Dr. Hsiao-Dong Chiang
(i) Prof. of School of Electrical and Computer Engineering, Cornell University, Ithaca, NY
(ii) President of BSI, Ithaca, NY
Avaiable Delivery Capability of Unbalanced Distribution Networks to
Support High DG Penetrations
Contents
• Transmission networks are highly-utilized while distribution networks are usualy under-utilized.
• New tools are required to assess the
capabilities of distribution networks to meet the continual growth of renewable energy (RE) and to withstand the stress caused by these intermittent, and often uncoordinated sources of power.
Contents
• To achieve the goal of 33% renewable energy, the delievry capability of distribution networks needs to be assessed and enhanced,
• a CDFLOW (continuation distribution power flow) tool will be presented along with some analytical results.
• Its comprehensive modeling capability is
useful to simulate real-life situations.
Test Systems
IEEE 8,500-node networks
A practical 1101-node 3-phase unbalanced network
Features of Distribution Network
Compared to the transmission power system: 1. Radial or weakly meshed networks, 2. High R/X ratios, 3. Multi-phases ,unbalanced three phases, two
phases and single phase, 4. Unbalanced loads and 5. A wide variety of Distributed generators
Special Features of Distribution System
Computation engine : continuation methods
Transmission networks
Continuation Power Flow Engine (CPFLOW, CPF)
Chiang et al. (1990)
Ajjarapu et al. (1992)
Cañizares et al. (1992)
Chiang, Flueck, Balu (1995)
Iba, Yorino
others
Unbalanced distribution networks (?)
CDFLOW - Applications
to analyze voltage problems due to load and/or generation variations.
to evaluate/maximize available delivery capability.
to simulate distribution system static behaviors due to load and/or generation variations with/without control devices.
to conduct coordination studies of control devices for steady-state security assessment.
CDFLOW can be used in a variety of application such as
2
1
3
4
CDFLOW - Functional Specification
Functional Specification of CDFLOW Deriv. Imple.
Network Configuration
Radial distribution network
Meshed distribution network
Analytical Capabilities
3-phase power flow calculation
Voltage profiles
Power flows on each phase
3-phase continuation power flow
P-V, Q-V, P-Q-V curves
Exact SNB point and eigen analysis
Units and Typical Values Used in the Program
“km” unit for length
Typical voltages for Japanese distribution network High voltages: 6.6kv, 22kv and 33kv Low voltages: 100v, 200v and 415v
No Default Value for Voltage
CDFLOW - Modeling Capabilities
Functional Specification of CDFLOW Deriv. Imple.
Modeling Capabilities
Power Source Model for Transmission network Modeling
Dispersed Generational Models
PV Bus
PQ Bus
Capacitor Models Single-phase
3-phase
Load Models ZIP model for single-phase 2-phase and 3-phase
Loop Controller Model: Phase Shifting Transformers DC Link, Electro-mechanical Link
Not likely used in Distribution Systems
FACTS: Series/Parallel Compensated Model
CDFLOW - Modeling Capabilities
Functional Specification of CDFLOW Deriv. Imple.
Modeling Capabilities
Transformer
Single-phase
2-wire
3-wire
3-Phase
Wye-Delta
Delta-Delta Ungrounded
Grounded
Wye-Wye
Open Delta
V Connection 3-wire
4-wire
Delta-Grounded Wye
Autotransformer
Regulator Single-phase / 3-pahse LTC LDC
Uniqueness of power flow solution
Miu and Chiang (1998)
For radial unbalanced networks, the number of feasible power flow solution is one.
Local Bifurcations calculated in CDFLOW
SIB
SNB
Saddle-Node Bifurcation The stable equilibrium point and another equilibrium point coalesce and disappear in a saddle-node bifurcation as parameter varies. The physical meaning of SNB is available delivery capability of a network configuration.
Structure-Induced Bifurcation The mathematical mechanism is the switching of system equations. It usually induced by certain resources reach its Q-limit, the terminal bus type switches from PV to PQ, then the associated power flow equations changed.
Two types of local bifurcations will occur in the quasi-static power system model.
Conditions of Saddle-node Bifurcation
(A1) Bifurcation point is also an equilibrium point of the dynamic system, (A2) The corresponding system Jacobian matrix has a simple zero eigenvalue with right eigenvector v and left eigenvector w, such that (A3) (A4)
Condition A3 and A4 are transversal conditions,which are usually satisfied for general nonlinear system
Here we will focus on the condition A1 and A2
0 0, 0f x
0 0det , 0xf x
0 0, 0fx
2
0 02 , 0fx
x
Conditions of Saddle-node Bifurcation
Condition A1 requires that SNB point is also an
equilibrium point
Condition A2 requires that there is only one simple zero eigenvalue when a
SNB encounters.
The following equations also ensure condition A1 and A2
, 0 , 00 0
1 1x x
T T
f x f x
wf or f v
ww vv
Structure Induced Bifurcation: SIB: (Dobson 1992, Li and Chiang 2005) A peculiar bifurcation in power networks at which the real part of eigenvalues is non-zero
V
V
Post QPost Q
Pre Q Pre Q
Structure Induced Bifurcation No stable equilibrium point after branch switch (corresponding to bus type switch from PV to PQ)
Structure Induced Exchange Process Reach a stable equilibrium point after branch switch (corresponding to bus type switch from PV to PQ)
CDFLOW - Formulation
01 1
01 1
cos sin 0
sin cos 0
N M
ji ij ij ij ij Lii Gij
N M
ji ij ij ij ij Li
k k k kk
ij
k
k k k kk
k
P V V G B P P
Q V V G B Q
01 1
0
01 1
, ,
cos sin 0
sin cos 0
N M
ji ij ij ij ij Lii Gk
ij
ii
N M
ji ij ij ij ij Liij
min i m
k k kk
k
k
ax iGi
k k kk
k
P V V G B P P
V V
Q V V G B Q
Q Q Q
For a PV node , the continuation power flow equations is
For a PQ node, the continuation power flow equations is
CDFLOW - Formulation
20 0 0
0
1.0 1,2,...,
ii i iiLi Li Li Li Li
i
i i i
VP jQ S V S S
V
i N
For a P-V node with a Q limit, say Qmin and Qmax, the 3-phases continuation power flow equation can be expressed as:
, 0
, 0
, 0
, 0
pre Q G max i ii
post Q G max i ii
f x Q Q V V
f x Q Q V V
The load model can be constant impedance, constant current, constant power or their combination:
In summary, a three-phase continuation power flow can be expressed in the following form:
0 ,f x f x b
CDFLOW - Architecture
The implementation of the four basic elements of the continuation method in CDFLOW
Predictor
Corrector
Step-size Control
Tangent predictor (phase-one)
Secant predictor (phase-two)
Nonlinear predictor (phase-three)
Parameterization
Arc-length
Pseudo-arc-length
Local parameter
Modified Newton method
Implicit Z-bus Gauss method
Adaptive Step-size Control
CDFLOW
CDFLOW - Parameterization
2 2 2
10
n
i ii
x x s s s
1. Arc-length Parameterization 2. Pseudo Arc-length Parameterization 3. Local Parameterization
1
0n
i i ii
x x s x s s s s
ˆ 0k kx x
kx
x
x
P-V Curve of the Weak Bus M1026328
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
Lambda
Voltage M
agnitude (
p.u
)
3-phase P-V Curves (abc)
BusM1026328
Vmaga
Vmagb
Vmagc
Dominant Eigenvalue of the Jacobian
0 0.5 1 1.5 2 2.5-4
-2
0
2
4
6
8x 10
-4
Lambda
Real P
art
P-Eigenvalue Curve
SNB
Stable Half Portion
Unstable Half Portion
Voltage Magnitudes of Weak Bus 1216
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
Lambda
Vol
tage
Mag
nitu
de (
p.u)
3-phase P-V Curves (abc)
Vmaga
Vmagb
Vmagc
Fig. 3. The voltage magnitudes at Bus1216 as parameter varies
Bus with DG
Mathematical description λ
value
Physical description MW
value
Voltage or thermal violation
(Vmin = 0.9 p.u)
M1209790 0 0 L3177894 M3016088
…
M1108378 2.302819 6.908457 L3082993 M3016088
…
M1026927 3.866104 11.598312 Bus L3216339 Bus L3225319
…
L2691967 8.070759 24.212277 * L_6102322_6100788
DG Penetration at Different Locations TABLE I
DG Penetration Capability with Load Variation = 0 +j 0
Bus with DG
Mathematical description λ
value
Physical description MW value
Voltage or thermal violation
(Vmin = 0.9 p.u)
M1209790 0 0 L3177894 M3016088
…
M1108378 0.141103 0.423309 L3195327 M1027114
…
M1026927 0.236323 0.708969 L3225319 L3048201
…
L2691967 0.255488 0.766464 L3216339 L3225319
…
DG Penetration at Different Locations TABLE II
DG Penetration Capability with Load Variation = Pd0 +j Qd0
Test Case 4: ZIP Load Model
TABLE III Voltage Stability Limit with different load models
Case No. SCENARIO Limit ( )
1 Half number of loads are constant impedance, other loads are constant PQ 1.47632457
2 Half number of loads are constant current, other loads are constant PQ 1.41906738
3 All loads are constant PQ 1.11565160
4 Half number of loads are 30% constant current, 30% constant current and 40% constant PQ, other loads are constant PQ
1.36719488
*
load margin of 1101-node practical distribution network with different load models
Dominant Eigenvalue of the Jacobian
0 0.2 0.4 0.6 0.8 1 1.2-4
-3
-2
-1
0
1
2
3
4
5
6
7x 10
-3
SNB
Lambda
Rea
l Par
tP-Eigenvalue Curve
Stable Half Portion
Unstable Half Portion
Fig. 5. The largest real part of eigenvalues of Jacobian matrix
012-Sequence Voltage of Weak Bus 1216
0 0.2 0.4 0.6 0.8 1 1.20
0.2
0.4
0.6
0.8
1
Lambda
Vol
tage
Mag
nitu
de (
p.u)
3-phase P-V Curves (012 sequence)
Vmag1
Vmag2
Vmag0
Fig. 6. The sequence voltage magnitudes at Bus1216 as parameter varies.
Voltage Stability
NYSEG 394 Bus System
The NYSEG 394 bus, 1103 node system is an unbalanced distribution network serving balanced and unbalanced loads. The substation bus delivers power to the full feeder and there is no DG in this network. In this test case, the load variation is same as the initial loading condition, and the following aspects are studied: • The properties of 3-phase P-V curves; • The weak node in the network • The trajectory of dominant eigenvalue along solution
curve, and • The effect of composite load model
Test Case 4: ZIP Load Model
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.2
0.4
0.6
0.8
1
Lambda
Vol
tage
Mag
nitu
de (
p.u)
3-phase P-V Curves (abc)
Pm2
Pm4
Pm3
Pm1Bus1102
Vmaga
Vmagb
Vmagc
Preventive and Enhancement Control
The key objective for development of enhancement control is to enlarge the available delivery capability and voltage stability load margin. The control actions includes: • Shunt capacitor; • ULTC tap changers; • ULTC phase-shifter; • Distributed generator terminal voltage control; • Load shedding; • Placement of new shunt capacitors;
There are two key steps in control design: • Determine a set of most effective controls; • Determine the amount of these controls;
Control #1: Load Shedding
Elements in left eigenvector corresponding to bus 671 are (0.042069, 0.031748), shedding load 1155+ j 660 kw at bus 671.
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
log10 (Real)
log10 (
Imag)
Eigenvalues of Jacobian before control Maximum load is 5.047896
Eigenvalues of Jacobian after control Maximum load is 5.980143
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
log10 (Real)
log10 (
Imag)
Thanks!
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