algorithm is also capable of handeling weakly meshed
and meshed DSs via a meshed RCM (mRCM) RCM or mRCM is the only matrix that
needs to be constructed in order to proceed with the iterative process During the data
preparation stage each RCM (or mRCM) row focuses only on a system bus and its
directly connected buses That is while building such a matrix there is no need to
inspect the entire system buses and sections Moreover no complicated node numbering
scheme is required The building block matrix is designed to have a small condition
number with a determinant and all of its eigenevalues equal to one to ensure its
invertibility By incorporating this matrix and its direct descendant matrices in solving
the power flow problem the CPU execution time is decreased compared with other
methods The FFRPF method is flexible in accommodating any changes that may take
place in an existing radial distribution system since these changes can be exclusively
incorporated within the RCM matrix The proposed power flow solution technique was
tested against other methods in order to judge its overall performance using balanced and
unbalanced DSs
In Chapters 4 and 5 the FFRPF method is incorporated within the developed FSQP
and HPSO algorithms in solving the optimal DG installation problem It is implemented
21
as a subroutine within the proposed algorithms to satisfy the equality constraints ie
solving the radial power flow equations
32 FLEXIBILITY AND SIMPLICITY OF FFRPF I N NUMBERING THE RDS
BUSES AND SECTIONS
The RDS is configured in a unique arborescent structure with the distribution substation
located at its root node from which all the active and reactive power demands as well as
the system losses are supplied The substation feeds one or more main feeders with
spurred out laterals sublaterals and even subsublaterals For this reason the substation is
treated as a swing bus during the power flow iterative procedure
Most radial power flow techniques proposed in the literature assign sophisticated
procedures for numbering the radial distribution networks in order to execute their
algorithms This is cumbersome when expanding andor modifying existing RDSs In
this section a very simple numbering rule for the RDS buses and sections is introduced
A section is defined as part of a feeder lateral or sublateral that connects two buses in the
RDS and the total Number of Sections (NS) is related to the total Number of Buses (NB)
by this relation (NS=NB -1 )
321 Bus Numbering Scheme for Balanced Three-phase RDS
A balanced radial three-phase RDS is represented by a single line diagram In such a
system a feeder or sub level of a feeder having more than one bus is numbered in
sequence and in an ascending order Consequently each section will carry a number
which is less than its receiving end bus number by one as shown in Figure 31
Therefore sections are numbered automatically once the simple numbering rule is
applied
22
Substation
Figure 31 10-busRDS
In numbering the RDS shown in Figure 31 the following was considered buses 1 -
4 form the main feeder and buses 2 - 6 and 3 - 10 are the laterals while the sublateral is
tapped off bus 5 Figure 32 and Figure 33 manifest the ease in renumbering the system
shown previously and the flexibility in adding any portion of RDS to the existing one
respectively In Figure 32a buses 1 -4 have a different path compared to Figure 31 and
are considered to be a main feeder buses 2 - 9 and 3 - 6 are the laterals whereas the
sublateral 7 - 10 is the section spurred off bus 7 Figure 32b shows another numbering
scheme The same system numbered differently would have the same solution when
solved by the FFRPF
Figure 33 illustrates the ease of numbering in the case of a contingency situation or
a switching operation that could cause the existing system to be modified andor to be
augmented with other systems The lateral 3 - 10 in Figure 31 was modified to be
tapped off bus 2 instead and a couple of radial portions were added to be fed from buses
6 and 4 as illustrated in the figure
23
Substation Substation
(a) (b)
Figure 32 Different ways of numbering the system in Fig 31
Figure 33 The ease of numbering a modified and augmented RDS
322 Unbalanced Three-phase RDS Bus Numbering Scheme
The three-phase power flow is more comprehensive and realistic when it comes to
finding the three-phase voltage profiles in unbalanced RDSs Figure 34 shows an
unbalanced three-phase RDS The missing sections and buses play a significant role in
the multi-level phase loading and in making the unbalanced state of such a three-phase
DS more pronounced
24
The RDS shown in Figure 34 includes 6 three-phase buses (3(j)NB = 6) 5 three-
phase sections (3(|)NS = 5) no matter how many phase buses or sections exist physically
As such it has 14 single-phase buses (1(|)NB = 14) and 11 single-phase sections (1lt|gtNS =
11) The relations expressed in Eq (31) govern the three-phase and single-phase buses
to their corresponding sections
3^NS = 3^NB-1
l^NS = l^NB-3 (31)
Figure 34 Three-phase unbalanced 6-bus RDS representation
It is simple to implement the numbering process in the three-phase system as was
done in the balanced case Any group of phase buses to be found along a phase feeder or
a sub level of a feeder is to be numbered in a consecutive ascending order Consequently
each phase section number will carry a number which is one less than its receiving end
bus number as shown in Figure 34 In other words the sections are numbered routinely
after the ordering of the three-phase RDS buses has been completed
To develop the building block matrix as will be shown shortly the unbalanced three-
phase system is redrawn by substituting for any missing phase section or bus using dotted
representation as depicted in the 6-bus RDS in Figure 35 By performing this step each
three-phase bussection in the RDS consists of a group of 3 single-phase busessections
a b and c including the missing ones for double and single-phase buses
25
l a
I (1) 2 a | (2) 3 a | (3) 4 a | (4)
Figure 35 Three-phase unbalanced 6-bus RDS with missing buses and sections
33 THE BUILDING BLOCK MATRIX AND ITS ROLE I N FFRPF
The proposed FFRPF procedure starts with a matrix that mimics the radial structure
topology called a system Radial Configuration Matrix (RCM) The inverse of RCM is
then obtained to produce a Section Bus Matrix (SBM) that will be utilized in summing
the section currents during the backward sweep procedure A Bus Section Matrix (BSM)
is next generated by transposing the SBM to sum up the voltage drops in the forward
sweep process Therefore the only input data needed in the solution of an existing
modified or extended RDS other than the system loads and parameters is the RCM
It is worth mentioning that the inversion and transposition operations take place only
once during the whole process of the proposed FFRPF methodology for a tested RDS
whereas other methods like the NR technique invert the Jacobian matrix in every single
iteration The following subsections demonstrate the building of a three-phase RCM and
elucidate the role of both SBM and BSM in solving the radial power flow problem
331 Three-phase Radial Configuration Matrix (RCM)
The only matrix needed to be built for an unbalanced three-phase RDS is the RCM
Whatever changes need to be accommodated as a modification in the existing structure or
an addition to the existing network would be performed through the RCM only The
26
other matrices utilized in the backwardforward sweep are the direct results of the RCM
and no other built matrix is needed to perform the FFRPF
Such a matrix exploits the radial nature of such a system The RCM is of 3(3())NB x
3(|)NB) dimension in which each row and column represents a single-phase bus For a
balanced three-phase RDS represented by a single line diagram the RCM dimension is
(NBxNB) The RCM building algorithm for the unbalanced three-phase RDS case is
illustrated as follows
1 Construct a zero-filled 3(3(|)NB x 3(|)NB) square matrix
2 Change all the diagonal entries to +1 every diagonal entry represents sending
missing or far-end buses
3 In each row if the column index corresponds to an existing receiving single-phase
bus its entry is to be changed to - 1
4 If a single-phase bus is missing or is a far-end bus the only entry in its
corresponding row is the diagonal entry of+1
The above RCM building steps are summarized in the following illustration
Columns Description
RCMbdquo
if is either
a - sending phase bus b - far-end phase bus c - missing phase bus (32)
-1 jkl if jkI are receiving phase buses
connected physically to phase bus 0 otherwise
The [abc] matrix is defined as a zero (3 x 3) matrix with the numbers a b and c as
its diagonal elements eg [ I l l ] is the identity (3 x 3) matrix while [110] is the identity
matrix with the third diagonal element replaced by a zero By following the preceding
steps and utilizing the [abc] definition the RCM for the unbalanced three-phase RDS
shown in Figure 35 is to be constructed as shown in (33)
27
[Ill] [000] [000] [000] [000]
[000]
-[111] [111]
[000] [000] [000]
[000]
[000] -[111]
[111] [000] [000]
[000]
[000] [000]
-[110] [111]
[000] [000]
[000]
[000] [000]
-[010] [111]
[000]
[000] [000]
-[on] [000] [000]
[111]
Because of the nature of the RDS the RCM has three distinctive properties The first
is that the RCM is sparse the second is that RCM is a strictly upper triangular matrix
and thirdly such a matrix is only filled by 0 +1 or - 1 Such a real matrix makes the data
preparation easy to handle and less confusing Figure 36 shows the sparsity pattern plots
of the RCM matrices for unbalanced three-phase 25-bus [48] and balanced 90-bus [38]
radial systems
RCM for 25-Bus unbalanced 3-phase RDS RCM for 90-Bus balanced 3-phase RDS
nz = 131 nz = 179
Figure 36 RCM matrices sparsity plots for the 25 and 90 bus RDSs
3311 Assessment of the FFRPF Building Block RCM
The RCM is well-conditioned and should have a small Condition Number (CN) and a
non-zero determinant The CN measures how far from singularity any matrix is It is
defined as
28
cond(A) = A jjA-l (34)
where ||A|| is any of the 3 types of norm formulations of matrix A 1-norm 2-norm or oo-
norm An ill-conditioned matrix would have a large CN while a CN of 1 represents a
perfectly well-conditioned matrix By definition a singular matrix would have an infinite
CN [81] Having all the matrix eigenvalues to be equal to 1 and a determinant value of 1
safeguard the RCM against singularity For this reason the RCM is not only invertible
but also its inverse is an upper triangular matrix filled only with 0 and +1 digits and no
other numbers would appear in RCM-1
332 Three-phase Section Bus Matrix (SBM)
The SBM for the three-phase RDS is obtained by performing the following steps
1 Remove the corresponding substation rows and columns from the RCM ie the
first three rows and columns The reduced version of the RCM is labeled as
RCM
2 Invert the RCM to obtain the SBM as shown in (35) and more explicitly in
Figure 37
To clarify the two rows and the two columns outside the matrix border shown in
Figure 37 are the three-phase buses and sections ordered respectively The dimension of
the unbalanced three-phase system SBM is 3(3lt|)NS x 3(|gtNS) For the balanced case the
SBM dimension is (NSxNS)
[Ill] [000]
[000] [000] [000]
[111] [111]
[000] [000] [000]
[110] [110]
[111] [000] [000]
[010] [010] [010] [111] [000]
[011] [011] [000] [000] [111]
29
1 a
1 b
1 c
2 a
2 b 2 c
SBM = 3 a
3 b
3 c
4 a
4 b
4 c
5 a
5 b
5 c
2 a
0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 b
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
2 c
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
3 3 a b
1 0 0 1 0 0 i o 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 c
0 0 1 0 0
1 0 0 o 0 0
o 0 0 0
4 a
1 0 0 1 0
o 1 0
o 0 0
o 0 0 0
4 b
0 1 0 0 1 0 0 1 0 0 0 0 0 0 0
4 c
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
5 a
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
5 b
0 1 0 0 1 0 0 1 0 0 1 0 0 0 0
5 c
0 0 0 0 0 o 0 0
o 0 0 1 0 0 0
6 a
0 0 0 0 0
o 0 0
o 0 0 o 1 0 0
6 b
0 1 0 0 1 0 0 0 0 0 0 0 0 1 0
6 c
0 1 0 0 1 0 0 0 0 0 0 0 0 1
Figure 37 SBM for three-phase unbalanced 6-bus RDS
By inspecting Figure 35 it is noted that any single-phase section is connected
downhill to a single-phase far-end bus or buses through a Unique Set of Phase Buses
(USPB) xt bull By inspecting Figure 35 and Figure 37 the USPB for the following
single-phase sections la lb lc 3b 4b are^0 [ gt xlgt x respectively as explained
in (36)
=2 f l 3bdquo4a x=2b3b4b5bA]
Xl=2cA) Xl=) (36)
X=5b]
In the SBM the single-phase section is represented by a row i and will have entries
of ones in all the columns where their indices represent single-phase buses that belong to
the section USPB xf bullgt a s illustrated in (37)
SBMrmt =
Columns Description
c bullgt bull a - Id - buses e r ~ - 1 ijk- ijk-- are either w (37)
lb - diagonal entry 0 other columns otherwise
30
333 Three-phase Bus Section Matrix (BSM)
The BSM is the transpose of the SBM as shown in (38) In the BSM the rows represent
the RDS single-phase buses excluding the substations and all the sections are
represented by the BSM columns Each single-phase bus is connected uphill through a
Unique Set of Phase Sections (USPS) yf to a substation single-phase bus The USPS
for buses 2deg 3a 4b 5b 6C are y ydeg y y yc6 respectively as depicted in Figure 35
and Figure 37 and demonstrated in (39)
BSM
[111] [000] [000] [000] [000]
[111] [111] [000] [000] [000]
[110] [110] [111] [000] [000]
[010] [010] [010] [111] [000]
[011] [011] [000] [000] [111]
(38)
V H U V3=1gt2 y=b2b yb
5=lb2bdquoAb (39)
lt=1C2C5C
In the BSM a single-phase bus i is represented by a row and will have entries of
ones in all the columns where their indices represent single-phase sections that belong to
the bus USPS yf as equivalently shown in (310)
Columns Description
BSMrmi =
( gt - [a-l^-sectionse yf ~ i m
1 ijk ijk are either lt Y Y (310) lb - diagonal entry
0 other columns otherwise
34 FFRPF APPROACH AND SOLUTION TECHNIQUE
The FFRPF methodology is based simply on Kirchhoff s voltage and current laws which
are used in performing the backwardforward sweep iterative process By utilizing the
direct descendant matrices of the RCM ie SBM and BSM the RDS buses complex
31
voltages are calculated in every iteration until convergence criteria are met The next
subsections illustrate the proper usage of such matrices in the proposed FFRPF method
through appropriate modeling of the unbalanced multi-phase RDS section impedances
341 Unbalanced Multi-phase Impedance Model Calculation
Figure 38 shows a three-phase section model that is represented by two buses (sending
and receiving ends) of an overhead three-phase four-wire RDS with multi-grounded
neutral The assumption of a zero voltage drop across the neutral in a three-phase two-
phase and single-phase RDS is found to be valid [4582] Such a configuration is widely
adopted in North Americas distribution networks [8384]
V Sa
_ bull
V Ra
V Sb ab
^VWVVYgt
V Sc Z be
bn
^AAArmdashrYYYV bull
V s n en
ll1 Sa
s b
La
Lb
Lc
bull r Figure 38 Three-phase section model
In the proposed radial power flow solution method each of the three-phase lines is to
be modeled appropriately and mutual coupling effects between phases are not neglected
The primitive impedance matrix for such a four-wire system is a square matrix with a
dimension equal to RDS utilized number of phase and neutral conductors For a system
consisting of three-phase conductors and a neutral wire the section primitive impedance
matrix is expressed as shown in (311)
32
Zaa
ha
Zca
zna
Kb
Kb
Kb
Kb
Ke Kc Ke
nc
art
Zbn
en
nn
where
Z bull primitive impedance matrix
RDS section length
z per unit length self-impedance of conductor i
z per unit length mutual-impedance between conductors andy
zu and zy are calculated according Carsons work [85] and its modifications [86-88] as
illustrated by the following equations
where
k
GMRj
Dbdquo
v GMR
bulli J
zu=rt+rd+ja)k
zv=rd+jltok
resistance of conductor i
earth return conductor resistance
inductance multiplying constant
distance between overhead and its earth return counterpart and it is a
function of both earth resistivity and frequency
geometric mean radius of conductor i
distance between conductors i andj
(312)
(313)
The parameters used in (312) and (313) are shown in Table 31 for both operational
frequencies 50Hz and 60Hz in both metric and imperial units
33
Table 31 cok rj and De Parameters for Different Operation Conditions
De = 2160 Ij (ft)
cok rd
p = 100 Qm
p = 1000 Qm
Metric Units RDS operating frequency 50 Hz 60 Hz
006283km
0049345 QJ km
931 m
29443 m
007539 km
005921412km
850 m
26878 m
Imperial Units RDS operating frequency 50 Hz 60 Hz
010111mile
00794 QI mile
30547f
96598 ft
012134mile
009528 QI mile
27885
88182
Since the neutral is grounded the primitive impedance matrix Zsec can be
transformed into a (3 x 3) symmetrical impedance matrix Zsae
c by utilizing Krons
matrix reduction method The resultant section three-phase impedance matrix is
expressed mathematically in (314) and the three-phase section model is represented
graphically in Figure 39
7 abc
aa
zba Zca
Zab
^bb
Zcb
zac zbc Zee
(314)
VSn
mdash bull i 7 T
i ah bull-sec a
zbdquobdquo bull A V W Y Y Y V
v izK
^WW-rrYYv -+bull
I
bull
vR
Figure 39 The final three-phase section model after Krons reduction
If the RDS section consists of only one or two phase lines its primitive impedance
matrix is transformed by Krons matrix reduction to (laquo-phase x w-phase) symmetrical
impedance matrix Next the corresponding row and column of the missing phase are
replaced by zero entries in the (3x3) section impedance matrices Zsae
c For a two-phase
34
section its impedance matrix Z^c is demonstrated below
Z_a
zci Kron h-gt ZZ za
zbdquo zai
zaa o zac
0 0 0
z_ o zbdquo
Underground lines such as concentric neutral and tape shielded cables are typically
installed in the RDS sections For underground cables with m phases and n additional
neutral conductors the primitive impedance of each section is a (2m+n x 2m+n) matrix
with the entries computed as illustrated in [89-92]
Usually the RDS is modeled as a short line ie less than 80 km and the charging
currents would be neglected by not modeling the line shunt capacitance as depicted in
Figure 38 However under light load conditions and especially in the case of
underground cables the line shunt capacitance needs to be considered in order to obtain
reasonable accuracy ie use nominal 7i-representation The rc-equivalent circuit consists
of a series impedance of the section and one-half the line shunt admittance at each end of
the line Figure 310 (a) (b) and (c) show the RDS 7i-model representation The shunt
admittance matrix for an overhead three-phase section is a full (3x3) symmetrical
admittance matrix while it is a strictly diagonal matrix for the underground RDS cable
section That is the self admittance elements are the only terms computed [92] For the
unbalanced three-phase section eg one or two phases the non-zero elements of shunt
admittances are only those corresponding to the utilized phases
[zic] -AAVmdashrwvgt
T yabc 1 |_ sec J [ yaf tc |
sec J
(a)
35
Lsec J _
2
s
1
Yaa
Yba
Yea
Yab
Ybb
Ycb
Yac
Ybc
Ycc
zaa
zba
tea
zab
zbb
zcb
zac
zbc
zcc
P yabc ~|
lgtlt 2 - =
Yaa
Yba
Yca
Yab
Ybb
Ycb
R
1 1
Yac
Ybc
Ycc
(b)
[ yabc 1 sec J
s 1 1
Yaa
0
0
0
Ybb
0
0
0
Ycc
zaa
zba
zca
Zab
zbb
zcb
zac
zbr
zcc
V yabc ~j
L sec 2 - =
Yaa
0
0
0
Ybb
0
R
1 1
0
0
Ycc
(c)
Figure 310 Nominal 7i-representation for three-phase RDS section
(a) Schematic drawing for the single line diagram of 3(|gt RDS (b) Matrix equivalent for overhead section (c) Matrix equivalent for underground cable section
By applying Kirchhoff s laws to the three-phase system section k the relationship
between the sending and receiving end voltages for medium and short line models and
the voltage drop across the same section in the latter model are expressed in Eq (315)-
(316) and Eq (317) respectively
36
rabc S rabc S
14 L sec Jax3 L rabc
3x3 L secgt J3x3
[C]3 [4 [ yabc~ |~ yabc 1
sec J3x3 L sec J3 [4
zt 1 L sec J3x3
f yabc ~| [~ yaampc 1
L sec J3x3 L sec h
bull R rabc
(315)
rrabc VS rabc
S
1 J3x3 L sec J
[degL [L abc R
(316)
where
TT-afec rrabc S ^ R
rabc rabc S XR
rabc
AK
13x3
aAc sect
rabc see
KrH^ic] three-phase sending and receiving end voltages
three-phase sending and receiving end section currents
three-phase shunt admittance of section k
(3gtlt3) identity matrix
(3gtlt3) zero matrix
voltage drop across three-phase section k
section k three-phase currents
(317)
It is worth noting that Eq (315) is reduced to Eq (316) in the case of short line
modeling since YS^C 1 = 0 The voltage drop across the three-phase section k in the short
line model is expressed in Eq (317) and its corresponding sending end phase voltages
can be expressed in expanded forms as follows
V = V + Ta 7 + 1 7 + Tc 7 rS yR ^1secZjaa ^ Isec^ab ^rlsecpoundjac
v =v +r 7 +17 +r 7 y S y R ^ l sec^ab ^ 1 sec^bb ^ 1 sec^Ac
S mdashyR+ Ktc^ca + sec^cb + sec^a
(318)
(319)
(320)
Equations (317)-(320) show that the voltage drop along any phase in a three-phase
section depends upon all the three-phase currents
37
342 Load Representation Accurate and proper load modeling is of significant concern in power distribution
systems as well in its transmission systems counterpart [8693] Loads in electric power
systems are usually expressed by adequate representations so as to mimic their effects
upon the system The load dependency on the operating bus voltage and on system
frequency is among those representations
Static load models are often utilized in the power flow studies since they relate the
apparent power active and reactive directly to the bus operating voltage A static load
model is used for the static load components ie resistive and lighting load and as an
approximation to the dynamic load components ie motor-driven loads [93] Generally
static loads in DS are assumed to operate at rated and fixed frequency value [94-96]
Loads in the DS are usually expressed as function of the bus operating voltage and
represented by exponential andor polynomial models
The exponential model is shown in (321) and (322)
P = Pbdquo
Q = Q0 vbdquo
(321)
(322)
where
V0 nominal bus voltage
V operating bus voltage
P0 real power consumed at nominal voltage
Q0 reactive power consumed at nominal voltage
Exponents a and fi determine the load characteristics and certain a and values lead to a
specific lode model Therefore
1 If a = P mdash 0 the model represents constant power characteristics ie the load is
constant regardless of the voltage magnitude
2 If a = P = 1 the model represents constant current characteristics ie the load is
proportional to the voltage magnitude
3 If a = P = 2 the model represents constant impedance characteristics ie the load is
38
a quadratic function of the voltage magnitude
As indicated in [97] the exponents could have values larger than 2 or less than 0 and
certain load components would be represented by fractional exponents
The constant current model is considered to be a good approximation for many
distribution circuits since it approximates the overall performance of the mixture of both
constant power and constant impedance models [98] However representing loads with
the constant power model is a conservative approach with regard to voltage drop
consideration [99] and consequently this model will be used in this thesis
Loads can also be represented by a composite model ie the polynomial model The
polynomial model is expressed in (323) and (324)
P = Pbdquo
Q = Q0
(
a p
V
r
V
V
K
V
v0
2
2
V
K
V
+CP
J
)
(323)
(324)
where ap + bp + cp = 1 and aq + b + cq = 1
The polynomial model is also referred to as a ZIP model since it combines all the
three exponential models constant impedance (Z) constant current (I) and constant
power (P) models The ZIP model needs more information and detailed data preparation
The load models can be used in the FFRPF solution method during its iterative
process where flat start values are initially assumed to be the load voltages The three-
phase load voltages are changed during each iteration and consequently the three-phase
currents drawn by the constant current constant impedance andor ZIP three-phase load
models will change accordingly
Different shunt components like spot loads distributed loads and capacitor banks are
customarily spread throughout the RDS In power flow studies spot and distributed
loads are typically dealt with as constant power models while shunt capacitors are
modeled as constant impedances [94 100 101]
The uniformly distributed loads across RDS sections can be modeled equivalently by
either placing a single lumped load at one-half the section length or by placing one-half
the lump-sum of the uniformly distributed loads at each of the section end buses
39
[99 102] The former modeling approach has the disadvantage of increasing the
dimension of the RCM SBM and the BSM since more nodes would be added to the
existing RDS topology In the proposed FFRPF technique the distributed load is
modeled using the latter approach while the three-phase shunt capacitor banks are
modeled as injected three-phase currents [101] as schematically shown in Figure 311
and mathematically represented by Eq (325) and (326)
Qk Cap
^
CCap a Cap
(a) (b)
Figure 311 Three-phase capacitor bank (a) The schematics (b) The Modeling
O3 = poundCap
Qo Capa vbdquo
T-34 _ 1Cap ~
V
a Capbdquo
SQL M Cap
V
filt bullCapo
F
JQ( Cap
(325)
(326)
343 Three-phase FFRPF BackwardForward Sweep
The FFRPF technique employs the SBM in performing the current summation during the
backward sweep and the BSM in updating the RDS bus complex voltages during the
forward sweep as demonstrated in the following subsections
3431 Three-phase Current Summation Backward Sweep DS loads are generally unbalanced due to the unbalanced phase configuration the double
and single-phase loadings as well as the likelihood of unequal load allocation among the
three-phase configuration For the loads they could be represented as constant power
40
constant current constant impedance or any combination of the three models [97 103]
The three-phase load currents at three-phase bus i drawn by a three-phase load eg of a
constant impedance load model is mathematically expressed as shown in Eqs (327) and
(328)
jabc U ~
7e a gt
va
(si) ~
(327)
where
ctabc
o
V
K
2
rft
0
v K
2
K v
2
(328)
where Sf represents the load apparent power at single-phase bus lt|gt As shown in the
preceding equations each load current is a function of its corresponding bus voltage For
Eq (327) if the a phase bus is missing its corresponding phase load current is
eliminated and its corresponding position in the three-phase current vector is replaced by
a zero entry As an illustration and by assuming that there are loads connected to all
existing buses the three-phase load current vector for the system shown in Figure 35 is
expressed as follows
jabc V ja rb jc ra rb re ra rb Q Q rb Q Q rb re l l
The charging currents at the RDS three-phase buses are not to be neglected when
dealing with sections modeled as The shunt admittance at bus is obtained by
applying the following relation
where
Ysh^ bull total three-phase shunt admittance at bus
[l if section k attached to bus i
[0 otherwise
The three-phase shunt currents at bus is as shown in Eq (330)
tabc jrabc 1ch ~~ 1Anbus y i (330)
41
The 3-ltj) bus current is sum of both the 3-sect load current and the 3-cj) charging current as
expressed mathematically in Eq(331)
jabc jabc jabc ) T I 1busl ~ 1Li
+Ich V-gt )U
where 1^ is bus three-phase currents In the case of modeling a three-phase section as
a short line its charging currents are neglected ie I^c = 0 and the bus current will be
represented by the load currents only
The backward sweep sums the phase load currents in the corresponding phase
sections starting from far-end phase buses and moving uphill toward the substation phase
buses The current in phase (j) and section p is computed by utilizing the USPB
principle xp gt during the backward sweep as expressed in (332)
lt = E lt ^here = j 0 ^ J (332)
where
I current through single-phase section and phase ^ (^ =a b or c) SQCp
j current at bus and phase ltb bus x
The SBM is utilized in obtaining the system three-phase section currents in matrix
representation by performing the relation in Eq (333)
[G] = [SBM][lpound] (333)
where the I^ is a 3-(3())NS)-order column vector For a balanced short line RDS
model Eq (333) can be expressed as
[CS] = [SBM][IL] (334)
3432 Three-phase Bus Voltage Update Forward Sweep
The voltage at each phase bus is determined through the forward sweep procedure by
subtracting the sum of the voltage drops across the bus corresponding USPS from the
substation nominal complex voltage The voltage drop across three-phase section k is
calculated as in Eq (317) The voltage drop across all sections of the three-phase RDS
can be obtained by utilizing the SBM as shown in matrix notation via Eq (335)
42
[AKbdquo]=[zr][c] (335)
[ A ^ ] = [ Z s r ] [ 5 5 M ] [ C ]
where ZtradeS J is a diagonal matrix with an 3(3c|)NS x 3ltj)NS) dimension in which its
diagonal entry k corresponds to section k impedance and AV3^ is the computed three-
phase voltage drop values across all the RDS sections as shown below
A ^ = AVa AVb AVC bullbullbull AVb AVC T s e c l s e c l s e c l sec3ltWS sec3tNS J
For calculating the RDS voltage profiles the FFRPF solution method starts by asshy
suming the initial values for all bus voltages to be equal to the substation complex
voltage As a flat start the initial phase voltages at bus will be as follows
2TT 2TT
ya Jdeg vb e~^ VC e+JT V ss e V sisK v sis e (336)
where Vls is the substation complex phase voltage
For the voltage at bus m and phase (j) to be determined at iteration v the calculation is
performed as follows
= amp - pound r A lt wherer = trade lt (337)
The RDS voltage profiles at the system phase buses are obtained by utilizing the BSM as
shown in following matrix representation
[Vi] = [Vsy[BSM][AV^] (338)
where V^fs 1 and V3A are respectively the substation nominal three-phase voltage
column vector and the resultant three-phase bus voltage solution column vector and each
has a dimension of 3(3lt|gtNS)
3433 Convergence Criteria
The bus complex voltage is obtained after every backwardforward sweep After each
iteration all the bus voltage magnitudes and angles are compared with the previous
iteration outcomes The power flow process is concluded and a solution is reached if the
complex voltage real and reactive oo-norm mismatch vector is less than a certain
43
predetermined empirical tolerance value e The convergence criterion is expressed
mathematically as shown in Eq (339)
+i
([gt]w) A a ( |y f lts
where th
i iteration A
(339)
and symbol
||J| vector oo-norm ||x II = max fix IV II lloo l l c 0 1=12NBV I
5H (bull) real part of complex value
3 (bull) imaginary part of complex value
3434 Steps of the FFRPF Algorithm
The FFRPF iterative process can be summarized as follows
Step 1 Begin FFRPF by choosing a test RDS
Step 2 Number and order RDS buses and sections
Step 3 Construct RCM
Step 4 Obtain both SBM and BSM
Step 5 Select load model
Step 6 Start the iterative procedure by assuming flat start voltages for all buses
Step 7 Calculate load currents
Step 8 Start the backward sweep process by calculating section currents using SBM
Step 9 Start the forward sweep process by determining the bus complex voltages
using BSM
Step 10 Compare both magnitudes and angles of the RDS bus voltages between the
current and previous iterations
bull If the co-norm of their difference is lt st
o Solution is reached
44
o Stop and end FFRPF procedure
o Obtain bus voltage profiles section currents and power losses
etc
bull If not utilize the outcome of this iteration (bus complex voltages)
to start a new one by going back to Step 7
The FFRPF solution method is illustrated by the following flow chart shown in Figure
312
45
i laquo - i +1
Calculate Load and leakage
currents
I Start Backward
sweep process by calculating section
currents using SBM
Start Forward sweep process by determining bus
complex voltages V[+1] using BSM
V[+1] Section currents
Section Power Losses Etc
Start FFRPF
Read the test RDS data
Number and order RDS Buses and
Sections
I Construct RCM
Remove the substation
corresponding rows and columns
from RCM to Obtain RCM
Obtain RCM1
To Get SBM
Z Transpose SBM to
get BSM
Calculate RDS section
Impedance and Shunt admittance
Matrices
Select load model
Assuming a flat start voltages for
all buses V[]=10 =0
Figure 312 The FFRPF solution method flow chart
46
344 Modifying the RCM to Accommodate Changes in the RDS The FFRPF technique deals with changes in the network such as the addition of a
transformer by adjusting the original RCM to incorporate its conversion factor (c)
Subsequently the SBM and BSM are obtained accordingly and used in the
backwardforward sweep procedure If a three-phase transformer is incorporated in a
three-phase RDS between buses m and n at section n - 1 the modified BSM entries are
located at the intersection of the matrix rows and columns defined by Eq (340)
BSMZ EzL~-inBSMZ euro lt _ (340)
The affected rows and columns of the modified BSM are those belonging to the
sections USPB and the sending buss USPS respectively For demonstration purposes
the 10-bus balanced RDS shown in Figure 31 will be utilized If two transformers with
conversion factors cfj and cS are to be added within sections 3 and 6 respectively the
original RCM is modified to accommodate such additions as illustrated in (341) Thus
instead of filling -1 for the receiving end bus entry the negative of the conversion factor
is the new entry The process is repeated rc-times for -installed transformers The
corresponding modified SBM and BSM are to be obtained as demonstrated in Section
33
10
RCM^ =
1
2
3
4
5
6
7
8
9
10
1
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
-cfi 1
0
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
-cf2
0
1
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
-1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
-1
1
(341)
The affected entries of the new BSM are obtained by applying the relation in (340) as
follows
47
[BSMZ eZ s ^nBSMJ euro ^ 3 = 40(12=[(4l)(42)]
(laquoSWe^)nJM6^)=J7 )8)njl I4=gt[(7 )l)(7 )4)(8l) )(M)]
The matrix shown in (342) shows the final B S M after including the transformers in the
10-bus R D S Such a procedure is easily extended to the unbalanced three-phase RDSs
1
1
1
CJ
1
1
cfi
cf2
1
1
2
0
1
ch 0 0
0
0
1
1
It is worth mentioning that by integrating the cf for any transformer configuration
into the RCM building block in the FFRPF technique another light is shed on the
flexibility criterion of the proposed method
35 FFRPF SOLUTION METHOD FOR MESHED THREE-PHASE DS
In practical DS networks alternative paths are typically provided to accommodate for
any contingency incidents that might take place eg feeder failure Therefore it is not
unusual for meshed distribution networks to be part of the DS topology in order to make
the system more reliable The loop analysis approach as well as the graph theory
technique are used to study and analyze the behavior of meshed DS The loop analysis
technique basically applies Kirchhoff s voltage law principle to solve for the fundamental
loop currents in both planar and nonplanar networks while the graph theoretic
formulation preserves the network structure properties [104]
A meshed DS can be viewed from a graph theory perspective as an oriented looped
graph that preserves the network interconnection properties whereas a DS that has no
0
0
0
1
1
cfi
cf2
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
(342)
48
loops is considered a tree In graph theory terminology line segments that connect
between buses in a loopless DS tree are called twigs branches or sections (represented by
solid line segments in Figure 313) while those which do not belong to the tree are
known as links (represented by dotted line segments in Figure 313) Links are segments
that close loops in a DS tree thereby composing a co-tree Removal of all co-tree links
results in a strictly radial system Links are usually activated by closing their
corresponding Normally Open (NO) switches Whenever a link is added to a RDS
network a loop is formed and as a result the system will have as many fundamental
loops as the number of links A fundamental loop is a loop that contains only one link
besides one or more sections Segments are used here to name sections and links
together It is noted that the number of fundamental loops is significantly less than the
number of buses in the meshed DS which makes the loop analysis a more appropriate
method in dealing with such systems than other circuit analysis methods like nodal
voltage method [105]
The current directions in the meshed DS sections and links are arbitrarily chosen to
be directed form a lower bus index to a higher one and the positive direction of loop
current is assumed to in the same direction of that of the link as illustrated in Figure 313
The number of segments in a meshed DS is equal to the sum of the total number of its
corresponding graph tree sections and its co-tree links For a meshed DS with NB buses
and mNS segments (total number of sections and links in the meshed DS) the number of
links nL and the number of the fundamental loops as well are obtained according to the
following relation
laquoL=mNS-NB + l (343)
49
Substation 2 Imdash 31 4 1
^ -gtT-gtL- -
Figure 313 10-bus meshed distribution network
351 Meshed Distribution System Corresponding Matrices The FFRPF solution method can deal with the DS meshed networks through modifying
the original RCM Discussion is now focused on the balanced three-phase meshed DS
which can easily be extended to the unbalanced three-phase DS networks
Meshed RCM The meshed RCM (wRCM) order is ((NB + wL) (NB + nVj) and the
mRCM building algorithm is as follows
1 Remove links from the meshed DS and build the RCM for the resulting network
tree as demonstrated earlier in section 331
2 Add nL rows and columns toward the end of the RCM ie each link is represented
by a row and a column attached to the end of the RCM
3 In each link column there are 3 non-zero entries and are to be filled in the following
manner
a -1 at the row which corresponds to the lower index terminal of the link
b +1 at the row which corresponds to the higher index terminal of the link
c +1 at the link diagonal entry
For the 10-bus system shown in Figure 31 three directed links Li L2 and L3 are added
to the DS tree through connecting the following buses 4 - 5 6 - 8 and 4 - 1 0
respectively The system mRCM is constructed as illustrated in (344)
50
10
mRCM (13x13)
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
0
0
-1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
-1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
-1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
-1
0
1
0
0
0
1
0
0
0
0
-1
0
0
0
0
0
1
0
0
1
(344)
Remove the substation corresponding rows and columns from the mRCM to produce the
mRCM The mRCM for the 10-bus system is shown in (345)
10
mRCM (12x12)
1
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
-1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
0
0
0
0
-1
0
1
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
0
-1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
-1
1
0
0
0
0
0
1
0
0
0
0
0
0
-1
0
1
0
0
0
1
0
0
0
-1
0
0
0
0
0
1
0
0
1
(345)
Meshed SBM The meshed SBM (mSBM) is then obtained by inverting the mRCM As
51
an illustration the 10-bus meshed network mSBM is obtained as shown in (346)
10
mSBM (12x12)
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
1
0
0
1
0
1
0
0
0
0
0
0
1
0
0
1
0
1
1
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
1
1
0
0
0
i deg 1
1
-1
1 deg 0
o 0
0
i
o o
0
0
0
0
1
-1
-1
0
0
0
1
0
0
0
1
0
0
0
0
-1
-1
0
0
1
(346)
Let the mSBM be partitioned into two submatrices mSBMp and C so that the mSBMp
submatrix corresponds to the DS tree sections and the second submatrix C to the
fundamental loops or links as shown in (347)
wSBM = SBM
6 [cl (mNSxnL) = [mSBMp C]
JmNSx(NB-l)
The dotted line shown in the above relation implies matrix partitioning
(347)
Fundamental loop matrix The second submatrix in (347) ie C is the fundamental
loop matrix which governs the direction of currents in each of fundamental loop sections
and links The fundamental loop matrix C can be partitioned into two submatrices [Csec]
and [I] as demonstrated below
M_ where [Csec] is an ((NB - 1) x (laquoL)) matrix and [7] is an nL square identity matrix The
former matrix corresponds to the tree loop sections while the latter corresponds solely to
c-- (348)
52
the co-tree links
By inspecting the fundamental loop matrix C it is noted that each row represents a
section or a link and each column represents a loop Each column entry in the C matrix
CM will have one of the following values
1 Qy = +1 if section k belongs to and is oriented in the same direction of loop
2 Cki = - 1 if section k belongs to and is oriented in the opposite direction of loop
3 Claquo = 0 if section k is not in the loop
By inspecting the 10-bus DS mSBM illustrated in (346) the first loop which is represhy
sented by the tenth column of the matrix is comprised of three sections in addition to the
link The current in two of these sections runs in the same direction as their correspondshy
ing fundamental loop ie sections 2 and 3 while the third one ie section 4 has an
opposite orientation This can be easily verified by tracing the first loop in the meshed
DS single line diagram One can also note that two loop currents pass through the third
section in an additive manner
Meshed BSM The meshed BSM (mBSM) is the transpose of mSBM as illustrated in
(349)
[ B S M I 0](mNS-nL)mNS
L J(nLxmNS)
The second submatrix C[ is the transpose of the fundamental loop matrix as manifested in
(350)
mBSM = [mSBM] = mBSMr
c (349)
C=L
1
0
0
0
2 3
1 1
0 0
0 1
4
-1
0
0
5
0
1
0
6
0
-1
0
7
0
-1
0
g
0
0
-1
9
0
0
-1
h 1
0
0
h 0
1
0
h (f 0
1
(350)
Meshed DS impedance matrices The (laquoLx riL) loop-impedance matrix Zioop is
formulated as follows [106]
KHc]|Xf][c] (35D
53
where [ztradegS ] is defined as a non-singular diagonal (mNS x wNS) segment-impedance
matrix that contains all the meshed DS segment impedances (tree sections and links)
along its main diagonal The matrix [Z^fM is formulated as two diagonal submatrices
as follows
|~ rymDS I
L eg J
Zl
0
^
0
0
7
Zk
0
0
0
raquoL
|gtr ] | o o |[zr] (352)
where Z^S J is the (NB - 1 x NB - 1) loopless tree section-impedance diagonal square
matrix and Z^k 1 is the (raquoL x nL) links impedance diagonal matrix
352 Fundamental Loop Currents The algebraic sum of voltage drops AV around any fundamental loop is zero according
to Kirchhoff s voltage law (KVL) and this can be mathematically expressed in terms of
the fundamental loop matrix C as follows
[C][AF] = 0 (353)
The voltage drop across the meshed DS segments is determined by the following
relations
[W] = [zf][mSBM][mILL]
where
Lm seg J bull (wNS x l) segment currents column vector of the meshed DS network
jW iZJ (mNS x l) meshed DS bus Loads and Link currents column vector
(354)
In order to account for the link currents in the meshed network the segment currents
column vector and the meshed DS bus loads and links currents column vector are
54
respectively partitioned into two subvectors as defined below
[ jtree 1
J(mNSxl)
J((JVB-l)xl)
Jloop[ J(nLxl)
(355)
[mILL l(mNSxl)
L L J((MJ-l)xl)
Jloop J (wLxl)
(356)
where
[Cr J ((NB - 1) x 1) tree section currents column vector
[lL] ((NB - 1) x 1) RDS bus load currents column vector
j 1 (nL x 1) fundamental loop current vector which is also the meshed DS link
currents column vector
By multiplying both sides of Eq (354) by [C] the left-hand-side will equal to zero
according to KVL as shown in (353) and by employing Eqs (351) and (356) Eq (354)
is reformulated as
[C][AV] = [c][z^][mSBM[mILL]
0 = [c f [z f ] [mSBM | C ] L op]
bull = [Cr[zS]([raquoSBM][ l ]+ [C][ 1 ] )
0 = [C] [^ ] [ -raquoSBM] [ I ] + [cr [zbdquor][C][U]
-[c] [zf ][c][J=[c] [ Z - ^ S B M J ]
-[^][^]-[cT[zS][laquoSBM][I] Thus the (nL x 1) fundamental loop currents vector in the meshed DS loop frame of
reference can mathematically be expressed as
[4 J - -K]1 [Cj [z^JmSBMr][lL] (357)
Eq (357) can be expressed in terms of the RDS matrices ie SBM and [ Z ^ J by
performing the following operation
55
[ ^ ] = -[z f c J1[cr[zS8][laquoSBM][L]
[ 2 T ] I 0
o [[z^f] =-[^r[[c118ri[]] SBM
0 [h]
=-[zY[ic-l i]] [zr][SBM]
6 [h]
Finally the fundamental loop currents vector is formulated in terms of the RDS matrices
as follows
[ 4 ] = -[zi00PT lCJ [ z r ] [ S B M ] [ J (358)
Calculating the fundamental loop current vector utilizing Eq (358) involves less-
dimensioned matrices than that of Eq (357) which in turn requires less memory storage
and makes it a better candidate for performing the meshed DS FFRPF method
353 Meshed Distribution System Section Currents To express the meshed DS section current vector in terms of a fundamental loop current
vector the fundamental cut-set principle is utilized A fundamental cut-set contains only
one tree section and if any one or more links Once a cut-set is removed from the
network at least one bus will be separated from the rest of the system That is the
removal of a cut-set will basically result in two separate systems or graphs [107] As an
illustration Figure 314 shows several cut-sets for the meshed 10-bus DS
56
bull0D H
Figure 314 Fundamental cut-sets for a meshed 10-bus DS
All the fundamental cut-sets are arranged in an ((NB - 1) x mNS) matrix ie B The
fundamental cut-set for the meshed system exemplified in Figure 314 is constructed as in
(359)
B
1
1 0 0
-1
-1
1
0
0
0
0
0
0
0
0
0
-1
1
1
0
0
0
0
- ]
0
0
0
0
1
1
(359)
The first (NB - 1) columns of B constitute an identity matrix whereas the remaining
nL columns form an ((NB - 1) x nL) co-tree cut-set matrix The first submatrix
corresponds to the tree sections while the second to the links in the meshed DS The cutshy
set matrix B is expressed as follows
B = m (NB-l) [ rtLinks 1 4 s e c J((Affl-l)xnL) (360)
If the section which constitutes a fundamental cut-set does not belong to a loop its
57
corresponding entry in the [fi^fa] matrix row is set to zero The links in a cut-set would
either be +1 -1 or 0 according to the following algorithm
1 +1 if the link belongs to the cut-set and is oriented in the same direction as its cutshy
set
2 -1 if the link belongs to the cut-set and is oriented in the opposite direction of its
cut-set
3 0 if the link does not belong to the cut-set
By inspecting the 10-bus meshed DS B matrix one notices the first section has a cutshy
set that does not have link element meaning that its corresponding row entries in the
second submatrix C are all zeros It is also worth mentioning that the number of all the
cut-sets is equal to (NB-1) which is basically the number of rows in matrix B
The relationship between the fundamental loop and cut-set matrices is given by the
following relation [107]
[B][C] = 0 (361)
By utilizing Eq (348) (360) in expanding (361) the [Csec] can be expressed in terms
of the co-tree submatrix of fundamental cut-set matrix | B^ as follows
[B][C] = 0
[Csec]~ [MI [Cfa]] M = 0
[Qec] = [C f a ] (3-62)
Usually directly finding the fundamental cut-set matrix is avoided and relation (362) is
usually utilized instead since [Csec ] is easier to obtain by inspection
The algebraic sum of section currents for any cut-set is zero according to Kirchhoff s
Current Law (KCL) and can be expressed in terms of the fundamental cut-set matrix as
follows
58
[ 5 ] [ lt e g ] = 0 (363)
By utilizing the submatrices of the fundamental cut-set matrix and the meshed DS
segment currents column vector one can relate the fundamental loop currents (which are
also the link currents) to the tree section currents by performing the following steps
[59108109]
~[c]~ [MI [Cfa]]
ltoopj - 0
[C]+[iCb][4] = o and finally
[C] = -[Cfa][4laquo] (3-64) By integrating the results obtained by Eq (362) into Eq (364) the tree section currents
can be expressed as
[ C ] = [pound][] (3-65)
The entry Itrade in |s^ | represents the algebraic sum of loop currents passing through
the tree section k Substituting for [ ] from Eq (358) in Eq (365) the tree section
currents vector ie | ^ e l can be expressed in terms the RDS SBM and bus load
currents vector as follows
[C] = -K][Zl00PT [Cj [zr][SBM][J (366)
354 Meshed Distribution System BackwardForward Sweep During the backward sweep the overall net meshed DS section currents are calculated
using Eqs (333) and (366) as follows
= [SBM][L]-[C1Be][zlB(p]-I[C1M][zr][SBM][pound] (367)
= ([ V t ) -C-lZ-V lC-l [Z])lSBM][h]
59
where [ pound f ] was defined in Eqs (332) and (334) and [](Aaw) is an ((NB - 1) x (NB -
1)) Identity matrix The voltage drops across the tree sections of the meshed DS and the
bus voltage profiles vector are obtained during the forward sweep by performing Eq
(368) and (369) respectively
[ A F - ] = [ z r ] [ J 068)
[ye J = [ j s ] - [BSM][AF m ^] (369)
It is worth reiterating that the matrices needed during the FFRPF solution method for
solving both radial and meshed DSs are RCM SBM and BSM and they are computed
just once at the start of the solution technique
36 TEST RESULTS AND DISCUSSION
The proposed FFRPF method presented in this chapter utilizes the building block
matrices RCM or mRCM and their sequential matrices SBM BSM mSBM and mBSM
in solving power flow problems for different balanced and unbalanced three-phase radial
and meshed distribution systems The relating matrices are shown for the first case study
of each section That is the involved matrices for the tested DSs will be shown for the
31-bus balanced three-phase RDS 28-bus balanced weakly meshed three-phase DS and
for 10-bus unbalanced three-phase RDS The FFRPF simulations were carried out within
the MATLABreg computing environment using HPreg AMDreg Athlonreg 64x2 Dual Processor
5200+ 26 GH and 2 GB of memory desktop computer
361 Three-phase Balanced RDS
In order to investigate the performance of the proposed radial power flow three case
studies of three-phase balanced radial systems were tested The power flow solution of
the proposed method was tested and compared with two radial power flow techniques as
well as with four other different methods The radial distribution power flow methods
utilized in the comparisons are those proposed by Shirmohammadi et al [39] and by
Prasad et al [49] The other four methods are the Gauss iterative method using Zbus
[110] GS NR and FD [111] methods
The following comments are made regarding the preceding four methods used in
60
assessing the proposed radial method The substation is considered to be the reference
while building the Zbus matrix to be used later in the Gauss iterative method When
applying the GS technique the best acceleration factor was carefully chosen to produce
the least number of iterations and minimum execution time to make for a fair
comparison When solving using NR method the Jacobian direct inverse is avoided
especially for those systems with large CNs instead it is computed using the method of
successive forward elimination and backward substitution ie Gaussian elimination For
the FD method as a result of the high RX ratio the technique diverged in all the tested
systems indicating that the conventional decoupling simplification assumption of the
Ybus is inapplicable in the RDS
The comparison between all the methods and the proposed FFRPF technique is in
terms of the number of iterations before converging to a tolerance of 00001 and in terms
of the CPU execution time in milliseconds (ms) The Reduction in CPU execution Time
(RIT) between the proposed method and other methods is calculated as follows
(Other method time - Proposed method time) RIT = plusmn - z xl00 (370)
Other method time
All the FFRPF steady state complex bus voltage results are found to be in agreement
with those of the converged other methods The tested cases are 31-bus 90-bus 69-bus
and 15-bus RDSs The 90-bus case radial system is of a very radial topology in nature
while the 69-bus is configured of more than the conventional one main feeder connected
to the main distribution substation The 15-bus RDS test case is a practical DS that
consists of several modeled sections The results obtained are briefly described in the
following sections
3611 Case 1 31-Bus with Single Main Feeder RDS
This test system is a 31-bus single main feeder with 6-laterals shown in Figure 315 Bus
No 1 is a 23kV substation serving a total real and reactive load of 15003 kW and 5001
kvar respectively The system detailed line and load data is obtained form [112] Figure
316 shows the 31-bus RDS building block bus-bus oriented matrix ie RCM while
Figure 317 shows its inverse The CN of the system RCM is 2938 where the Jacobian
CN used in the first iteration of the NR solution method is 1581 and worsens to 2143 in
61
the last iteration Figure 318 shows the SBM which is basically the RCM1 after removshy
ing the first row and column from it ie the substation corresponding row and column
Figure 319 shows the BSM which is the system SBM transpose Table 32 tabulates the
FFRPF iterative procedure results for the tested RDS while Table 33 tabulates the
resultant RDS line losses The FFRPF was also used to solve for the voltage profiles of
three load models to show that the proposed method is capable of handling different load
characteristics Table 34 shows the FFRPF voltage profile results for the constant
power constant current and constant impedance load models
Table 35 reveals the comparison between the three different models results in terms
of maximum and minimum bus voltages and real and reactive power losses By
inspecting Table 34 and Table 35 the constant power load model has the largest power
loss and voltage drop while the constant impedance model has the lowest Table 36
shows a comparison between the performance of the proposed method and other
techniques The proposed method converged much faster than all the methods in terms
of CPU execution time With regard to the iteration number the proposed power flow
converged faster than [39] and GS methods and had comparable iteration number to [49]
and NR methods
Substation 29
bull m bull bull laquoe bull
22 30
31
Figure 315 31-busRDS
62
1 2 3 4 5 6 7 8 9 10 11 12 13
RCM= 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
mdash 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
bull
0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
in
0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CD
0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1^
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
O)
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
o CM 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0
CM CM
0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
CO CM
0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0
I - -CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0
CO CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0
CM
0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
o CO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0
CO
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1
Figure 316 TheRCMofthe 31-busRDS
63
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
T -
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
bull0-
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
lO
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
co
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
l-~
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
oo
0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
C)
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
o CM
0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0
CM CM
1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
CO CM
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
CM
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
in CM
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
CO CM
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
CO CM
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0
CM
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
o CO
r 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
CO
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
Figure 317 The RCM1 of the 31-bus RDS
64
2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
bull bull
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
bull
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
N-
0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
00
0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
ogt
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
o CM
0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0
CM CM
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
co CM 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
-tf-CM 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
CM
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
CD CM
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0
CM
1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
oo CM 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0
en CM 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
o CO r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0
co 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
Figure 318 The SBM of the 31 -bus RDS
65
2 3 0 0 1 0
0 0 0 0 0 0
4 0 0 0
0 0 0 0 0 0 0 0 0
5 0 0 0 0
0 0 0 0 0 0 0 0 0
6 0 0 0 0 0
0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
o
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
~ 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
bull0-
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
m
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
CD
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
h-
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
CO
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0
agt
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0
o CM 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
CN CN
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0
CO CN
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0
CD CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0
CM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
co CM 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
en CM
o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
o co 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Figure 319 The BSMofthe 31-busRDS
66
Table 32 FFRPF Iteration Results for the 31-Bus RDS
Bus No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
First Iteration
|V| 10
09731
09665
09533
09387
09261
09076 08947
08818
08736
08659
08582
08516
08469
08447
08787
08756
08741
09043 09019
09003 09072
09478
09430
09378
09326
09298 09274
09717
09663
09635
Angle(deg)
0 02399
03496
-00369
-04082
-07388 -09802 -11549
-13347
-14530
-15649
-16789
-17792
-18501
-18845
-14253
-14705 -14917 -10725
-11403 -11611
-09921
-01999 -03471
-04804 -06152
-06894
-07204
02633
02023
01715
Second Iteration
|V| 10
09707
09635
09487
09319
09173 08961
08810 08659
08561
08470
08379
08300
08245
08218
08623
08587
08570 08923
08896
08879 08956
09428
09376
09320
09265 09234
09208
09693
09636
09608
Angle(deg)
0 02858
04150
00019
-03975 -07561
-10010 -11791 -13634
-14851
-16008
-17189
-18233 -18972
-19332
-14628
-15095
-15313 -11001
-11730 -11942
-10138
-01697
-03248
-04649 -06066
-06847 -07164
03098
02456 02132
Third ]
|V| 10
09704
09630
09480
09310
09161
08943
08789 08634
08534
08440
08347
08266
08209 08182
08597 08561
08543 08905
08878 08861
08938
09421
09369
09313 09257
09226
09199
09689
09633 09604
teration
AngleO 0
02896
04207
00019 -04050
-07710 -10209
-12033 -13922
-15173
-16363
-17580
-18655 -19418
-19789 -14938
-15415
-15638 -11215
-11955 -12171
-10339
-01710 -03273
-04685 -06114
-06902 -07221
03135
02489
02163
Fourth Iteration
|V| 10
09703
09629
09479
09308
09159 08941
08785 08630
08529
08436 08342
08260
08203
08176
08593
08556
08539 08903
08875 08858
08936
09420 09368
09312
09255
09225
09198
09689
09632 09604
Angle(deg)
0 02906
04221
00028 -04048
-07715 -10215
-12040 -13930
-15182
-16373 -17591
-18667 -19431
-19802
-14948
-15425
-15649 -11223 -11964
-12179
-10345
-01703
-03267
-04680 -06110
-06898
-07218
03146
02499 02172
Fifth Iteration
|V| 10
09703
09629
09479 09308
09158 08940
08785 08630
08529
08435 08341
08259 08202
08175
08593
08556
08538 08902
08874 08857
08935
09420
09368
09311
09255 09225
09198
09689 09632
09604
Angle(deg)
0 02907
04223
00028 -04050
-07719 -10220
-12046 -13938
-15190
-16382 -17601
-18678 -19442
-19814
-14956
-15434
-15657 -11228
-11969 -12185
-10350
-01703
-03267
-04681
-06111
-06900
-07219
03147
02500
02173
67
Table 33 The 31-bus RDS Section Power Losses Obtained by the FFRPF Method
Section From-To
1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9
9-10 10-11 11-12 12-13 13-14 14-15 9-16
T Losses
Power Losses (kW)
519104 112642 165519 117899 90321 143635 72780 72780 30790 26754 26754 20143 9839 2295 5593
1526706
(kvar) 89800 6056
163655 10248 78508 80912 40998 40998 17345 15071 15071 11347 5542 1293 4861
765194
Section From-To
16-17 17-18 7-19 19-20 20-21 7-22 4-23 23-24 24-25 25-26 26-27 27-28 2-29
29-30 30-31
Power Losses (kW) 4158 0901 5889 3143 0901 0097
25827 20675 12860 12860 3848 2140 4414 9708 2434
(kvar) 2342 0507 5119 2732 0508 0085
25537 20442 11178 11178 3345 1205 0237 5469 1371
68
Table 34 FFRPF Voltage Profiles Results for the Three Different Load Models
Bus No
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Constant Power Model
V __
100 09703 09629 09479 09308 09158 08940 08785 08630 08529 08435 08341 08259 08202 08175 08593 08556 08538 08902 08874 08857 08935 09420 09368 09311 09255 09225 09198 j 09689 09632 09604
AngleO
0 02907 04223 00028 -04050 -07719 -10220 -12046 -13938 -15190 -16382 -17601 -18678 -19442 -19814 -14956 -15434 -15657 -11228 -11969 -12185 -10350 -01703 -03267 -04681 -06111 -06900 -07219 03147 02500 02173
Constant Current Model
JV 100
09732 09666 09533 09385 09258 09073 08943 08813 08730 08653 08575 08508 08462 08439 08782 08750 08735 09039 09014 08999 09068 09478 09429 09377 09325 09297 09273 09718 09663 09636
Angle(deg)
0 02578 03733 -00004 -03522 -06639 -08765 -10283 -11846 -12865 -13827 -14807 -15668 -16275 -16570 -12700 -13100 -13286 -09647 -10294 -10482 -08880 -01606 -03052 -04348 -05660 -06380 -06672 02809 02188 01876
Constant Impedance Model
YL 100
09752 09691 09570 09438 09326 09162 09048 08935 08863 08796 08729 08671 08631 08612 08907 08879 08866 09131 09108 09095 09158 09518 09472 09424 09375 09349 09326 09738 09685 09659
AngleO
0 02357 03403 -00015 -03163 -05918 -07800 -09123 -10480 -11354 -12177 -13012 -13744 -14258 -14507 -11228 -11578 -11741 -08596 -09179 -09348 -07904 -01519 -02874 -04082 -05303 -05972 -06242 02579 01981 01680
69
Table 35 Comparisons between 31-Bus RDS Exponential Model Results
Constant Power Model
Constant Current Model
Constant Impedance Model
Maximum Bus Voltage (pu)
09703
09732
09752
Minimum Bus Voltage (pu)
08175
08439
08612
Power Loss
kW
152650
117910
97208
Kvar
76507
58178
47394
Voltage Drop
1825
1561
1388
Table 36 31-bus RDS FFRPF Results vs Other Methods
FFRPF RPFby [391 RPFby [49] Gauss Zbus Newton-Raphson Gauss-Seidel
No of Iterations 5 8 5 5 4
102
Execution Time (ms) 8627 11376 15013 18553 167986 242167
RIT
2416 4254 535
9486 9644
3612 Case 2 90-bus RDS with Extreme Radial Topology
The 90-bus test system is extremely radial as shown in Figure 3 20 and it is tested here to
show the performance of the proposed power flow method in dealing with such types of
RDS The system data is provided in [38] In order to test the limits of the proposed
power flow algorithm the RX ratio was set to be 15 times the RX ratio of the original
data Such a ratio represents the RDS steady state stability limit The minimum voltage
magnitude of 08656 is obtained at bus No 77 for the modified system The radial
system real and reactive power losses for the 15 RX are 0320 pu and 0103 pu while
those for the original RX ratio system are 0019 pu and 0091 pu respectively The CN
of the 90-bus RDS RCM is found to be 4087 and the system Jacobian CN in the first
and the last NR iterations are 25505 and 26141 Both NR and GS diverged with the 15
RX system which has a Jacobian CN of 31818 in the first iteration The 90-bus system
power flow comparison results are presented in Table 37
70
Substation
Figure 3 20 90-BusRDS
Table 37 90-bus RDS FFRPF Results vs Other Methods
FFRPF RPFby [39] RPFby [491 Gauss ZBus Newton-Raphson Gauss-Seidel
No of Iterations Original
RX
3 4 4 3 3
509
15 RX
5 6 6 5
Diverged Diverged
CPU Execution Time (ms) Original
RX
11028 12958 15455 36463
227798 1674626
15 RX
12675 15113 16002 42373
Diverged Diverged
RIT Original
RX
1489 2864 6976 9516 9934
15 RX
1613 2079 7009
3613 Case 3 69-bus RDS with Four Main Feeders
This 11 kV test system consists of a main substation that supports a total real and reactive
load of 4428 kW and 3044 kvar respectively and a 69-bus distributed among four main
feeders and their laterals All four main feeders are connected to a main distribution
substation as shown in Figure 321 The original 70-bus system [113] consists of two
substations each connected to two main feeders whereas in this research the original
configuration is altered to join the four main feeders to one substation to increase the
71
complexity level as well as to show how robust the power flow can be when dealing with
multi-main feeders connected to one main substation The RX ratio was raised to 45
times the original RX beyond which all conventional power flow methods diverged
This was done to increase the ill-conditioned level of the tested system With such an
increase in the RX ratio the Jacobian CN increased from 1403 for the original system to
8405 for the 45 RX ratio in their final iteration On the other hand the RCM CN for
same system is 2847
Even though the number of iterations in the original RX ratio was equal for all
methods except for the GS and [39] approaches the proposed radial power flow was the
fastest in providing the final solution The number of iterations varied among the
different methods used however the proposed method still had the least CPU execution
time as shown in Table 38 Convergence was achieved even though the bus voltage was
as low as 0506 pu at bus No 69
Substation
1 ^ ^ ^ ^ ^ M
2(
3lt
4lt
5lt 6(
1 6 T mdash
9
MO
H2
113
gt14
(15
18
22
32
34
36
29 49
30 50
3 1 51 39
40l
53
59
42 46
43 k47 63
48 64
69
62
Fieure321 69-bus multi-feeder RDS
72
Table 38 69-bus RDS FFRPF Results vs Other Methods
FFRPF RPFby [391 RPFby [491 Gauss Zbus Newton-Raphson Gauss-Seidel
No of Iterations Original
RX 4 5 4 4 4 61
45 RX
11 24 31 31 8
309
CPU Execution Time (ms) Original
RX
11562 12924 14982 29719 203868 224871
45 RX
17646 20549 31102 37161
272708 728551
RIT Original
RX
1054 2283 6110 9433 9486
45 RX
1413 4326 5251 9353 9758
3614 Case 4 15-bus RDS-Considering Charging Currents
The 66 kV 15-bus distribution network is a real practical RDS that has several n-
represented sections in its topology Such balanced RDS is a part of the Komamoto area
of Japan and the system data is provided in [114] and shown in Figure 329 The RDS
has 14 sections 7 of which are modeled as a nominal n The main substation serves a
total load of 6229 kW and 2624 kvar The proposed FFRPF converged in less CPU
execution time than all other methods as shown in Table 39 Considering the effect of
charging currents by representing some of the RDS sections by 7i-model the system
becomes more practical and realistic As a result the oo-norm of the voltage profiles
decreased from 00672 when not considering the charging current effects to 00545 when
their effects are considered
12 13
T T T T -U
T
14 15
i li ill ill il 7 8 T 9 T 1 0 T T~11
Figure 322 Komamoto 15-bus RDS
73
Table 39 Komamoto 15-bus RDS FFRPF Results vs Other Methods
FFRPF RPFby [391 RPFby [49] Gauss Zbus Newton-Raphson Gauss-Seidel
No of Iterations 4 4 5 4 3
287
Execution Time (ms) 10322 12506 14188 29497 88513 147437
RIT
1746 2725 6501 8834 9300
362 Three-phase Balanced Meshed Distribution System
Three meshed distribution networks are tested by the proposed technique for meshed DSs
that was presented in Section 35 Topology-wise the tested systems are categorised as
weakly meshed meshed and looped (or tightly meshed) networks By applying the
proposed solution method on such a variety of topologies the FFRPF method is proven
to be robust and an appropriate tool to be utilized in distribution planning and operation
stages
3621 Case 1 28-bus Weakly Meshed Distribution System The first test case is an 11 kV 28-bus weakly meshed DS with 27 sections and 3 links
The total served real and reactive loads are 1900 kW and 1070 kvar respectively The
RDS data is available in [115] Three new branches were added to the network to form
three extra loops as shown in Figure 323 The mRCM C and mSBM are shown in
Figure 324 -Figure 326 Table 311 highlights the fast criterion of the FFRPF proposed
method since it had the least execution time compared to the other methods While the
proposed distribution power flow converged in the same number of iterations as that of
the Zbus method all other methods converged within a higher number
74
22
2 3 4 5 6 7 8 9 10 11 v 13 14 15 16 17 18
Figure 323 28-bus weakly meshed distribution network
mRCM =
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 L1 L2 L3
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
12 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
13 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
19 20 21 22 23 24 25 26 27 28 L1 L2 L3
o o o o o o o o o o| o o o - 1 0 0 0 0 0 0 0 0 0 0 0 0 - 1 0 0 0 0 o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 - 1 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 - 1 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1
Figure 324 mRCM for 28-bus weakly meshed distribution network
75
mSBM =
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 L1 U2 L3
2 3 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
7
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
10
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
13
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15
0 0 0 0 0 0 0 0 0 0 0 0
6 0 0
16
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 18 19 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0
21 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0
22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0
23 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
24 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
25 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
26
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
27 28 L1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0
0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 1 0 0
2 L3 0 0 0 0 -1 0 -1 0 -1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 -1 -1 0 -1 0 -1 0 0 1 0 0 1
Figure 325 mSBM for 28-bus weakly meshed distribution network
c 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 - 1 - 1 - 1 - 1 o o o o o oi 1 o o 00-1-1-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1-1 0 0J01 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 -1-1-11001
Figure 326 C for 28-bus weakly meshed distribution network
76
Table 310 Voltage Profiles for Radial and Meshed 28-bus Distribution Network
Meshed Distribution System
Bus No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Voltage (pu)
1000
09604
09310
09200
09134
08915
08805
08761
08706
08668
08668
08681
08754
08689
08663
08661
08688
08724
09377
09296
09149
08909
09168
09064
08903
08888
08849
08816
AngleO
0
02444
04357
05363
05924
07789
08633
09068
09849
1052
10798
10699
0996
11268
11678
11643
10949
10365
05268
06284
08123
11121
05867
06906
08661
08317
08852
09318
77
Table 311 28-bus Weakly Meshed DS FFRPF Results vs Other Methods
FFRPF
RPFRef[391
Gauss Zbus
Newton-Raphson
Gauss -Seidel
No of Iterations
4
4
4
3
258
Execution Time (ms)
16120
20157
23189
148858
228665
RIT
2003
3048
8917
9295
3622 Case 2 70-Bus Meshed Distribution System This 11 kV network is a meshed DS with 70-buses 2 substations 4 feeders 69 sections
and 11 links The real and reactive load supplied by the distribution substations are 4463
kW and 2959 kvar respectively The system single line diagram is shown in Figure 327
and the topology data as well as the served loads are available at [113] Table 312
shows that the proposed method converged faster than the other used methods
Hi Hi H i -
(D (0
4mdash I I
4 laquo _
t
_- mdash mdash
M bull bull m 8 -0 f 9
mdashbullmdash S
CO
~4 1
) bull
U )
-T
ft bull bull 1 bull
^
raquo1
8 S S
8 -
r laquo
1 i p 1
bull s
s s f-
1
1
bull
w
_ i
1
IS
1
I
1
5
5
^ s 0
Figure 327 70-bus meshed distribution system
78
Table 312 70-bus Meshed DS FFRPF Results vs Other Methods
FFRPF
RPFby [391
Gauss Zbus
Newton-Raphson
Gauss -Seidel
No of Iterations
4
5
4
3
427
Execution Time (ms)
25933
51745
77594
355264
1253557
RIT
4988
3331
9270
100
3623 Case 3 201-bus Looped Distribution System Endeavoring to assess the proposed FFRPF proposed solution technique in dealing with
an extremely meshed distribution network an augmented looped system is tested This
system is a hybrid network comprised of one 70-bus [113] two 33-bus [116] and one 69-
bus [43 117] meshed systems The new system consists of 201-buses 200 sections and
26 links as shown in Figure 328 The real and reactive served loads are 15696 MW and
10254 Mvar respectively Table 313 shows how robust the proposed technique is in
dealing with highly spurred and looped distribution system In spite of a comparable
number of iterations among all methods the FFRPF method converged in less time than
all the other methods used for comparison It is noticed that the GS method diverged
when dealing with the looped 201-bus tested system
79
SS-1
122
121 i l
120
119o
118 I |
117
116
116
114
113 I I
T1Z 111
110
109
108 J I
106
105
104
103
133^
132
1311
130lt
128
127
yenraquo
125
124
123
V=
SS-2
91 I 92 bull 93 1 -
I I
100
^101
f 7 2 73 74
is f76
77
78
479 89
bullgt 81
82
8 3
f 84
85
199 bull 1201
198 bull | bull 2M 146 149
laquo raquo raquo
Figure 328 201-bus hybrid augmented test distribution system
Table 313 201-bus Meshed DS FFRPF Results vs Other Methods
FFRPF
RPFby [39]
Gauss Zbus
Newton-Raphson
Gauss -Seidel
No of Iterations
7
7
7
6
mdash
Execution Time (ms)
57132
79743
1771397
2261549
Diverged
RIT
2835
9678
9747
~
363 Three-phase Unbalanced RDS
Three unbalanced three-phase RDSs were tested All have unbalanced loading conditions
and have three-phase double-phase and single-phase sections throughout the system
layout The proposed solution method is compared to the three-phase radial distribution
power flow developed by [52] and to Gauss Zbus iterative method
80
3631 Case 1 10-bus Three-phase Unbalanced RDS The first system is 10-bus three-phase RDS with 20 single-phase buses (3^NB = 10) and
17 single-phase sections (3^NS = 9) as shown in Figure 329 [118] The 866 kV
substation serves total real and reactive power of 825 kW and 475 kvar respectively It is
noted that phase a in this system suffers a heavy loading condition of 450 kW which is
more than half of the total load supplied by the substation Such an unbalanced loading
in the tested system resulted in large voltage drops A voltage drop of 81 is found at
bus No 7 terminals accompanied by the lowest bus voltage magnitude of 0919 pu
Figure 330-Figure 333 show the 10-bus three-phase RDS corresponding RCM RCM1
SBM and BSM Table 314 shows the performance of the FFRPF methodology in
handling such systems against all the other techniques
Figure 329 10-bus three-phase unbalanced RDS
81
1 a
1 b
1 c
2 a
2 b
2 c
3 a 3 b 3 c
4 a 4 b 4 c
5 a 5 b 5 c
6 a
6 b
6 c 7 a
7 b
7 c
8 a
8 b
8 c
9 a 9 b 9 c
10 a 10 b 10 c
1 1 1 a b c 1 0 0 0 1 0 0 0 1
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0 0 0 0
_ bdquo
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
2 2 2
a b c - 1 0 0 0 - 1 0 0 0 - 1
1 0 0
0 1 0
0 0 1
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0
h o o o]
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
3 3 3 a b c 0 0 0 0 0 0 0 0 0
- 1 0 0
0 - 1 0
-P9mdash-1 1 0 0 0 1 0 0 0 1
0 0 0 0 0 0
PQP 0 0 0 0 0 0 0 0 0
4 4 4
a b c
0 0 0
0 0 0
L9P9H h o o o 0 0 0
0 0 0
- 1 0 0 0 0 0 0 0 - 1
1 0 0 0 1 0
L9PL h o o oH
0 0 0 0 0 0
0 0 0| 0 0 0
0 0 Oj 0 0 0
o o oi o o o b 6 oT 6 d o o o oi o o o o o o o o o o 6 bj 6 o o o o oi o o o o o oi o o o 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
5 5 5 6 6 6 a b c a b c 0 0 Oj 0 0 0 0 0 OJ 0 0 0 0 0 Oi 0 0 0
7 7 7 a b c 0 0 0 0 0 0 0 0 0
b 6 of -i d 6 6 6 6 o o o i o - 1 o o o o o o oi o o o o o o 0 0 Oj 0 0 0 0 0 Oj 0 0 0 0 0 0| 0 0 0
d 6 d[ 6 6 6 o o oi o o o 0 0 -11 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 6 6| 6 6 6 6 6 6 0 1 OJ O O O O O O o o i i o o o o o o 0 0 Oj 1 0 0
0 0 Oj 0 1 0
o o oi o o 1 o 6 o[ 6 d 6 0 0 0 0 0 0
o o o[ o o o
- 1 0 0 0 0 0 0 0 0
1 0 0
0 1 0
0 0 1
b 6 b i 6 6 6 6 6 6 0 0 0 0 0 Oj 0 0 0
0 0 0 0 0 0 0 0 0 O O O j O O O j O O O o o o j o o o j o o o o o o j o o o j o o o 6 6 d[ 6 6 6[ 6 6 6 o o o j o o o j o o o o o o j o o o j o o o
8 8 8 a b c 0 0 0 0 0 0 0 0 0
0 0 0
0 - 1 0
0 0 - 1
9 9 9
a b c 0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 Oj -1 0 0
0 0 Oj 0 -1 0
o o oj o o o 0 0 0 0 0 0 0 0 OJ 0 0 0 o o o o o o b 6 o 6 d 6 o o 0 o o o o o oi o o o 0 0 Oj 0 0 0 0 0 Oj 0 0 0 0 0 Oj 0 0 0
0 0 0
0 0 0
_9_q_o 1 0 0
0 1 0
0 0 1
0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
______ 0 0 0
0 0 0
0 0 0
1 0 0 0 1 0 0 0 1
0 0 0
0 0 0
0 0 0
a b c 0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 - 1 0 0 0 0
1 0 0
0 1 0
0 0 1
Figure 330 The 10-bus three-phase unbalanced RDS RCM
82
1 1 a b 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 c 0 0 1 0 0 o 0 0 0 0 0 0 0 0 o 0 0 o 0 0 0 0 0 0 0 0 o 0 0 0
2 a 1 0 o 1 0 o 0 0 0 0 0 0 0 0 o 0 0 o 0 0 0 0 0 0 0 0 o 0 0 0
2 b 0 1 o 0 1 o 0 0 o 0 0 0 0 0
bullf 0 o 0 0 0 0 0 0 0 0 o 0 0 0
2 c 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0
6 0 o 0 0 o 0 0 o 0 0 o 0 0 0
3 a 1 o o 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 b 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 c 0 0 1 0 0 1 0 0 1 0 0 0 0 0
bull1 0 0 0 0 o 0 0 0 0 0 o 0 0 0
4 a 1 0 o 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 b 0 0 o 0 0 o 0 0 0 0 1 0 0 0 o 0 0 o 0 0 o 0 0 o 0 0 o 0 0 0
4 c 0 0 1 0 0 1 0 0 1 0 0 1 0 0 o 0 0 o 0 0 o 0 0 0 0 0 o 0 0 0
5 a 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 b 0 0 o 0 0 o 0 0 o 0 0 0 0 1
-P 0 0 o 0 0 o 0 0 o 0 0 o 0 0 0
5 c oi o 11 o oi ii degi oi ii o Oi 1j Oi oi 1 o oi oi oi o Oj 0| oi oi o oi oi oi 0 o
6 a 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 b 0 1 o 0 1 0 0 0 0 0 0 0 0 0 o 0 1 0
o 0 0 0 0 o 0 0 o 0 0 0
6 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 0
7 a 1 0 o 1 0 o 0 0 0 0 0 0 0 0 o 1 0 o 1 0 0 0 0 0 0 0 o 0 0 0
7 b 0 0 o 0 0 o 0 0 0 0 0 0 0 0 o 0 0 o 0 1 0 0 0 0 0 0 o 0 0 0
7 c 0 0 o 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
8 a 0 0 o 0 0 o 0 0 o 0 0 0 0 0 o 0 0 0 0 0 0 1 0 0 0 0 o 0 0 0
8 b 0 1 o 0 1 o 0 0 o 0 0 0 0 0 o 0 0 o 0 0 o 0 1 0 0 0 o 0 0 0
8 c 0 0 1 0 0 1 0 0 0 0 0 o 0 0 0 0 0 0 0 0 o 0 0 1 0 0 0 0 0 0
9 a 1 0 o 1 0 0 1 0 0 0 0 o 0 0 0 0 0 0 0 0 o 0 0 0 1 0 0 0 0 0
9 b 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
9 c 0 0 o 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 o 0 0 0 0 0 1 0 0 0
o o o a b c 0 0 0 0 1 0 0 0 0 6 6 b 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
6 6 o 0 0 0 0 0 0
h 6 oo 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
r 6 6 b 0 0 0 0 0 0 6 6 o 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
r i 6 b 0 1 0 0 0 1
Figure 331 21 The 10-bus three-phase unbalanced RDS RCM1
83
2 a 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 b 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2 3 c a 0 1 0 j 0
-US-Oil oi o oi o 0| 0
o i o oi o oTo oi o 0 | 0 bi 6 oi o oi o oi o oi o oi o oTo 0 | 0 o i o bi o oi o oi o OJ 0 o i o oi o
3 b 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 c 0 0 1 0 0 1 0 0 0 0 0
o 0 0 0 0 0 0 0 0
o 0 0 0 0 0 0
4 a 1 0
--0 0 1 0 0 0 0
i 0 0 0 0 0 0 0
i-0 0 0 0 0
4 b 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 c 0 0
i 0 1 0 0 1 0 0
o 0 0 0 0 0 0 0 0
o 0 0 0 0 0 0
5 a 0 0
o 0 0 0 0 0 0 1 0
pound 0 0 0 0 0 0 0
pound 0 0 0 0 0
5 b 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 c 0 0
0 1 0
o 1 o 0
0 0 0 0 0 0 0
i 0 0 0 0 0
6 a 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 b 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
6 7 c a oi 1 oj o o o bi 6 oi o oi o Oj 0
oi o 0| 0
bid 0 j 0 o o ci i f oi o 1 i o 011 0| 0
oi o bid oj o oi o b i d oi o oi o OJ 0 oi o oi o
7 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
7 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
8 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
8 b 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
8 c 0
o
i 0
oi o 0 0
oi 0
i 0 0 0 0 0 0 0
bullh 0 0 0
o 0
9 a 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
9 b 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
9 c a
oi o 0 j 0 _OPbdquo 0| 0 o i o oi o 0 0
oi o oi o oio oi o
0| 0 oi o oi o 0 0
oi o oi o oio oi o qpbdquo 0 i 0 oi o 1 0 0 1
oi o oi o
o b 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0
o c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Figure 332 The 10-bus three-phase unbalanced RDS SBM
84
BSM 3
1 a
2 a 2 b 2 c 3 a 3 b 3 c 4 a 4 b 4 c 5 a 5 b 5 c 6 a 6 b 6 c 7 a 7 b 7 c 8 a 8 b 8 c 9 a 9 b 9 c 10 a 10 b 10 c
1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0
1 b 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0
1 2 c a 0 0 1 0 0 1 0 0 1 0 0 1 0 0 o 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 1 0 o 1 0 0 0 0 0 0 0 o 0 0 0 0 0 0 1 0 0 0 0 0
2 b 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0
2 3 c a 0i 0 oi o oi o oio 0 0 1|0 oi 1 oi o i i o 0| 0 oi o i i o oio o o 00 oi o oi o oi o o o oi o
Q|o 0 0 o o o o 0- 0 oi o oi o
3 b 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 4 c a OJ 0 oi o oi o oio o o 0- o oi o oi o 1 0 o 1 0J 0 i i o oio 0 0 OIO oi o oi o oi o 0J 0 OJ 0
4-deg~ 0 0 0 0 0 0 oi 0 oi 0 oi 0
4 b 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 5 c a OJ 0 oi 0 oi 0 oio 0 0 oi 0 oi 0 oi 0 oi 0 0 0 OJ 0 i i 0 oi 1 0 0 oi 0 oi 1 oi 0 oi 0 Oi 0 oi 0
4~deg-0 0 0 0 oi 0 oi 0 oi 0 oi 0
5 b 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
5 6 c a 0 0 Oj 0 oi 0 oio 0 0 00 oi 0 oi 0 0 0 Oj 0 0 0 oi 0 oio 0 0 110 0| 1 oi 0 0 0 0 0 OJ 0
4-deg~ 0 0 0 0 Oj 0 0| 0 oi 0 oi 0
6 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
6 7 c a 0 0 oi 0 oi 0 oio 0 0
40 0- 0 oi 0 oi 0 0 0 oi 0 oi 0 oio 0 0 OIO oi 0 oi 0 i i 0 0 1 oi 0
4_o 0 0 oi 0 oi 0 oi 0 oi 0 oi 0
7 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
7 8 c a 0 0 0
0 0 0
oio 0 0 oi 0 0 0 0 0 0 0
0 0 0 0 0 0
oio 0 0 oi 0 0 0
0 0
oi 0 0 0 0 1
0 0
Oil 0 0 0 0 0
0 0 0 0 0
8 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0
8 9 c a 00 oi 0 oi 0 oio 0 0 oi 0 oi 0 oi 0 oi 0 OJ 0 OJ 0 oi 0 oio 00 OIO oi 0 oi 0 oi 0 OJ 0 OJ 0
4__ 0 0 0 0 110 oi 1 oi 0 0 0
9 9 b c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1
Figure 333 The 10-bus three-phase unbalanced RDS BSM
Table 314 10-bus 3lt|gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods
FFRPF
RPFby [52]
Gauss Zbus
No of Iterations
4
6
4
CPU Execution Time (ms)
41621
70266
115378
RIT
4077
6393
3632 Case 2 IEEE 13-bus Three-phase Unbalanced RDS The IEEE 13-bus three-phase RDS is the second system to be tested in this category It
consists of 39 single-phase buses (3^NB =13) and 36 single-phase sections (3^NS = 12)
two transformers 115416 kV AGY main substation and 416048 kV in-line GYGY
distribution transformer besides a voltage regulator Different load configurations such
as A and Y as well as unbalanced spot and distributed connected loads were installed
85
throughout the system with all combinations of load models Three-phase and single-
phase shunt capacitors are utilized in the system The RDS topology consists of both
overhead lines and underground cables The basic system topology is shown in Figure
334 and its data is available at [119] Table 315 manifests how the FFRPF outperforms
the other methods in terms of the CPU execution time That is the proposed technique
converged in half the number of iterations required by [52] radial method and the RIT
was nearly 43 Although the FFRPF converged in the same number of iterations with
the Gauss Zbus method the time consumed by the proposed technique was 60 less
646 645 mdash bull -
611 684
652
650
671
632 633 634
v 692 675
680
Figure 334 IEEE 13-bus 3ltgt unbalanced RDS
Table 315 IEEE 13-bus 3lt|gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods
FFRPF RPFby [52] Gauss Zbus
No of Iterations
4 8 4
CPU Execution Time (ms)
49252 86191 123747
RIT
4286 6020
3633 Case 3 26-bus Three-Phase Unbalanced RDS An extremely unbalanced loaded three-phase 26-bus RDS is tested for the robustness of
the proposed poWer flow This system consists of 62 single-phase buses (3^NB = 26)
86
with 59 single-phase sections (3^NS = 25) and has a 416 kV substation as its root node
while supporting real and reactive loads of 21825 kW and 1057 kvar respectively [48]
The systems three-phase sections are not symmetrically coupled due to the lack of
transposition in the distribution system lines and bus 26 suffers from an extremely
unbalanced loading As a result the ill-conditioned system causes the voltage drop at
phase a of bus No 26 to be 282 with a voltage magnitude of 0718pu
The comparison between the FFRPF [52] and the Gauss iterative Zbus methods in
dealing with all the unbalanced three-phase RDSs are tabulated in Table 316 All system
voltage profiles obtained by the proposed method were in agreement with the other two
methods results The CPU execution time was in the vicinity of 40 and 60 less than
that consumed by [52] and the Gauss iterative methods respectively
Table 316 26-bus 3lt|gt RDS FFRPF Results vs Ref [52] and Gauss Zbus Methods
FFRPF RPFby [521 Gauss Zbus
No of Iterations
4 8 3
CPU Execution Time (ms)
103357 185816 273114
RIT
4438 6216
37 SUMMARY
In this chapter a fast and flexible radial distribution power flow method was presented It
was tested over several balanced and unbalance radial and meshed distribution systems
The proposed FFRPF technique offers attractive advantages over the other power flow
techniques It does not employ complicated calculations ie the derivatives of the power
flow equations It is flexible and easily accommodates changes that may occur in any
RDS These changes could be modifications or additions of either transformers other
systems or both to the current DS The proposed method starts by constructing only the
building block unit RCM or mRCM which exploits the radial structured system No
other constructed matrix is needed during the data entry when solving for the power flow
problem Such a matrix is proved to be easily inverted and then transposed to produce
the other two matrices utilized in solving the backwardforward sweep process Such
matrix operations are conducted only once at the initialization stage of the proposed
87
FFRPF method
This would tremendously ease system data preparation efforts making it fast and
flexible to deal with The FFRPF technique is easy to program and has the fastest CPU
computation time when compared to other radial and conventional power flow methods
Such advantages make the FFRPF method a suitable choice for planning and real-time
computations The computational time consumed by other methods like NR and GS was
extremely excessive while the FD method diverged because of the significant high RX
value in the RDS Convergence for well and ill-conditioned test cases was robustly
achieved The convergence number of iterations was found to be comparable to the NR
method and to some extent independent of the radial system size
88
CHAPTER 4 IMPROVED SEQUENTIAL QUADRATIC
PROGRAMMING APPROACH FOR OPTIMAL DG SIZING
41 INTRODUCTION
Integrating DG into an electric power system has an overall positive impact on the
system This impact can be enhanced via optimal DG placement and sizing In this
chapter the location issue is investigated through an All Possible Combinations (APC)
search approach of the distribution network The DG rating on the other hand is
formulated as a nonlinear optimization problem subject to highly nonlinear equality and
inequality constraints Sizing the DG optimally is performed using a conventional SQP
method and an FSQP method The FSQP is an improved version of the conventional
SQP method that incorporates the FFRPF routine which was developed in Chapter 3 to
satisfy the power flow requirements The proposed equality constraints satisfaction
approach drastically reduces computational time requirements The results of this hybrid
method are compared with those obtained using the conventional SQP technique and the
comparison results favor the proposed technique This approach is designed to handle
optimal single and multiple DG sizing with specified and unspecified power factors
Two distribution networks 33-bus and 69-bus RDSs are used to investigate the
performance of the proposed approach
42 PROBLEM FORMULATION OVERVIEW
There are two main aspects to the optimal DG integration problem the first is the optimal
DG placement while the second is the optimal DG sizing The criterion to be optimized
in the process of choosing the optimal bus and size is minimizing the distribution network
real power losses The search for appropriate placement of the DG to be installed is
performed via the APC search technique Theoretically the APC method of choosing n-
buses at a time out of NB-bus distribution system with irrelevant orders is computed as
follows
r NBl
m n(NB-n)
As an illustration if three DG units were to be installed in a 69-bus system the number
89
of possible bus selections would be as large a number as 50116 combinations Though
this process is tedious and lengthy it is utilized here as an attempt to find the global
optimal placement for single and multiple DG units which are consequently to be size-
optimized and installed That is the DG size will be optimized in every single
combination using both deterministic methods ie SQP and FSQP The results obtained
are used as a reference guide when employing the developed HPSO technique in Chapter
5 The APC simulations are also used in the comparison between the two
aforementioned deterministic methods in terms of their corresponding CPU convergence
times This process sometimes results in an unrealistic time frame as will be seen in
subsequent sections which paves the way towards the HPSO being a better alternative in
tackling the DG integrating problem
43 DG SIZ ING PROBLEM ARCHITECTURE
Optimal DG sizing is a highly nonlinear constrained optimization problem represented by
a nonlinear objective function that is subject to nonlinear equality and inequality
constraints as well as to boundary restrictions imposed by the system planner The
detailed formulation of the DG optimization problem is presented in the following
sections
431 Objective Function The objective function to be minimized in the DG sizing problem is the distribution
network active power losses formulated as
Minimize ^W(x) (41) xeM
PRPL is the real power losses of NB-bus distribution system and is expressed in
components notation as
NB ( NB
v J-1 (42)
where
pG generated power delivered to DS bus if the DG is to be installed at bus i the
real and reactive DG generated powers are respectively modeled as P^G =
90
-SG PDG a n d
QDG =-SZG PDG tan(acos(7D O ))
PL load power supplied by DS bus
Yv magnitude of the ifh element of admittance bus matrix Y
ytJ phase angle of YtJ = YyZry
Vt magnitude of DS bus complex voltage
Sj phase angle of yi=ViA5i
NB number of DS buses
Equations (43) and (44) present another form of the real power losses written in
components notation as well
1 NB NB
PraquoL = ~ItIty9[v+VJ2-2VlVJc0s(St-6J)] (43)
1 i=l 7=1
NB NB
PRPL = I I gt U [ ^ 2 + ^ 2 - 2 ^ ^ COS(^-^ ) ] (44) (=1 jgti
where ytj is the line section if admittance The real power losses expression in Eq (44)
would require half the function evaluations of that of Eq (43) hence the second formula
is preferable in terms of computational time
Distribution network real power losses can be also expressed in matrix notation as
i ^ L = ( V Y V ) (45)
where
bull transpose of vector or matrix
bull complex conjugate of vector or matrix
V (1 x NB) DS bus Thevenin voltages
Y (NB x NB) DS admittance matrix
Although the reactive power losses are not to be ignored the major component of power
loss is due to ohmic losses as this is responsible for reducing the overall transmission
efficiency [120]
91
432 Equality Constraints The equality constraints are the nonlinear power flow equations which state that all the
real and reactive powers at any DS bus must be conserved That is the sum of all
complex powers entering a bus should be zero as
A ^ = 0 z = 23NB (46)
A Q = 0 i = 23NB (47)
Where
APj real power mismatch at bus i
AQ reactive power mismatch at bus i
NB
7=1
NB
Aa=ef-ef-^Z^[Gsin(^-^)-^cos(^-^)] 7=1
Y(i=Yu(cosyy+jsmyy) = Gu+jBv
433 Inequality Constraints There are two sets of inequality constraints to be satisfied The first set is boundary
constraints imposed on the system and they consist of the DS bus voltage magnitudes and
angles and the DG power factor The bus voltage magnitudes and phase angles are
bounded between two extreme levels imposed by physical limitations It is customary to
tolerate the variation in voltage magnitudes in the distribution level to be in the vicinity
of plusmn10 of its nominal value [121 122] The DG power factor is allowed for values
within upper and lower limits determined by the type and nature of the DG to be installed
in the distribution network Such restrictions are expressed mathematically as shown in
Eqs(48)-(410)
V- lt Vt lt V+ (48)
S-lt8ilt8+ (49)
Pf^^Pfoa^Pf^ (4-10)
where
92
maximum permissible value
minimum permissible value
DG operating power factor
Limiting the DG size so as not to exceed the power supplied by the substation and
restricting the power flow in feeders to ensure that they do not approach their thermal
limits are another set of inequalities imposed on the distribution system Such nonlinear
constraints are expressed mathematically as
nDG
IXo ^S s s (411)
S AS J 7 ltS^ (412)
where
S^j DG generated apparent power
SsS main DS substation apparent power
r scalar related to the allowable DG size
Stradeax apparent power maximum rating for distribution section if
StJ apparent power flow transmitted from bus to busy
^ = - 3 ^ ^ - ^ [ laquo ^ s i n ( lt y lt - lt y y ) - 3 j j r c o s ( lt y lt - ^ ) ]
434 DG Modeling Different models were proposed in the literature to represent the DG in the distribution
networks The most common representations for conventional generating units used are
the PV-controlled bus and the PQ-bus models A DG could be modeled as a PV bus if it
is capable of generating enough reactive power to sustain the specified voltage magnitude
at the designated bus The CHP type of DG has the capability of satisfying such a
requirement However it is reported that such an integration may cause a problematic
voltage rise during low load intervals in the distribution system section where the DG is
Rfi DG
93
integrated [123] The IEEE Std 1547-2003 stated that The DR shall not actively
regulate the voltage at the point of common coupling (PCC) that is at the bus to which
the DG is connected [12] This implies that the DG model is represented by injecting a
constant real and reactive power at a designated power factor into a distribution bus
regardless of the system voltage [14] ie as a negative load [16] The PQ-model is
widely used in representing the DG penetration into an existing distribution grid [124-
127] Most DGs customarily operate at a power factor between 080 lagging and unity
[28128]
44 T H E DG S I Z I N G PROBLEM A NONLINEAR CONSTRAINED
O P T I M I Z A T I O N PROBLEM
Optimization can be defined as the process of minimizing an objective function while
satisfying certain independent equality and inequality constraints The target quantity
that is desired to be optimized minimized or maximized is called the objective function
A general constrained optimization problem is mathematically expressed as in (413)
Minimize f(x) xeR
subject to hj(x) = 0 = l2m
gj(x)lt0 j = l2p (413)
X~ lt X lt X(+
X mdash ^Xj X^ bull bull bull Xn J
where ( x ) h((x) and g (x) are the objective function and the imposed equality and
inequality constraints respectively x is the vector of unknown variables and m is less
than n Whenever the objective function andor any function of the equality and the
inequality constraints sets is nonlinear the optimization problem is classified as a
nonlinear optimization problem The DG sizing problem is a nonlinear constrained
optimization problem that minimizes the real power losses subject to both equality and
inequality sets of constraints All elements of the DG sizing optimization problem
functions ie objective equality and inequality are both continuous and differentiable
The DG sizing optimization problem can be written in vector notation as
94
Minimize m(x) xeR
subject to h(x) = 0
g(x)lt0 (414)
X lt X lt X+
X = L ^ l X2 raquo bull bull bull 5 bullbulllaquo J
where ^ (x ) ls t n e DS real power losses The objective function variables vector x
encompasses dependent (state) and independent (control) variables The DS complex
voltage magnitudes and angles are examples of the former type of variables while the
DG (or multiple DGs) real and reactive output power as well as the DGs power factor
are variables of the latter type Eq (414) shows that the problem solution feasible set is
closed and bounded That is the solution vector feasible set is bounded between upper
and lower real values and also includes all its boundary points
Nonlinear constrained optimization problems are dealt with in the literature using
direct and indirect methods Indirect methods transform the constrained optimization
problem into an unconstrained optimization problem before proceeding with a solution
Therefore they are referred to as Sequential Unconstrained Minimization Techniques
(SUMT) Such methods augment the objective function with the constraints through
penalty functions and transform the new objective function into an unconstrained
optimization problem and solve it accordingly The penalty functions are presented to
penalize any constraint violations On the other hand direct solution methods deal
explicitly with the nonlinear constraints when solving the constrained nonlinear
optimization problems The exterior penalty function method and the Augmented
Lagrange Multiplier (ALM) method are examples of SUMT while the Sequential Linear
Programming (SLP) Sequential Quadratic Programming (SQP) and Generalized
Reduced Gradient (GRG) methods are examples of direct methods Schittkowski [129]
and Hock and Schittkowski [130] tested the SQP algorithm against several other methods
like SUMT ALM and GRG using an excessive number of test problems and found out
that it outperformed its counterparts in terms of efficiency and accuracy
Most general purpose optimization commercial software utilizes the SQP algorithm
in solving a large set of practical nonlinear constrained optimization problems due to its
excellent performance [131] MATLABreg [132] SNOPTreg NPSOLreg [133] and SOCSreg
95
[134] are examples of commercial software that utilize the SQP method in solving large-
scale nonlinear optimization problems The DG sizing problem is handled via SQP
methodology that solves the original constrained optimization problem directly
45 THE CONVENTIONAL SQP
The following SQP deterministic optimization method material presented in this section
is based on references [129135-142]
The SQP method deals with the constrained minimization problem by solving a
Quadratic Programming (QP) subproblem in each major iteration to obtain a new search
direction vector d The search direction obtained along with an appropriate step size
scalar a constitutes the next approximated solution point that would be utilized in
starting another major SQP iteration The new feasible solution estimate point x(+1) is
related to the old solution point x( through the following relationship
x ( w ) = x W + A x W
xlt i + 1gt=xlaquo+adlaquo ( 4 1 5 )
where k is the iteration index For xlt4+1) to be accepted as a feasible point that would start
a new SQP iteration the objective function evaluated at the new point must be less than
that evaluated at the preceding one Eq (415) can be rewritten in an individual
component notation as
x^=xf+akdf
The SQP algorithm has two stages the first is finding the search direction via the QP
subproblem and the second is the step size (or length) determination via a one-
dimensional search method
451 Search Direction Determination by Solving the QP Subproblem
In the QP subproblem a quadratic real-valued objective function is minimized subject to
linear equality and inequality constraints The QP subproblem at iteration k is formulated
by using the second-order Taylors expansion in approximating the SQP objective
function and the first-order Taylors expansion in linearizing the SQP equality and
i = l2 raquo (416)
96
inequality constraints at a regular point x(k) A regular point is a solution point where
both equality and active inequality constraints are satisfied and the gradient vectors of
the constraints are linearly independent ie gradients are not to be parallel nor can they
be expressed as a linear combination of each other By employing the curvature
information provided by the Hessian (H) matrix in determining the search direction the
SQP algorithms rate of convergence is improved The QP subproblem is formulated as
Minimize xeK
subject to h(x) = 0
g(x)lt0
x lt x lt x
Approximation bull H
where
Vtrade(xw)
d
fiW
Vh(x(i))
~(k)
Vg(xlaquo)
Minimize xsH
subject to
rW laquo V 1 i ( ) m ( x ^ + V w t (x w ) d + ^ d l F d
h w ( d ) h ( x w ) + Vh(xw)d = 0
g w (d ) g (x w ) + Vg(xW)dlt0
x lt x lt x
(417)
gradient of the objective function at point x w
(laquox l ) search direction vector
(nxri) Hessian symmetric matrix at point x w
first-order Taylors expansion of the equality constraints at point xw
(nm) Jacobian matrix of the equality constraints at point xw
first-order Taylors expansion of the inequality constraints at point xw
(np) Jacobian matrix of the inequality constraints at point xw
Equation (417) is rewritten in component notation as follows
Minimize ^ ( x ) w + xeR x~ dx
-j[d d2 J lx= fi)
cbc
dn
d
dxbdquo v laquo
97
subject to h(x)
K (x)
+
x=xlaquo
d (x) dh^ (x)
dxx dXj
d (x) 5^ (x)
dx2 dx2
d (x) 5jj (x)
g laquo
ftW
+
laquo
5xbdquo
3amp(x) cbCj
^ ( x ) dx2
fc00
abdquo
3g2(x) dxi
3g2(x)
a2
5g2(x)
dx
^ (x) 3x2
^ m (x) dxn
x=xlaquo
A
= 0
3xbdquo 9xbdquo
lt9xj
Sgp(x)
Sx2
5g(x)
5xbdquo x=x
J2
d - n _
lt0
where the columns of Vh and Vg matrices represent the gradients of equality and
inequality functions
4511 Satisfying Karush-Khun-Tuker Conditions SQP methodology solves the nonlinear constrained problem by satisfying both the
Karush-Khun-Tuker (KKT) necessary and sufficient conditions That is at an optimal
solution both KKT necessary and sufficient optimality conditions are to be met The
SQP solution method transforms the constrained nonlinear optimization problem to a
Lagrangian function and subsequently applies the KKT necessary and sufficient
conditions to solve for the optimal point that would achieve the minimum value of the
approximate objective function while satisfying all constraints
The SQP method applies the Lagrange multipliers method to the general constrained
optimization problem expressed in Eq (414) by first defining the problem Lagrange
function at a given approximate solution point xw then by applying KKT first-order
optimality conditions to the Lagrange function and finally by applying Newtons method
to the Lagrange function gradient to solve for the unknown variables
The Lagrange function is written in components and compact notations as follows
98
m p
pound (x X P) = f^ (x) + pound M (x) + J pgj (x) (418) bull=1 M
pound(x X p) = f^ (x) + 1 h(x) + plt g(x) (419)
where Xi and j are the individual equality and inequality Lagrange multiplier scalars X
and on the other hand are m-dimensional and 7-dimensional equality and inequality
Lagrange multiplier column vectors h gh h g are the individual and vector
representations of the nonlinear constraints The Lagrange function is namely the
nonlinear objective function added to linear combinations of equality and inequality
constraints
The KKT first-order necessary conditions state that the Lagrange function gradients
at the optimal solution are equal to zero and by solving the necessary condition set of
equations the stationary points are obtained The KKT sufficient condition assures that
the stationary points are minimum points if the Hessian of the Lagrange function is
positive definite that is d H d gt 0 for nonzero d The KKT first-order-necessary
conditions are
V x r ( x A p ) = VxfiJi(x) + Vh) + VgP = 0 (420)
h(x) = 0 (421)
Pg(x) = 0 (422)
Pgt0 (423)
The SQP algorithm deals with inequality constraints by implementing the active set
strategy When solving for the search direction only active s-active and violated
inequality constraints are considered in that major iteration Inactive active s-active and
violated inequality constraints are expressed as follows
g(x)lt0 it A (424)
g(x) = 0 ieA (425)
gl(plusmn)pound0bxit g(x) + s gt 0 ieA (426)
ft()gt0 ieA (427)
where e is a predefined small tolerance number and A is the active set By using the
99
active set principle only the equality constraints and those inequality constraints that are
not inactive ie Eqs (425)-(427) will be included in the active set The Lagrange
multipliers in the Lagrange function that correspond to the inactive inequalities are set to
zero The resultant active set at iteration k will be included in the Lagrange function as
equality constraints and the optimization problem will be solved so as to satisfy the KKT
conditions In another SQP iteration eg k+r the active set elements might change that
is some of the previously inactive inequality constraints might become either active e-
active or violated inequality at the new approximate solution xk+r and consequently are
to be included in the new active set Conversely some of the previously active e-active
or violated inequality constraints in the preceding iterations active set might be dropped
off from the current SQP iterations active set list due to its present inactive status
Both the number of gradient evaluations and the subproblem dimension are
significantly reduced by incorporating the active set strategy which only includes a
subset of the inequality constraints in addition to the equality constraints The number of
the nonlinear equations to be solved in order to satisfy the KKT first-order necessary
conditions is
(n + m + a)
where
n is the number of the gradients of Lagrange function with respect to the solution
vector elements V^ pound(x I p) VXi Z(x k p) V^ pound(x I p)
m is the number of all equality constraints
a out of the original inequality constraints a is the number of inequality constraints
that satisfy Eqs (425)-(427) at the current iteration ie number of the active set
equations
By considering all the active set constraints the Lagrange function can be rewritten as
^(xAP) = m ( x W ) + h(xW) + Pg^(xW) (428)
where gA is the vector of the active inequality constraints at iteration k
KKT first-order optimal necessary conditions imply that the Lagrange function gradient
with respect to decision vector x and Lagrange multipliers X and p are equal to zero as
100
()
illustrated in Eq (429)
vxr(xAP) V x r (x ^ p ) =0 (429)
_vpr(xxp)_
The resultant nonlinear set of equations of the Lagrange gradients is expanded and
represented in components compact and vector notations as illustrated in Eqs (430)-
(432)
V ^ x ^ P )
Vx-(x)p)
()
mdash
0
0
0
0
0
0
0
0
_0_
KM 8AI()
SAIW
8M()
Vxr(xAP) h(x)
g^W
F(XltUlaquo
n+m+a)x
bull ( )
J(n+m+a)xl
pw) = o
= 0 (431)
(432)
4512 Newton-KKT Method The Newton method is utilized in order to solve the KKT first-order optimality condition
equations in (430) (431) or (432) By using Taylors first-order expansion at assumed
solution point to be an estimate of (xA|3 j the Newton-KKT method
is developed as follow
(x ( i U W P W ) + VF(xWAW pW)[AxW Alaquo AP () = 0 (433)
101
Vx^(x3p)
h(x)
g^O)
()
+ V h(x)
Ax
Ak
AP
()
= 0 (434)
V ^ ( x ) p ) Vh(x) Vg^(x)
Vh(x) 0 0
Vg^(x) 0 0
V2Mr(xAP) Vh(x) Vg^(x) Vh(x) 0 0
() Ax
Ak
gtP
()
= -
()
Vg^(x) 0 0
() Ax
AX
gtP
()
= -
Vxr(xX h(x)
V^(x) + Vh(x)X + Vg^(x)P
h(x)
g ^ laquo
(435)
(k)
(436)
V^(xXP) Vh(x) Vg^(x) Vh(x) 0 0
V g raquo 0 0
w x(k+l) _x(k)
p(+l)_p()
VWi(x) + Vh(x)X + Vg^(x)p
h(x)
() (437)
Eq (437) can be further simplified hence the Newton-KKT solution is expressed as
V ^ x ^ p ) Vh(x) Vg^(x)
Vh(x) 0 0
Vg^(x) 0 0
(k) - d w jj+l)
p(+0
= -
v^00 h(x)
s^x) _
-()
(438)
The new vector ldw X(A+1) p(t+1l obtained by solving the Newton-KKT system is the
solution of the QP subproblem It gives the search direction and new values for the
Lagrange multipliers in order to be utilized in the next iteration It is worthwhile to
mention that the search direction obtained would be the QP subproblem unique solution
if the KKT sufficient conditions are satisfied ie Vj^pound(xgtp) is a positive definite as
well as both constraint Jacobians Vh(x) and Vg(x) are of full row ranks ie
constraint gradients are linearly independent
Expanding Eq (438) results in the following formulae
VBPL (x(i)) + V ^ (x X p f dW + Vh(x(t))X(ft+1) + Vg^ (xW)P(t+I) = 0
h(xW) + Vh( (x ( t ))dw = 0 (439)
g^(xW) + V g ^ ( x laquo ) d laquo = 0
It can be seen that Eq (439) is the solution for the QP subproblem mathematically
102
expressed in Eq (440) which minimizes a second-order Taylor expansion of the
Lagrange function over first-order linearized equality and active inequality constraints
Minimize xeE
subject to
Wi(xw) + V^(xlaquo)d + idV^(iXgtpf)d
h(d w ) h (x w ) + Vh (x w )d w =0
^ ( d W ) g ^ ( x W ) + Vg^(
(440) ^ ( d W ) g ^ ( x ( V V g ( x laquo ) d laquo = 0
J x lt x lt x
where V^JC- (xXp) is the Hessian of the Lagrange function and is expressed in Eq
(441) Since the Lagrange function is the objective function in the SQP method the SQP
method is also called the projected Lagrangian method
a ^ x ^ p ) d2ltk)(xip) d2ltkxxV) dx2
a^O^P) dx2dx1
d2^k)X$) dxndxx
dxxdx2
a2^(x^p) dx2dx2
d2^k)(XV) dxndx2
dx1dxn
mk)(w) dx2dxn
Mk)(hD dx2
n
(441)
4513 Hessian Approximation The KKT second-order sufficient condition requires that besides being positive definite
the Hessian of the Lagrange function is to be calculated in every iteration Evidently the
explicit calculation of the second-order partial derivative of the Lagrange function ie
the Hessian matrix is cumbersome and rather time consuming to calculate Therefore the
quasi-Newton method is used instead Rather than explicitly calculating the Lagrange
function Hessian matrix the second-order partial derivatives matrix is approximated by
another matrix using only the first-order information of the same Lagrange function
Moreover the Lagrange function first-order information can be obtained using the finite
difference approximation method ie forward backward or central approximation This
approximate Hessian is updated iteratively in every major iteration of the SQP process
starting from a positive definite symmetric matrix
BFGS is a well known quasi-Newton method for approximating and updating the
103
Hessian matrix The four letters in the BFGS formula correspond to the last names of its
developers Broydon Fletcher Goldfarb and Shanno The BFGS formula was further
modified by Powell to ensure the Hessian symmetry and positive defmiteness during the
iterative process The modified BFGS approximation is expressed by
H(+0 _ | |() | w w H Ax Ax H Axlaquowlaquo AxlaquoHlaquoAxlaquo C -
where
H the approximate of Lagrange function Hessian matrix V ^ (xX p)
Ax the change in solution point vector Ax = akltvk
y The change in the Lagrange functions between two successive iterations
yW =VZ ( i+ )(xAp)-V^ )(xAp)
w wk)=ekyk)+(l-dk)H
k)Axk)
1 Ax W y W gt02Ax W HlaquoAxlaquo
0= 08(AxlaquoHWAxW)
[AxWHlaquoAxW)-(Axlaquoylaquo otherwise
The second and third terms in the BFGS formula are the Hessian update matrices
while the ^-dimension identity matrix is its initial start As noted from the BFGS
formula only the change in the solution points in two successive SQP iterations along
with the change in their corresponding Lagrange function gradients are employed in
approximating the Hessian Lagrange function
452 Step Size Determination via One-Dimensional Search Method
Once the QP subproblem in the SQP kx iteration yields a search direction the transition
to a new iteration k + 1 will not inaugurate until a search for a suitable step size is
performed in order to enhance the change in the decision variable vector making it yield
a better feasible point That is between the SQP old and the new QP subproblem
solution points the attempt to find a step length that would lead to an improved decision
point will take place
104
The procedure of determining the step length scalar is called a line or one-
dimensional search which tries to find a positive step size a that would minimize an
appropriate merit or descent function over both equality and inequality constraints The
line search as an iterative procedure demands the descent function evaluated at the new
computed step size be reduced further until the reduction value is less than or equal a preshy
selected tolerance
Two types of line search procedures are available in the literature exact and inexact
line search methods Examples of the exact line search methods are golden section and
quadratic and cubic polynomial interpolation methods Exact line search methods
especially for large scale engineering problems are often criticized for excessive
computational efforts and consequently are time consuming Inexact line search methods
assure sufficient decrease in the descent function during an iterative process Such
methods attempt to produce an acceptable step size not too small and not too large
while searching for the optimum a
A descent function used to test the step size obtained is in general a combination of
the optimization objective function and other terms that penalize any kind of constraint
violation In other words the descent or merit function is a trade-off between the
minimization of the objective function and the violation of the imposed constraints
Practical descent functions such as those proposed by Han [143] and Powell [144] and
Schittkowski [145] are widely implemented in SQP solution methods
453 Conventional SQP Method Summary In summary the SQP algorithm models the Lagrange function of the constrained
nonlinear optimization problem by a QP subproblem The transformed subproblem is
solved at a given approximate solution xk to determine a search direction at each major
iteration The step size a calculated by minimizing a descent function along the search
direction is joined with the QP subproblem solution to construct a new iterate with a
better solution xk+x The process is repeated iteratively until an optimal solution x is
reached or certain convergence criteria are satisfied Figure 41 shows the conventional
SQP algorithm in simplified steps SQP is also sometimes called Recursive Quadratic
Programming (RQP) or Successive Quadratic Programming In a nutshell the SQP
105
solution method is not a single algorithm but rather a sophisticated collection of
algorithms that collaborate endeavoring to search for an optimal solution that minimizes
a nonlinear objective function over both equality and inequality nonlinear constraints
106
The Conventional SQP Algorithm
1- State the constrained nonlinear programming problem by defining the foil owing
Minimize fwi(x)
subject to h(x) = 0
g(x)fpound0
x lt x lt x
X = [j X2 Xn ]
2- Set SQP Iteration counter to k=0 Estimate initial values for the following
1- Solution variables x(0) A(0) and p(0gt
2- Convergence tolerance E-I
3- Constraints violation tolerance e2
4- Hessian matrix n-dimensional Identity matrix 3- Form The QP subproblem
a- State the objective function and the equality and inequality constraints as follows
i- Objective function ^ = pound lf-pf- ^ ^ c o s ^ -9+y
wfl [i = 2 3
^^G-^-^l^oos(8-8y-y)=o j lt = u NB
Aef-ei-^poundf(sin(8-8-Ti) = 0
bull Equal ity constrai nt functions
NB
NB-
1 = 23 NB
= NBNB + 2NB-2
iii- Inequality constraint functions I
Vtrade ltVb ltVtrade 1 = 23JVB
4 ltlt ltlt5trade i = 23 Areg
PmT ^ J00 pound gtm^ ( = 12 npoundgtG
sSASjltsr
b- Evaluate w i(xlt) V ^ f x ) h(xgt) g(xltraquogt) Vh(xm) Vg(xlt) c- Apply active set strategy The inactive inequal ity constraints in this iteration are not to be considered d- Formulate the Lagrange function
r(xXp) = Bpl(x()) + lh(^ )) + P^(x ( ))
e- Obtain a new search direction d(k) by solving the following QP subproblem
Minimize RPi(x( i )) +V^L (xlaquo)d + ~ d V ^ ( x J p f d
subject to h(dw) = h ( x w ) + V h W = 0
iAdW) = g4(W) + Vg^(x w )d w lt 0
x lt x lt x
The QP subproblem solution is obtained by solving the following Newton-KKT system
V ^ ( x X P ) Vh(x) Vg^(x)
Vh(x) 0 0
Vg^(x) 0 0
4- Check if all stopping criteria are satisfied ie ||d|k)||Spoundi then STOP otherwise continue
5- Determine an appropri ate step size ak that would cause a sufficient dec rease in a chosen merit function
6-Setx (k+1)=x (k )+akd (k )
7- Update the Hessian matrix H = V^zr(xXp) using the modified BFGS updating method
ltgt bull d w bull
iltgt
p(raquolgt = -
v^W h(x)
fc00
Hgt H^WW1
8- Update the counter k=k+1 and GOTO step 3
Figure 41 The Conventional SQP Algorithm
107
4 6 FAST SEQUENTIAL QUADRATIC PROGRAMMING (FSQP)
The nonlinear power flow equality constraints in the DG sizing problem are a mixture of
nonlinear terms and trigonometric functions as shown in Eqs (46) and (47) When
solving the DG sizing problem via the conventional SQP such equations are linearized
and augmented to the Lagrange function Their Jacobian matrix as well as their
corresponding elements in the Hessian matrix are evaluated and updated during each
major iteration in the SQP algorithm These computationally expensive operations result
in longer execution times for the problem to converge
In Chapter 3 of this thesis a FFRPF method was developed for strictly radial weakly
meshed and looped distribution networks The FFRPF solution method is employed in
solving the power flow equality constraints that govern the DG-integrated DS The
developed distribution power flow method is incorporated as an intermediate step within
the SQP algorithm and consequently eliminates the use of the derivatives and their
corresponding Jacobian matrix in solving the power flow equations since it mainly relies
on basic circuit theorems ie Ohms law and Kirchhoff voltage and current laws The
cause-effect relationship between installing one or more DGs in a DS and its
corresponding resultant complex bus voltage state variables is exploited in developing a
Fast SQP (FSQP) algorithm to solve for the optimal DG size
For single and multiple DGs to be installed in the DS the variables to be optimized
in the conventional SQP and the proposed FSQP algorithms for solving its corresponding
nonlinear constrained programming problem are as follows
For single DG with specifiedpf case
= K - VSBgt laquoi - ampmgt DGJ[ (443)
For single DG with unspecifiedpf case
= Pigt - Vm 8bdquo - 5NB DGsize pfm] (444)
For multiple DGs with specifiedpfs case
i = fr - Vm 815 - 5 ^ DGV raquo DGnDG] (445)
For multiple DGs with unspecified case
108
where
laquoDG total number of DGs
nuDG total number of the unspecified pf DGs
The search space of the solution vector x is defined as x e M1 and its dimension
i-e- dimension s obtained according to the following
xdimension = ( 2 bull N B + 2 bull (No- o f unspecified DGs) + 1 bull (No of specified DGs)) (447)
During the QP subproblem iterative process where the search direction finding
procedure is taking place the FFRPF technique is employed to solve the DG-integrated
DS power flow to obtain its corresponding bus complex voltage profiles That is in the
kth iteration of the SQP method the QP subproblem starts with a new solution point x(
and obtains the DG-integrated DS voltage profiles by utilizing the FFRPF algorithm The
FFRPF solution within the current QP subproblem is actually based on the DG size and
power factor proposed by current iterate of xreg The DS voltage profiles are then passed
to the QP subproblem as a set of simple homogeneous linear equality constraints along
with the imposed nonlinear inequality constraints in order to determine a better search
direction d(k) The FSQP iteration k equality constraints are simply the vector difference
between the current FFRPF bus voltage profiles obtained and the FSQP estimated
complex voltage values The FSQP equality constraints at the A iteration are formulated
as follows
K K
h nNB
h
h nNB+2
_ 7NB _
() X
x2
XNB
XNB+
XNB+2
X2NB
() V y FFRPF M
^FFRPFb2
yFFRPF bNB
FFRPF M
FFRPF b2
^ FFRPF bNB _
() o 0
0
0
0
0
(448)
where
FFRPF A voltage magnitude of bus i obtained by the FFRPF technique
109
ampFFRPF bull vdegltage phase angle of bus i obtained by the FFRPF technique
The expanded form of the linear equality constraints shown in Eq (448) can be rewritten
in vector notation as
hW[^LD-[VtradegL=raquo (4-49gt It is worth mentioning that the equality constraints introduced by the FFRPF to the QP
subproblem are linear functions ie without any trigonometric or nonlinear terms These
linear equality constraints will contribute a (n x m)-dimension matrix with a unity main
diagonal elements U that replaces Vh(x) in the QP subproblem Newton-KKT system
shown in Eq (438) as illustrated in Eq (450) That is in each QP subproblem
formulation the time consuming Jacobian evaluation of the nonlinear equality constraints
is avoided and a constant real matrix is utilized instead
~Vlr(xlV) U Vg^(x)
U 0 0
Vg^(x) 0 0
The FSQP is concluded once both necessary and sufficient KKT conditions as well
as other stopping criteria are satisfied Otherwise the FSQP process continues by
performing a line search to find an appropriate step size aamp that would cause a sufficient
decrease in the utilized merit function Both a and d ( are combined to predict the next
estimate of the solution point x ( W ) Subsequently the Lagrange function Hessian matrix
is updated by the modified BFGS to start a new FSQP iteration
In the next FSQP algorithm iteration the new solution point x( i+1 includes an
updated estimate of the DG size and its corresponding power factor The equality
constraints in the new QP subproblem will be again solved by the developed FFRPF
technique based on the new DG parameters presented by x( +1) and on the new state
variables estimate as the new FFRPF flat start bus voltage variables In other words the
equality constraints function formulation is dynamic they are different in each iteration
Each FSQP iteration has its updated version of the equality constraints based on the new
estimate of the DG parameters in the solution vector obtained
In Chapter 3 the FFRPF was proven to use less CPU time than any other
w d w
^(+l)
laquo(+)
= -
VWL(x) h(x)
g^w
w (450)
110
conventional and distribution power flow method since it is a matrix-based methodology
and relies mainly on basic circuit theorems The FSQP is a hybridization of the
conventional SQP algorithm and the developed FFRPF solution method By solving the
highly nonlinear equality constraints via the developed radial distribution power flow as a
subroutine within the conventional SQP structure the reduction of CPU computational
time was a plausible merit and a noticeable advantage Figure 42 shows the detailed
steps of the FSQP algorithm
I l l
The Fast SQP (FSQP) Algorithm 1- State the constrained nonlinear programming problem by defining the following
Minimize xeR
subject to
2- Set SQP Iteration counter to k
AraW
h(x) = 0 g(x)lt0
x lt x lt x
x = [xbdquox2xbdquo]
=0 Estimate initial values for the following
1- Solution variables x1 A(u) and p1 2- Convergence tolerance e 4- Hessian matrix n -dimensional Identity matrix 3- Constraints violation tolerance pound2
3- Form The QP subproblem
a- State the objective function and the equality and inequality constraints as follows
i- Objective function fmL (V d) = ]T JT 9 y t [ ( f + vf - 2V VJ cos(S - Sj)]
ii- Equality constraint functions bull Call the FFRPF method subroutine with the DG installed on the selected location bull The DG size value is substituted from the current iterate of x bull Solve the FFRPF accordingly to obtain the DS voltage profiles vector VFFRPF bull The equality constraints are formulated as follows
x2
XNB-l
XNB
XNB+1
XWB-1^
[) VI 1 FFRPFh
v 1 FFBPFh
v 1 WFRPF^
regFFRPFbt
degFFWFtl
degFFRPFM
- ) 0
0
0
0
0
0
iii- Inequality constraint functions
Vtrade lt Vhi i Ktrade i = 23 NB
Sf ZS^ZSZ 1 = 23NB
Pfpound s Pff Pfpound = U bull bull bull nDG MDG
b-Evaluate m(xlaquogt) Y ^ x 1 ) h(xgt) g(x(laquo) Vg(xlt) c- Apply active set strategy The inactive inequality constraints in this iteration are not to be considered d- Formulate the Lagrange function
r(xiP) = w i (xlaquo) + gth(xlaquo) + P^ ( x W )
e- Obtain a new search direction dltk) by solving the following QP subproblem
Minimize I 6 R
subject to
^ ( x ) + V^(x( i ))d + i d V ^ ( x J flf d
h ( d w ) = h (x w ) + U d ( ) = 0
^ ( d ( ) = g ^ ( ^ ) + Vg^(xltgt)dW lt0 x lt x lt x
The QP subproblem solution is obtained by solving the following Newton-KKT system
V 2bdquo^(xJ P) U V g ^ x )
U 0 0
Vg^(x) 0 0
() d w J_(raquo+l)
Q ( - H )
= - h(x)
84 0 0
i()
4- Check if all stopping criteria are satisfied ie Ild^yse then STOP otherwise continue
5- Determine an appropriate step size ak that would cause a sufficient decrease in a chosen merit function
6-Set xltk1) = x(k)+akd(lcgt
7- Update the Hessian matrix H = V^ (x X p ) using the modified BFGS updating method
Hlt H W A X W A X ^ H
Axww l A x w H w A x w
8- Update the counter k=k+1 and GOTO step 3
Figure 42 The FSQP Algorithm
112
47 SIMULATION RESULTS AND DISCUSSION
Incorporating single and multiple DGs at the distribution level is investigated using two
DSs The DG sizing nonlinear constrained optimization problem was solved using both
the SQP and the FSQP algorithms Using the APC search process optimal DG sizing is
computed via SQP and FSQP for all possible bus combinations and CPU computation
time was recorded for each case The simulations were carried out at a dedicated
personal computer that runs only one simulation at a time with no other programs running
simultaneously Moreover the PC is rebooted after each simulation operation Such
measures were assured during the experimentations of both SQP and FSQP solutions in
order to make the record of consumed CPU time as realistic as possible The time saved
by the proposed FSQP method is computed as follows
Time Saved By FSQP = SregPme ~ FSQPtttrade x 100 (451) SQPtime
Simulations were carried out within the MATLABreg computing environment using an
HPreg AMDreg Athlonreg 64x2 Dual Processor 5200+ 26 GH and 2 GB of memory desktop
computer
471 Case 1 33-bus RDS The first test system is a 1266 kV 33-bus RDS configured with one main feeder and
three laterals with a total demand of 3715 kW and 2300 kvar The detailed system data is
provided in the appendix [116] A single line diagram of the 33-bus system is shown in
Figure 43 The constrained nonlinear optimization DG sizing problem for the 33-bus
RDS is solved using both SQP and FSQP methodologies To search for the optimal
location to integrate single and multiple DGs into the distribution network the APC
method is utilized in the investigation
113
Substation
19
20
21
22
26
27
28
29
30
31
32
33
4 _
5 mdash
6 ^
7
8
9
10
11
12
13 14
15
16
17
mdash 2 3
mdash 2 4
_ 2 5
bull18
Figure 43 Case 1 33-bus RDS
4711 Loss Minimization by Locating Single DG A single DG is to be installed at 33-bus RDS with unspecified power factor by using the
APC method The APC procedure was performed by installing a single DG at every bus
and the optimal DG size that minimized the real power losses while satisfying both
equality and inequality constraints were presented That is all combinations were tried to
find the optimal location for integrating a DG unit with an optimal size
The optimization variables in the deterministic methods utilized ie SQP and FSQP
are the RDS bus complex voltages the DG real power output and its corresponding
power factor The number of variables optimized in the 33-bus RDS constrained single
unspecified pf DG sizing optimization problem is 68 variables Table 41 shows the
single DG unit optimal size and location profiles as well as the CPU execution time for
the two deterministic solution methods Both SQP and FSQP procedures resulted in the
same solutions and both obtained the optimal DG size and its corresponding power factor
to be 15351 kW and 07936 respectively However as shown in the same table the
FSQP algorithm used much less time than that consumed by the SQP algorithm Table
42 shows the values of all the DG optimal size and power factors and their
corresponding real power losses at all the tested system buses Figure 44 shows the
114
corresponding real power losses for placing an optimal DG size at each of the test system
buses This confirms that system losses may increase significantly with the installation of
DG at non-optimal locations Placing the DG at bus 30 yielded the least real power
losses while satisfying all the constraint requirements If bus 30 happened to be
unsuitable for hosting the proposed DG unit the same figure shows alternative bus
locations with comparable losses Figure 45 shows the relation between the DG power
factor and real power losses for each corresponding optimal DG rating at bus 30 By
installing a DG with an optimal size at an optimal location the RDS voltage profiles are
improved as shown in Figure 46
It is noted that by installing a single DG in the 33-bus RDS the real power losses are
reduced from 210998 kW to 715630 kW This is a 6608 reduction in distribution
network losses By installing the single DG in the system the co-norm of the deviation of
the system bus voltage magnitudes from the nominal value IIAFII = max (|Fbdquo-K|)
was reduced from 963 to 613 in the unspecifiedcase and to 586 in the fixed
case
Table 41 Single DG Optimal Profile at the 33-bus RDS
No of Combinations
SQP Method CPU Time (sec)
FSQP Method CPU Time (sec)
Single Run
APC
Single Run
APC
Optimal Placement Bus Optimal DG Size (kW) Optimal DG Power Factor Minimum Real Power Losses (kW)
W x (pu)
Single DG Profile-Unspecified pf
C =32 32 -l J Z
35807
925390
06082
21067
30 15351 07936 715630
00613
115
Table 42 Optimal DG Profiles at all 33 buses
Bus No
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
D G P (kW)
19580000
19356000
19254000
19158000
18968000
18963000
18029000
15808000
14178000
13927000
13456000
11879000
11388000
10877000
10262000
9340800
8862300
17189000
4824400
4255600
3377700
19362000
17211000
13070000
18961000
18954000
18405000
16396000
15351000
13677000
13163000
12581000
D G Q (kvar)
12189000
12072000
12018000
11967000
11803000
11793000
11534000
9857700
8681400
8498500
8156000
7086200
6761900
6421200
6030900
5490600
5209900
10351000
2525800
2198900
1785800
12076000
9979200
7439600
11799000
11796000
11784000
11772000
11769000
11034000
10618000
10180000
PLoss (kW)
2010700
1561200
1357600
1166800
785090
776110
828280
888200
930810
938760
955900
1019800
1042700
1077300
1121400
1194900
1235700
2045200
2077100
2078700
2083100
1573500
1615700
1692500
771460
758250
732370
715670
715630
820270
857570
910130
A F (pu)
00946
00858
00794
00727
00563
00492
00459
00505
00539
00544
00554
00587
00597
00608
00621
00640
00650
00948
00958
00959
00960
00858
00871
00893
00563
00563
00570
00598
00613
00645
00657
00671
D G Power Factor
08489
08485
08483
08481
08490
08492
08424
08485
08528
08536
08552
08588
08599
08611
08621
08621
08621
08567
08859
08884
08841
08485
08651
08691
08490
08490
08422
08123
07936
07783
07784
07774
116
13 17 21
33-Bus RDS Bus No
33
Figure 44 Optimal real power losses for placement of optimal DG size at all the 32 buses using APC method
02 03 04 05 06 07 08 09
DG Power Factor at Bus 30
Figure 45 Optimal real power losses vs different DG power factors at bus 30
117
bull No DG installed bull Single DG at Bus 30
13 17 21
33-Bus RDS Bus No
33
Figure 46 Bus voltages improvement before and after installing a single DG at bus 30
4712 Loss Minimization by Locating Multiple DGs Installing a single DG can enhance different aspects of the RDS However multiple DGs
installations can further improve such aspects The multiple DG optimal sizing
constrained problem is solved using both deterministic methods SQP and FSQP
procedures The number of decision variables in the double DG three DG and four-GD
cases are 70 72 and 74 variables respectively The DG placement is carried out using
the APC search method The searching process investigates the real power losses by
placing a combination of two three and four DGs at a time in the tested 33-bus RDS
The number of combinations is found to be 496 4960 and 35960 for sitting the two three
and four DG units respectively Table 43 shows the optimal placement and sizing
results for the multiple DG cases which are investigated next
118
Table 43 Multiple DG Installations in the 33-bus RDS with Unspecified Power Factor
No of Combinations
SQP Method CPU Time
FSQP Method CPU Time
Single Run
APC
Single Run
APC
Optimal Placement Buses
Optimal DG Size (kW)
Optimal DG Power Factors
Minimum Real Power Losses (kW) AF a (pu)
Double DGs Profile
32C2=496
106770 sec
37150653 sec (619178 min)
12532 sec
6083348 sec (101389 min)
DG1 Bus= 14 DG2 Bus= 30
DG1P = 78841 DG2P= 10847
DG1 pf= 09366 DG2 pf= 07815
311588
0020675
Three DGs Profile
32C3=4960
136669 sec
550055760 sec (15 hrs 16758
min)
20681 sec
121133642 sec (3 hrs 21888 min)
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30
DG1P = 67599 DG2P = 35373 DG3 P = 84094
DGl= 09218 DG2= 09967 DG3= 07051
263305
0020477
Four DGs Profile
32 C4 =35960
184498 sec
350893908 sec 974705 hrs
(4 days 1 hr 26 min)
25897 sec
67509755sec (18 hrs 45180 min)
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30 DG4 Bus= 32
DG1 P = 6728343 DG2P = 3533723 DG3P = 5118179 DG4P = 3318699 DGl= 09201 DG2gt= 09968 DG3^= 06296 DG4= 08426
247892
0020474
Double DG Case By optimally sizing two DG units at the optimal locations (buses 14
and 30) in the 33-bus RDS the real power losses are reduced and consequently the
system bus voltage profiles are also improved Any other combination of locations
would not cause the real power losses to be as minimal The total power losses are
reduced from 210998 kW prior to DG installation to 3115879 kW which represents an
8523 reduction With respect to the single-DG case the real power losses were
reduced from 715630 kW to 3115879 kW Thus by installing a second DG the losses
were reduced by a further 5646 Figure 47 shows the 33-bus RDS voltage magnitude
comparisons among the original system single-DG and double-DG cases It is worth
mentioning that the deviation infinity norm of the voltage magnitudes after optimally
119
installing the DGs is reduced from 963 in the case of no DG and 613 in the single-
DG case to 207
bull No DG installed bull Single DG at Bus 30 A Double DGs at Buses 14 and 30
101
-3-Q
bulllaquo
i 3
I (0 E sectgt amp p gt
099-
097-
095 -
093 -
091 -
089
t i ^ - bull bull bull bull bull A A bull bull bull 11 bull bull
bull bull bull bull + bull bull bull bull
11 16 21
33-Bus RDS Bus No
26 31
Figure 47 Voltage profiles comparisons of 33-bus RDS cases
Table 43 also shows that the FSQP method executed the 33-bus RDS double-DG
APC installation procedure in one sixth the time that was consumed by the SQP method
By studying the 496 output results of the SQP method it was found that 15 out of the 496
combinations cycled near the optimal solution As a result those 15 combinations were
running until the maximum function evaluation stopping criterion was reached The
aforementioned combinations are shown in Table 44 On the other hand all 496 FSQP
combinations converged to their optimal DG size solution before reaching the maximum
function evaluation number This sheds some light on the robustness and efficiency of
the FSQP method of dealing with such situations
120
Table 44 SQP Method Double-DG Cycled Combinations
DG1 Bus
28
24
5
4
5
DG2 Bus
30
31
32
31
11
DG1 Bus
14
12
9
17
7
DG2 Bus
30
30
29
28
32
DG1 Bus
3
3
8
23
2
DG2 Bus
31
11
21
25
21
Three DG Case The distribution network real power losses in the three-DG cases were
reduced even more when compared to the double-DG case The loss reduction in the
three DG case was 8752 6321 1550 compared to the pre-DG single DG and
double DG cases respectively Figure 48 shows the improvement in the system voltage
profiles of the three DG case when compared to that of the pre-DG single-DG and
double-DG cases
The APC search process revealed that the three optimal locations for the three-DG
case are buses 14 25 and 30 In addition Table 43 shows that approximately 78 of the
CPU time was saved by the FSQP APC method compared to that of the SQP algorithm
Of the 4960 output results of the SQP method 226 combinations cycled near the optimal
solution On the contrary all 4960 of the FSQP method combinations converged to
optimal DG size solutions in less CPU time than that of the SQP procedure It can be
concluded therefore that the FSQP algorithm is faster in terms of CPU execution time
and more robust and efficient than the conventional SQP
121
bull No DG installed bull Single DG at Bus 30 A Double DGs at Buses 14 and 30 x Three DGs at Buses 14 25 and 30
101
099
mdash 097 dgt bulla
i O) 095 Q
s o ogt 8 093
gt 091
089
A A A A A A A
^ i i x x x x x bull
A A
X X
bull I f
bull
A A bull - 1 bdquo X IB R X X X
X X
bull bull bull bull
11 16 21
33-Bus RDS Bus No
26 31
Figure 48 Voltage improvement of the 3 3-bus RDS due to three DG installation compared to pre-DG single and double-DG cases
Four DG Case Additional installation of a DG at an optimal location also caused the
real power losses to decline The losses and the maximum voltage deviation from the
nominal system voltage are 58536 and 0015 less than those of the three-DG case
Such a percentage is to be investigated for its practicability by the distribution planning
working group when the decision to go from a three DG to a four DG case is to be made
Figure 49 shows the improvement of the bus voltages resulting from adding a fourth DG
unit to the distribution network Investigating the optimal locations for the four-DG case
took a very long time utilizing the SQP method ie in the vicinity of a four day period
compared to the proposed FSQP method which took approximately 18 hours
Fixed Power Factor DGs Simulations of the multiple DG cases were repeated but this
time the power factor was fixed at a practical value of 085 Table 45 shows the results
of all the optimal multiple DG installations with specified power factors The maximum
difference between the specified and the unspecified power factor cases with respect to
the real power losses is in the vicinity of 5 as depicted in Table 46 Moreover
choosing DG units of a specified power factor of 085 saved simulation CPU time when
compared to the unspecified cases Therefore it might be a practical decision to proceed
with such a suggested power factor value
122
Table 45 Single and Multiple DG Installations at Pre-specified 085 Power Factor
No of Combinations
SQP Method CPU Time
FSQP Method CPU Time
Single Run
APC
Single Run
APC
Optimal Placement Buses
Optimal DG Size (kW)
Minimum Real Power Losses (kW)
l|AK|L (pu)
Single DG Profile
C = 32 32 W bull-
2148 sec
567081 sec
050843
117532 sec
30
17795232
735821
00586
Double DGs Profile
32C4=496
45549 sec
13573060 sec (226218 min)
07691 sec 2761264 sec (46021 min) DG1 Bus= 14 DG2 Bus= 30
DG1P = 6986784 DG2P = 11752222
328012
00207
Three DGs Profile
32C4=4960
59627 sec
172360606 sec (4 hrs 472677 min)
14107 sec 37316290 sec
(2 hrs 21938 min) DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30
DG1 P = 6504360 DG2P = 3216023 DG3P = 9006118
293056
00202
Four DGs Profile
32C4 =35960
77061 sec 1420406325 sec
(394557 hrs) (1 days 15 hr 273439 min)
18122 sec 326442210sec
(9 hrs 40703 min) DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30 DG4 Bus= 32
DG1 P = 6280595 DG2P = 2751438 DG3 P = 4962089 DG4P = 4713174
277073
00199
Table 46 Loss Reduction Comparisons for all DG Cases
Single DG Case
Double DG Case
Three DG Case
Four DG Case
UnSpec pf DG
085 pf DG
UnSpec pfDG
085 pf DG
UnSpec pf DG
085 pf DG
UnSpec pf DG
085 pf DG
of Losses
Pre-DG Case
660836
654637
852327
844543
875210
861110
882515
868685
Single DG Case
mdash
mdash
564596
549873
632065
597843
653603
619776
Reduction Compared to
Double DG Case
564596
549873
mdash
mdash
154958
106569
204424
155297
Three DG Case
632065
584120
154958
106569
mdash
mdash
58537
54540
Four DG Case
653603
619776
204424
155297
58537
54540
mdash
mdash
123
bull No DG installed
x mree DGs at Buses 1425 and 30
bull Single DG at Bus 30
x Four DGs at Buses 142530 and 32
A Double DGs at Buses 14 and 30
102
I deg9 8
ogt bullo 3 096 E en n E 094 laquo S o 092
09
088
bull bull A A X X X X X
IK
bull bull
x x x
II
A laquo
X X bull
-flN ampbull X
x t 1 x x X x x
bull bull +
11 16 21
33-Bus RDS Bus No
26 31
Figure 49 Voltage profiles improvement in 33-bus RDS for all DG cases
472 Case 2 69-bus RDS The second distribution network investigated is a 69-bus RDS test case Figure 410
shows its corresponding single line diagram topology This practical system is derived
from the PGampE distribution network provided in [43] It encompasses one main feeder
and seven laterals with a total real and reactive power demand of 380219 kW and
269460 kvar respectively The substation is taken as a slack bus with a nominal voltage
of 1266 kV The constrained nonlinear optimization DG sizing problem for the 69-bus
RDS is performed utilizing both SQP and FSQP methodologies while the optimal DG
placement in the 69-bus RDS is investigated via the APC search process In subsequent
subsections locating and sizing single and multiple DGs in the tested network are
presented examined and analyzed
124
Figure 410 Case 2 69-bus RDS test case
4721 Loss Minimization by Locating a Single DG By installing an optimal sized DG at the most suitable bus in the distribution system the
real power losses will be minimal Thus the APC procedure was performed by installing
a single DG at every bus The network losses are computed according to the optimal DG
size obtained from the utilized deterministic solution methods Figure 411 shows the
corresponding real power losses of the installed optimal sized DG at all of the 68-buses
The figure shows that placing the DG at bus 61 has the minimal value of the objective
function It also shows near optimal bus locations for the DG to be installed as
alternative placements with comparable losses
125
ampuj -
200
f 175 2
I 150 (0 o - 1 125 o i o 100 a T5 _bdquo 2 75
50
25
0
bull bull bull bull bull bull bull bull bull bull bull
bull bull bull bull bull
bull
bull bull bull
bull
bull bull
bull bull bull bull
bull bull
bull bull
bull
bull
12 17 22 27 32 37 42 47 52 57 62 67
69-Bus RDS Bus No
Figure 411 Optimal power losses obtained using APC procedure
Results from locating and sizing a single DG unit in the 69-bus RDS are presented in
Table 47 The simulations were performed for two cases In the first case the DG
power factor was unspecified in order to investigate the optimal size of the proposed DG
in terms of its real power output and its corresponding power factor In the second case
the first case simulations were repeated with a proposed power factor value of 085 Both
the SQP and FSQP were utilized in the simulations The CPU time was obtained for
running the APC search process using both deterministic methodologies Results of the
proposed DG as well as the simulated CPU execution times are also shown in Table 47
In the first case of simulations the DG power factor as well as the DG size is
optimized during the real power loss minimization process By locating a single DG with
an output of 18365 at 083858 power factor at bus 61 the real power losses are
minimized from 225 kW to 23571 kW Integration of a single DG in the 69-bus RDS
with optimal size and placement causes the magnitude of the new network real power
losses to be 1048 of that of the original DS The main distribution substation output is
decreased from 4901206 kVA to 2711194 kVA in the unspecified power factor case and
to 2710846 kVA in the 085 power factor DG case This means that on average 45 of
substation capacity is released Such a release may be of benefit if the existing
126
distribution network is congested or desired to be expanded Figure 412 shows the
relation between the DG power factors against the real power losses for every
corresponding optimal DG rating The voltage profiles are also improved as one of the
benefits of installing the DG as shown in Figure 413 For example their deviation from
the nominal values is reduced from 908 to 278 in the unspecified case
In the unspecified power factor DG case the CPU execution time for finding the
optimal solution in a single simulation was 205434 seconds and that of the APC
simulations lasted for 191867 minutes respectively using the SQP optimization
technique By utilizing the proposed FSQP the execution time was significantly reduced
to 24871 seconds for calculating the single simulation and 13514 minutes for
performing the APC search method calculations The CPU execution time is reduced to
around 90 using the proposed FSQP method with the same exact results
In the second case it is assumed that the DG to be installed at bus 61 has a lagging
power factor of 085 The optimal DG size that kept the real power losses at a minimum
is 19038 kW Figure 414 illustrates the changes in the system real power losses as a
function of the bus 61 DG real power output The DG addition to the network improved
the voltage profiles and reduced the real power losses from 225 kW to 23867 kW This
is approximately a 90 decrease in the losses compared to the pre-DG case The
difference in terms of losses between the two single DG power factor cases (specified and
unspecified) is insignificant As a result choosing a specified power factor DG of 085
lagging is a practical decision to proceed with
127
Table 47 69-bus RDS Single DG Optimal Size and Placement Profiles
No of Combinations
SQP Method CPU Time
FSQP Method CPU Time
Single Run
APC
Single Run
APC
Optimal Placement Buses Optimal DG Size (kW) Optimal DG Power Factor Minimum Real Power Losses (kW)
AK M (pu)
Single DG Profile Unspecified pf
68^1 = 6 8
205434 sec
11511998 sec (191867 min)
21770 sec
810868 sec (13514 min) DGBus=61
DGP= 18365 DG= 08386
23571
002782
Single DG Profile Specified pf
68C =68
102126 sec
6761033 sec (112684 min)
15117 sec
396650 sec
DGBus=61 D G P = 19038 DG=085
23867
002747
01 02 03 04 05 06
DG Power Factor
07 08 09
Figure 412 Real power losses vs DG power factor 69-bus RDS
128
bull No DG Installed bull Single DG at Bus 61
I I
101
1
099
098
097
096
095
094
093
092
091
09
t bull raquo
bullbullbullbullbullbullbullbullbullbullbulllt
bullbullbull
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
Figure 413 Bus voltage improvements via single DG installation in the 69-bus RDS
C- 200 -
CO
sect 150 -_ l
5 ioo-
Q
2 50
0 -
^ ^ _ _ mdash mdash
I I I I
500 1000 1500
DG Power Output (kW)
2000 2500
Figure 414 Variation in power losses as a function of the DG output at bus 61
473 Loss Minimization by Locating Multiple DGs Sometimes the optimal single DG size is either unrealistic in physical size or smaller DG
alternatives are available at cheaper prices It is emphasized here that the total real power
129
of the multiple DGs is not to exceed that of the main distribution substation The APC
procedure is performed on the 69-bus RDS using both algorithms ie SQP and FSQP
methods and their corresponding CPU execution time is recorded The multiple DG
location and sizing optimization problem is investigated with fixed and unspecified
power factor DGs
Double DG case The CPU simulation time for an unspecified power factor case is
nearly twice that of the pre-specified case simulation This is because the number of the
optimization variables in the unspecified power factor is x e R142 while in the pre-
specified power factor case the number of variables to be optimized is decreased to
x e R140 Table 48 shows that the proposed FSQP CPU execution time is very fast
compared to the conventional SQP method The reduction in simulation time between
the two techniques is approximately 90 on average for both the specified and
unspecified power factor cases Installing double DG units caused the real power loss
value to drop to 1103 kW with an unspecified power factor and to 1347 kW with 085
DG power factor This is approximately a 95 reduction in losses compared to the
original system and a 43-53 reduction with respect to single DG cases In addition to
reducing the losses significantly the substation loading is reduced from 4901206 kVA to
1905919 kVA in the unspecified power factor case and to 1907828 kW in the 085
power factor DG case This means that around 61 of substation capacity is released
and can be benefited from in future planning Moreover the voltage profiles are
enhanced and maintained between acceptable limits
Figure 415 shows the improvement in the 69-bus RDS voltage magnitudes in the pre-
DG single-DG and double-DG cases Based on Table 48 the optimal size of the two
DGs have power factors of 083 and 081 Thus a power factor of 085 would be an
appropriate and practical choice with which to proceed
130
Table 48 Optimal Double DG Profiles in the 69-bus RDS
No of Combinations
SQP Method CPU Time
FSQP Method CPU Time
Single Run
APC
Single Run
APC
Optimal Placement Buses
Optimal DG Size (kW)
Optimal DG Power Factor
Minimum Real Power Losses (kW)
AF x (pu)
Double DGs Profile Unspec pf
68 C2 = 2 2 7 8
254291 sec
476977882 sec (13 hrs 14963min)
34446 sec
38703052 sec (1 hr 4505 lmin) DGBuses=2161 DG1 P = 3468272 DG2P= 15597838 DG1 pf= 08276 DG2= 08130
110322
001263
Double DGs Profile Specified pf
68 C2 =2278
123328 sec
256528600 sec (7 hrs 75477 min)
15814 sec
16291569 sec (271526 min)
DGBuses=2161 DG1P = 3241703 DG2P= 15836577
DGl=085 DG2 pf= 085
134672
001351
bull No DG Installed bull Single DG at Bus 61 A Double DGs at Buses 21 and 61
101
I
nitu
de
D) ra E
Vo
ltag
e
1
099
098
097
096
095
094
093
092
091
bullbullbullbull-
09
bull pound$AAAAAAAAAAAAAAAA bull bull bull bull bull a
A A A i j A lt
bull bull bull bull bull
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
Figure 415 Bus voltage magnitudes of the original 69-bus RDS and Single DG and
double DGs cases
131
Three DG case In this scenario the DG sizing constrained minimization problem is
performed using the conventional and the proposed deterministic methods Both methods
yielded the same solutions and proved that by integrating three DG units in the 69-bus
RDS the real power loss magnitude is decreased The proposed FSQP method CPU
simulation time is lower than that of the conventional SQP as shown in Table 49 The
same table also shows the three-DG integration profiles and their effect on both losses
and the 69-bus RDS voltage profiles The improvement regarding the system voltage
magnitudes is shown through Figure 416 It is found that the losses in the three-DG case
are less than that of the both single and multiple DG case However the losses incurred
by installing more than two DGs in the system did not reduce the real power losses
significantly The loss reduction caused by the multiple DG installations ranges from
436 to 58 when compared to the single DG cases When considering the pre-
specified and unspecified DG power factor cases between two and three DG installations
the difference in the amount of losses for each power factor case is in the vicinity of
couple of kilowatts Consequently one can argue that the decision to be made is whether
or not to proceed with installing more than two DGs Table 410 shows the real power
loss reduction comparison among all the DG installations in the system tested
It is worth mentioning that bus No 61 in the PGampE practical radial system is the
designated bus for placing a single DG as well as being a common placement bus in all
cases of multiple DGs By inspecting the considered RDS it is noted that this bus is the
site of the largest load of the system Since the objective target of installing DG(s) is to
minimize the real power losses such heavy loaded bus(es) are to be strongly
recommended for being DG candidate locations
132
Table 49 Optimal Three DG Units Profiles in the 69-bus RDS
No of Combinations
SQP Method CPU Time
FSQP Method CPU Time
Single Run
APC
Single Run
APC
Optimal Placement Buses
Optimal DG Size (kW)
Optimal DG Power Factor
Minimum Real Power Losses (kW)
AV x (pu)
Three DGs Profile Unspecified pf
68C3 =50116
363232 sec
12398664174 sec (14 days 8 hrs 244464 min)
49091 sec
1587661933 sec (1 day 20 hrs 61032 min)
DGBuses=216164 DG1 P = 3463444 DG2 P= 12937085 DG3P= 2661795 DG1 pf= 08275 DG2 pf= 08264 DG3 =07491
102749
00108798
Three DGs Profile Specified pf
68C3 =50116
172362 sec
5471670576 sec (6 days 7hrs 5945 lOmin)
25735 sec
580575800 sec (16 hrs 76266 min)
DGBuses=216164 DG1P = 3191431 DG2 P= 12883908 DG3 P= 2998994
DGl pf=QS5 DG2=085 DG3 p=085
126947
0012296
bull No DG Installed bull Single DG at Bus 61 A Double DGs at Buses 21 and 61 x Three DGs at Buses 2161 and 64
101
1
099
1 deg 98
bullsect 097
1 096 Dgt
| 095
O) 094
| 093
092
091
faasa
09
bull gt i i i i i i i K lt x raquo i _ bull bull bull bull bull bull bull bull bull
bullbullbullbull bull bull
bull bull
bull bull bull bull laquo bull bull raquo bull lt
bull bull bull bull bull
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
Figure 416 69-bus RDS bus voltage magnitudes improvement in all DG cases
133
Table 410 Loss Reduction Comparison for all DG Installations in the 69-bus RDS
Single DG Case
Double DG Case
Three DG Case
UnSpec pf DG 085 pf DG UnSpec pf DG 0857DG UnSpec pf DG 085 pf DG
of Losses Reduction Compared to Pre-DG
Case
895243 893927 950969 940147 954335 943581
Single DG Case
mdash mdash
531957 435738 564087 468106
Double DG Case
531957 435738
mdash mdash
68649 57363
Three DG Case
564087 468106 68649 57363
mdash mdash
474 Computational Time of FSQP vs SQP Optimizing the DG sizes located at the optimal buses of the 33-bus and the 69-bus RDSs
was executed twice in order to emphasize the time saved by implementing the FFRPF
into the conventional SQP ie FSQP The first instance was executed using the
conventional SQP which deals directly with highly non-linear power flow equality
constraints through gradients and their corresponding Jacobian matrices All the same
problems were again simulated using FSQP that incorporates the FFRPF to take care of
the distribution network power flow equality constraints It is found that by utilizing the
FSQP technique the execution time reduction in the 33-bus RDS case ranges from 75
to 88 when compared to the time it took the conventional SQP to converge For the
69-bus RDS the time saved by implementing the proposed FSQP method is 85 to 94
compared to that of the SQP method Table 411 and Table 412 show the time (in
seconds) saved by executing the proposed method for the 33-bus and 69-bus RDSs
respectively
134
Table 411 33-bus RDS CPU Execution Time Comparison
33-Bus RDS
Single DG
Double DG
Three DG
Four DG
pf=0Z5
Unspec pf
N)85
Unspec pf
pfplusmn0S5
Unspec
gtK)85
Unspec
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
SQP CPU Time (sec)
22623
612968
35807
925390
45549
13573060
106770
37150653
59627
172360606
136669
550055760
77061
1420406325
184498
3508939080
FSQP CPU Time (sec)
05637
144847
06082
210670
07691
2761264
12532
6083348
14107
37316290
20681
121133642
18122
326442210
25897
675097550
Time Saved BxFSQP
750816
763696
830145
772345
831147
796563
882626
836252
763413
783499
848678
779779
764836
770177
859637
807606
Table 412 69-bus RDS CPU Execution Time Comparison
69-Bus RDS
Single DG
Double DG
Three DG
pfrO5
Unspec
j^085
Unspec pf
pf=0Z5
Unspec pf
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
Single Run
APC
SQP
CPU Time (sec)
102126
6761034
205435
11511998
123328
2565286
254291
476977882
172361
5471670576
363232
1239866417
FSQP
CPU Time (sec)
15117
39665
21771
810868
15814
16291569
34446
38703052
25735
5805758
49092
1587661933
Time Saved
By FSQP
851979
941333
894027
929563
871774
936492
864541
918858
850691
893894
864847
871949
135
475 Single DG versus Multiple DG Units Cost Consideration In this chapter it was shown that by installing multiple DG units at different locations in
the tested DSs the active network losses were minimized and the system voltage profiles
were also improved From a practical point of view cost considerations have to be
considered when the decision is to be made whether to proceed with installing single or
multiple DG sources and the number thereof The decision maker needs to consider the
following
bull The direct (purchase) cost of a single DG unit vs the direct cost of the multishy
ple DG units
bull The cost of installing and decommissioning a single unit at single bus locashy
tions vs that of multiple units at different locations within the system
bull Suitability of bus site for installing DG This involves space and municipal
zoning constraints that may involve environmental and aesthetic issues
bull The cost of operating and monitoring a single unit vs multiple units dispersed
in the system
bull The cost of maintaining a single DG unit at one place vs maintaining multiple
units installed at different locations
Such cost considerations are part of any practical evaluation regarding installing single or
multiple DG units in the concerned distribution network Minimizing the real power
losses of the network and the overall cost as well as improving the voltage profiles are to
be considered when a practical judgment is to be taken In this study the objective is to
minimize the overall real power losses of the tested distribution network as well as
improve its voltage profiles
48 SUMMARY
In this chapter optimally placing and sizing single and multiple DGs at the distribution
level were considered and studied Comparisons between the installation of single and
multiple DGs with pre-specified and unspecified power factors were performed and
tested on 33-bus and 69-bus distribution networks It is confirmed that the real power
losses depend highly on both the DG location and its size Integrating the DG optimally
in the network reduced real power losses of the system to its optimum state improved the
136
voltage profiles and released the substation capacity allowing for future expansion
planning Multiple DG installations decreased the losses more than that of a single DG
installation However the losses reduced by installing more than two DGs in the 69-bus
RDS and more than three DG in the 33-bus RDS were comparable to those of the double
and triple DG installation cases respectively This chapter shows that beyond a certain
limit the decrease in power loss is insignificant furthermore DG integration may result
in unnecessary additional cost and possible technical difficulties From the perspective of
real power losses the results of installing single and multiple DGs with specified power
factors were practically comparable to the unspecified power factor DG installation
outcomes The reductions in power losses in the unspecified power factor cases were
insignificant when compared with their counterparts The proposed FSQP approach
reduced the computation execution time significantly
137
CHAPTER 5 PSO BASED APPROACH FOR OPTIMAL
PLANNING OF MULTIPLE DGS IN DISTRIBUTION NETWORKS
51 INTRODUCTION
This chapter presents an improved PSO algorithm HPSO to solve the problem of
optimal planning of single and multiple DG sources in distribution networks This
problem can be divided into two subproblems - determining the location of the optimal
bus or buses and the optimal DG size or sizes that would minimize the network active
power losses The proposed approach addresses the two subproblems simultaneously by
using an enhanced PSO algorithm that is capable of handling multiple DG planning in a
single run The proposed algorithm adopts the distribution power flow algorithm
developed in Chapter 3 to satisfy the equality constraints ie the power flow in the
distribution network while the inequality constraints are handled by making use of some
of the PSO intrinsic features To demonstrate its robustness and flexibility the proposed
algorithm is tested on the 33-bus and 69-bus RDSs Two scenarios of each DG source
are tested The first considers the DG unit with a fixed power factor of 085 while the
second has unspecified power factor These different test cases are considered to validate
the proposed metaheuristic approach consistency in arriving at the optimal solutions
52 PSO - THE MOTIVATION
Deterministic optimization techniques which traditionally are used for solving a wide
class of optimization problems involve derivative-based methods Momoh et al
[146147] reviewed and summarized most of these methods For these problems to be
solved by any of the deterministic methods their objective functions and their
corresponding equality and inequality constraints have to be differentiable and
continuous Derivative information is usually employed by deterministic methods to
explore local minima or maxima of the objective the function However unless certain
conditions are satisfied these techniques cannot guarantee that the solution obtained is a
global one Instead they are prone to be trapped in local minima (or maxima)
Expensive calculations and consequently increasing computational complexity pose other
impediments to deterministic optimization methodologies The need to overcome such
138
shortcomings motivated the development of metaheuristic optimization methods The
PSO method is the metaheuristic technique that is adopted in this chapter to solve the DG
sizing and placement problem in the distribution systems
The metaheuristic term has its roots in Greek terminology It is comprised of two
Greek words meta and heuristic The prefix term- meta is interpreted as beyond in
an upper level and the suffix word- heuristic stands for to find Metaheuristic
methods are iterative practical optimization methods that deal virtually with the whole
spectrum of optimization problems [148] They sometimes outperform their
deterministic methods counterparts Metaheuristic methods are non-calculus-based
methods that are capable of solving multimodal non-convex and discontinuous functions
Not only are they capable of searching for local minima but depending on the problems
searching space they are also capable of searching for global optimal solutions as well
[149] PSO ant colony optimization genetic algorithm and simulating annealing are
examples of the metaheuristic optimization class
53 PSO - AN OVERVIEW
The PSO method is a relatively new optimization technique introduced by Kennedy and
Eberhart in 1995 [150] Their initial target was to try to graphically simulate the social
behavior of birds in flocks and fish in schools during their search for food andor
avoiding predators Their work was influenced by the work of Reynolds [151] and
Heppner and Grenander [152] The former was interested in simulating the bird flocking
choreography while Heppner and Grenander developed an algorithm that mimics the
way birds fly together synchronously behave unsystematically due to external
disturbances like gusty winds and change directions when spotting a suitable roosting
area Kennedy and Eberhart noticed that birds and fish species behave in an optimal way
during the food hunt the search for mates and the escape from predators that mimics
finding an optimal solution to a mathematical optimization problem They also realized
that by modifying the Heppner and Grenander algorithm objective from a roost finding
goal to food searching the PSO can serve as new simple powerful and efficient
optimization tool
139
While the PSO was initially intended to handle continuous nonlinear programming
problems in 1997 Kennedy and Eberhart developed a version of PSO that deals solely
with discrete and binary variables [153] and discussed the integration of binary and
continuous parameters in their book [154] The PSO algorithm has advanced and been
further enhanced over the years becoming capable of handling a wide variety of
problems ranging from classical mathematical programming problems like the traveling
salesman problem [155 156] and neural network training [154 157] to highly specialized
engineering and scientific optimization problems such as biomedical image registration
[158] Over the last several years the PSO technique has been globally adopted to
handle single and multiobjective optimization problems of real world applications [159]
Moreover the PSO algorithm was even utilized in generating music materials [160]
Figure 51 shows the progress of PSO in terms of the number of publications in two
major databases the IEEEIET and ScienceDirect since the year 2000 References
[159 161-163] shed more light on recent advances and developments in the PSO method
BScienceDirect Data Base bull IEEEIET Data Base
1000 -I 900
ID 800
bullI 7 0deg SS 6 0 0 -
bullg 500-
pound 400
d 300 Z 200
100
H ScienceDirect Data Base
bull IEEEIET Data Base
2000
0
8
2001
2
10
bull^ 2002
5
31
bull 2003
4
64
J 2004
13
143
bull J 2005
23
217
1 J 2006
59
440
bull
J J 2007
106
647
bull bull bull
J I 2008
201
978
Publication Year
Figure 51 Number of publications in IEEEIET and ScienceDirect Databases since the year 2000
140
531 PSO Applications in Electric Power Systems PSO as an optimization tool is widely adopted in dealing with a vast variety of electric
power systems applications It was utilized as an optimization technique in handling
single objective and multiobjective constrained optimization of well-known problems in
power system areas such as economic dispatch optimal power flow unit commitment
and reactive power control to name just a few
El-Gallad et al used the PSO method to solve the non-convex type of the Economic
Dispatch problem (ED) In their work the practical valve-effect conditions as well as the
system spinning reserve were both incorporated in the formulation of the linearly
constrained ED [164] In [165] they incorporated the fuel types with the traditional ED
cost function and used the PSO method to solve a piecewise quadratic hybrid cost
function with local minima Chen and Yeh [166] also solved the ED problem with valve-
point effects using several modified versions of the standard PSO method Their
proposed PSO modifications mainly contributed to the position updating formula Kumar
et al [167] and AlRashidi and El-Hawary [168169] used PSO to solve the emission-
economic dispatch problem as a multiobjective optimization problem The former joined
the emission and the economic objective functions into a single objective function
through a price penalty factor while the latter solved the same multiobjective problem
through the weighting method and consequently obtained the trade-off curves of the
emission-economic dispatch problem
The PSO technique was also applied to solve the Optimal Power Flow (OPF)
optimization problem in the electric power systems Such a highly nonlinear constrained
optimization problem was first solved utilizing the PSO method by Abido [170] The
PSO was applied to optimize the steady state performance of IEEE 6-bus [171] and IEEE
30-bus [170] transmission systems while satisfying nonlinear equality and inequality
constraints Abido used the PSO to solve single objective and multiobjective OPF
problems The former type of OPF minimized the total fuel cost objective function
while the latter augmented the total fuel cost the improvement of the system voltage
profiles and the enhancement of the voltage stability objective functions with weighting
factors AlRashidi and El-Hawary [172] used a hybrid version of the PSO methodology
to minimize objective functions that included fuel emission fuel cost and the network
141
real power losses In their approach the nonlinear equality constraints were handled via
the Newton-Raphson method and their version of the PSO method was tested on the
IEEE 30-bus transmission system
Gaing [173] integrated the lambda-iteration deterministic method with the Kennedy
and Eberhart binary PSO algorithm in solving the unit commitment problem Ting et al
[174] hybridized the binary code and the real code PSO algorithms in their approach to
solve the unit commitment problem
Yoshida et al [175 176] presented a mixed-integer modified version of PSO to solve
for reactive power and voltage control problems and they tested the proposed algorithm
on the IEEE 14-bus transmission system beside two other practical power systems
Mantawy and Al-Ghamdi [177] applied the same technique to optimize the reactive
power of the IEEE 6-bus transmission power system Miranda and Fonseca [178 179]
applied a modified version of the classic PSO to solve the voltagevar control problem as
well as the real power loss reduction problem They hybridized the PSO method with
evolutionary implementations superimposed upon the swarm particles That is they
implemented some of the evolutionary strategies like replications mutations
reproductions and selection For attention-grabbing reasons they gave this hybridization
such an interesting name as Best of the Two Worlds
Wu et al [180] solved the distribution network feeder reconfiguration problem using
binary coded PSO to minimize the total line losses during normal operation Chang and
Lu [181] also used the binary coded PSO to solve the same problem to improve the RDS
load factor Zhenkun et al [182] employed a hybrid PSO algorithm to solve the
distribution reconfiguration problem and applied it to a 69-bus RDS test case Their
proposed hybrid PSO approach is a combination of the binary PSO and the discrete PSO
algorithms AlHajri et al [183] applied a mixed integer PSO method for optimally
placing and sizing a single DG source in a 69-bus practical RDS as well as to solve for
the capacitor optimal placement and sizing problem in the same system [184]
Minimizing the real power losses of the tested RDS was used as the optimization
objective function subject to nonlinear equality and inequality equations Khalil et al
[185] used the PSO metaheuristic method to optimize the capacitor sizes needed improve
142
the voltage profile and to minimize the real power losses of a 6 bus radial distribution
feeder
532 PSO - Pros and Cons PSO just like any other optimization algorithm has many advantages and disadvantages
It has many key features over deterministic and other metaheuristic methodologies as
well They are summarized as follows
bull Unlike deterministic methods PSO is a non-gradient derivative-free method
which gives the PSO the flexibility to deal with objective functions that are not
necessarily continuous convex or differentiable
bull PSO does not use derivative information ( 1 s t andor 2nd order) in its search for an
optimal solution instead it utilizes the fitness function value to guide the search
for optimality in the problem space
bull PSO by utilizing the fitness function value eliminates the approximations and
assumption operations that are often performed by the conventional optimization
methods upon the problem objective and constraint functions
bull Due to the stochastic nature of the PSO method PSO can be efficient in handling
special kinds of optimization problems which have an objective function that has
stochastic and noisy nature ie changing with time
bull The quality of a PSO obtained solution unlike deterministic techniques does not
depend on the initial solution
bull The PSO is a population-based search method that enables the algorithm to
evaluate several solutions in a single iteration which in turn minimizes the
likelihood of the PSO getting trapped in local minima
bull The PSO algorithm is flexible enough to allow hybridization and integration with
any other method if needed whether deterministic or heuristic
bull Unlike many other metaheuristic techniques PSO has fewer parameters to tune
and adjust
bull Overall the PSO algorithm is simple to comprehend and easy to implement and to
program since it utilizes simple mathematical and Boolean logic operations
On the other hand PSO has some disadvantages that can be summarized as follows
bull There is no solid mathematical foundation for the PSO metaheuristic method
143
bull It is a highly problem-dependent solution method as most metaheuristic methods
are for every system the PSO parameters have to be tuned and adjusted to ensure
a good quality solution
bull Other metaheuristic optimization techniques have been commercialized through
code packages like Matlab GADS Toolbox for GA [186] GeaTbx for both GA
and Evolutionary Algorithm (EA) [187] and Excel Premium Solver for EP [188]
however PSO- to the knowledge of the author- has not commercialized yet
bull Compared to GA EP algorithms PSO has fewer published books and articles
54 PSO - ALGORITHM
The PSO searching mechanism for an optimal solution resembles the social behavior of a
flock of flying birds during their search for food Each of the swarms individuals is
called an agent or a particle and the latter is the chosen term to name a swarm member in
this thesis The PSO search process basically forms a number of particles (swarm) and
lets them fly in the optimization problem hyperspace to search for an optimal solution
The position and velocity of the swarm particles are dynamically adjusted according to
the cooperative communication among all the particles and each individuals own
experience simultaneously Hence the flying particle changes its position from one
location to another by balancing its social and individual experience
The PSO particle represents a candidate potential solution for the optimization
problem and each particle is assigned a velocity vector v as well as a position vector Xj
For a swarm of w-particles flying in W hyperspace each particle is associated with the
following position and velocity vectors
s = [ x x2 bullbullbull xn~] i = l2m (51)
v = [vj v2 bullbullbull vm] (52)
where i is the particle index v is the swarm velocity vector and n is the optimization
problem dimension For simplicity the particle position vector is hereafter represented
by italic font The particles new position is related to its previous location through the
following relation
SW = M+VW (53)
144
where
s(k+l) particle i new position at iteration k+1
s(k) particle old position at iteration k
v(k+1) particle i new velocity at iteration k+1
Eq (53) shows that positions of the swarm particles are updated through their own
velocity vectors The velocity update vector of particle is calculated as follows
vfk+1) = w v f ) + c 1 ri[pbestlk)-sik)) + c2 r2[gbestk) - sk)) (54)
where
VM the previous velocity of particle
w inertia weight
Cj c2 individual and social acceleration positive constants
f r2 random values in the range [01] sampled from a uniform distribution ie
i r 2 ~ pound7(01)
pbest bull personal best position associated with particle i own experience
gbesti bull global best position associated with the whole neighborhood experience
541 The Velocity Update Formula in Detail The velocity update vector expressed in Eq (54) has three major components
1 The first part relates to the particles immediate previous velocity and it consists
of two terms particle last achieved velocity v^ and the inertia weight w
2 The second part is the cognitive component which reflects the individual s own
experience
3 The third part is the social component which represents the intelligent exchange
of information between particle i and the swarm
The velocity update vector can be rewritten in an illustrative way as
vf+1gt= w v f +clrx[pbestf)-sf)Yc2r2[gbestk)-sk)) (55) Previous Velocity ~ ~ X bdquo ~77
Component Cognitive Component Social Component
145
Without the cognitive and social components in the particles velocity update formula
the particle will continue flying in the same direction with a speed proportional to its
inertia weight until it hits one of the solution space boundaries So unless a solution lies
in same path of the previous velocity no solution will be obtained It is the second and
the third components of Eq (54) that change the particles velocity direction in addition
to its magnitude The optimization process is based on and is driven by the three
components of the velocity update formula added altogether
Different versions of the PSO algorithm were proposed since it was first introduced
by Kennedy and Eberhart namely the local best PSO and the global best PSO The main
difference between the two models is the social component of the velocity update
formula The local best PSO model divides the whole swarm into several neighborhoods
and the gbest of particle is its neighborhoods global value Whereas the global best
model deals with the overall swarm as one entity and therefore the PSO particles gbest
is the best value of the whole swarm In general the global model is the preferred choice
and the most popular metaheuristic version of the PSO since it needs less work to reach
the same results [189190] It is noteworthy to mention that the PSO global best model
algorithm is the one that was applied to solve electric power system problems covered in
section 531 This model is the one that is utilized in this thesis to deal with the DG
placement and sizing problem
5411 The Velocity Update Formula - First Component The first segment of the velocity-updating vector is the previous velocity memory
component It is also called the inertia component It is the one that connects the particle
in the current PSO iteration with its immediate past history ie serving as the particles
memory It plays a vital role in preventing the particle from suddenly changing its
direction and allows the particles own knowledge of its previous flight information to
influence its newer course
Inertia Weight (w) The first version of the velocity-update vector introduced by
Kennedy and Eberhart did not contain an inertia weight in other words the inertia
weight was assumed to be unity The inertia weight was first introduced by Shi and
Eberhart in 1998 to control the contribution of the particles previous velocity in the
current velocity decision making which consequently led to significant improvements in
146
the PSO algorithm [191] Such a mechanism decides the amount of memory the particle
can utilize in influencing the current velocity exploration momentum When first
introduced static inertia weight values were proposed in the range of [08-12] and [05-
14] Large values of w tend to broaden the exploration mission of the particles while
small values will localize the exploration Several dynamic inertia weight approaches
were proposed in the literature such as random weights assigned at each iteration [192]
linear decreasing function [191 193 194] and nonlinear decreasing function [195] The
formulations of the aforementioned inertia weights are respectively expressed as follows
wW=ClrW+c2r2W (56)
(k) M (I) (nk) nt bull ^
laquo j (57)
)_)(bdquo it) wM) = [- j^mdashL (58)
where
w(k) inertia weight value at iteration k
nk bull maximum number of iterations
WM inertia weight value at the last iteration nk
Shi and Eberhart [196] suggested 09 and 04 as the initial and final inertia weight
values respectively They asserted that during the decrease in the inertia weight from a
large value to a small one the particles will start searching globally for solutions and
during the due course of the PSO run they will intensify their search in a local manner
Constriction Factor () Clerc [197] and Clerc and Kennedy [198] suggested a
constriction factor similar to the inertia weight approach that aims to balance the global
exploration and the local exploitation searching mechanism It was shown that
employing the constriction factor improves convergence eliminates the need to bound
the velocity magnitude and safeguards the algorithm against explosion (divergence) [199-
201] The proposed approach is to constrict the particles velocity vector by a factor
as expressed in Eq (59)
147
vf+ 1) = x (vlaquo + c r (^5f - sf) + c2 r2 [gbestk) - sreg )) (59)
where
2
2-(|gt-Vlttraquo2-4ltt) (510)
lt|gtgt4
The constriction factor is a function of cx and c2 and by assigning a common value of
41 to lt|) and setting c = c2=205 x will have the value of 072984 The value obtained is
equivalent to applying the static inertia weight PSO with w= 072984 and c = c2=14962
The constriction factor is sometimes considered as a special case of the inertia weight
PSO algorithm because of the constraints imposed by Eq (510) The constriction factor
X controls the particles velocity vector while the inertia weight w controls the
contribution of the particles previous velocity toward calculating the new one
Though utilizing the constriction factor eliminates velocity clamping Shi and
Eberhart [202203] suggested a rule of thumb strategy that would result in a faster
convergence rate The strategy is to constrain the maximum velocity value to be less than
or equal to the maximum position once the decision to use the constriction factor model
has been made or to use the static inertia weight PSO algorithm with w cx and c2 to be
selected according to Eq (510)
5412 The Velocity Update Formula - Second Component The second component is the cognitive component of the velocity update equation The
tQtmpbest in the cognitive component refers to the particles best personal position that it
has visited thus far since the beginning of the PSO iterative process That is each
particle in the swarm will evaluate its own performance by comparing its own fitness
function value in the current PSO iteration with that evaluated in the preceding one If
the fitness function is of ^-dimension space Rd -raquo R the pbest] given that its
pbest] is the best personal position so far is defined as
148
Eq (511) in a way implies that the particle performs book-keeping for its personal
best position achieved thus far to make it handy when performing the velocity update in
a future PSO iteration In other words each particle remembers its optimal position
reached and the overall swarm pbest vector is updated after each PSO iteration with its
vector entries either updated or remaining untouched Furthermore the cognitive part of
the velocity update equation diversifies the PSO searching process and helps in avoiding
possible stagnation
5413 The Velocity Update Formula-Third Component The third component of the velocity update vector represents the social behavior of the
PSO particles The gbest term in the social component refers to the best solution
(position) achieved among all the swarm particles Namely particle now evaluates the
performance of the whole swarm and stores the best value obtained in the gbest That is
whenever the best solution among the whole body of the swarm is achieved such
valuable information is directly signaled and delivered to all peers as shown in Figure
52 The gbest should have the optimal fitness value among all the particles during the
current PSO iteration as defined in the following equation
gbest^=minf(s^) (gt) - (laquo) (512)
where flsk I is particle fitness value at iteration k and m is the swarm size
149
Particle with gbest
Figure 52 Interaction between particles to share the gbest information
5414 Cognitive and Social Parameters The pbest and gbest in the second and third parts are scaled by two positive acceleration
constants c and c2 respectively [204] c and c2 are called the cognitive and social
factors respectively The trust of the particle in itself is measured by c while c2
reflects the confidence it has in its neighbors A value of 0 for both of them leaves the
particle only with its previous velocity memory to proceed with in updating its new
velocity and subsequently its new position A cx value of 0 would eliminate the
particles own experience factor in looking for a new solution while assigning 0 to the
social factor would localize the particles searching process and eliminate the exchange
of information between the PSO particles A value of 2 for both of them is the most
recommended value found in the literature In a way cx and c2 are considered as the
relative weights of the cognitive and social perspectives respectively r andr2 are two
random numbers in the range of [01] that are sampled from a uniform distribution The
150
PSO method has a stochastic exploration nature because of the randomness introduced by
rx and r2 All three parts of the velocity update vector constitute the particles new
velocity which when combined together determines a new position
Figure 53 illustrates the velocity and position update mechanism for a single PSO
particle during iteration k Figure 54 on the other hand is a virtual snapshot that
demonstrates the progress of particle movement during two PSO consecutive iterations
k and k+l with an updated values of the pbest and gbset
pbesti
Figure 53 Illustration of velocity and position updates mechanism for a single particle
during iteration k
151
Figure 54 PSO particle updates its velocity and position vectors during two consecutive iterations k and k+
542 Particle Swarm Optimization-Pseudocode The standard PSO algorithm generally could be summarized as in the following
pseudocode
Step 1 Decide on the following
1 Type of PSO algorithm
2 Maximum number of iterations nk
3 Number of swarm particles m
4 PSO dimension n
5 PSO parameters cvc2w
Step 2 Randomly initialize ^-position vector for each particle
Step 3 Randomly initialize m-velocity vector
Step 4 Record the fitness values of the entire population
Step 5 Save the initial pbest vector and gbest value
152
Step 6 For each iteration
Step 7 For each particle
bull Evaluate the fitness value and compare it to its pbest
if(f4)) lt fpbest^)=gt pbestreg = sreg
else
if f(sreg)gt f(pbestf-l))=gt pbestreg = pbestf-x)
end For each particle
bull Save the pbest new vector
gbestreg=minf(sreg) ( laquo ) - (laquo)
bull Update velocity vector using Eq (54)
bull Update position vector using Eq (53)
bull Reinforce solution bounds if violation occurs
Step 8 if Stopping criteria satisfied then
bull Maximum number of iterations is reached
bull Maximum change in fitness value is less than s for q iterations
f(gbestreg)-f(gbestk-h))lte h = l2q
=gt Stop-end For each iteration
Otherwise GOTO to Step 6
55 PSO APPROACH FOR OPTIMAL DG PLANNING
The PSO method is employed here to deal with DG planning in the distribution networks
When DGs are to be deployed in the grid both the DG placement and the size of the
utilized DG units are to be carefully planned for The DG planning problem consists of
two steps finding the optimal placement bus in the DS grid as well as the optimal DG
size
The DG sizing problem formulation was tackled in Chapter 4 of this thesis The DG
to be installed has to minimize the DS active power losses while satisfying both equality
and inequality constraints The sizing problem was handled previously by the
153
conventional SQP method as well as the proposed FSQP method developed in the last
chapter
In this chapter the PSO metaheuristic method is used to solve for the optimal
placement and the DG rating simultaneously to reveal the optimal location bus in the
tested DS and optimal DG rating for that location In the PSO approach the problem
formulation is the same as that presented in the deterministic case with the difference
being the addition of the bus location as a new optimization variable
The DG unit size variables are continuous while the variables that represent the DG
placement buses are positive integers The DG source optimized variables are its own
real power output PDG along with the its power factor pfm and they are expressed as
PDG G Rgt PDG = |_0 PDT J ~ ~
PDG e R Pfaa = [0 l]
The corresponding reactive power produced by the DG is calculated as follows
eDGeR
A DG with zero power factor is a special case that represents a capacitor The variables
that represent the eligible DS bus locations are stated as
^ e N + w h e r e laquo = [ gt pound pound] (514)
where the main distribution substation is designated as bm = 1
The developed PSO is coded to handle both real and integer variables of the DG
mixed-integer nonlinear constrained optimization problem The PSO position vector
dimension depends on the number of variables present If the proposed DG has a
prespecified power factor then the dimension will be two variables per DG installed (the
positive integer bus number and the DG real power output) Moreover for multiple DG
units (nDG) to be installed in the grid the swarm particle i position vector will have a
dimension of (l x 2laquoDG) as illustrated below
DGl DG2 nDG
QDG=PDGtanaC0S(pf))gt W h e r e
S = VDG^DG) K^DG^DG) DGgtregDG) (515)
154
On the other hand if the DG power factor was left to be optimized there will be three
variables per DG in the particles position vector To clarify for nDG to be planned for
deployment their corresponding particle position vector is
DG DG2 nDG
S = DG PJ DG bullgt regDG ) VDG PJDG gt ^DG ) DG PDG ^DG ) (516)
551 Proposed HPSO Constraints Handling Mechanism Two types of constraints in the PSO DG optimization problem are to be handled the
inequality and the equality constraints in addition to constrain the DS bus location
variables to be closed and bounded positive integer set The following subsections
discuss them in turn
5511 Inequality Constraints The obtained optimal solution of a constrained optimization problem must be within the
stated feasible region The constraints of an optimization problem in the context of EAs
and PSO methods are handled via methods that are based on penalty factors rejection of
infeasible solutions and preservation of feasible solutions as well as repair algorithms
[205-207] Coath and Halgamuge [208] reported that the first two methods when utilized
within PSO in solving constrained problems yield encouraging results
The penalty factor method transforms the constrained optimization problem to an
unconstrained type of optimization problem Its basic idea is to construct an auxiliary
function that augments the objective function or its Lagrangian with the constraint
functions through penalty factors that penalize the composite function for any constraint
violation In the context of power systems Ma et al [209] used this approach for
tackling the environmental and economic transaction planning problem in the electricity
market He et al [210] and Abido [170] utilized the penalty factors to solve the optimal
power flow problem in electric power systems Papla and Erlich [211] utilized the same
approach to handle the unit commitment constrained optimization problem The
drawback of this method is that it adds more parameters and moreover such added
parameters must be tuned and adjusted in every single iteration so as to maintain a
quality PSO solution A subroutine that assesses the auxiliary function and measures
155
the constraint violation level followed by evaluating the utilized penalty function adds
computational overhead to the original problem
Rejecting infeasible solutions method does not restrict the PSO solution method
outcomes to be within the constrained optimization problem feasible space However
during the PSO iterative process the invisible solutions are immediately rejected deleted
or simply ignored and consequently new randomly initialized position vectors from the
feasible space replace the rejected ones Though such a re-initialization process gives
those particles a chance to behave better it destroys the previous experience that each
particle gained from flying in the solution hyperspace before violating the problem
boundary [204206] Preserving the feasible solutions method on the other hand
necessitates that all particles should fly in the problem feasible search space before
assessing the optimization problem objective function It also asserts that those particles
should remain within the feasible search space and any updates should only generate
feasible solutions [206] Such a process might lead to a narrow searching space [208]
The repair algorithm was utilized widely in EAs especially GA and they tend to restore
feasibility to those rejected solutions which are infeasible This repair algorithm is
reported to be problem dependent and the process of repairing the infeasible solutions is
reported to be as difficult and complex as solving the original constrained optimization
problem itself [212213]
In this thesis the DG inequality constraints concerning the size as stated in Chapter
4 and the bus location as stated in section 55 are to be satisfied in all the HPSO
iterations The particles that search for optimal DG locations and sizes must fly within
the problem boundaries In the case of an inequality constraint violation eg the particle
flew outside the search space boundaries the current position vector is restored to its
previous corresponding pbest value By asserting that all particles are first initialized
within the problem search space and by resetting the violated position vector elements to
their immediate previous pbest values the preservation of feasible solutions method is
hybridized with the rejection of infeasible solutions method That is while preserving
the feasible solutions produced by the PSO particles the swarm particles are allowed to
fly out of the search space Nevertheless any particle that flies outside the feasible
solution search space is not deleted or penalized by a death sentence but in a way they
156
are kept energetic and anxious to continue the on-going optimal solution finding
journey starting from their restored best previously achieved feasible solution AlHajri
et al used the hybridized handling mechanism in the PSO formulation to solve for the
DG optimal location and sizing constrained minimization problem [183190]
5512 Equality Constraints The power flow equations that describe the complex voltages at each bus as well as the
power flowing in each line of the distribution network are the nonlinear equality
constraints that must be satisfied during the process of solving the DG optimization
problem One of the most common ways to compute the power flow is to use the NR
method This method is quite popular due to its fast convergence characteristics
However distribution networks tend to have a low XR ratio and are radial in nature
which poses convergence problems to the NR method Thus a radial power flow
method the FFRPF that was developed in Chapter 3 is adopted within the proposed PSO
approach to compute the distribution network power flow A key attractive feature of
this method is its simplicity and suitability for distribution networks since it mainly relies
on basic circuit theorems ie Kirchhoff voltage and current laws The PSO algorithm is
hybridized with the FFRPF solution method to handle the nonlinear power flow equality
constraints Hence FFRPF is used as a sub-routine within the PSO structure
By hybridizing the classic PSO with 1) the hybrid inequality constraints handling
mechanism and 2) with the FFRPF technique for handling the equality constraints the
resultant Hybrid PSO technique (HPSO) is used in tackling the DG optimal placement
and sizing constrained mixed-integer nonlinear optimization problem
5513 DG bus Location Variables Treatment The DG bus location is an integer variable previously defined in Eq (514) To ensure
that the bus where the power to be injected is within its imposed limits a rounding
operator is incorporated within the HPSO algorithm to round the bus value to the nearest
real positive integer That is in each HPSO iteration the particle position vector element
that is related to the DG bus is examined If it is not a positive integer value then it is to
be rounded to the nearest feasible natural number The included rounding operator is
mathematically expressed as in Eq (517) to ensure that the HPSO bus location random
157
choice when initialized is a positive integer and bounded between minimum and
maximum allowable location values
roundlbtrade + (random)x[btrade -btrade))) (517)
During the HPSO iterations the obtained particle position vector elements related to the
DG bus locations are examined to be within limits and subsequently processed as shown
in Eq (518) to assure its distinctive characteristic ie positive integer value
round(b^) (518)
The proposed HPSO methodology is summarized in the flowchart shown in Figure 55
158
HIter Iter+lj^mdash
i - bull I Particle = Particle+l |
Update particle vectors
Apply FFRPF to satisfy the equality
constraints
Restore previous pbest
Save the pbest new vector Record
swarm gbest and its I fitness value
Determine number V ofDGs J
Decide on the following bull No of iterations bull No of swarm particles m bull PSO dimension n bull PSO parameters CiC2w
Randomly initialize feasible bull n-dimension position vector bull m-dimension velocity vector
Apply FFRPF to satisfy the equality
constraints
lt0 Compute the following
PLOSSM for all particles
Record gbest and pbest Set Iteration and Particle
counter to 0
Figure 55 The proposed HPSO solution methodology
159
5 6 SIMULATION RESULTS AND DISCUSSION
The HPSO algorithm is used in solving the DG planning problem The metaheuristic
technique is utilized to optimally size and place the DG units in the distribution network
simultaneously ie in a single HPSO run the optimal size as well as the bus location are
both obtained for every DG source
The same test systems used in the previous chapter are tested here via the HPSO
approach and the results obtained are presented and compared to those obtained by the
FSQP deterministic method The FSQP was chosen for comparison since it was proven
that it has the lowest simulation CPU time when compared with the conventional SQP
The deviation of losses calculated by the HPSO method from that determined by the
FSQP is measured as
bullpFSQP _ jyHPSO
APLosses = to- mdash x 100 (519)
Losses
where P ^ is the mean value of HPSO simulation results of the DS real power losses
and P ^ is the real power loss determined by the FSQP deterministic method A
negative percentage indicates higher losses obtained by the proposed method while a
positive percentage implies higher losses associated with the FSQP method
As was performed in the deterministic case the DG unit or units are optimally sized
and placed in the DS network with a specified power factor (pf) and with unspecified pf
That is the HPSO method is utilized in optimally placing and sizing a DG unit with a
specified power factor of 085 and with the power factor treated as an unknown variable
in all the tested DSs
Though the linear decreasing function is found to be popular in the PSO literature
the inertia weight is found to be best handled with the nonlinear decreasing function
expressed in Eq (58) The initial and final inertia weight values as well as the velocity
minimum and maximum values are set to [0904] and [0109] respectively The
other HPSO parameters for both models eg maximum number of iterations number of
swarm particles and acceleration constants are problem-dependent and they are to be
160
tuned for each case separately The HPSO simulations for each tested case are executed
at least 20 times to check for consistency with the best answer reported in the
comparison tables
561 Case 1 33-bus RDS The 33-bus RDS was tested in the last chapter by applying the APC using the developed
FSQP and conventional SQP optimization methods The same system is tested here via
the HPSO method for single and multiple DGs cases The following subsections present
and discuss corresponding simulation results
5611 33-bus RDS Loss Minimization by Locating a Single DG A single DG source is to be installed in the 33-bus RDS and the HPSO is used in
investigating the optimal DG size and bus location simultaneously The HPSO maximum
number of iterations swarm particles and acceleration constant parameters are tuned for
both of the pf cases and recorded in Table 51 The obtained HPSO results for both
cases are tabulated in Table 52 and Table 55 Table 53 and Table 56 present the
descriptive statistics for obtained HPSO solutions ie mean Standard Error of the Mean
(SE Mean) Standard Deviation (St Dev) Minimum and Maximum values The
comparison between the FSQP method outcome and the proposed HPSO method results
for the fixed and unspecified pf cases are presented in Table 54 and Table 57
respectively The HPSO method obtained both the single DG optimal bus location and
rating simultaneously It returned a different bus location for the DG to be installed in
bothcases than that of the deterministic method The HPSO proposed bus No 29 for
the single fixed and unspecified pf DG while the bus location obtained by the
deterministic method is No 30 The mean value of the real power losses for both pf
cases is comparable to that of the deterministic method for both cases ie HPSO losses
are lower by 1 in the fixed pf case and lower by 08 for the other case The
simulation time of the HPSO method to reach both location and sizing results
simultaneously outperforms that of its counterpart The convergence characteristic of the
proposed HPSO in the fixed pf single DG case is shown in Figure 56 for a maximum
HPSO number of iterations of 30 Figure 57 shows that even when the number of the
iterations is increased the HPSO algorithm is already settled to its final value Figure
161
58-Figure 515 show the clustering behavior of the swarm particles during the HPSO
iterations of the fixed pf case
Table 51 HPSO Parameters for the Single DG Case
No of Iterations
Swarm Particles
lt
C2
Fixed pf 30 10
20
20
Unspecified pf 40 15
25
25
Table 52 33-bus RDS Single DG FixedCase 20 HPSO Simulations
HPSO-PLoss (kW)
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
728717
Bus No
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
D G P (kW)
17795654
17795656
17795656
17795656
17795656
17795657
17795656
17795656
17795656
17795656
17795656
17795656
17795656
17795656
17795655
17795658
17795652
17795654
17795656
17795656
AF m (pu)
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
00586
Table 53 Descriptive Statistics for HPSO Results for the Fixedpf Cass
Variable HPSO-PLoss
N 20
Mean 72872
SEMean 0
StDev 0
Minimum 72872
Maximum 72872
Table 54 HPSO vs FSQP Results 33-bus RDS-Single DG-FixedgtDG Case
Optimal Placement Bus Optimal DG Size (kW) Optimal DG Power Factor Minimum Real Power Losses (kW)
AF w (pu)
Simulation Time (sec)
Single DG Profile HPSO
29 17795654
085 728717
00586
04984
Single DG Profile FSQP
30 17795232
085 735821
00586
Single Run APC
05084 117532
Table 55 33-bus RDS Single DG Unspecifiedpf Case 20 HPSO Simulations
HPSO-PLoss (kW)
710126
710124
710122
710360
710122
710159
710123
710124
710122
710131
710123
710122
710129
710123
710122
710125
710122
710122
710123
710122
Bus No
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
DG P (kW)
16482970
16425300
16446070
16163350
16448400
16356250
16442840
16467950
16448340
16500830
16445120
16444730
16482140
16446770
16447630
16457710
16451710
16444840
16456960
16453560
DGpf
07816
07802
07807
07774
07807
07775
07804
07813
07808
07819
07810
07808
07822
07812
07808
07803
07808
07808
07810
07808
AF x (pu)
00467
00587
00585
00590
00586
00585
00585
00585
00599
00583
00583
00585
00584
00587
00578
00588
00583
00585
00584
00584
Table 56 Descriptive Statistics for HPSO Results for an UnspecifiedpCase
Variable HPSO-PLoss
N 20
Mean 71014
SE Mean 000119
StDev 000531
Minimum 71012
Maximum 71036
Table 57 HPSO vs FSQP Results 33-bus RDS-Single DG-UnspecifiedpCase
Optimal Placement Bus Optimal DG Size (kW) Optimal DG Power Factor Minimum Real Power Losses (kW)
AF bdquo (pu)
Simulation Time (sec)
Single DG Profile HPSO
29 1644763 07808 710122
005783
07307
Single DG Profile FSQP
30 15351 07936
715630
00613
Single Run APC
06082 21067
Maximum HPSO Iterations =30
13 15 17 19
HPSO Iteration No
23 25 27 29
Figure 56 Convergence characteristic of the proposed HPSO in the fixed pf single DG case HPSO proposed number of iterations = 30
164
Maximum HPSO Iterations =50
re amp 727
19 22 25 28 31
HPSO Iteration No
Figure 57 Convergence characteristic of the proposed HPSO in the fixedsingle DG case HPSO extended number of iterations = 50
Swarm Particles at Iteration 1
13 17 21
33-Bus RDS Bus No
33
Figure 58 Swarm particles on the first HPSO iteration
165
Swarm Particles at Iteration 5
13 17 21
33-Bus RDS Bus No
33
Figure 59 Swarm particles on the fifth HPSO iteration
Swarm Particles at Iteration 10
13 17 21
33-Bus RDS Bus No
25 29 33
Figure 510 Swarm particles on the tenth HPSO iteration
166
Swarm Particles at Iteration 15
1 L
5 o Q 0)
gt -M
lt O Q
2000 - 1800 1600 1400
1200
1000 -
800
600 400 -200 -
0-| 1 1 5 9 13 17 21 25 29 33
33-Bus RDS Bus No
Figure 511 Swarm particles on 15 HPSO iteration
Swarm Particles at Iteration 20
2000
V )J
1 pound s +
$ n a
1800
1600
1400 1200 1000
800
600 400 200 0
1 5 9 13 17 21 25 29 33
33-Bus RDS Bus No
Figure 512 Swarm particles on the 20 HPSO iteration
Swarm Particles at Iteration 25
13 17 21 25 29 33
33-Bus RDS Bus No
th Figure 513 Swarm Particles on the 25 HPSO iteration
Swarm Particles at Iteration 30
13 17 21 25 29 33
33-Bus RDS Bus No
Figure 514 Swarm Particles on the last HPSO iteration
168
Swarm Particles at Iteration 30
f P
ower
(I
Act
ive
a
1780
1775
1770
1765
1760
1755
1750
1 5 9 13 17 21 25 29 33
33-Bus RDS Bus No
Figure 515 A close-up for the particles on the 30th HP SO iteration
5612 33-bus RDS Loss Minimization by Locating Multiple DGs Optimally locating and sizing more than a single DG unit minimizes the DS network real
power losses HPSO is used to solve the multiple DG installations scenario double DG
three DG and four DG cases The proposed HPSO parameters are tuned for the multiple
DG cases to obtain consistent outcomes Two three and four DG cases are tested in the
33-bus RDS with both fixed and unspecified pf cases In the unspecified pf case each
DG unit has two variables to be optimized at the optimal chosen bus location the real and
the reactive power outputs
Double DGs Case The tuned HPSO parameters for both DG cases are shown in
Table 58 The proposed HPSO algorithm was utilized to optimally size and place two
DG units in the 33-bus RDS Table 59 and Table 510 present the fixeddouble DG
case results for 20 simulations of the HPSO and their corresponding descriptive statistics
The first table shows that the HPSO consistently chooses buses 30 and 14 for the two
optimally sized DG units to be installed Unlike the FSQP method the HPSO metaheu-
ristic technique obtained the optimal DG locations and sizes simultaneously The
corresponding HPSO results are compared to those of the FSQP deterministic method as
shown in Table 511 The HPSO real power losses results are close to the deterministic
obtained result ie HPSO losses are higher by 04
169
On the other hand the proposed HPSO method assigned a different bus location for
the first DG unit in the unspecifiedcase as shown in Table 512 By selecting bus No
13 instead of bus No 14 the DS network real power losses were reduced by
approximately 75 when compared to the losses of the FSQP method as shown in
Table 514 For both double DG cases the DS bus voltages range not only within limits
but their deviation from the nominal value is minimal ie 0021 and is similar to that of
the FSQP method
Table 58 HPSO Parameters for Both Double DG Cases
No of Iterations Swarm Particles
cx C2
Fixed pf
100 40
20
20
Unspecified pf
100 60
25
25
170
Table 59 33-bus RDS Double DG Fixedpf Case 20 HPSO Simulations
HPSO-PLoss (kW)
329458
329553
329514
329371
329374
329372
329374
329572
329748
329373
329372
329371
329510
329370
329372
329385
329377
329583
329431
329370
Bus 1 No
30
30
30
30
30
14
14
30
30
30
30
14
14
14
14
30
30
14
30
14
DGlP(kW)
11792350
11540020
11572230
11679170
11666120
6969715
6982901
11532080
11734750
11675020
11673750
6968644
7063828
6960787
6952874
11649680
11719790
7118906
11775930
6964208
Bus 2 No
14
14
14
14
14
30
30
14
14
14
14
30
30
30
30
14
14
30
14
30
DG 2 P (kW)
6856625
7108923
7074405
6969823
6982871
11679170
11666100
7116907
6891157
6973904
6975254
11680310
11581830
11688180
11696040
6999170
6929218
11529730
6873075
11684790
AKjpu)
002072
002084
002125
002072
002074
006172
005636
002073
006871
002078
005383
002075
002073
002073
002082
009058
002072
002113
002094
002072
Table 510 Descriptive Statistics for HPSO Results for Fixed Double DG Case
Variable HPSO-PLoss
N 20
Mean 32944
SE Mean 000235
StDev 00105
Minimum 32937
Maximum 32975
171
Table 511 HPSO vs FSQP Results 33-bus RDS-Double DGs-FixedgtCase
Optimal Placement Bus
Optimal DG Size (kW)
Optimal DG Power Factor Minimum Real Power Losses (kW)
AF x (pu)
Simulation Time (sec)
Double DGs Profile HPSO
DG1 Bus =14 DG2 Bus =30
DG1 P= 6964208 DG2P= 11684795
085 329370
0020724
421998 sec
Double DGs Profile FSQP
DG1 Bus =14 DG2 Bus =30
DG1P = 6986784 DG2P= 11752222
085 328012
0020679
Single Run
APC
07691 2761264
46021 min
Table 512 33-bus RDS Double DG UnspecifiedpCase 20 HPSO Simulations
HPSO-PLoss (kW)
288541
288142
288136
288243
288350
288128
288141
288138
288144
288182
288177
288146
288229
288130
288479
288168
288124
288457
288284
288124
Busl No
13
13
30
13
30
13
30
30
30
13
30
30
30
30
30
13
30
13
13
30
DG1P (kW)
8367509
8047130
10593890
7953718
10436980
8081674
10587578
10583572
10585108
8018625
10718348
10572279
10492694
10622907
10380291
8139958
10636739
8338037
8168418
10630855
DG1 Pf
09006
08957
07046
08947
07000
08972
07073
07058
07042
08930
07109
07045
07026
07067
06979
08949
07073
09048
09015
07074
Bus 2 No
30
30
13
30
13
30
13
13
13
30
13
13
13
13
13
30
13
30
30
13
D G 2 P (kW)
10362222
10683717
8137192
10777377
8293669
10649219
8143482
8147280
8145345
10712187
8012494
8158742
8238406
8108039
8350740
10591136
8094357
10392766
10542577
8100245
DG2
Pf
06989
07095
08984
07123
08994
07070
08974
08990
08995
07111
08971
08999
09003
08964
09042
07055
08980
06992
07035
08974
ampv II l loo
(pu) 002010
002010
004289
001934
001998
002015
001963
002010
002010
003371
002011
002016
001996
002007
003796
002007
002019
001923
002178
002054
172
Table 513 Descriptive Statistics for HPSO Results for Unspecified^Double DG Case
Variable HPSO-PLoss
N 20
Mean 28822
SE Mean 000293
StDev 00131
Minimum 28812
Maximum 28854
Table 514 HPSO vs FSQP Results 33-bus RDS-Double DGs-UnspecifiedpCase
Optimal Placement Bus
Optimal DG Size (kW)
Optimal DG Power Factor
Minimum Real Power Losses (kW)
AF x (pu)
Simulation Time
Double DGs Profile HPSO
DGlBus=13 DG2 Bus =30
DG1 P= 8100245 DG2P= 10630855
DG1 pf= 08974 TgtG2pf= 07074
288124
002054
51248 sec
Double DGs Profile FSQP
DG1 Bus= 14 DG2 Bus= 30
DG1P = 78841 DG2P= 10847 DG1 pf= 09366 DG2 pf= 07815
311588
002067
Single Run
APC
12532 sec 6083348 sec (101389 min)
Three DGs Case The proposed HPSO tuned parameters for the two cases under
consideration are shown in Table 515 Table 516 and Table 519 show the 20 HPSO
simulations for the three DG cases ie fixed pf and unspecified pf cases while Table
517 and Table 520 show their corresponding descriptive statistics respectively The
HPSO results for both three DG cases are compared with the FSQP method outcomes
correspondingly and tabulated in Table 518 and Table 521
The placement bus locations and their corresponding DG sizes are determined
simultaneously by the proposed HPSO The bus placements recommended by the
proposed metaheuristic method are the same as those suggested by the FSQP APC
method However while the mean value of real power losses obtained by the HPSO is
similar to that of the FSQP result in the fixed pf case (HPSO losses value is lower by
07) the mean value of the real power losses in the unspecified pf case is soundly
improved by approximately 19 when compared to its FSQP counterpart Not only did
the proposed HPSO simultaneously provide both optimal placements and sizes for the
multiple DG cases but the resultant losses were either better or at least comparable with
173
those of the deterministic solution The RDS bus voltages obtained are within allowable
range and both solution methods returned similar results
Table 515 HPSO Parameters for Both Three DG Cases
No of Iterations
Swarm Particles
lt
c2
Fixed
150 50 30
30
Unspecified pf 100 70
25
25
Table 516 33-bus RDS Three DG Fixed pf Case 20 HPSO Simulations
HPSO-PLoss (kW)
290829
290829
290829
290829
290831
290832
290868
291026
291045
290833
290838
290972
290883
290924
290886
290831
290831
290837
290845
290829
Bus 1 No
30
30
14
30
30
30
14
14
25
25
30
14
25
14
30
25
14
14
14
25
DG1P (kW)
9905706
9905813
6173596
9905707
9906686
9889657
6168332
6059714
2599472
2642328
9944151
6177179
2608769
6187166
9893877
2632592
6171492
6198642
6219215
2647290
Bus 2 No
14
14
30
25
14
14
30
30
14
14
14
30
30
30
14
14
30
30
30
30
DG2P (kW)
6173451
6173443
9905309
2647769
6173055
6190620
9831444
9849325
6342238
6155639
6147817
9751556
9862118
10020660
6253967
6172385
9926226
9867430
9878060
9905713
Bus 3 No
25
25
25
14
25
25
25
25
30
30
25
25
14
25
25
30
25
25
25
14
DG3P (kW)
2647344
2647246
2647596
6173026
2646709
2646213
2726669
2817194
9784792
9928535
2634534
2797767
6255500
2518655
2578624
9921524
2628784
2660429
2629227
6173499
II Moo
(pu)
002057
002057
002101
002478
002079
002115
002091
002121
002215
002066
002046
002120
002166
002699
002047
002051
002033
002069
002062
002057
174
Table 517 Descriptive Statistics for HPSO Results for Fixedpf Three DG Case
Variable HPSO-PLoss
N 20
Mean 29087
SE Mean 000151
StDev 000676
Minimum 29083
Maximum 29104
Table 518 HPSO vs FSQP Results 33-bus RDS-Three DG-FixedgtCase
Optimal Placement Bus
Optimal DG Size (kW)
Minimum Real Power Losses (kW)
AF a (pu)
Simulation Time
Three DGs Profile HPSO
DG1 Bus =14 DG2 Bus =25 DG3 Bus =30
DG1 P= 86173499 DG2 P= 2647289 DG3P= 9905713
2908291
002057
56878 sec
Three DGs Profile FSQP
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30
DG1 P = 6504360 DG2P = 3216023 DG3P = 9006118
293056
002016
Single Run
APC
14107 sec 37316290 sec
(2 hrs 21938 min)
Table 519 33-bus RDS Three DGs UnspecifiedpfCase 20 HPSO Simulations
HPSO-PLoss (kW)
210728 210815 210849 210963 211095 211367 211827 215061 215235 215578 217387 210732 210931 211454 211509 211833 212073 214234 215059 215705
Bus 1 No
14 25 25 30 30 30 25 14 30 14 25 14 25 25 14 30 30 30 14 14
DG1 P (kW)
6935008 3467525 3536383 8035969 8080569 8493449 3936378 6148109 8892055 6238083 3404837 6842756 3777428 3541335 6827243 8763070 7857500 8494754 6209975 6123234
D G l p
08889 06446 06467 06158 06258 06362 06743 08809 06495 08754 06611 08862 06575 06298 08987 06643 06106 06594 08426 08459
Bus 2 No
30 30 30 25 14 14 14 30 25 30 30 30 30 30 25 25 25 25 30 25
DG2 P (kW)
8359344 8245928 8587325 3733524 6790160 6443962 6699864 9215988 3921948 7906602 8338033 8398375 8140127 8520024 3706051 3391292 3881093 3737682 9153850 3659521
DG2pf
06282 06241 06433 06702 08787 08633 08833 06770 07328 06125 06261 06304 06276 06531 06969 06314 07146 07248 06629 06652
Bus 3 No
25 14 14 14 25 25 30 25 14 25 14 25 14 14 30 14 14 14 25 30
DG3P
3411993 6992347 6577857 6936881 3835565 3768424 8053757 3315200 5889225 4558759 6723886 3459190 6787794 6636993 8170680 6550687 6955221 6415564 3327371 8801864
DG 3 pf
06437 08897 08759 08862 06951 06976 06282 06241 08589 06902 09096 06516 08842 08837 06235 08568 08786 08576 07045 06631
l A K L 001515 001507 001681 001638 006399 001723 001725 001839 001741 002235 002626 001899 001727 001887 001890 002195 001558 002012 001632 006178
175
Table 520 Descriptive Statistics for HPSO Results for the Fixedpf Three DG Case
Variable HPSO-PLoss
N 20
Mean 21272
SE Mean 00485
StDev 0217
Minimum 21073
Maximum 21739
Table 521 HPSO vs FSQP Results 33-bus RDS-Three DG-UnspecifiedCase
Optimal Placement Bus
Optimal DG Size (kW)
Optimal DG Power Factor
Minimum Real Power Losses (kW)
AK^Oro)
Simulation Time
Three DGs Profile HPSO
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30
DG1P = 6935008 DG2P = 3411993 DG3P = 8359344 DG1 pf= 08889 DG2= 06437 DG3 pf= 06282
210728
001515
51435 sec
Three DGs Profile FSQP
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30
DG1P = 67599 DG2P = 35373 DG3 P = 84094 DGl pf= 09218 DG2 pf= 09967 DG3 pf= 07051
263305
002048
Single Run
APC
20681 sec 121133642 sec
(3 hrs 21888 min)
Four DGs case The proposed HPSO is used for installing four DG units with and
without specifying their pfs in the tested 33-bus RDS with the chosen tuned parameters
shown in Table 522 Table 523 and Table 526 show a sample run of 20 simulations of
the HPSO results their corresponding descriptive statistics are displayed in Table 524
and Table 527 The best HPSO results for both DG cases are compared with those
obtained with the FSQP APC technique and are presented in Table 525 and Table 528
The HPSO real power losses for the four DGs with fixed pf case were found to be
comparable to those obtained by the FSQP method however the HPSO proposed several
bus location combinations for the units to be seated Of the 20 HPSO simulations 9 of
them gave the same bus combinations as of the deterministic method ie bus No 14 25
30 and 32 As to the other bus location combinations they produced comparable losses
when optimal sizes were installed The unspecifiedcase real power losses mean value
obtained by the proposed HPSO was around 23 lower than that of FSQP method The
176
HPSO solution for the second case delivered several bus location combinations for the
four DG units to be installed
Choosing 4 DG locations out of 32 bus locations resulted in a large number of
combinations ie 35960 and the HPSO solution method provided diverse bus location
combinations with losses either comparable to the deterministic case as in the first pf
case or even better as in the second pf case That consequently would introduce
flexibility in making the proper decision to place DGs in the distribution network It is
noteworthy that buses 25 and 30 are the most common locations in both cases 100
swarm particles were used to solve such complex problems and although such a size is
not frequently used in literature Hu and Eberhart support increasing the swarm size when
dealing with complex problems [207]
Table 522 HPSO Parameters for the Four DG Case
No of Iterations Swarm Particles
cx C2
Fixed pf 150 100
20
20
Unspecified pf 300 100
25
25
177
Table 523 33-bus RDS Four DG FixedCase 20 HPSO Simulations
HPSO-PLoss (kW)
277083
279546
276120
275513
279060
277060
278930
275691
275490
275503
275567
275511
276301
276967
275505
276793
280457
277035
276955
277083
Busl No
30
30
14
32
30
30
14
32
30
30
14
30
30
10
25
30
30
16
30
30
DG1P (kW)
9418793
8899458
5902035
3533880
8850666
9431930
5138807
3258655
6240482
6283890
6130877
6113547
6097041
3161291
2652935
9345404
9230294
3760506
9347878
9418793
Bus 2 No
15
9
25
14
14
10
30
30
14
14
32
14
25
25
32
25
25
25
25
15
DG2P (kW)
3855380
3803090
2860738
6148504
4965770
2978961
9152690
6571557
6186146
6172676
3538431
6155489
3028569
2201454
3526143
2301409
2245170
2331059
2305772
3855380
Bus 3 No
25
15
30
30
8
25
8
14
25
32
25
25
14
15
14
16
8
30
15
25
DG3P (kW)
2122888
4066616
6449916
6389478
2945827
2225142
2315442
6145663
2648659
3560187
2767195
2699495
6121276
3896900
6165658
3639479
1685866
9263938
3925357
2122888
Bus 4 No
10
25
32
25
25 J
15
25
25
32
25
30
32
32
30
30
10
14
10
10
10
DG4P (kW)
3310235
1925458
3494606
2635434
1945033
4071263
2100357
2731420
3632004
2690543
6270793
3738765
3460409
9447651
6362560
3421004
5545966
3351793
3128289
3310235
llAFll II Moo
(PU)
002886
002221
002493
002007
002252
002118
002180
002021
001998
002008
002031
002014
002071
002115
002004
002165
002180
002183
002157
002886
Table 524 Descriptive Statistics for HPSO Results for Fixed Four DG Cases
Variable HPSO-PLoss
N 20
Mean 27703
SE Mean 00342
StDev 0157
Minimum 27549
Median 27695
Maximum 28046
178
Table 525 HPSO vs FSQP Results 33-bus RDS-Four DG-FixedCase
Optimal Placement Bus
Optimal DG Size (kW)
Minimum Real Power Losses (kW)
A^ M ( pu )
Simulation Time
Four DG Profile HPSO
DG1 Bus =14 DG2 Bus =25 DG3 Bus =30 DG4 Bus =32
DG1P= 6186146 DG2 P= 2648659 DG3 P= 6240482 DG4P= 3632004
275490
0019975
141003 sec
Four DG Profile FSQP
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30 DG4 Bus= 32
DG1 P = 6280595 DG2P = 2751438 DG3 P = 4962089 DG4P = 4713174
277073
0019902
Single Run
APC
18122 sec 326442210sec
(9 hrs 40703 min)
Table 526 33-bus RDS Four DG Unspecifiedpf Case 20 HPSO Simulations
HPSO-Ploss (kW)
191111
189348 194469 196670 189306 191996 191344 190727 195886 190710 189593 188979 192809 192432 189050 191777 190604 191085 189123 190001
Busl No
10 14 30 14 14 30 16 9 17 9 14 14
25 30 8
25 30 16 25 15
DG1P (kW)
3918316 5741913 7809755 6479723 5728568 7883676 3497597 2841444 2412423 3806417 5818800 5632387 3261018 7264036 3491641 3576640 7914267 3807784 2999968 4713467
DG1 pf
08240
09178 06244 08701 09173 06217 09013 07409 08711 08238 09189 09140 06250 06000 07624 06615 06366 09200 06049 09112
Bus2 No
30 30 8
30 25 10 30 15 11 25 8 8 15 9
25 10 8
25 30 9
DG2P (kW)
7712309 7600806
1543993 5809869 3082108 3654257 7794761 4826099 4984319 3250157 2558453 2833733 4968191 3092430 2909138 3772454 2351115 3128447 7507225 3329225
DG2
Pf 06129 06226 06001 06000 06149 08108 06168 09135 08637 06270 06736 07139 09325 07697 06000 08150 06322 06048 06172 07888
Bus3 No
16 25 25 25 8
25 25 25 30 30 25 25 10 15 30 16 15 30 8
25
DG3P (kW)
3832397 2928746 2728126 3519895 2527884 3502500 3204443 3021292 7578558 7325065 3115176 2970421 2520139 4543564 7189997 3772939 5401385 7821754 2564003 3031467
DG3p
09170 06017 06000 06469 06737 06517 06145 06042
06001 06001 06187 06007 07206 09015 06010 09145 09163 06168 06802 06031
Bus4 No
25 8 14 32 30 16 10 30 25 15 30 30 30 25 15 30 25 10 14 30
DG4P (kW)
3235923 2427474 6617070 2888865 7360383 3658059 4201858 8010100 3723588 4317304 7205534 7262401 7949596 3798891 5108167 7576905 3032087 3940762 5627747 7624785
DG4 pj
06232
06543 09201 07740 06098 09085 08434 06331
06784 08973 06052 06048 06232 06852 09102 06055 06101 08243 09135 06142
mi 001551 001544 001497 002604 001615 001554 001537 001484 001506 001598 001585 001617 001531
001723 001623 001638 001641 001518 001588 001568
Table 527 Descriptive Statistics for HPSO Results for Unspecified Four DG Case
Variable HPSO-PLoss
N 20
Mean 19154
SE Mean 00462
StDev 0236
Minimum 18898
Maximum 19667
179
Table 528 HPSO vs FSQP 33-bus RDS-Four -UnspecifiedpCase
Optimal Placement Bus
Optimal DG Size (kW)
Optimal DG Size (kW)
Minimum Real Power Losses (kW)
AKM(pu)
Simulation Time
Four DG Profile HPSO
DG1 Bus =8 DG2Bus=14 DG3 Bus =25 DG4 Bus =30
DG1P= 2833733 DG2P= 5632387 DG3 P= 2970421 DG4P= 7262401 DGl= 07139 DG2= 09140 DG3= 06008 DG4 pf= 06048
188979
001617
230804 sec
Four DG Profile FSQP
DG1 Bus= 14 DG2 Bus= 25 DG3 Bus= 30 DG4 Bus= 32
DG1 P = 6728343 DG2P = 3533723 DG3P = 5118179 DG4P = 3318699 DGlgt= 09201 DG2= 09968 DG3= 06296 DG4 pf= 08426
247892
002047
Single Run
APC
25897 se 67509755sec
(18 hrs 45180 min)
562 Case 2 69-Bus RDS The 69-bus RDS is the second network to be tested by the proposed HPSO method The
same system was tested previously by the FSQP using the APC method in the previous
chapter The proposed metaheuristic method is applied to find out the optimal placement
and size of single double and three DG units simultaneously The DG unit planned to be
installed is dealt with either as a fixed pf and consequently its real power output is the
variable to be optimized by the proposed HPSO or as an unspecified in which the DG
unit real and reactive output powers are both to be optimized
5621 69-bus RDS Loss Minimization by Locating a Single DG The HPSO method was used in obtaining single DG optimal placement and size of fixed
and unspecified pf Table 529 shows the tuned HPSO parameters for both DG cases
The HPSO simulations results consistently picked bus No 61 for the optimal size of both
DG cases as shown in Table 530 and Table 533 Their corresponding descriptive
characteristics are shown in Table 531 and Table 534 The HPSO results for both
cases are compared to those obtained by the FSQP APC method and are recorded in
180
Table 532 and Table 535 The proposed HPSO method obtained both the optimal bus
location and the DG size that will cause the losses to be minimal simultaneously The
real power losses obtained by the HPSO are similar to those obtained by the FSQP
method The proposed HPSO convergence characteristics in the 69-bus fixed pf single
DG case are shown in Figure 516 when the maximum number of iterations is set to 15
Figure 517 shows HPSO particles at an extended number of the iterations ie 50 to
further examine its behavior Figure 518-Figure 522 show the swarm particles
clustering during the HPSO iterations of the fixed 69-bus pf DG case
Table 529 HPSO Parameters for 69-bus RDS Both Single DG Cases
No of Iterations Swarm Particles
ci
C2
Fixed DG pf 15 30
25
25
Unspecified DGpf 30 30
20
20
181
Table 530 69-bus RDS Single DG FixedpfCase 20 HPSO Simulations
HPSO-PLoss (kW)
238672
238672
238672
238673
238672
238673
238672
238672
238672
238672
238673
238672
238672
238672
238672
238672
238672
238672
238672
238672
DG Bus No
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
DG P (kW)
19043802
19041194
19043107
19038901
19044055
19052963
19044591
19042722
1904215
19041093
19047545
19045601
1904287
19045675
19046072
19043069
19045721
19044829
19043677
19042638
AFJpu)
002746
002748
002746
007578
002746
00277
002704
002746
00275
002731
002744
002795
002759
002706
002752
002746
003021
002808
002812
002747
Table 531 Descriptive Statistics for HPSO Results for the Fixedpf Single DG Case
Variable HPSO-PLoss
N 20
Mean 23867
SE Mean 0
StDev 0
Minimum 23867
Maximum 23867
Table 532 HPSO vs FSQP Results 69-bus RDS-Single DG-FixedgtDG Case
Optimal Placement Bus Optimal DG Size (kW) Minimum Real Power Losses (kW)
AKw gt(pu)
Simulation Time (sec)
Single DG Profile HPSO
61 19043069 238672
002746
0626260
Single DG Profile FSQP
61 19038
238670
002747
Single Run APC
15117 396650
182
Table 533 69-bus RDS Single DG UnspecifiedCase 20 HPSO Simulations
HPSO-PLoss (kW)
231718
231718
231719
231719
231727
231720
231719
231727
231752
231719
231720
231731
231718
231719
231718
231718
231719
231718
231718
231880
Bus No
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
DG P (kW)
18286454
18276258
18302607
18284797
18234223
18262366
18272948
18314543
18363127
18297682
18308059
18280884
18286849
18270745
18285174
18286025
18274493
18278084
18280971
18131141
GGpf
08149
08148
08152
08151
08143
08148
08148
08145
08173
08149
08154
08161
08149
08148
08149
08149
08149
08147
08149
08093
AF x (pu)
002753
002754
002752
002753
002756
002755
002754
002750
002750
002752
002752
002755
002753
002754
002753
002753
002754
002753
002753
002757
Table 534 Descriptive Statistics for UnspecifiedSingle DG Case
Variable HPSO-PLoss
N 20
Mean 23173
SE Mean 000081
StDev 000361
Minimum 23172
Maximum 23188
183
Table 535 HPSO vs FSQP Results 69-bus RDS-Single DG-Unspecified^DG Case
Optimal Placement Bus Optimal DG Size (kW) Optimal DG Power Factor Minimum Real Power Losses (kW)
AKB(pu)
Simulation Time
Single DG Profile HPSO
61 18285174
08149 231718
002753
098187
Single DG Profile FSQP
61 18365 08386 23571
002782
Single Run
APC
21770 sec 810868 sec (13514 min)
Maximum HPSO Iterations =15
7 9
HPSO Iteration No
15
Figure 516 Convergence characteristics of HPSO in the 69-bus fixed pf single DG case HPSO proposed number of iterations =15
184
Maximum HPSO Iterations = 50
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
HPSO Iteration No
Figure 517 Convergence characteristics of HPSO in the 69-bus fixedsingle DG case HPSO proposed number of iterations = 50
Swarm particles at Iteration 1
2000
1800
f 1600
~ 1400
1200
Q 1000
bullg 800
lt 600
sect 400
200
0
---
bull -
~_ -
bull
bull
bull
bull
bull bull
bullbull bull bull
bull
bull bull
bull
bull
bull bull
bull
bull
bull bull bull bull
bull t
bull
bull bull
5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
Figure 518 Swarm particles distribution at the first HPSO iteration
185
Swarm Particles at Iteration 5
bullsect 750
^ 500
deg 250
5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
th Figure 519 Swarm particles distribution at the 5 HPSO iteration
Swarm particles at Iteration 10
2500
2000
5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
th Figure 520 Swarm particles distribution at the 10 HPSO iteration
186
Swarm Particle at Iteration 15
2000 -
3 1500 ogt 5 pound 1000 0)
tgt o lt 500 O Q
0 J 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
Figure 521 Swarm particles distribution at the 15th HPSO iteration
Swarm Particle at Iteration 15
I i
Act
ive
Pow
er
O Q
1909 -
1907
1905
1903 -
1901
1899
1897
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
69-Bus RDS Bus No
Figure 522 Close up of the HSPO particles at iteration 15
5622 69-bus RDS Loss Minimization by Locating Multiple DGs Double DG Case The proposed HPSO tuned parameters utilized in optimally placing
and sizing double DG units with proposed fixedpf and unspecifiedare shown in Table
536 Table 537 and Table 540 show 20 simulation results of the proposed HPSO
method for both DG cases and their corresponding descriptive data are tabulated in Table
538 and Table 541 The comparison results between the metaheuristic and deterministic
methods are shown in Table 539 and Table 542 For the fixed pf case the HPSO
187
proposed the same bus locations as the FSQP with comparable distribution real power
losses However in the second double DG case where the pfs are to be optimized in
addition to the DG real power outputs the metaheuristic method proposed two different
bus locations alternatively along with bus 61 ie 17 and 18 That is the HPSO method
chose either busses 17 and 61 or 18 and 16 to host the DG units while the deterministic
method chose buses 21 and 61 The mean value of the real power losses of the second
case when optimal sized DGs were installed at the optimal locations proposed by HPSO
is approximately 10 lower than that of the FSQP method
Table 536 HPSO Parameters for 69-bus RDS the Double DG Cases
No of Iterations Swarm Particles
c i
C2
Fixed 100 50
205
205
Unspecified pf 100 60
21
21
188
Table 537 69-bus RDS Double DG FixedpCase 20 HPSO Simulations
HPSO-PLoss (kW)
134738
134677
134708
134676
134674
134673
134767
134694
134674
134793
134673
134706
134701
134728
134911
134673
134673
134679
134673
134707
Bus 1 No
21
21
61
61
61
21
21
21
21
61
21
21
21
21
61
21
61
61
61
21
DG1 P (kW)
3325027
3265562
15774943
15853625
15846278
3242582
3341803
3197361
3255470
15723766
3239613
3185220
3297781
3318475
15694493
3241813
15836767
15846228
15834565
3302481
Bus 2 No
61
61
21
21
21
61
61
61
61
21
61
61
61
61
21
61
21
21
21
61
DG 2 P (kW)
15753239
15812718
3303337
3224654
3232001
15835697
15736477
15880899
15822809
3354514
15838666
15893055
15780495
15759802
3381832
15836464
3241510
3231851
3243715
15775799
AF x (pu)
001381
001359
001373
001345
001348
001351
001387
001335
001356
001391
001350
001331
001371
001378
001402
001351
001351
001348
001352
001373
Table 538 Descriptive Statistics for HPSO Results for Fixed j^Double DG Case
Variable HPSO-PLoss
N 20
Mean 13471
SE Mean 000130
StDev 000583
Minimum 13467
Maximum 13491
189
Table 539 HPSO vs FSQP Results 69-bus RDS-Double DG-FixedpDG Case
Optimal Placement Bus
Optimal DG Size (kW)
Minimum Real Power Losses (kW)
AFM(pu)
Simulation Time
Double DG Profile HPSO
DGlBus=21 DG2 Bus= 61
DG1P= 3243716 DG2P= 15834565
134673
001352
53339
Double DG Profile FSQP
DGlBus=21 DG2 Bus= 61
DG1P = 3241703 DG2P= 15836577
134672
001351
Single Run
APC
15814 sec 16291569 sec (271526 min)
Table 540 69-bus RDS Double DG Unspecifiedpf Case 20 HPSO Simulations
HPSO-PLoss (kW)
98350
98355
98355
98375
98377
98377
98417
98483
98504
98597
98615
98642
98700
98714
98737
98935
98967
99208
99530
99817
Bus 1 No
17
17
61
17
61
61
17
17
61
61
61
17
61
18
61
17
61
61
18
61
DG1P (kW)
3635963
3603665
15478880
3616139
15508060
15503850
3522418
3766853
15285240
15629720
15594800
3410166
15213880
3503923
15195080
3888970
15652210
15614700
3804638
15830600
DG1 Pf
07182
07171
07807
07215
07815
07817
07054
07290
07767
07829
07851
06961
07780
06805
07764
07499
07909
07820
07598
07921
Bus 2 No
61
61
18
61
18
18
61
61
18
18
17
61
17
61
17
61
17
17
61
18
DG2P (kW)
15420040
15452330
3577076
15439680
3547943
3552105
15533580
15289140
3770750
3426158
3460978
15645820
3841595
15550060
3860893
15161870
3403486
3416307
15240540
3224263
DG2 Pf
07798
07798
07119
07814
07092
07127
07818
07767
07382
06997
06864
07842
07397
07840
07315
07757
06789
06740
07655
06441
IIAFII (Pu) II II00 v
001058
001047
001115
001032
000988
001037
001023
001094
001097
001017
001377
001105
001278
001023
001113
001131
001025
001034
001058
001031
190
Table 541 Descriptive Statistics for HPSO Results for Unspecified^Double DG Case
Variable HPSO-PLoss
N 20
Mean 98703
SE Mean 000915
StDev 00409
Minimum 98350
Maximum 99817
Table 542 HPSO vs FSQP Results 69-bus RDS-Double-UnspecifiedjpDG Case
Optimal Placement Bus
Optimal DG Size (kW)
Optimal Power factor
Minimum Real Power Losses (kW)
AF w (pu)
Simulation Time
Double DG Profile HPSO
DGlBus=17 DG2Bus=61
DG1P = 3635963 DG2P= 15420037
DGl pf= 07182 DG2 pf= 07800
983501
001058
83609
Double DG Profile FSQP
DGlBus=21 DG2 Bus= 61
DGl P = 3468272 DG2P= 15597838
DGl pf= 08276 DG2= 08130
110322
001263
Single Run
APC
34446 sec 38703052 sec
( lh r 4505 lmin)
Three DG case The tuned HPSO parameters for both cases of the three DG installations
are shown in Table 543 The HPSO results of installing three DG units with their pfs
fixed and unspecified are shown in Table 544 and Table 547 respectively Table 545
and Table 548 display the corresponding descriptive statistics of the HPSO simulations
Optimal results obtained by the proposed HPSO for bothcases of the three DG sources
are compared with those attained by the FSQP method and tabulated in Table 546 and
Table 549 The results of the fixed pf case is similar to that of the FSQP method
outcomes however the time consumed by the HPSO to reach both optimal locations and
sizes is drastically less than that of the FSQP APC method The HPSO method proposed
a different bus set for the unspecifiedunits The metaheuristic method bus location
solution sets are 17 61 and 64 or 18 61 and 64 while the FSQP APC technique optimal
locations are 21 61 and 64 The former bus location sets resulted in lower real power
losses than that of the deterministic method ie approximately 12 compared to its
191
FSQP counterpart All the bus voltages of the 69-bus RDS are within limits and their
deviation from the nominal value is similar to that of the FSQP method
Table 543 HPSO Parameters for Both 69-bus RDS Three DG Cases
No of Iterations Swarm Particles
lth C2
Fixed DG^
175 150
20
20
Unspecified DG
100 100
20
20
Table 544 69-bus RDS Three DG Fixedpf Case 20 HPSO Simulations
HPSO-Ploss (kW)
126920
126920
126920
126920
126920
126920
126920
126920
126920
126920
126920
126921
126923
126924
126925
126926
126929
127187
126920
126920
Bus 1 No
61
21
64
21
64
21
64
61
21
21
64
64
64
61
64
64
21 64
64
64
DG1P (kW)
12811740
3247850
3013549
3247568
3012648
3248786
3011778
12808460
3247902
3252575
3024740
2988894
3080030
12738410
3055250
3097303
3277815
3463001
3014590
3014261
Bus 2 No
64
61
21
64
61
64
21
64
61
61
61
21
21
21
21
21
64
61
61
21
DG2P (kW)
3014639
12811530
3247541
3016126
12813680
3013724
3249259
3016429
12820630
12819490
12795680
3254458
3243396
3255536
3267854
3242037
2991308
12461850
12811840
3248069
Bus 3 No
21
64
61
61
21
61
61
21
64
64
21
61
61
64
61
61
61 21
21
61
DG3P (kW)
3247955
3014953
12813240
12810640
3248007
12811820
12813300
3249439
3005797
3002270
3253914
12830980
12750910
3080382
12751230
12734990
12805210
3149486
3247907
12812000
llAKll (pu) II llco V1
001208
001208
001208
001208
001208
001208
001207
001207
001208
001206
001206
001042
001210
001205
001200
001210
001197
001243
001208
001208
192
Table 545 Descriptive Statistics for HPSO Results for FixedThree DG Case
Variable HPSO-PLoss
N 20
Mean 12693
SE Mean 000133
StDev 000595
Minimum 12692
Maximum 12719
Table 546 HPSO vs FSQP Results 69-bus RDS-Three DGs-FixedjCase
Optimal Placement Bus
Optimal DG Size (kW)
Minimum Real Power Losses (kW)
AF x (pu)
Simulation Time
Three DG Profile HPSO
DGlBus=21 DG2Bus=61 DG3 Bus= 64
DG1 P = 3247907 DG2P = 12811836 DG3P = 3014590
126917
001208
34137497 sec
Three DG Profile FSQP
DGlBus=21 DG2Bus=61 DG3 Bus= 64
DG1P = 3191431 DG2 P= 12883908 DG3 P= 2998994
126947
001230
Single Run
APC
25735 sec 580575800 sec
(16 hrs 76266 min)
193
Table 547 69-bus RDS Three DG Unspecifiedpf Case 20 HPSO Simulations
HPSO-Ploss (kW)
90618
90618
90618
90620
90621
90626
90627
90627
90628
90629
90630
90630
90632
90632
90642
90645
90649
90649
90656
90657
Bus 1 No
64
64
18
18
18
18
61
64
17
61
64
17
61
17
64
17
61
61
17
18
DG1P (kW)
2892620
2884199
3624913
3644557
3619850
3624040
12535890
2911554
3625999
12535820
2894295
3637839
12570950
3657899
2702745
3639403
12638440
12376520
3692494
3667257
DG1 Pf
08139
08133
07167
07191
07170
07171
07723
08153
07172
07696
08131
07188
07732
07202
07949
07185
07755
07684
07241
07227
Bus 2 No
61
18
64
61
64
61
64
61
64
64
61
64
17
64
61
64
64
64
61
64
DG2P (kW)
12530550
3625321
2899040
12502150
2825088
12649170
2887758
12503590
2856924
2894843
12572390
2831037
3600503
2888943
12735400
3059250
2741028
2983367
12395320
2688736
DG2 Pf
07723
07173
08133
07715
08067
07751
08138
07717
08106
08274
07736
08076
07148
08138
07772
08313
07956
08224
07691
07926
Bus 3 No
18
61
61
64
61
64
17
17
61
18
18
61
64
61
18
61
18
18
64
61
DG3P (kW)
3629152
12542800
12528370
2905612
12607390
2779116
3628678
3637178
12569400
3621582
3585635
12583450
2880873
12505480
3614176
12353670
3672854
3692438
2964511
12696330
DG3 Pf
07177
07725
07727
08153
07743
08029
07175
07181
07732
07196
07137
07734
08129
07716
07163
07671
07191
07236
08193
07772
llA1 II Moo
(pu)
000947
000947
000947
000945
000948
000947
000947
000946
000947
000950
000958
000946
000954
000944
000948
000945
000940
000940
000940
000944
Table 548 Descriptive Statistics for HPSO Results for UnspecifiedpThree DG Case
Variable
HPSO-PLoss
N
20
Mean
90633
SE Mean StDev
0000279 000125
Minimum 90618
Maximum 90657
194
Table 549 HPSO vs FSQP Results 69-bus RDS-Three DGs-Unspecified^Case
Optimal Placement Bus
Optimal DG Size (kW)
Optimal DG Power Factor
Minimum Real Power Losses (kW)
AF K (pu)
Simulation Time
Three DG Profile HPSO
DG1 Bus= 18 DG2Bus=61 DG3 Bus= 64
DG1P = 3629152 DG2P =12530552 DG3P = 2892612 DGl pf= 07177 DG2 pf= 07723 DG3 pf= 08139
906180
0009467
105018 sec
Three DG Profile FSQP
DGlBus=21 DG2 Bus= 61 DG3 Bus= 64
DGl P = 3463444 DG2 P= 12937085 DG3P= 2661795 DGl p=08275 DG2 pf= 08264 DG3=07491
102749
001088
Single Run
APC
25735 sec
580575800 sec (16 hrs 76266 min)
563 Alternative bus Placements via HPSO In practice not all buses can necessarily accommodate the DG source If the optimal set
of bus locations is not suitable to host the DG units alternative bus locations can also be
proposed via the HPSO method That is by relaxing the HPSO parameters ie not
optimally tuned suboptimal solutions will be obtained instead However the suboptimal
proposed DG locations and sizes might yield a good-enough solution and is left as a
suggestion for the distribution system planner to consider As an example if alternative
bus locations are needed for the fixed pf three DGs instead of the optimal bus placement
set of 21 61 and 64 reducing the number of iterations andor tuning any of the HPSOs
other parameters suboptimally as shown in Table 550 will obtain different bus location
sets within reasonable real power loss levels compared to its optimal case counterpart
The last column of the table shows the percentage of the real power losses obtained by
the suboptimal solutions compared with the optimal real power losses obtained from
Table 546 The percentage is calculated as follows jySubOptimal -nOptimal
0 p Losses Losses 1 fi orLosses ~ -pSubOptimal (520)
Losses
195
Table 550 20 HPSO Simulations of the 69-bus RDS Three DG Fixed Case with Suboptimal Tuned Parameters 50 Iterations and 50 Swarm Particles
HPSO-PLoss
(kW)
128607
133509
135925
133760
133202
130080
130620
131654
129292
129840
135013
133163
127482
129346
127684
127210
129930
132025
138624
133856
Busl
No
64
22
61
22
23
61
21
22
64
21
62
61
64
64
64
61
64
61
61
17
DG1P
(kW) 1651962
2446599
15155360
1247132
2806169
14825300
3243916
3324601
4519564
2994546
7020292
15723540
3802847
1746433
2224049
12218480
1732514
10721640
15256200
1476435
Bus 2 No
22
61
59
61
61
65
61
61
61
64
61
18
21
21
21
64
18
22
15
61
D G 2 P
(kW) 3264935
15819390
779523
15929380
14532960
1095336
14876490
15038080
11208700
1646331
8850952
1206409
3300895
2938428
3156370
3568548
3641291
3049827
2403629
15428600
Bus 3 No
61
17
22
18
65
21
64
64
20
61
21
22
61
61
61
21
61
64
24
21
DG3P
(kW) 14157330
807880
3138036
1897812
1670272
3152199
952623
711351
3345310
14403870
3202974
2144132
11970570
14384650
13687960
3286823
13700420
5293711
1331709
2169113
llAKJI 11 1 loo
00124
00136
00137
00108
00139
00127
00136
00131
00160
00129
00129
00128
00119
00132
00123
00119
00332
00128
00156
00148
Losses
1312
4936
6625
5114
4716
2429
2833
3596
1835
2249
5995
4688
0441
1876
0599
0229
2317
3867
8443
5182
57 SUMMARY
This chapter presents a new application of PSO in optimal planning of single and
multiple DGs in distribution networks The proposed HPSO approach hybridized PSO
with the developed FFRPF method to simultaneously solve the optimal DG placement
and sizing problem A hybrid constrint handling mechanism was utilized to deal with the
constrained mixed-integer nonlinear programming problems inequality constraints
Many overall positive impacts such as reducing real power losses and improving
network voltage profiles can be encountered once an optimal DG planning strategy is
implemented This can improve stability and reliability aspects of power distribution
systems HPSO performance and robustness in its search for an optimal or near optimal
solution is highly dependant on tuning its parameters and the nature of the problem at
196
hand The 33-bus RDS as well as the 69-bus RDS had been used to validate the
proposed method Results of the HPSO method were compared to those obtained by the
FSQP APC technique The comparison results demonstrate the effectiveness and
robustness of the developed algorithm Moreover the results obtained by the proposed
HPSO method were either comparable to that of the deterministic method or better
197
CHAPTER 6 CONCLUSION
61 CONTRIBUTIONS AND CONCLUSIONS
Integrating DG within electric power system networks is gaining popularity worldwide
due to its overall positive impact The DG is different from large-scale power generation
in its energy efficiency capacity and installation location Technological advancement is
allowing such generating units to be economically feasible to be built in different sizes
with high efficiency and efficient sources of electricity that would support the distribution
system Located at or near the load DG helps in load peak shaving and in enhancing
system reliability when it is utilized as a back-up power source should a voluntary
interruption be scheduled The DG can defer costly upgrades that might take place in the
transmission and distribution network infrastructure and decrease real power losses
Having a minimal environmental impact and improving the DS voltage profiles are
additional merits of such addition to the network
Distribution networks where the DG is usually deployed are different from the
transmission and sub-transmission system in many ways For the DS rather than being
networked as in its transmission system counterpart they are usually configured in a
radial or weakly meshed topology The DS is categorised as a low voltage system that
have feeders with low XR ratios It has large number of sections and buses that are
usually fed by a main distribution substation located at its root node
In this thesis the optimal DG placement and sizing problems within distribution netshy
works were investigated by utilizing deterministic and heuristic methods A FFRPF
method for balanced and unbalanced three-phase DSs was developed in Chapter 3 This
proposed power flow algorithm was incorporated within the conventional SQP determishy
nistic method as well as in the HPSO metaheuristic method to satisfy the nonlinear
equality constraints as discussed in Chapters 4 and 5
The FFRPF was developed based on the backwardforward sweep technique where
the load currents summation process takes place during the backward sweep and the bus
voltages are updated during the forward sweep The unique structure of the RDSs was
exploited in developing RCM for strictly radial topology and mRCM for meshed systems
198
in order to proceed with the solution This matrix which represents the DS topology is
designed to be an upper triangular matrix with unity determinant magnitude and all of its
eigenvalues are equal to 1 in order to insure its invertiblity Besides the DS parameters
only the RCM (or mRCM) is needed to carry out the FFRPF method The backward
forward sweep process is carried out by using two matrices ie SBM and BSM (or
wSBM and mBSM) which are direct descendents of their corresponding building block
matrix That is the RCM (or mRCM) is inverted to obtain the SBM (or mSBM) and is
consequently utilized in the backward sweep to sum the distribution load currents The
SBM (mSBM) is transposed and the resulting BSM (or mBSM) is used to update the bus
voltage during the forward sweep The FFRPF is tested on small large strictly radial
weakly meshed and looped DSs (10 DSs were tested in total) The FFRPF is proven to
be robust and to have the lowest CPU execution time when compared with other
conventional and distribution power flow methods
The DG sizing problem is formulated as a constrained nonlinear programming optishy
mization problem with the DS real power losses as the objective function to be
minimized The optimal DG rating problem was solved by both the SQP and the develshy
oped FSQP methods In the developed FSQP methodology the FFRPF was incorporated
within the conventional SQP method to satisfy the nonlinear equality constraints By
employing the FFRPF as a subroutine to satisfy the power flow requirements the compushy
tational time was reduced drastically compared to that consumed by the SQP
optimization method Optimally installing single and multiple DGs with fixed and
unspecified pfs throughout the DS were studied thoroughly utilizing both methods The
APC search method was utilized to find the optimal DG placement and sizing in the
tested distribution networks these results were subsequently compared to those obtained
by the HPSO heuristic method
The HPSO was utilized to optimally locate and size single and multiple DGs with
specified and unspecified pfs The DG integration problem was formulated as a conshy
strained mixed-integer nonlinear optimization problem and was solved via the developed
HPSO method The output solution of the developed HPSO optimization method is
expected to deliver both the DG location bus as a positive integer number and its correshy
sponding rating as real value in a single run That is both optimal DG placement and
199
sizing are obtained simultaneously The HPSO method developed in this thesis is an
advanced version from the classical PSO The developed FFRPF technique was incorposhy
rated within the HPSO method to take care of the distribution power flow equality
constraints Two constraint handling methodologies were hybridised together in order to
satisfy the requirements of the HPSO inequality constraint requirements ie the preservshy
ing feasible solutions method is hybridized with the rejecting infeasible solutions method
That is while the HPSO method initially emphasizes all of the population to be a feasible
set of solutions the particles are allowed to cross over the boundaries of the problem
search space However whenever infeasible solutions are encountered they are rejected
and replaced by their previous preserved feasible values and no further reinitializing is
required
In this research it is shown that proper placement and sizing of DG units within the
DS networks generally minimized the real power losses improved the system voltage
profiles and released the substation capacity The DG also decreased the feeders
overloading consequently allowing more loads to be added to the existing DS in future
planning without the need to build costly new infrastructure
It is also shown that the active distribution power losses are decreased further when
more than one DG unit is optimally integrated within the DS However beyond a certain
number of DGs the decrease in power losses is insignificant Therefore the power
distribution planner should pay more attention to the expected decrease in power losses if
additional DG units are to be installed
Deploying single and multiple DG units within the DS network are examined with
fixed and unspecified pfs In the latter case the power factor variables are also optimized
along with their corresponding sizes and placements in the hopes of searching for the best
combinations that would cause the losses to be minimal The fixed pf cases showed that
their resultant real power losses are comparable to that of the unspecified cases Thus a
fixed power factor DG unit to be installed at or near the load center is a practical and
suitable choice for the system planner
200
62 FUTURE WORK
The analysis of optimal DG placement and sizing problems and the proposed solution
methods presented in this thesis can be further extended and enhanced The following
subjects may shed some light on the intended work extensions
bull A constant power representation was used in modeling the DS loads Differshy
ent load models as well as more precise practical modeling can be studied to
examine their effect on the DG integration problem
bull Several heuristic tools have evolved or been introduced during the last few
years that have shown the capability of solving different optimization probshy
lems that are difficult in nature or even impossible to solve by conventional
deterministic methods Examples of such techniques are the bacteria swarm
foraging optimization method the bee algorithm and the ant colony optimizashy
tion The DG placement and sizing problem can be further tackled by such
methods and their obtained results can be compared with that of the proposed
HPSO method presented in this thesis
bull The effect of the developed FFRPF method in handling the equality conshy
straints in the aforementioned heuristic tools can be studied when applied to
solve the DG mixed-integer nonlinear optimization problem
bull The possibility of hybridizing the developed FFRPF within the GRG nonlinshy
ear programming method can be examined and its impact can be analysed as
done in the FSQP method
bull Incorporating harmonic aspects in the developed FFRPF method for both balshy
anced and unbalanced three-phase distribution networks is a task that can
further extend the scope of the proposed version of the FFRPF method
bull The developed distribution power flow can be extended to accommodate PV
bus types and to examine its efficiency in solving the transmission system
power flow by comparing its outcomes with that of conventional methods
bull The fuzzy set theory can be incorporated in the DG optimal placement and in
the sizing optimization problem formulation as well as in modeling the DS
load uncertainties
201
bull Tuning the HPSO parameters using statistical generalized models where the
errors are not necessarily normally distributed is an interesting research area
202
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219
APPENDIX
Section No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29 30
Ta
From
1
2
3
4
5
6
7
8
9
10
11
12
13
14
9
16
17
7
19
20
7
4
23
24
25
26
27
2
29 30
bleAl 31-
To
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Bus Balanced R D S Data
R(Q)
0896
0279
0444
0864
0864
1374
1374
1374
1374
1374
1374
1374
1374
1374
0864
1374
1374
0864
0864
1374
0864
0444
0444
0864
0864
0864
1374
0279
1374
1374
X (Q)
0155
0015
0439
0751
0751
0774
0774
0774
0774
0774
0774
0774
0774
0774
0751
0774
0774
0751
0751
0774
0751
0439
0439
0751
0751
0751
0774
0015
0774
0774
P(kW)
0
522
0
936
0
0
0
0
189
0
336
657
783
729
477
549
477
432
672
495
207
522
1917
0
1116
549
792
882
882 882
Q (kvar)
0
174
0
312
0
0
0
0
63
0
112
219
261
243
159
183
159
144
224
165
69
174
639
0
372
183
264
294
294
294 Sbase = 1000 kVA Vbase = 23 kV
220
Table A2 90-Bus Balanced RDS Data Section No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
From
1
2
3
4
5
6
7
8
9
10
11
12
12
4
5
6
7
18
18
8
9
22
23
23
22
10
11
3
29
30
31
32
33
33
30
31
37
To
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
R(Q)
00002
00004
00003
000002
00004
00001
00007
00012
0002
00009
00017
00013
00017
00001
00002
00002
00005
00004
00002
0001
00015
00002
00015
00012
0001
00007
00015
00001
000015
00004
00001
000015
00002
00003
0001
00002
00015
X (Q)
00015
00019
0002
000005
00008
00007
00012
00021
0008
00021
00027
00023
00025
00012
00001
00008
0001
00008
0001
00072
00025
00009
00092
00072
0007
00014
00028
00009
00008
00009
00003
000045
00009
00016
0004
00008
00017
P(kW)
0
0
0
0
0
0
0
0
0
0
0
0012
0123
0165
0066
0076
0
0231
0078
0234
0
0
0088
0067
0243
0123
0045
0
0
0
0
0
0028
0123
0181
0
0245
Q (kvar)
0
0
0
0
0
0
0
0
0
0
0
0009
0054
0091
0023
0034
0
0123
0035
0115
0
0
0033
0024
0124
0076
0021
0
0
0
0
0
0017
0051
0067
0
0123
221
Section No
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
From
37
32
29
41
42
43
44
44
43
42
48
48
41
51
52
53
54
54
53
52
58
58
51
61
61
2
64
65
66
67
68
69
70
70
65
73
74
75
To
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69 70
71
72
73
74
75
76
R ( Q )
0001
00001
000001
000004
00002
00012
00025
00015
00001
00001
00001
00002
00001
00004
00002
00004
00005
00003
00001
00002
00001
00002
00002
00003
00005
00005
00003
0009
00002
00001
00015
00009
00001
00006
000015
00012
00012
00025
X (pound1)
00025
00004
000005
000009
00007
00075
00085
00079
00009
00006
00005
00008
00012
00007
00008
00007
00009
0001
00009
00006
00007
00005
00007
00008
00012
00021
0001
0031
00015
00005
00025
00021
00004
0001
00021
00076
00095
00087
P ( k W )
0014
0013
0
0
0
0
0045
0013
0089
0
0091
0123
0
0
0
0
0088
0077
0098
0
0024
0124
0
0035
0032
0
0
0
0
0
0 0
0016
0017
0
0
0
0062
Q (kvar)
0011
0011
0
0
0
0
0019
0009
0034
0
0045
0067
0
0
0
0
0054
0052
0067
0
0013
0057
0
0012
0014
0
0
0
0
0
0
0
0012
0011
0
0
0
0034
222
Section No
76
77
78
79
80
81
82
83
84
85
86
87
88
89
From
75
74
73
64
80
81
81
80
66
85
85
67
68
69
To
77
78
79
80
81
82
83
84
85
86
87
88
89
90
R(Q)
00128
0002
000012
0001
00015
00017
00016
00001
00085
00012
00015
00003
00002
00003
X (Q)
00425
0009
00003
0005
00075
00082
0008
0007
00125
00075
00161
00025
00006
00015
P ( k W )
034
0082
0123
0
0
0087
0067
0012
0
0023
0024
0025
0034
0029
Q (kvar)
012
0032
0071
0
0
0045
0023
0006
0
0017
0018
019
0014
0019
All Section Impedance and Power Values are in pu
223
Table A3 69-Bus Balanced RDS Section No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
From
1
2
3
4
5
6
7
8
4
10
11
12
13
14
7
16
1
18
19
20
21
22
23
19
25
26
27
28
29
30
1
32
33
34
35
36
37
To
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
R(Q)
1097
1463
0731
0366
1828
1097
0731
0731
108
162
108
135
081
1944
108
162
1097
0366
1463
0914
0804
1133
0475
2214
162
108
054
054
108
108
0366
073
0713
0804
117
0768
0731
X (Q)
1074
1432
0716
0358
179
1074
0716
0716
0734
1101
0734
0917
055
1321
0734
1101
1074
0358
1432
0895
0787
111
0465
1505
111
0734
0367
0367
0734
0734
0358
0716
0716
0787
1145
0752
0716
P(kW)
100
60
150
75
15
18
13
16
20
16
50
105
25
40
100
40
60
40
15
13
30
90
50
60
100
80
100
100
120
105
80
60
13
16
50
40
60
Q (kvar)
90
40
130
50
9
14
10
11
10
9
40
90
15
25
60
30
30
25
9
7
20
50
30
40
80
65
60
55
70
70
50
40
8
9
30
28
40
224
Section No
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
From
38
39
34
41
42
43
44
42
46
44
37
49
50
51
1
53
54
55
56
57
54
59
60
61
57
63
64
65
64
67
68
To
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
R(X2)
1097
1463
108
054
108
1836
1296
1188
054
108
054
108
108
108
0366
1463
1463
0914
1097
1097
027
027
081
1296
1188
1188
081
162
108
054
108
X (Q)
1074
1432
0734
0367
0734
1248
0881
0807
0367
0734
0367
0734
0734
0734
0358
1432
1432
0895
1074
1074
0183
0183
055
0881
0807
0807
055
1101
0734
0367
0734
P(kW)
40
30
150
60
120
90
18
16
60
60
90
85
100
140
60
20
40
36
30
43
80
240
125
25
10
150
50
30
130
150
25
Q (kvar)
30
25
100
35
70
60
10
10
35
40
70
55
70
90
40
11
30
24
20
30
50
120
110
10
5
130
30
20
120
130
15 Sbsae = 1000 kVA Vbase = 11 kV
225
Section No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
From
1
2
3
4
5
4
7
8
9
10
3
12
13
14
Table A4
To
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Komamoi
R(Q)
000315
000033
000667
000579
001414
000800
000900
000700
000367
000900
002750
003150
003965
001607
to 15-Bus
X (fi)
007521
000185
003081
001495
003655
003696
004158
003235
001694
004158
012704
008141
010298
000415
Balanced RDS
12 B
0
000150
003525
000250
0
003120
0
000150
000350
000200
0
0
0
0
P(kW)
208
495
958
132
442
638
113
323
213
208
2170
29
161
139
Q (kvar)
21
51
98
14
45
66
12
33
22
29
2200
3
16
14
Sbsae = 10000 kVA Vbase = 66 kV
226