Multi-Objective Differential Evolution for Voltage

doi 10.1515/ijeeps-2013-0086

International Journal of Emerging Electric Power Systems 2013; aop

Research Article J. Preetha Roselyn*, D. Devaraj, and Subhransu Sekhar Dash

Multi-Objective Differential Evolution for Voltage Security Constrained Optimal Power Flow in Deregulated Power Systems Abstract: Voltage stability is an important issue in the planning and operation of deregulated power systems. The voltage stability problems is a most challenging one for the system operators in deregulated power systems because of the intense use of transmission line capabilities and poor regulation in market environment. This article addresses the congestion management problem avoiding offline transmission capacity limits related to voltage stability by considering Voltage Security Constrained Optimal Power Flow (VSCOPF) problem in deregulated environment. This article presents the application of Multi Objective Differential Evolution (MODE) algorithm to solve the VSCOPF problem in new competitive power systems. The maximum of L-index of the load buses is taken as the indicator of voltage stability and is incorporated in the Optimal Power Flow (OPF) problem. The proposed method in hybrid power market which also gives solutions to voltage stability problems by considering the generation rescheduling cost and load shedding cost which relieves the congestion problem in deregulated environment. The buses for load shedding are selected based on the minimum eigen value of Jacobian with respect to the load shed. In the proposed approach, real power settings of generators in base case and contingency cases, generator bus voltage magnitudes, real and reactive power demands of selected load buses using sensitivity analysis are taken as the control variables and are represented as the combination of floating point numbers and integers. DE/randSF/1/bin strategy scheme of differential evolution with self-tuned parameter which employs binomial crossover and difference vector based mutation is used for the VSCOPF problem. A fuzzy based mechanism is employed to get the best compromise solution from the pareto front to aid the decision maker. The proposed VSCOPF planning model is implemented on IEEE 30-bus system, IEEE 57 bus practical system and IEEE 118 bus system. The pareto optimal front obtained from MODE is compared with reference pareto front and the best compromise solution for all the cases are

obtained from fuzzy decision making strategy. The performance measures of proposed MODE in two test systems are calculated using suitable performance metrices. The simulation results show that the proposed approach provides considerable improvement in the congestion management by generation rescheduling and load shedding while enhancing the voltage stability in deregulated power system. Keywords: voltage stability, congestion management, load shedding, generation rescheduling, differential evolution

*Corresponding author: J. Preetha Roselyn, Department of EEE, SRM University, Kattankulathur, Chennai 603 203, India, E-mail: [email protected] D. Devaraj, Department of CSE, Kalasalingam University, Kalasalingam College of Engineering, Krishnankoil, Srivilliputhur, Tamil Nadu 626190, India, E-mail: [email protected] Subhransu Sekhar Dash, Department of EEE, SRM University, Kattankulathur, Chennai 603 203, E-mail: [email protected]

1 Introduction In the electricity market, generator buses and load buses are designated as Generation companies (GENCOs) and Distribution companies (DISCOs) and transmission system is operated by independent system operator (ISO). In the PoolCo model, ISO has to establish equitable and fair transmission services in open market structure to provide reliable and secure power systems [1] to cope with stability related problems. In the open electricity market, the system is congested [2] because of inadequate transmission capacity to meet the demands of customers and expensive generating units and hence electricity markets will not be able to operate under system security. The conventional methods to relieve network congestion are not suitable for the new competitive environment. Voltage stability is an important issue in the planning

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J. P. Roselyn et al.: MODE for VSCOPF in Deregulated Power Systems

and operation of power systems. Voltage stability [3] is the ability of the power system to maintain acceptable voltage profile under normal condition and after being subjected to disturbances. Voltage stability can be assessed using static and dynamic approaches. In this work, L-index [4] one of the static voltage stability index is used for assessing voltage stability of the system. Actions have to be taken if the voltage security level of the system is found to be low. Voltage security enhancement can be achieved through preventive or corrective control. Preventive control is applied so as to ensure that operating point is away from point of collapse in anticipation of incredible contingencies. The corrective control action on the other hand is activated only after the occurrence of contingency. Corrective control is considered as economic one in the market environment, nevertheless preventive control is also needed to reduce system interruption. To achieve voltage security enhancement, the contingency state voltage stability index is included in the formulation of OPF problem [5–7] along with the contingency state constraints. A lot of works have been carried out to include voltage stability constrained optimal power flow [8] for optimal power dispatch with operational and system constraints in deregulated environment. The main contribution of this article is to include voltage stability constrained OPF in congestion management technique which is not reported in many literatures. As the voltage stability problem cannot be effectively solved by the available reactive power resources and control actions, generation rescheduling is necessary to change the power flow pattern and to limit the power transfers in stressed condition and improving voltage stability. In the pool, ISO decides the power transaction scheme by minimizing the increased costs caused by generation rescheduling from preventive to post contingency corrective actions based on incremental and decremental price bids offered by the GENCOs from initial market clearing values. Devaraj et al [9, 10] proposed a single objective Improved Genetic Algorithm to solve the VSC-OPF problem. Wu et al [11] proposed and effective scheme of rescheduling pool generation and adjusting contract transactions to ensure adequate voltage stability margin. A congestion management approach using rescheduling of generators considering voltage security constraints is presented by Phichaisawat et al. [12]. Basu [13] presents a MODE to optimize cost of generation, emission and active power transmission loss of flexible ac transmission systems device equipped power systems. When the system is closer to the voltage collapse point, the corrective control actions may not be fast

enough to prevent the system from voltage collapse. Hence appropriate load shedding strategy [14, 15] which results in decrease in the system load is to be carried out at suitable locations. It is generally carried out to protect the system from deteriorating to an inextremis state from alert or emergency state. However, it is very important to define an optimum load shedding scheme at selected critical load buses. Subramanian [16] proposed sensitivity based approach for load shedding problems in which weighted error criterion is approached for limiting the size of the loads being dropped. In Echavarren et al. [17], a Linear Programming (LP) based optimization algorithm has been proposed to determine the amount and location of the minimum load shedding to improve the load margin to voltage collapse. Fu et al. [18] divided the load shedding problem into two sub problems: restoring solvability sub problem and improving voltage stability margin sub problem. LP based OPF is used to solve each sub problem. In this work, the most suitable locations for load shedding are decided based on minimum eigen value [19] which is derived from Jacobian of A.C load flow equations. The load buses having larger sensitivities are the most suitable locations for load shedding. Because of the presence of conflicting objectives, a Multi objective Optimization Problem (MOP) results in a number of optimal solutions known as pareto optimal solutions [20]. In multi-objective optimization, effort must be made in finding the set of trade off pareto solutions by considering all objectives to be important. The ability of evolutionary techniques like differential evolution to find multiple solutions in one single simulation run makes them unique in solving multi-objective optimizations. This article proposes adaptive MODE with selftuned parameters for VSCOPF problem. DE/randSF/1/bin scheme [21] is used for the OPF problem in which mutation scheme uses a randomly selected vector and only one weighted difference vector is used to perturb it. The mutation scheme is combined with binomial type crossover and with random scale vector. Due to the convergence speed, simplicity and robustness by MODE to reach the optimal solutions makes it suitable for large scale optimization problem like VSCOPF problem. This article considers day ahead electric energy market in the pool in which generators and consumers submit production and consumption bids to the ISO, which clears the market based on minimization of cost incurred for generation rescheduling and load shedding by solving optimal power dispatch including voltage stability subject to operational and system constraints. In this article,



traditional OPF problem is extended to include bidding cost of generation rescheduling adjustment of generators and load shedding cost along with improvement in system voltage stability and congestion relief which is not reported in many literatures. The article is organized as follows: The formulation of multi-objective congestion management problem in deregulated environment is presented in Section 2. Section 3 gives a brief introduction of MOP along with fuzzy decision making. In this section, the MODE algorithm along with performance measures are presented along with detailed discussion. The results and discussions showing the effectiveness of the proposed method in IEEE 30 bus, IEEE 57 bus and IEEE 118 bus test systems are presented in Section 4. The major contributions and conclusions are discussed in Section 5.

the system. The L-indices for a given load condition are computed for all the load buses and the maximum of the L-indices (Lmax) gives the proximity of the system to voltage collapse. The L-index has an advantage of indicating voltage instability proximity of current operating point without calculation of the information about the maximum loading point. Hence the minimization of L-index makes the system less prone to voltage collapse. The calculation of L-index is given in Appendix 1.

2.2 Minimization of congestion cost The second objective function is to minimize the total congestion cost which refers to generation rescheduling cost and load shedding cost to relieve congestion problem along with voltage stability improvement. The total congestion cost is as follows:

2 Problem formulation

Tc ¼ TGR þ TLS

In the OPF problem considered here, generator rescheduling and load shedding in the contingency state are considered for voltage security enhancement. The ISO decides the rescheduling of generators based on incremental and decremental bids submitted by each GENCO. Also, an appropriate load shedding strategy is carried out in severe contingency cases and under stressed system conditions. The optimal load shedding algorithm considered in this work consists of two parts: optimal location of buses to be identified by sensitivity matrix [19] based on minimum eigen value of the load flow Jacobian and the optimum load to be shed is formulated as an optimization problem. Minimization of contingency state Lindex (maximization of voltage security level) is considered as the objective to achieve voltage security enhancement. To achieve optimal security enhancement in addition to minimization of L-index, minimization of congestion cost which comprises of minimization of generation rescheduling cost and minimization of load shedding cost are also considered as the objectives. The price bids provided by the selected DISCOs are used to calculate the load shedding cost in the optimization problem. The objectives are defined as follows:

2.1 Minimization of L-index

ð$=hrÞ

ð1Þ

2.2.1 Minimization of generation rescheduling cost The optimal real power dispatch is calculated under base case and post contingent states by corrective generation rescheduling. The incremental and decremental bids provided by GENCOs are used to calculate the rescheduling cost. The optimization problem is formulated as minimization of generation rescheduling cost as follows:

TGR ¼

Ng X

Cjup ΔPgi þ Cjdown ΔPgi

ð$=hrÞ

ð2Þ

i¼1

2.2.2 Minimization of load shedding cost The DISCOs for load shedding are selected based on sensitivity matrix calculated using sensitivity analysis [19] by minimum eigen value of the load flow Jacobian and the optimum load to be shed in the selected DISCOs are calculated by proposed MODE. The DISCOs having larger sensitivities are selected for decreasing their load. The cost incurred for shedding the amount of load by selected DISCOs is given as follows: TLS ¼

L-index method [4] is a approximate measure of closeness of the system to voltage collapse. The bus with the highest L index value will be the most vulnerable bus in

3

Ns X

Cjbid ΔPDj

ð$=hrÞ

ð3Þ

j¼1

The above optimization problem is formulated subject to following operational and system constraints.


4


2.3 Equality constraints The real and reactive power balance equations are: Pi Vi

NB X

Vj ðGij Cosθij þ Bij Sinθij Þ ¼ 0; i ¼ 1; 2; . . . NB1

j¼1

ð4Þ Qi Vi

NB X

Vj ðGij Sinθij Bij Cosθij Þ ¼ 0;

a single solution which can optimize all objective functions does not exist. The best trade-offs solutions called the pareto optimal solutions in one single simulation run can best be approximated by Multi-Objective Evolutionary Algorithms (MOEAs). In this work, MODE is applied to obtain the pareto front. The above MOP is solved using MODE algorithm. The details of MODE are explained in the next section.

i ¼ 1; 2; . . . NPQ

j¼1

ð5Þ

2.4 Inequality constraints Real power constraint: Pgimin Pgi Pgimax

i 2 Ng

ð6Þ

i 2 NB

ð7Þ

AVR constraint: Vimin < Vi < Vimax

Generator reactive power generation constraint: Qgimin < Qgi Qgimax

i 2 Ng

ð8Þ

Congestion constraint: Sl < Smax l

l 2 Nl

ð9Þ

Generation rescheduling constraint: Pgimin Pgi ΔPgi Pgimax Pgi

i 2 Ng

ð10Þ

Load shedding constraint: min max PDi PD PDi

3 Multi-objective differential evolution The details of MODE are explained below: Differential evolution [21] is a population-based stochastic search algorithm that works in the general framework of evolutionary algorithms. Unlike traditional evolutionary algorithms, DE variants perturb the generation population members with the scaled difference of randomly selected and distinct population members. The optimization variables are represented as floating point numbers in the DE population. It starts to explore the search space by randomly choosing the initial candidate solutions within the boundary. Differential evolution creates new off springs by generating a noisy replica of each individual of the population. The individual that performs better from the parent vector (target) and replica (trial vector) advances to the next generation. This optimization process is carried out with three basic operations namely, mutation, crossover and selection.

3.1 Initialization of parameter vectors i NPQ

ð11Þ

An optimization problem in which more than one objective is involved is called as MOP. A MOP can be mathematically formulated as, Min FðxÞ ¼ ½f1 ðxÞ; . . . fm ðxÞ

ð12Þ

Subject to: gj ðxÞ ¼ 0

j ¼ 1; . . . M

hk ðxÞ 0

k ¼ 1; . . . K

Where F(x) consists of m conflicting objective functions, x is the decision vector, gj is the jth equality constraint and hk is the kth inequality constraint. In multi-objective optimization, the improvement of one objective may lead to deterioration of another. Thus,

DE begins with a randomly initiated population of NP real parameter vectors known as genomes/chromosome which forms a candidate solution to multidimensional optimization problem and is expressed as: Xi;G ¼ x1;i;G ; x2;i;G ; x3;i;G ; . . . xD;i;G Where G is the generation number and D is the problem’s dimension. For each parameter of the problem, there will be minimum and maximum value within which the parameter should be restricted. Hence the jth component of ith vector is initialized as follows: xj;i;0 ¼ xj;min þ randi;j ½0; 1: xj;max xj;min

ð13Þ

Where randi;j ½0; 1 is a uniformly distributed random number lying between 0 and 1.



5

3.2 Mutation with difference vectors

3.4 Selection

After the population is initialized, the mutation operator is in charge of introducing new parameters into the population. The mutation operator creates mutant vectors by perturbing a randomly selected vector (Xr1) with the difference of two other randomly selected vectors (Xr2 and Xr3). All of these vectors must be different from each other, requiring the population to be of at least four individuals to satisfy this condition. To control the perturbation and improve convergence, the difference vector is scaled by a user defined constant. The difference vector based mutation is believed to be the strength of DE because of the automatic adaptation in improving the convergence of the algorithm which comes from the idea of difference based recombination operator i.e. Blend crossover operator (BLX) [22]. The process can be expressed as follows:

To keep the population size constant over subsequent generations, the selection process determines which one of the target vector and trial vector will survive in the next generation and is outlined as follows:

Vi;G ¼ Xr1;G þ FðXr2;G Xr3;G Þ

ð14Þ

where F is scaling constant.

3.3 Crossover In this work, binomial crossover is performed on each of the D variables. If the value of the random number is less or equal to the value of the crossover constant, the parameter will come from the mutant vector, otherwise the parameter comes from the parent vector. The crossover operation maintains diversity in the population preventing local minima convergence. The crossover constant must be in the range from 0 to 1. If the value of crossover constant is one then the trial vector will be composed of entirely mutant vector parameters. If the value of crossover constant is zero then the trial vector will be composed of entirely parent vector. Trial vector gets at least one parameter from the mutant vector even if the crossover constant is set to zero. The scheme may be outlined as follows: Ui;j ðGÞ ¼

Vi;j ðGÞ if rand ð0; 1Þ CR or j ¼ q Xi;j ðGÞ otherwise ð15Þ

Where q is randomly chosen index in the D dimensional space. CR is crossover constant Xi,j(G) is parent vector Vi,j(G) is mutant vector

Xi ðG þ 1Þ ¼ Ui ðGÞ if f ðUi ðGÞÞ f ðXi ðGÞÞ; ¼ Xi ðGÞ if f ðXi ðGÞÞ < f ðUi ðGÞÞ

ð16Þ

Where f(X) is the objective function to be minimized. So if the new trial vector yields a better value of the fitness function, it replaces its target in the next generation; otherwise the target vector is retained in the population. This process is continued until the convergence criterion is satisfied. The termination condition is satisfied when the best fitness of the population does not change appreciably over successive iterations. The differential evolution algorithm explained above is modified so that it can be applied to solve the MOPs. The proposed modifications are presented here. First the algorithm checks each solution for its dominance in the population. Two solutions (x(1) and x(2)) are compared on the basis of whether one dominates the other solution or not. A solution x(1) is said to dominate the other solution x(2), if the following conditions are satisfied: (a) The solution x(1) is no worse than x(2) in all objec tives, or fi xð1Þ < fi xð2Þ for all i ¼ 1, 2,. M where M be the objective functions. (b) The solutions x(1) is strictly better than x(2) in at least one objective, or fi xð1Þ < fi xð2Þ for at least one j (j 2 f1; 2; . . . ; M g) If any of the above condition is violated, the solution x(1) does not dominate the solution x(2) (or mathematically x(1) ≤ x(2)) To a solution i, a rank equal to one plus the number of solutions ηi that dominate solution i is assigned: ri ¼ 1 þ ηi

ð17Þ

The non-dominated solutions are assigned a rank equal to 1, since no solution would dominate a non-dominated solution in a population. After ranking, raw fitness is assigned to each solution based on its rank by sorting the ranks in ascending order of magnitude. Then, a raw fitness is assigned to each solution by linear mapping function which is chosen to assign fitness between N (for best rank solution) and 1 (for worst rank solution). Thereafter, solutions of each rank are considered at a time and their averaged raw fitnesses are called assigned


6


fitness. Thus the mapping and averaging procedure ensures that the better ranked solutions have a higher assigned fitness. To maintain diversity in the population, niching is introduced among solutions of each rank. The niche count is calculated by summing the sharing function value as below: nci ¼

μðri Þ X

Shðdij Þ

ð18Þ

perturbed, 1 denotes the number of difference vectors considered for perturbation and bin stands for binomial type of crossover operator. After the population is initialized, the mutation operator is in charge of introducing new parameters into the population. In this work, DERANDSF (DE with random scale factor) is used in which the scaled parameter F is varied in a random manner in the range (0.5,1) by using the relation:

j¼1

F ¼ 0:5 ½1 þ randð0; 1Þ

Where μðri Þ is the number of solutions in a rank and Shðdij Þ is the sharing function value of two solutions i and j. The sharing function is calculated by using objective function as distance metric as: 8 9 dij α > < max Fi

Fi Fimax Fi

F max Fimin > > : i 0;

;

Fimin < Fi < Fimax

ð22Þ

Fi Fimin

Where Fimax and Fimin are the maximum and minimum value of the ith objective function among all non-dominated solutions. The above equation gives a degree of satisfaction for each objective function for a particular solution and map the objectives in the range of 0 to 1. The membership function for the non-dominated solutions in a fuzzy set is calculated as follows:


N obj P

μ ¼ k

μki i¼1 obj M N P P k¼1

i¼1

ð23Þ μki


7

Where M is the number of non-dominated solutions and Nobj is the number of objectives. Finally, the best compromise solution is the one achieving the maximum membership function.

Where PFknown is number of vectors in known pareto front, p ¼ 2 and di is the Euclidean phenotypic distance between each member, i of PFknown and closest member in reference pareto front to that member.

3.6 Performance measures

3.6.3 Spacing (Di)

The performance of proposed MODE is studied by comparing the reference pareto front which is obtained with multiple runs. In this article, the proposed algorithm with conventional weighted sum approach is used as reference pareto front. The reference pareto front is generated by considering the problem as a single objective optimization problem by linear combination of the objective functions as given below:

This describes the spread of the vectors in known pareto front.

Minimize f ¼ w1 Lmax þ w2 Congestion cos t

ð24Þ

Where w1 , w2 and w3 are weighting factors and the sum of weighting factor should be 1. The non-dominated solutions are obtained by applying the algorithm 20 times with different weighting factors between 0 and 1. The performance of proposed MODE is studied under various performance measures such as convergence metric, generational distance (GD) and spacing [20]. The details of the performance metrics are discussed below:

3.6.1 Convergence metric (γ) This metric finds an average distance between non-dominated solutions found and the actual pareto optimal front as follows: N P

γ¼

di

i¼1

N

ð25Þ

Where di is the distance between non-dominated solutions and actual pareto front and N is the number of solutions in the front.

3.6.2 Generational distance This reports how far on average pareto front known from reference pareto front and given as follows: n 1=p P p di i¼1 GD ¼ ð26Þ jPFknown j

Di ¼ minj f1i ðxÞ f1j ðxÞ þ f2i ðxÞ f2j ðxÞ

ð27Þ

4 Results and discussion The proposed algorithm has been tested using IEEE 30 bus, IEEE 57 bus and IEEE 118 bus systems in deregulated environment. The IEEE 30 bus system [24] consists of 6 generators, 24 loads and 4 transformers with off nominal tap ratio and 41 transmission lines. The lower voltage magnitude at all buses are 0.95 pu for all buses and the upper limits are 1.1 pu for generator bus and 1.05 pu for remaining buses. The active and reactive powers of system load are 283.4 MW and 126.2 MVAR respectively. The IEEE 57-bus system was chosen as the second test system to demonstrate the method’s usefulness on a large system. IEEE 57-bus system has 4 generators, 3 synchronous condensers, 50 load buses, 80 transmission lines and 16 tap changing transformers. The lower voltage magnitude at all buses is 0.94 pu and the upper limit is 1.06 pu for all buses. The IEEE 118 bus system which is chosen as the third large system has 54 generators, 64 load buses, 186 transmission lines. The MATLAB-code was developed for the Adaptive MODE and ran on Pentium IV, 2.4 GHz system. While solving the VSC-OPF problem, generators are modeled as PV buses with reactive power limits and the loads are represented by constant PQ loads. The power system is stressed by increasing the load and by simulating line outages. Case 1: Single line (N–1) contingency analysis is performed to identify the contingency with respect to voltage stability. From the (N–1) contingency analysis, the line outages (28–27) and (27–30) are identified as the most severe contingency cases with Lmax values of 0.6630 and 0.4319 in IEEE 30 bus system under 125% stressed system conditions. In IEEE 30 bus system, the problem is formulated as bi objective optimization problem with minimization of the sum of congestion cost


8


such as generation rescheduling cost and load shedding cost and Lmax in the contingency states under stressed system conditions satisfying the system equality and inequality constraints. The bidding incremental and decremental prices provided by the six GENCOs in the IEEE 30 bus system are given in Appendix 2. The generation rescheduling is carried out by the market operator from base case to contingency states under 125% heavily loaded conditions. The real power settings of the GENCOs are decided based on minimization of rescheduling cost provided to the suppliers. Additionally, to relieve congestion in deregulated environment, the most sensitive five DISCOs for load shedding in the three test systems are obtained using sensitivity analysis as explained in Section 2 and are tabulated in Table 1. The amounts of load to be shed in the selected DISCOs are decided by the optimization technique considering the minimization of load shedding cost based on the bidding cost provided by the DISCOs to reduce the amount of consumption of electricity. The bidding costs of the selected DISCOs are shown in Appendix 3. The control parameters are real power settings of the generator in base case, contingency case and real and reactive power demand of the sensitive buses. The best

Table 1

objective functions in contingency (28–27) under 125% loaded condition, the Lmax and total congestion cost comprising generation rescheduling cost and load shedding cost along with best compromise solution are shown in Table 2. The pareto optimal front along with reference front for the severe line outage (28–27) is displayed in Figure 1. The line flows before and after the proposed method under line outage (28–27) is shown in Table 3 which clearly shows that congestion has been totally relieved from the competitive market in the PoolCo model. The change in real power settings from base case to contingency (28–27) for the six GENCOs in the system is shown in Figure 2. Similarly, the system is studied under the next severe contingency (27–30) under 125% loaded condition and the optimization results are shown in Table 4. From this table, it is clear that the best compromise solution is closer to the individual best solutions. The pareto optimal front along with reference front for the severe line outage (27–30) displayed in Figure 3 clearly shows that the search space is well explored by the proposed approach. The line flows before and after the proposed method under line outage (28–27) is shown in Table 5 which clearly shows that congestion has been totally relieved from the competitive market in

Selection of sensitive buses for load shedding in IEEE 30 bus and IEEE 57 bus and IEEE 118 bus test systems IEEE 30 bus system

Bus no. 29 24 4 17 18

IEEE 57 bus test system

IEEE 118 bus system

Sensitivity value

Bus no.

Sensitivity value

Bus no.

Sensitivity value

0.8665 0.4555 0.4036 0.4026 0.3974

30 31 32 33 57

0.8543 0.6234 0.5016 0.4238 0.3875

39 45 63 86 115

0.9784 0.8742 0.7134 0.5243 0.3628

Table 2 Simulation results of best congestion cost, best L-index in IEEE 30 bus system with line outage 28–27 under 125% loaded condition Algorithm

MODE Congestion cost ($/hr) Lmax Reference pareto front Congestion cost ($/hr) Lmax

Objective functions Minimum congestion cost

Minimum Lmax

Best compromise solution

96.17 0.1871

2,216.3 0.1503

328.45 0.1530

94.3 0.1786

2,012 0.1432

– –



9

0.195 0.19

Reference pareto front MODE POF

0.185 0.18 Lmax

0.175 0.17


0.165 0.16 0.155 0.15 500

1,000

1,500 2,000 2,500 Congestion cost (S/hr)

3,000

3,500

Figure 1 Pareto optimal front of MODE under corrective control in line outage (28–27) under 125% loaded condition in IEEE 30 bus system

Table 3 Comparison of congested line flow for line outage (28–27) in IEEE 30 bus system Overloaded lines

Before rescheduling (MVA)

25–27 24–25 22–24 15–23 10–21 12–15

34.30 36.48 19.30 28.63 26.80 18.76

After Maximum line flow rescheduling limit (MVA) (MVA) 28.64 27.75 12.94 15.21 15.46 14.87

32 32 16 16 16 16

the PoolCo model. The change in real power settings in GENCOs under line outage (27–30) in IEEE 30 bus system is shown in Figure 4. The voltage profile improvements for the two severe contingencies after the application of the proposed optimization approach are displayed in Figures 5 and 6. From these figures, it is clear that by including, Lmax as an additional objective along with

Table 4 Simulation results of best congestion cost, best L-index in IEEE 30 bus system with line outage (27–30) under 125% loaded condition Algorithm

Objective functions


20 DP1G

0 Change in real power settings in GENCOs

–10

1

DP2G

DP5G

DP8G

DP11G

DP15G

–20 –30 –40 –50 –60


27.05

628.93

121.32

0.1934

0.1405

0.1595

26

600.67

–

0.1898

0.1327

–

congestion cost, it is proved that voltage stability is improved under most critical contingencies in the test system. The proposed method is also performed under multiple contingencies under 100% loaded condition and

30

10

Minimum Lmax

Minimum congestion cost

GENCOs

Figure 2 Generation rescheduling of GENCOs under contingency (28–27) in IEEE 30 bus system


10


0.2 MODE POF Reference POF

0.19 0.18 Lmax 0.17


0.16 0.15

0

200

800 400 600 Congestion cost (S/hr)

1,000

1,200

Figure 3 Pareto optimal front of MODE in line outage (27–30) under 125% loaded condition in IEEE 30 bus system



After rescheduling (MVA)

Maximum line flow limit (MVA)

17.39 23.11 36.14

15 14.36 15.5

16 16 32

29–30 27–29 6–8

the results are given in Table 6. The line flows before and after the proposed method under multiple contingency is shown in Table 7 which clearly shows that congestion has been totally relieved using the proposed method. The pareto optimal front and the voltage profile improvement

in multiple contingency case are given in Figures 7 and 8. Hence it is clear that the proposed method performs well under multiple contingency cases also. Case 2: In IEEE 57 bus test system, the proposed optimization method is applied under line outage (46– 47) under 150% loaded condition. Line outage (46–47) is found to be the most severe case with a Lmax value of 0.4598 under 150% stressed system conditions in IEEE 57bus test system. The bidding incremental and decremental prices provided by the six GENCOs in the IEEE 30 bus system are given in Appendix 4. The bidding costs of the selected DISCOs are shown in Appendix 5. The pareto optimal front along with reference front for the contingency is displayed in Figure 9. It can be seen that the obtained solutions are well distributed on trade-off surface, except some discontinuity, caused by the discrete decision variables. The best compromise solution along with best Lmax and best congestion cost is shown in Table 8. The extreme points of pareto optimal front by MODE and the best solutions of each functions optimized individually by DE are identical. Hence, it is verified that the proposed method is capable of exploring more efficient search space. With the proposed method, it is observed that MODE generates optimal result and hence improves the voltage stability of the system. The line flows before and after the proposed method under line outage (46–47) is shown in Table 9 which clearly shows that congestion has been totally relieved in the deregulated environment. The changes in real power settings in the GENCOs after generation rescheduling under the two contingencies are shown in Figure 10. The voltage violations in the load buses before optimization have been corrected in the system. With the proposed method, it is observed that MODE generates optimal result and hence improves

30 20 10 Change in real power settings in GENCOs

0

DP1G 1

DP2G

DP5G

DP8G

DP11G

DP13G

–10 –20 –30 –40

GENCOs no.

Figure 4 Generation rescheduling of GENCOs under contingency (27–30) in IEEE 30 bus system



11

1.2 1 0.8

Voltage magnitude 0.6 (pu)

Before optimization After optimization

0.4 0.2 0 3 4 6 7 9 10 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

DISCOs no. Figure 5 Voltage profile improvement under line outage (28–27) using the proposed algorithm

1.2 1 0.8

Voltage magnitude 0.6 (pu)

After optimization Before optimization

0.4 0.2 0

3 4 6 7 9 10 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 DISCOs no. Figure 6 Voltage profile improvement under line outage (27–30) using the proposed algorithm

Table 6 Simulation results of best congestion cost, best L-index in IEEE 30 bus system with double contingencies (28–27) and (27–30) under 100% loaded condition Algorithm

Overloaded lines



Table 7 Comparison of congested line flow for multiple contingencies (28–27) and (27–30) in IEEE 30 bus system

Best Minimum Lmax compromise solution

362.4

848.5

580

0.1876

0.1245

0.1437

360

830.7

–

0.1850

0.1239

–

22–24 24–25




22.74 21.67

11.72 15.4

16 16

voltage stability of the system along with congestion management in deregulated power systems. Case 3: The proposed method is also applied in IEEE 118 bus system under multiple contingencies (8–5) and (12–16) in 150% loaded condition. The bidding incremental and decremental prices provided by the six GENCOs in the IEEE 118 bus system are given in Appendix 6. The


12


0.19 Reference pareto front MODE POF

0.18 0.17 Lmax 0.16


0.15 0.14 0.13

0.12 350 400 450 500 550 600 650 700 750 800 850 Congestion cost(S/hr) Figure 7 Pareto optimal front of MODE under double contingencies (28–27) and (27–30) under 100% loaded condition in IEEE 30 bus system

1.2 1 0.8 Voltage magnitude 0.6 (pu)

Before optimization After optimization

0.4 0.2 0 3 4 6 7 9 10 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

DISCOs no. Figure 8 Voltage profile improvement under double contingencies (28–27) and (27–30) in IEEE 30 bus system using the proposed algorithm

0.37 MODE POF

0.36

Reference POF

0.35 0.34 0.33


Lmax 0.32 0.31 0.3 0.29 0

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 Congestion cost(S/hr)

Figure 9 Pareto optimal front of MODE in line outage (46–47) under 150% loaded condition in IEEE 57 bus test system



Table 8 Simulation results of best congestion cost, best L-index in IEEE 57 bus test system with line outage 46–47 under 150% loaded condition Algorithm



Minimum Lmax


8,157

1,199

0.2880

0.3150

8,006

–

0.2361

–

303.65 0.3512 289.57 0.3018



1–2 8–9 1–15 1–17 10–51 11–43

173.87 265.34 237.67 147.75 108.44 79.46

After Maximum line flow limit (MVA) rescheduling (MVA) 145.23 224.45 198.65 96.82 95.38 57.91

150 250 200 100 100 60

bidding costs of the selected DISCOs are shown in Appendix 7. The obtained pareto optimal front along with reference front is shown in Figure 11 which shows that the proposed method is well explored in large size systems also. The best compromise solution along with

best Lmax and best congestion cost is shown in Table 10. From this table, it is clear that the best compromise solution is closer to the individual best solutions. The line flows before and after the proposed method under multiple contingencies is shown in Table 11 which clearly shows that congestion has been totally relieved after the application of the proposed method. The performance measures namely, convergence metric, GD and spacing for the three test systems are computed for the best, mean and worst solutions for 20 simulation runs and are tabulated in Table 12. From this it is observed that all the three statistical performance measures have minimum mean values in the two test systems using proposed adaptive MODE.

5 Conclusion In this article, optimal proposed procedure for congestion management along with improvement in voltage stability in the deregulated power systems during emergency condition is achieved by implementing Adaptive MODE in which suitable corrective control strategies such as load shedding and generation rescheduling are used. The proposed technique results in both more economical and secure operating conditions by considering bidding cost to providers and suppliers and voltage stability than imposing offline thermal capacity limits. The proposed methodology provides the optimal location and amount of sheddable load power using appropriate load shedding strategy. Generation rescheduling is carried out from base case to contingency state. As the proposed problem includes stability constraints, possible enhancement in steady state and improved results in higher loading condition

100 80 60 40 Change in real 20 power settings 0 of generator –20

ΔP1G 1

ΔP2G

ΔP3G

ΔP6G

ΔP8G

ΔP9G

ΔPG12

–40 –60 –80

13

GENCOs no.

Figure 10 Generation rescheduling of GENCOs under contingency (46–47) in IEEE 57 bus test system


14


MODE POF 0.1040

Reference POF

0.1030 Lmax 0.1020 0.1010 Best compromise solution

0.1000

0

1,000

2,000

3,000

4,000

5,000

Congestion cost(S/hr)

Figure 11 Pareto optimal front of MODE in multiple contingencies (8–5) and (12–16) under 150% loaded condition in IEEE 118 bus system

Table 10 Simulation results of best congestion cost, best L-index in IEEE 118 bus system with multiple contingencies (8–5) and (12– 16) under 150% loaded condition

Table 12 Performance metrics of MODE in IEEE 30, IEEE 57 and IEEE 118 bus test systems

Algorithm

Performance metrics

Solution

Case1

Case 2

Case 3

Convergence metric

Mean Best Worst Mean Best Worst Mean Best Worst

0.1897 0.1438 0.2066 0.4532 0.3765 0.5043 0.0353 0.0214 0.0896

0.2034 0.1967 0.3024 0.5362 0.4987 0.6045 0.0345 0.0897 0.1145

0.2435 0.2071 0.3162 0.5269 0.4871 0.6178 0.0451 0.0845 0.1354

Objective functions Minimum Lmax


850

4,820

3,000

0.1040

0.09

0.1000

842

4,650

–

0.1036

0.08

–

Minimum congestion cost MODE Congestion cost ($/hr) Lmax Reference pareto front Congestion cost ($/hr) Lmax

Table 11

Generational distance

Spacing

Comparison of congested line flow for multiple contingencies in IEEE 118 bus system

Overloaded lines 4–5, 15–17, 17–18, 31–32, 42–49, 65–66 47–69, 49–69 6–7, 11–12, 2–12, 7–12, 11–13, 12–14, 13–15, 14–15, 27–28, 31–32, 45–46, 41–42, 40–42, 46–47 77–78, 89–90 8–9, 9–10, 30–17, 8–30, 68–69 15–19




148.2, 156, 232, 172, 134, 173, 159, 146 77, 145, 79, 104, 89, 120, 84, 71, 75, 91, 95, 91, 79, 85, 89, 89

121, 128, 126, 130, 124, 129, 118, 125 64, 69, 58, 65, 57, 65, 59, 60, 58, 51, 59, 60, 62, 60, 51, 45

130

654, 664, 861, 662, 655 326

589, 561, 600, 574, 600 298

600


65

300


is achieved. A fuzzy based mechanism is employed to extract the best compromise solution from the pareto front. The method does not impose any limitation on the number of objectives. The proposed approaches have been tested on IEEE 30 bus system, IEEE 57 bus test system and IEEE 118 bus system under normal and stressed system conditions. The proposed MODE is better in characterizing the pareto optimal front in solving the multi-objective OPF problem in deregulated environment by its well distribution and diversity characteristics.

Appendix 1 The L-index calculation for a power system is briefly discussed below: Consider a N-bus system in which there are Ng generators. The relationship between voltage and current can be expressed by the following expression:

IG IL

YGG YGL ¼ YLG YLL

VG VL

ð28Þ

where, IG, IL and VG, VL represent currents and voltages at the generator buses and load buses. Rearranging the above equation we get,

VL IG

¼

ZLL FLG KGL YGG

IL VG

δj Voltage phase angle of jth generator unit. Ng Number of generating units. The values of Fji are obtained from the matrix FLG.

Appendix 2

Generator data in IEEE 30 bus system

Generator Pgmin ðMW Þ Pgmax ðMW Þ Cjup ð$=MWhrÞ Cjdown ð$=MWhrÞ 1 2 5 8 11 13

Appendix 3

0 0 0 0 0 0

200 80 35 50 30 40

15 8 8 12 10 5

Demand data in IEEE 30 bus system Cjbid ð$=MWhrÞ

Customer 29 24 4 17 18

Appendix 4

35 15 15 30 25 15

22 23 24 21 24

Generator data in IEEE 57 bus test system

Generator Pgmin ðMW Þ Pgmax ðMW Þ Cjup ð$=MWhrÞ Cjdown ð$=MWhrÞ

ð29Þ

where FLG ¼ ½YLL 1 ½YLG

ð30Þ

1 2 3 6 8 9 12

0 0 0 0 0 0 0

575.88 100 140 100 550 100 410

44 43 42 43 42 44 44

41 39 38 37 39 40 41

The L-index of the jth node is given by the expression, Ng X Vi Fji ﬀðθji þ δi δj Þ L j ¼ 1 V j i¼1 Where Vi Voltage magnitude of ith generator. Vj Voltage magnitude of jth generator. θji Phase angle of the term Fji. δi Voltage phase angle of ith generator unit.

15

ð31Þ

Appendix 5

Demand data in IEEE 57 bus test system

Customer 30 31 32 33 57


Cjbid ð$=MWhrÞ 22 23 24 21 24

16


Appendix 6 Generator 1 4 6 8 10 12 15 18 19 24 25 26 27 31 32 34 36 40 42 46 49 54 55 56 59 61 62 65 66 69 70 72 73 74 76 77 80 85 87 89 90 91 92 99 100 103 104 105 107 110 111 112 113 116

Generator data in IEEE 118 bus system Pgmin ðMW Þ

Pgmax ðMW Þ

Cjup ð$=MWhrÞ

Cjdown ð$=MWhrÞ

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

100 100 100 100 550 185 100 100 100 100 320 414 100 107 100 100 100 100 100 119 304 148 100 100 255 260 100 491 492 805.2 100 100 100 100 100 100 577 100 104 707 100 100 100 100 352 140 100 100 100 100 136 100 100 100

43 43 43 43 44 43 43 43 43 43 43 44 43 43 43 43 43 43 43 42 43 43 43 43 39 39 43 44 44 45 43 43 43 43 43 43 44 43 43 45 43 43 43 43 43 42 43 43 43 43 42 43 43 43

39 39 39 39 41 40 39 39 39 39 39 41 39 39 39 39 39 39 39 38 39 40 39 39 38 38 39 41 41 42 39 39 39 39 39 39 41 39 39 42 39 39 39 39 39 38 39 39 39 39 38 39 39 39



Appendix 7

Demand data in IEEE 118 bus system

Customer

Cjbid ð$=MWhrÞ

39 45 63 86 115

22 23 24 21 24

Nomenclature TC TGR TLS Ng

Congestion cost Generation rescheduling cost Load shedding cost Number of generator buses

Cjup Cjdown ΔPgj Cjbid ΔPDj Pi Qi Vi Gij Bij θij NB–1 NPQ Pgi Qgi Sl Nl PD

17

Incremental bid coefficient Decremental bid coefficient Change in real power settings of generator Load shedding bid coefficient Amount of load shed in DISCOs. Real power in ith bus Reactive power in ith bus Voltage magnitude in ith bus Conductance in transmission line between i and j buses. Susceptance in transmission line between i and j buses. Angle between i and j buses Number of buses except slack bus Number of load buses Real power generation in ith generator bus Reactive power generation in ith generator bus Line flow in transmission line Number of transmission lines Real power demand in DISCOs

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Multi-Objective Differential Evolution for Voltage

Multi-Objective Differential Evolution for Voltage

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