Multi-Objective based Congestion Management using Generation Rescheduling and Load Shedding

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Transactions on Power Systems

1

Multi-Objective based Congestion Management

using Generation Rescheduling and Load SheddingS. Surender Reddy, Member, IEEE

AbstractThis paper proposes a new congestion managementapproach by using the generation rescheduling and load shed-ding, with the realistic voltage-dependent load modeling. Thepaper presents several objective functions such as Generationand Load shedding Cost Minimization (GLCM)/ Social WelfareMaximization including demand response offers (SWM), LoadShedding Minimization (PshdM), Load Served Maximization(LSM), and Load Served Error (LSE) minimization. To the bestof our knowledge, all previous congestion management effortsconsidered constant load models. Using voltage-dependent loadmodels, the paper clearly brings out the inappropriateness ofconventional single objectives for congestion management, such

as GLCM/SWM andPshd

M, due to the reduction of amount ofload served. Therefore, multi-objective optimization is requiredand the objectives can be judiciously combined depending on theloading condition. Multi-objective Strength Pareto EvolutionaryAlgorithm 2+ has been employed to solve the proposed conges-tion management problem. The effectiveness of the developedapproach is confirmed from the simulation results on IEEE 30bus test system.

Index TermsCongestion management, generation reschedul-ing, load shedding, load modeling, contingency, multi-objectiveoptimization.

NOMENCLATURE

ai, bi, ci Generator cost coefficients ofith generating

unit.a

k, b

k, c

k Demand response cost coefficients of kth

load bus.

Gij , Bij Transfer conductance and susceptance be-tween bus i and bus j .

NG Number of generating units.NBU S Total number of buses in the system.NL Number of demands/loads.NC Number of shunt capacitors.NT Number of regulating transformers.np, nq Voltage exponents.PGi Generation output of i

th generator.

Pshd,k Amount of load shed atkth bus.

P0Di, PcDi Nominal and actually supplied load activepower demand.

LS0 Nominal load served.Qci Reactive power injected by i

th capacitor

bank.

QGi Reactive power output ofith generator.PDi, QDi Active and reactive load demands.PminGi , P

maxGi Minimum and maximum generation capac-

ities ofith generator.

S Surender Reddy (email: [email protected]) is with theDepartment of Railroad and Electrical Engineering, Woosong University,Daejeon, Republic of Korea.

QminGi , QmaxGi Minimum and maximum reactive capacities

ofith generator.Sij MVA flow between busi and bus j .Smaxij Thermal limit of the line connected between

busi and bus j .Ti Tap settings of i

th transformer.

Pl0, Ql0 Nominal values of active and reactive powerloads.

V0 Nominal values of voltage magnitudes.Nobj Number of objective functions to be opti-

mized simultaneously.Meq, Nineq Set of equality and inequality constraints.M Number of non-dominated solutions.N Population size.N Archieve size.VGi Bus voltage ofith generator.VmaxGi , V

minGi Maximum and minimum limits of generator

bus voltage magnitudes.

VLi Bus voltage ofith load.VmaxLi , V

minLi Maximum and minimum limits of load bus

voltage magnitudes.

x Vector of decision variables.i, j Voltage angles at busi and busj .

I. INTRODUCTION

POWER system experiences variations in operating con-ditions all the time. Contingency situations may occurdue to sudden increase of electrical load, forced outage of

a transmission line or generator, or any defect in equipment

of the system. Optimal generation rescheduling and load

shedding during the contingency situations is one of the critical

issues in planning secured operation of power systems. Load

shedding is defined as the set of controls, which results in

a decrease of load demand in the power system in order to

achieve a new equilibrium state. Load shedding schemes have

become more important in deregulated power systems, where

there is lack of adequate spinning reserve (SR) or a shortageof tie line power capacity to make up for the lost generation

[1]. The load shedding schemes are necessary to prevent the

phenomena such as voltage collapse, line overload, etc., which

may lead to cascade outages and then black out. Load shedding

is considered as a powerful tool to avoid the system wide

blackouts [2].

A common characteristic of large regional electricity mar-

kets is that they have sub-hourly energy markets that are jointly

optimized with the ancillary services markets, in which all

generators and load demands can participate. In Pennsylvania-

New Jersey-Maryland Interconnection (PJM) energy market,





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the Curtailment Service Providers (CSPs) [3], and in California

Independent System Operator (CAISO) energy market, the

Demand Response Providers (DRPs) [4], submit demand/load

reduction bids individually, or in aggregated fashion. Cus-

tomers may have the capability to curtail their normal con-

sumption in order to participate in the market clearing [5]. A

client bids individually only when it is large enough to do so,

else a group of customers bid in aggregation. In CAISO, the

DRPs are designed to enable end use customers to contribute

energy load reduction. The integration of demand response

into PJM electricity markets recognizes the importance of load

response to a fully functioning market, and the effect of load

response on the reliability of electrical grid. PJM emergency

load response enables demand response resources that reduce

the load demand during the emergency situations, and in turn

to receive payment for those reductions [3]. Various classes of

demand-side resources are defined as follows:

Demand-side resources in the energy only option of

emergency load response are defined as the demand

resources that receive only an energy payment for the

load reductions. Demand-side resources in capacity only option of emer-

gency load response are defined as the demand response

resources that receive only a capacity payment for load

demand reduction.

Demand-side resources in the full emergency load re-

sponse are defined as the demand resources that receive

both an energy payment for load reductions and a capac-

ity payment.

Optimal Power Flow (OPF) is one of the most accurate

methods for the Congestion Management (CM) in power sys-

tem with existing transmission and operational constraints [6].

An OPF based method that minimizes the cost of congestionand the service costs has been described in [7]. A coordination

mechanism between the generating companies (GENCOs)

and the Independent System Operator (ISO) for congestion

management (CM) using Benders decomposition has been

discussed in [8]. From the point of view of operations, load

shedding plays a major role. As a traditional control action it

prevents occurrence and extension of voltage instability. It also

helps in over-load mitigation in the electrical power systems

[9]. Optimum steady state load shedding schemes are proposed

in [10]-[13]. The combined/composite effect of optimizing

economic dispatch (ED), fast action by the spinning reserves

and load shedding, in order to withstand sudden loss of

generation, without system collapse due to cascading effects,is presented in [14]. The utilization of an approximate event

cost technique in a load shedding strategy under emergency

condition is illustrated in [15].

An optimization approach to minimize the load curtailments

that are necessary to restore the equilibrium of operating point

with relaxation of restrictions is described in [16]. A simple

and straight forward approach using the generation reschedul-

ing and load shedding for CM in power system, is presented

in [17]. Minimization of total load shedding by considering

the system operating constraints, like, transmission line flows

and voltage deviation limits is solved in [18]. An algorithm

for CM based on Particle Swarm Optimization (PSO), which

minimizes the deviations of rescheduled values of generator

power outputs from the scheduled levels is proposed in [19].

Reference [20] proposes a cost efficient CM model for smooth

and non-convex cost functions using the multi-objective PSO

technique.

Mostly power system optimization problems involve simul-

taneous optimization of several objectives, which are conflict-

ing and competing with each other. In a single-objective opti-

mization problem there exists a global optimum, while in case

of multi-objective optimization (MOO), no optimal solution is

clearly defined; but a set of solutions called the Pareto optimal

front / Pareto optimal set is present. The main aim of MOO

approaches is to generate a set of non-dominated solutions as

an approximation to this Pareto optimal front. But, majority

of problems of this kind cannot be solved exactly because

of their highly complex and large search spaces. Recently,

several meta-heuristic techniques have become important tools

for solving the MOO problems encountered in the industry and

academia as well [21].

From the literature survey it is clear that most of thetraditional approaches convert the MOO functions into a single

objective function using weighting schemes or optimize single

objective function at a time. Therefore, these approaches do

not truly optimize the multiple objectives simultaneously. Due

to these inherent deficiencies of conventional algorithms, the

use of alternative non-traditional solution algorithms for such

complex problems has become more popular and is used

widely. Multi-objective evolutionary algorithms have received

wide acceptance due to their robustness and quality of solution.

This paper proposes a new congestion management (CM)

mechanism with generation rescheduling and load shedding

using the multi-objective optimization (MOO). Further, the

loads are modeled as voltage-dependent, which were hith-ero modeled as voltage-independent. Handling of demand

response is more complex under voltage-dependent load mod-

eling when compared to constant load modeling. This has

not been investigated so far. The proposed CM approach is

particularly suitable for stressed system-operating conditions,

where the demand elasticity alone cannot yield a feasible

optimal solution. The feasible solution is obtained by invoking

load reduction bids/demand response offers.

It is shown that the single objectives, like, Generation and

Load shedding Cost Minimization (GLCM)/ Social Welfare

Maximization including demand response offers (SWM) and

Load Shedding Minimization (PshdM) are not suitable with

this voltage-dependent load model, due to the reduction ofamount of load served (LS). But, these two objectives can

be combined to get the compromised solution with constant

load modeling. This work then proposes that the MOO is

important to do the justice to this complex optimization

problem. The importance of using the multiple objectives, i.e.,

GLCM/SWM,PshdM, Load Served Maximization (LSM) andLoad Served Error (LSE) minimization are highlighted. This

paper then highlights the requirement for selecting judiciously,

a combination of objective functions best suited for the CM.

In this paper, Strength Pareto Evolutionary Algorithm 2+ is

considered as one such suitable MOO algorithm. The major





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contributions of the paper are as follows:

It has been shown that the use of conventional single

objective of cost minimization / social welfare maximiza-

tion, while using voltage-dependent load models leads to

an unrealistically low loads being served, due to voltage

reduction inherent in achieving this objective.

In the light of above, a multi objective optimization

is suggested to tackle the above mentioned problem.

This is achieved by adding Load Served Error (LSE)

Minimization as an objective to the original one. The

numerical results confirm the benefit of the proposed

approach.

In critical situations when either conventional gener-

ation reserves are insufficient or cannot be deployed

fast enough due to Generation Rate Constraints (GRCs),

demand-side reserves are being increasingly used despite

their much higher costs. This is because the latter are

not only very fast in response, but also strategically

very well distributed throughout the system. Although,

all these things are very well known, how to handle them

in the context of voltage-dependent load modeling hasnot been investigated so far. Customers providing this

service are to be told, what actual load relief is required

at a specific voltage (obtained from optimization), so that

the difference between nominal load relief and the actual

load relief is clear. This is quite complex, and unlike the

simple, single load relief quantum instruction given, with

constant load model.

The remainder of the paper is organized as follows. Section

II describes the problem formulation. Section III presents

the description about multi-objective congestion management

using generation rescheduling and load shedding. Section IV

addresses the simulation results and discussion. Finally, Sec-

tion V brings out the contributions with concluding remarks.

I I . CONGESTION M ANAGEMENT(CM): PROBLEM

FORMULATION

Elimination/ alleviation of transmission lines overload and

maintaining voltages within stipulated limits, i.e., congestion

management (CM), in contingency/emergency situations, by

means of generators rescheduling and load shedding is formu-

lated as a non-linear optimization problem. Here, we discuss

possible primary as well as supplementary objectives. Sup-

plementary objectives are the ones which can not be used in

isolation. They need to be coupled with the primary objective

function in order to formulate a multi-objective formulation.

Various choices for the optimization problem are presented, as

follows:

A. Generation and Load shedding Cost Minimization (GLCM)

In this case, demand is inelastic to the price hence, the main

objective is to reduce the generation and load shedding cost

(GLC), and it is formulated as,

Minimize,

GLC=

NGi=1

[ai+ bi(PGi) + ci(PGi)2]

+

NLk=1

[ak+ b

k(Pshd,k) +c

k(Pshd,k)2] (1)

B. Social Welfare Maximization including demand response

offers (SWM)

In the presence of demand elasticity, the market is set-

tled with social welfare maximization as objective function.

Presently, most of the electric power markets have introduceddemand-side bidding in the market clearing process. The con-

cept of maximizing social welfare can be applied for the cen-

tralized market with demand elasticity. This traditional social

welfare includes the total surplus of generators and customers.

In this case, the system operator optimally dispatches the

generators in such a way that the social welfare is maximized

while satisfying the operation and security related constraints.

The CSPs or DRPs can bid into the market in terms of fixed

bids, linear bids or quadratic bids. In case of CAISO, the

demand response offers (load reduction bids) exactly similar

to generation bids are solicited. Hence, the modified social

welfare is the traditional social welfare including demand

response offers. This can be formulated as follows,

Maximize,

modified SW=NL

k=1

[BDk(PDk )] NGi=1

[CGi(PGi)]

NLk=1

[ak+ b

k(Pshd,k) +c

k(Pshd,k)2] (2)

where

NLk=1

[BDk(PDk)] =

NLk=1

[dk ek(PDk) fk(PDk )2] (3)

NGi=1

[CGi(PGi)] =

NGi=1

[ai+ bi(PGi) + ci(PGi)2] (4)

i=1,2,...,NG. PDk and Pshd,k are demand bids and amountof load reduction/demand response at bus k, BDk(PDk) isdemand cost function at bus k, CGi(PGi) is cost functionfor generating real power PGi; dk, ek and fk are demandcoefficients ofk th load bus.

C. Amount of Load Shedding Minimization (PshdM)

The amount of load shedding is the sum of difference

between nominal load, and actually supplied active power

demands, and it is formulated as,

minimize Pshd=

NBUSi=1

(P0Di Pc

Di) (5)

D. Load Served Error (LSE) Minimization

This objective function is formulated as follows,

minimize LSE= (LS LS0)2 (6)

whereLS0 is nominal load served i.e., amount of load servedwith the constant load modeling. LS is the amount of load





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served with the voltage-dependent load modeling, and it is

formulated as,

LS=

NBUSi=1

Pil0

ViV0i

np(7)

Equation (7) is also considered as a single objective function

to be optimized, and it is called as Load Served Maximization

(LSM). The practical loads/demands are function of voltages,therefore voltage-dependent load modeling is used. Improve-

ment of system voltage increases the amount of load served

(LS). The LSE minimization objective function will never

be used as an independent objective. LSE will be used as

a supplementary objective, to ensure that load served (LS)

reduction is prevented to the extent possible.

The above objectives are achieved by the optimal determi-

nation of control variables. The control variables considered

in this problem are shown in Figure 1. where PcGi, Pc

Di are

Fig. 1. Control Variables for Proposed Congestion Management (CM)Approach

the active power generation and load active power demand in

contingency/ emergency state. The equality and inequality con-

straints for the proposed optimization problem are described

next:

E. Equality Constraints

1) Nodal Power Balance Constraints: These constraints are

typical load flow equations, and they include the active and

reactive power balances.

PGi PDi = Vi

NBUSj=1

Vj (Gijcos ij+ Bijsin ij ) (8)

QGi QDi= Vi

NBUSj=1

Vj (Gijsin ij Bijcos ij ) (9)

In Equations (8) and (9),i=1,2,...,NBUS. Whereij =i jis the voltage phase angle difference between bus i and busj .

F. Inequality Constraints

These constraints represent system operating limits.1) Generation Constraints: The outputs of generating units

are restricted by their minimum and maximum limits, and they

are represented as,

PminGi PGi Pmax

Gi (10)

The reactive power limits of generator are expressed as,

QminGi QGi QmaxGi (11)

The generator bus voltage limits are represented by,

VminGi VGi Vmax

Gi (12)

2) Constraints on Control Variables: The limits on control

variables are represented by,

PminGi Pc

Gi Pmax

Gi (13)

PminDi Pc

Di P0

Di (14)

where P0

Di is load active power demand in the normal stateand PminDi is amount of load which must be supplied.

3) Constraints on Switchable var Sources: The switchable

var sources are restricted by,

Qminci Qci Qmaxci i= 1, 2,...,NC (15)

4) Constraints on Demand response offers/ Amount of load

shed: This constraint provides relation between Pshd,k andPDk.

0 Pshd,k (PDk Pmin

Dk ) (16)

that is0 Pshd,k P

maxshd,k (17)

wherePmaxshd,k is the amount of load shed provided by the loaddemands.

5) Security Constraints: The security constraints are ex-

pressed as follows:

The load bus voltage limits are represented by,

VminLi VLi Vmax

Li i= 1, 2,...,NL (18)

The line flows are restricted by,

|Sij| Smaxij (19)

The transformer taps have minimum and maximum setting

limits as,

Tmini Ti Tmax

i i= 1, 2,...,NT (20)

The proposed CM problem with the above objectives is

solved using Genetic Algorithm (GA). The variables have been

represented in binary strings and the corresponding description

about their representation, encoding of chromosomes and

genetic operators can be found in [22]. A penalty function [23]

is added to the objective function if the functional operatingconstraints violate any of their limits.

It is worth noting that there is a conflict between the two

objectives (i.e., GLCM/SWM and PshdM) presented in thispaper. For example, when GLCM/SWM is optimized indepen-

dently, then the cost is minimum/ social welfare is maximum

but the amount of load shedding is more. On the other hand,

when the amount of load shedding minimization is optimized

independently, then the amount of load shed is minimum but

GLC is more/ social welfare is less. Hence, multi-objective

based CM approach using generation rescheduling and load

shedding is proposed in this paper.





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G. Voltage-dependent Load Modeling

The power systems in the developed world were earlier

conservatively designed. This was due to relative abundance

of resources with the utilities till 1970s. The situation changed

dramatically during 1980s due to various factors such as:

deteriorating economic health of utilities, Right of Way (ROW)

restrictions, and environmental considerations. Deregulation,

which started a little later, compounded the problems signifi-cantly. Economic compulsions resulted in power flow patterns

for which the system was never designed. The effects were

seen in the large number of voltage collapses in power system

all over the world. In short, the pre 1980 networks were

strong, robust and maintaining near nominal voltage profile

was easy. This made the voltage-dependent load representation

almost redundant. Post 1980 power systems were subjected to

operations scenarios with low voltages, including the extreme

ones threatening voltage collapse. As mentioned in [24]-[26],

voltage-dependent load models are essential in such situations

of power systems operation, without which the results will be

impractical.

The load modeling has a major influence on the operationof electrical power systems. Generally, the power system

loads are modeled as the constant loads. But, this kind of

model is not suitable for practical voltage-dependent loads.

This is even more true for the aggregated load representation

seen from the EHV buses. This is due to the fact that the

effects of sub-transmission and distribution system are also

reflected in this equivalent load representation. More realistic

and practical approach is brought into this paper by modeling

the loads/demands as the voltage-dependent [26]-[27]. For the

steady state power system analysis, exponential load model or

constant Impedance-Current-Power (ZIP) load model [27] are

used.

In this paper, exponential load model is used and the activeand reactive powers of the load bus are related to the bus

voltage through an exponential function,

PDi = Pi

l0

Vi

V0i

np(21)

QDi= Qil0

ViV0i

nq(22)

In Equations (21) and (22),i=1,2,...,NBU S. Wherenp and nqare voltage exponents, and they depends on the composition

and type of load demand. PDi and QDi are the active andreactive power demands at ith bus; Pil0, Q

il0 and V0i are the

nominal values of the active, reactive power demands and the

voltage magnitude at ith bus, respectively.

III. MULTI-OBJECTIVE C ONGESTION M ANAGEMENT

USINGG ENERATION R ESCHEDULING ANDL OAD

SHEDDING

In this paper, single objective based CM problem is solved

by using a Genetic Algorithm (GA). In order to find the

optimal decision variables to optimize an objective function

and to satisfy the constraints, the variables are represented

in binary strings. Most of the real-world problems naturally

involve multiple and conflicting objectives to be optimized si-

multaneously. Defining multiple objectives often gives a better

idea of the problem. Generally, a single attribute that is most

essential and appropriate for a particular operating condition,

has been used as an objective function for the optimization

problem. All other major attributes are incorporated in the

mathematical formulation as constraints with correctly chosen

limits. Many times selecting these limits is not so simple. This

is a rigorous approach and does not provide the room for trade-

off between various attributes, which can be beneficial form

the system operation point of view. However, such a flexibility

is contributed by the multi-objective optimization (MOO). The

MOO problem is formulated as,

M inimize/Maximize fi(x) i= 1, 2,...,Nobj (23)

subjected to gj(x) = 0 j = 1, 2,...,Meqhk(x) 0 k= 1, 2,...,Nineq

Essentially, there are two different methods for solving the

MOO problem. The first method reduces the MOO problem

to a single objective optimization problem, by generating

a composite objective c(x), from a linear sum of multipleobjective functionsfi(x).

c(x) =min

Nobji=1

[Wifi(x)] (24)

The above objective function can be optimized by using the

existing single objective optimization algorithms. However,

the weights Wi (which by convention are non-negative andthe sum is equal to 1) must be pre-set. The solution to

this optimization problem will then be a single vector ofcontrol variables, rather than entire Pareto optimal set. This can

have undesirable consequences: setting the weights implicitly,

introduces the designers preconceptions about the relative

trade-off between the objectives. The practical world prob-

lems can produce surprising Pareto optimal fronts/sets which

may profoundly affect design decisions, and the potential to

generate novel designs is a key benefit of optimization [28].

The second method for solving the MOO problem, is to search

directly for the entire Pareto optimal set. This can be achieved

in a number of ways, and needs modification to the existing

single objective optimization algorithms.

Multi-objective evolutionary algorithms can yield a whole

set of potential solutions, which are all optimal, in somesense. The multi-objective evolutionary algorithms were first

proposed by Schaffer [29]. The principle of an ideal MOO pro-

cedure is to determine the multiple trade-off optimal solutions

with a wide range of values for objective functions and then

choose one of the solutions using higher level information. In

this paper, Strength Pareto Evolutionary Algorithm 2+ (SPEA

2+) has been used to solve the proposed CM problem, which

provides a set of points on the Pareto optimal front. The

user/System Operator can then select a point which is suitable

to his needs best. In this paper, a best compromise solution

can be determined through a fuzzy min-max approach [30].





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A. Multi-objective Optimization (MOO) using Strength Pareto

Evolutionary Algorithm 2+ (SPEA 2+)

The MOO problem is minimization/ maximization of mul-

tiple evaluation criteria that conflict with each other. The

solution which is an optimum for one criterion is may not

be an optimal solution for MOO, because the multiple criteria

have trade-off relationships with each other. The SPEA 2+ is a

multi-objective genetic algorithm (MOGA) that improves thesearch performance of SPEA 2. The SPEA 2+ is SPEA 2 with

the addition of following three mechanisms [31]:

Mating Selection: This reflects all good chromo-

somes/solutions preserved in the archive.

Neighborhood Crossover: This crossover allow crossing

over individuals located close to each other in the objec-

tive space.

The application of two archives to maintain diverse

solutions in the objective space and design variable space.

The description of Mating selection and Neighborhood

crossover are presented in [31]-[32]. The algorithm for im-

plementation of SEPA 2+ is presented next:

Step 1: Generate the initial population (P0). Empty theobjective archive population (OA0), and design variablearchive population(V A0). Initially, the generation countk=0.

Step 2: The fitness values of all individuals Pk, OAk

and V Ak are determined using the SPEA 2s fitnessassignment technique [33].

Step 3:All the non-dominated solutions/individuals in the

Pk, OAk and V Ak are copied to OAk+1 and V Ak+1.If the number of individuals ofOAk+1 andV Ak+1 haveexceeded the archive size, archive truncation in objective

space is applied to the individuals inOAk+1 , and archive

truncation in variable space is V Ak+1

to reduce thenumber of individuals. If the number of individuals of

OAk+1 or V Ak+1 is less than archive size, individualswith good fitness from Pk, OAk and V Ak are used tofill OAk+1 and V Ak+1.

Step 4: Stop the search if maximum number of gener-

ations are reached or other termination conditions are

satisfied.

Step 5: Pk+1 is generated by copying OAk+1. Theneighborhood crossover and mutation operations are per-

formed. Go to Step 2, and increase k to k+1.

B. Best Compromise SolutionAfter determining the Pareto optimal set of non-dominated

solutions (i.e., Pareto optimal front) using the SPEA 2+

approach, the fuzzy min-max approach [30] provides the best

compromise solution to the system operator/decision maker.

Due to the imprecise nature of the decision makers judgement,

the ith objective function Fi is expressed using the fuzzymembership functioni, and is expressed as,

i=

1 if Fi Fmin

iFmaxi Fi

Fmaxi Fmini

if Fmini < Fi< Fmax

i

0 if Fi Fmaxi

(25)

where Fmaxi and Fmin

i are the maximum and minimum

values of the ith objective function among all non-dominatedsolutions, respectively. For each non-dominated solution k, thenormalized membership function (k) is determined using,

k =

Nobji=1

kiM

k=1

Nobji=1

ki

(26)

The best compromise solution is the one having maximumvalue ofk. The description of fitness function evaluation ispresented in [34], [35].

IV. RESULTS ANDD ISCUSSION

In this paper, IEEE 30 bus test system [36] is used to test the

effectiveness of the proposed congestion management (CM)

approach. The single line diagram of IEEE 30 bus system is

depicted in Figure 2. This system consists of 6 generating

units, 21 load demands and 41 lines/branches. Among these

41 branches, 4 branches are tap setting transformer branches.

Buses 10, 12, 15, 17, 20, 21, 23, 24 and 29 have been

selected as shunt compensation buses. It is assumed that, the

system operator receives generator bids and load shedding costoffers from customers to perform the congestion management

analysis [17], [20]. The Genetic Algorithm (GA) encoding

is accomplished by using different gene/chromosome lengths

for each set of control variables, depending on the desired

accuracy level. All the optimization programs are coded in

MATLAB and implemented on a PC-Core2 Quad computer

with 3.24GB of RAM.

A. Study 1: Simulation Results Considering Demand is Inelas-

tic to the Price

In this Study, 6 generator active power outputs, 21

power demands are selected as the control variables. Thegene/chromosome length for the unit of generation or load

power is 12 bits and they are considered as the continuous

controls. In view of this, the chromosome length for proposed

congestion management approach is (6*12)+(21*12) = 324.

In this paper, exponential load modeling with np = 1 andnq = 2 are used [26]. The emergency/contingency situationis obtained by taking line 36 out (i.e., line connecting the

buses 27 and 28 in IEEE 30 bus system). It is considered that

PminDi = 0.7 P0

Di for all load buses. This equation means

that, load shedding at bus i cannot be greater than 30 percentof load demand in this bus. The generator and load shedding

cost coefficients are taken from [17]. The CM problem by

generation rescheduling and load shedding is first solved byoptimizing the single objective at a time, and later it is solved

by using the multi-objective SPEA 2+ technique, considering

appropriate multiple objective functions to be optimized. The

effect of realistic/practical voltage-dependent load modeling

on the same is evaluated. In each case, the optimization

algorithm is stopped when all the chromosomes/population

members assume similar fitness values.

The value of a particular objective function shows a ten-

dency to move away from the optimum value, if the problem

is optimized with respect to some other objective function.

When MOO is performed, it leads to the formation of a





7

Fig. 2. Single Line Diagram of IEEE 30 Bus Test System.

Pareto optimal front/set. In this paper, a multi-objective based

SPEA 2+ technique has been used to solve the proposed CM

problem. The GA and SPEA 2+ parameters used are shown

in Table I. The justification of chromosome length has been

explained earlier. The population and archieve sizes have been

considered for this optimization problem, after some trials.

These are not case dependent or operating condition depen-

dent. The last 4 parameters in Table I are considered based

on the experience of many researchers, for a wide variety of

problems [29], [31]-[33], [37]-[39]. A set of strong dominatedsolutions is selected from population of chromosomes to form

the Pareto optimal set. If the Pareto optimal set size exceeds

maximum size, a hierarchical clustering technique is used to

limit its size.

TABLE IGA A ND SPEA 2+ PARAMETERS.

Paramaters GA SPEA 2+

Chromosome length 324 324

Population size (N) 60 100

Archieve size (N) 100

Reproduction Operator Roulette Wheel Mating

Mutation operator, Rate Bitwise, 0.001 Bitwise, 0.001

Crossover operator Uniform Neighborhood

Maximum iterations/generations 200 100

Two different cases - one for constant load modeling and

the other for voltage-dependent load modeling have been

simulated. The simulation results for Study 1 are presented

next:

1) Study 1 - Case 1: Congestion Management (CM) by opti-

mal generation rescheduling and load shedding with constant

load modeling: Table II presents the control variables and

objective function values when the individual and combined

objective functions are optimized, considering the constant

load modeling. The variables are generator active power

outputs and load active power demands in contingency state

at corresponding buses. In the table, GC is the generation cost

and LC is the load shedding cost. When Generation and Load

shedding Costs (GLC) is optimized independently, then the

optimum value obtained is 550515.5076 Rs/h, but the amount

of load shedding is 11.0192 MW. When the amount of load

shedding minimization (PshdM) is optimized independently,its value is restricted to 3.7831 MW, but the GLC has

increased to 568757.7146 Rs/h. This shows that when conges-

tion management (CM) problem with one objective function

is optimized, then the other objective function value deviates

from the optimum value. Thus, there is a scope for solving

a trade-off between conflicting objectives. The CM problem

with these conflicting objective functions, should be solved

using the multi-objective optimization (MOO) algorithms.

Hence, in this case, Generation and Load shedding Cost

Minimization (GLCM) and amount of load shedding mini-

mization (PshdM) objectives are combined to get the compro-mise solution. In this paper, multi-objective SPEA 2+ approach

is employed to get the Pareto optimal front, and fuzzy min-maxapproach is used to get the best compromise solution. The best

compromise solution has the GLC of 556432.4382 Rs/h, and

amount of load shed (Pshd) of 7.3691 MW. Figure 3 depictsthe Pareto optimal front of GLCM and PshdM with constantload modeling. The computational time required when GLCM

andPshdM objectives are optimized independently using GAare 55.2809s and 61.4772s, respectively. When GLCM and

PshdM objectives are optimized simultaneously using multi-objective SPEA 2+ approach, the computational time required

is 126.5041s, and this has been shown in Table II.

5.54 5.56 5.58 5.6 5.62 5.64 5.66 5.68

x 105

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

X: 5.564e+005Y: 7.369

Cost minimization ($/hr)

Loadshedminimization(MW)

Cost and Load Shed Minimization

Fig. 3. Pareto optimal front of GLCM and PshdM with constant load

modeling for Study 1.

The results of CM with constant load modeling are also

compared using Interior Point Method (IPM). Table III

presents the simulation results using IPM. When GLCM

objective is optimized independently using IPM, then the

optimum generation and load shedding cost (GLC) obtained

is 550844.4960 Rs/h, which is higher compared to the value

presented in Table II using GA (i.e., 550515.5076 Rs/h).

When the amount of load shedding minimization (PshdM)objective is optimized independently using IPM, then the

optimum value of load shed obtained is 4.8262MW, which


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is less than the nominal load demand of 283.4 MW. Hence,

GLCM and PshdM are not appropriate multiple objectives tobe optimized with voltage-dependent load modeling. But, in

case of constant load modeling, the GLCM and PshdM areappropriate multiple objective functions to be optimized.

Hence, to get the better compromise solution GLCM,

PshdM and LSE minimization objectives are optimized si-multaneously. Then the compromise solution has GLC of

550935.1685 Rs/h, Pshd of 7.0274 MW and amount of loadserved (LS) is 283.4312 MW. Here, the LSE minimization

objective is combined with GLCM and PshdM to get theload served (LS) approximately equal to the nominal load

served. Figure 4 depicts the Pareto optimal front of GLCM

and PshdM objectives, where as Figure 5 depicts the Paretooptimal front of GLCM, PshdM and LSE minimizations withvoltage-dependent load modeling.

4.9 4.95 5 5.05 5.1 5.15 5.2 5.25 5.3 5.35

x 105

5.1

5.15

5.2

5.25

5.3

5.35

X: 4.976e+005Y: 5.284


Loadshedminimization(MW

)

Cost and Load Shed Minimization

Fig. 4. Pareto Optimal Front of GLCM and PshdM with Voltage-dependentLoad Modeling for Study 1.

5.4 5.45

5.55.55

5.65.65

x 105

5

6

7

8

90

0.2

0.4

0.6

0.8


Cost, Load Shed and LSE minimization

X: 5.509e+005Y: 7.027Z: 0.031

Load shed minimization (MW)

LSEminimization(MW)

Fig. 5. Pareto Optimal Front of GLCM, PshdM and LSE Minimizationswith Voltage-dependent Load Modeling for Study 1.

B. Study 2: Simulation Results Considering Demand is Elastic

to the Price

In this Study also, two different cases - one with constant

load modeling and the other with voltage-dependent load

modeling have been simulated at stressed loading conditions

with 140% loading (emergency situation). This emergency

situation can be because of increased load demand, generator

outage and transmission line outage etc. Here, 140% loading

is assumed only to show how the demand response offers

are utilized to relieve the congestion. It is observed that, the

demand elasticity bids alone can not yield a feasible optimal

solution. Load reduction bids/demand response offers have

used to obtain the same. The simulation results for Study 2

are described next:

1) Study 2 - Case 1: CM by Optimal Generation Reschedul-

ing and Load Shedding with Constant Load Modeling: Table

V presents the optimum objective function values obtained

when Social Welfare Maximization including load reduction

cost/demand response offers (SWM) is optimized, considering

the constant load model. The variables are the generator active

power outputs, load demands, and the demand response offers

at corresponding buses.

When SWM objective is optimized, the optimum social

welfare (SW) obtained is 37492.5024 Rs/h, and the amount

of load shed (Pshd) is 17.6443 MW. The net amount of load

served is 369.7271 MW. The net amount of load supplied is thedifference between the total demand supplied and the amount

of load reduced.

2) Study 2 - Case 2: CM by Optimal Generation Reschedul-

ing and Load Shedding with Voltage-dependent Load Model-

ing: Any voltage between the minimum and maximum limits

is acceptable from operations point of view. However, with

voltage-dependent load modeling it can be seen that an attempt

to maximize the social welfare will result in load served

reduction through voltage reduction. Hence, social welfare

maximization can not be the sole objective for these type of

loads. There are two alternative approaches to prevent this load

reduction. The first is through the appropriate enforcement of

hard constraints on voltages, and the second being throughthe proposed multi-objective optimization approach. The first

approach requires that the minimum voltage limit should be

nominal voltage of the load bus so that, satisfaction of the same

will not allow load reduction. Apparently this logic is perfect.

However, achieving the same through the proposed multi-

objective optimization alternative provides us with significant

advantages. Distributed nature of the reactive resources do

not normally allow perfect voltage control at all the buses

simultaneously. In such situations, the former optimization

technique may simply result in an infeasible solution. The

proposed approach however attempts to keep the loads near

to their nominal values, thereby having better chances of pro-

viding a feasible solution. The proposed approach is flexible,attempting to strike a compromise between two conflicting

requirements. Such a compromise by very nature, results in a

feasible solution, unlike that in the previous approach.

Table VI presents the optimum objective function values,

when individual and combined objectives are optimized, con-

sidering voltage-dependent load models. When social welfare

maximization is the sole objective, voltage profile is pushed

down to maximize the social welfare, as a result the net amount

of load served (LS) is decreased. The obtained optimum values

are: social welfare is 41953.1932 Rs/h, amount of load shed

(Pshd) is 14.7613 MW, and the net amount of load served


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TABLE VICM WITHD EMANDR ESPONSEOFFERSC ONSIDERINGVOLTAGE-DEPENDENTLOAD M ODELING USINGGA A ND SPEA 2+ F OR S TUDY2 .

Obj ective Fun ction Valu e SWM S WM & LSE min.

Social Welfare ( Rs/h ) 41953.1932 38967.0596

Amount of Load Shed (MW) 14.7613 14.8829

Generation Supplied (MW) 371.3211 382.5908

Demand Supplied (MW) 372.3524 386.1622

Net Load Supplied (MW) 357.5911 371.2793

V. CONCLUSIONS

The paper proposes a new multi-objective based conges-

tion management approach by generation rescheduling and

load shedding, with realistic voltage-dependent load modeling.

The objectives considered in this paper are Generation and

Load shedding Cost Minimization (GLCM)/ Social Welfare

Maximization including demand response offers (SWM), Load

Shedding Minimization (PshdM), Load Served Error (LSE)minimization and Load Served Maximization (LSM). The

simulation studies on IEEE 30 bus test system show the

suitableness of choice of multiple objective functions to be

used for given load modeling. For constant load modeling,GLCM and PshdM are appropriate multiple objective func-tions to be optimized. But, when the loads are modeled as

the voltage-dependent, then it is shown that GLCM/ SWM,

PshdM are not valid single or multiple objectives with thisload model, due to the reduction in the amount of load served

(LS). With voltage-dependent load modeling, GLCM, PshdMand LSE minimizations; SWM and LSE minimization are best

suited multiple objectives to be optimized simultaneously. The

Pareto curve/Pareto optimal front provided by the SPEA 2+,

allows the system operator/decision maker to make a better

informed decision, regarding the compromise between the

various conflicting objectives.

ACKNOWLEDGMENTS

The author would like to thank Prof. P.R. Bijwe and Dr.

A.R.Abhyankar, Department of Electrical Engineering, Indian

Institute of Technology Delhi, India, for their guidance and

support for completing this work.

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S. Surender Reddy (S12-M15) received the Ph.D. degree in electricalengineering from the Indian Institute of Technology, New Delhi, India, in2013.

He was a Post-Doctoral Researcher at Howard University, Washington,DC, USA, from 2013 to 2014. He is currently an Assistant Professor withthe Department of Railroad and Electrical Engineering, Woosong University,Daejeon, Republic of Korea. His current research interests include powersystem restructuring issues, ancillary service pricing, real and reactive powerpricing, congestion management, and market clearing, including renewableenergy sources, demand response, smart grid development with integration ofwind and solar photovoltaic energy sources, artificial intelligence applicationsin power systems, and power system analysis and optimization.

Multi-Objective based Congestion Management using Generation Rescheduling and Load Shedding

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