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Maximum loadability limit of power system using hybrid differential evolution with particle swarm optimization
Electrical Power and Energy Systems 43 (2012) 150–155
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Maximum loadability limit of power system using hybrid differential evolution with particle swarm optimization K. Gnanambal a,⇑, C.K. Babulal b a b
Department of Electrical and Electronics Engineering, K.L.N. College of Engineering, Pottapalayam 630 611, Tamilnadu, India Department of Electrical and Electronics Engineering, Thiagarajar College of Engineering, Madurai 625 015, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 15 May 2010 Received in revised form 27 April 2012 Accepted 29 April 2012 Available online 18 June 2012 Keywords: Maximum loadability limit Particle swarm optimization Differential evolution
a b s t r a c t Differential evolution (DE), a simple evolutionary algorithm which shows superior performance in global optimization. Since it utilizes the differential information to get the new candidate solution, sometimes it results in instability of performance. Particle swarm optimization (PSO) is widely used to solve the optimization problems as it can converge quickly. But PSO easily gets stuck in local optima. Hybridization of DE and PSO (DEPSO) eliminates the disadvantages of both. This paper presents the application of DEPSO algorithm to determine the maximum loadability limit of power system. It is tested on Matpower 30 bus and IEEE 118 bus systems. To compare the performance of this DEPSO algorithm with other evolutionary algorithms like DE and Multi Agent Hybrid PSO, statistical measures like best, mean, standard deviation of results and average computation time over 20 independent trials are considered here. The results show the better performance of DEPSO algorithm to solve the maximum loadability problem. DEPSO algorithm provides high maximum loading point in reduced time. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The power systems are heavily stressed due to the increased loading and this leads to the voltage stability problem in power systems. Recently, various studies for detecting loadability limits of power systems have been proposed in relation to voltage stability and security monitoring. The maximum loadability limit is the margin between the operating point of the system and the maximum loading point. Various methods have been proposed to determine maximum loadability limit [1–4]. Among these methods, the continuation power flow (CPF) technique has been widely used. This method fails to give the accurate result if the step length is more. Even though Interior Point method (IP) is efficient to solve the maximum loading problem, this method has the limitation of starting and terminating condition. The Sequential Quadratic Programming (SQP) algorithm includes the differentiation of the constraints. This method is very slow as it involves many matrices during the iteration process. Fuzzy logic has been used to find the loadability limit in [5], this algorithm does not give global optima. Evolutionary algorithms have been applied to solve this problem. PSO is a computational intelligence-based technique that is not largely affected by the size and nonlinearity of the problem, ⇑ Corresponding author. E-mail addresses: [email protected] (K. Gnanambal), [email protected] (C.K. Babulal). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.033
and can converge to the optimal solution in many problems [6]. PSO performance is improved by incorporating breeding of genetic algorithm into PSO [7]. This Hybrid PSO has been used to solve the maximum loadability problem [8]. This paper does not consider the voltage limit and this algorithm is not suitable for large scale system (limited to 14 bus system). Multi Agent Hybrid PSO (MAHPSO) has been developed and applied to determine the maximum loadability limit in [9]. This MAHPSO has the advantages of both HPSO and MAPSO. The optimal allocation of generators at this maximum loading point is determined in [10]. In this algorithm, the load is uniformly increased in all the load buses until the voltage limits are violated. Maximum loadability limit is identified using genetic algorithm in [11] and using CAPSO in [12]. Here the maximum limit is the voltage stability limit where the voltage magnitude is much lower than the lower limit. Power system loadability discussed in [13] considers the steady state and dynamic security constraints. One of the most powerful algorithms of evolutionary computation is differential evolution (DE) because of its excellent convergence characteristics and a few control parameters. The computational algorithm of DE is simple to understand and implement. Only a few parameters are required to be set by the users [10–12]. Even though DE is shown to be precise, fast as well as robust, the faster convergence yields in a higher probability searching towards a local optimum or getting premature convergence. Recently hybrid algorithms that combine the advantages
K. Gnanambal, C.K. Babulal / Electrical Power and Energy Systems 43 (2012) 150–155
of DE and PSO were developed [13–17]. But there is no published work on hybrid DE to solve the maximum loadability problem. This paper proposes a DEPSO algorithm to determine the maximum loadability limit. In this approach, the powerful DE algorithm is combined with PSO to get the advantages of both. The main operator is DE and its search capability is improved by PSO. The proposed algorithm is successfully tested on Matpower 30 bus and IEEE 118 bus systems. The simulation results are compared with the results obtained by MAHPSO and DE. The performance of DEPSO is compared with DE on statistical measures like best, mean, standard deviation of results and average computation time, over 20 independent trials. Results show the superiority of DEPSO over other evolutionary algorithms on accuracy and consistency. The organization of the paper is as follows: Section 2 presents the formulation of maximum loadability problem as an optimization problem. Sections 3 and 4 explain the overview of DE and PSO. In Section 5 the algorithm of DEPSO in solving the maximum loadability problem is presented. To check the validity of the DEPSO algorithm, it is tested on different systems and the results are discussed in Section 6. Finally conclusions are drawn in the last section. 2. Formulation of maximum loadability limit Maximum loadability can be formulated as an optimization problem. The objective of this problem is to determine the maximum load increase in a power system without violating the voltage limits. This can be mathematically formulated as follows:
Maximize f ¼
X k
ð1Þ
i
ieN PQ NPQ – load buses Subjected to the constraints: 1. Equality constraints:
0 ¼ PGi PDi V i
X
0 ¼ Q Gi Q Di þ V i
V j Y ij cosðhij þ dj di Þ;
X
V j Y ij sinðhij þ dj di Þ;
ieN 0
ð2Þ
ieNPQ
ð3Þ
N0 ¼ NPQ þ N PV For i
e NPQ ; PDi ¼ PD0 þ k
ð4Þ
j 2 nb
Real and reactive power limits on PV buses
Pgi min 6 Pgi 6 Pgi max
QGi – reactive power generation at bus i. QDi – reactive power demand at bus i. The real power demand is randomly increased in all the load buses for the given generation. For this loading condition, the power flow Eqs. (2) and (3) are solved using Newton Raphson method, if the voltages are within the limits, the increase in load is considered otherwise the load increase is 0. 3. Overview of differential evolution 3.1. Initialization This is the first step in DE. Typically, each decision parameter in every vector of the initial population is assigned a randomly chosen value from within its corresponding feasible bounds. ðG¼0Þ
xj;i
¼ xmin þ randj ½0; 1 ðxmax xmin Þ j j j
where i = 1, . . . , NP and j = 1, . . . , D. xj,i(G = 0) is the initial value (G = 0) of the jth parameter of the ith individual vector. xmin and j xmax are the lower and upper bounds of the jth decision parameter, j respectively. Once every vector of the population has been initialized, its corresponding fitness value is calculated and stored for future reference. 3.2. Mutation The DEA optimization process is carried out by applying the following three basic genetic operations; mutation, recombination (also known as crossover) and selection. After the population is initialized, the operators of mutation, crossover and selection create the population of the next generation P(G+1) by using the current population P(G). At every generation G, each vector in the populaðGÞ tion has to serve once as a target vector X i , the parameter vector has index i, and is compared with a mutant vector. The mutation ðGÞ operator generates mutant vectors (V i ) by perturbing a randomly selected vector (Xr1) with the difference of two other randomly selected vectors (Xr2 and Xr3).
V Gi ¼ X Gr1 þ FðX Gr2 X Gr3 Þ; i ¼ 1; 2; . . . ; Np
3.3. Crossover
ð6Þ
In this step, crossover operation is applied in DEA because it helps to increase the diversity among the mutant parameter vectors. At the generation G, the crossover operation creates trial vectors (Ui) by mixing the parameters of the mutant vectors (Vi) with the target vectors (Xi) according to a selected probability distribution. ðGÞ Ui
¼
ðGÞ uj;i
¼
8 < v ðGÞ
if randj ð0; 1Þ 6 CR or j ¼ s
: xðGÞ
otherwise
j;i
j;i
NPV – PV buses. Vi – magnitude of voltage at bus i. di – voltage angle at bus i. Yij – magnitude of Yij element in bus admittance matrix. hij – angle of Yij element in bus admittance matrix. PGi – real power generation at bus i. PDi – real power demand at bus i.
ð8Þ
Vector indices r1, r2 and r3 are randomly chosen, which r1, r2 and r3 {1, . . . , NP} and r1 – r2 – r3 – i. Xr1, Xr2 and Xr3 are selected anew for each parent vector. F is scaling mutation factor.
Q gi min 6 Q gi 6 Q gi max : Voltage limits on PQ buses & Q limits on PV buses.
ð7Þ
ð5Þ
2. Inequality constraints Voltage limit on all PQ and PV buses
V min 6 V max ; j j
151
ð9Þ
The crossover constant CR is usually selected from within the range [0, 1]. The crossover constant controls the diversity of the population and aids the algorithm to escape from local optima. randj is a uniformly distributed random number within the range (0, 1) generated anew for each value of j. s is the trial parameter with randomly chosen index e {1, . . . ,D}, which ensures that the trial vector gets at least one parameter from the mutant vector.
152
K. Gnanambal, C.K. Babulal / Electrical Power and Energy Systems 43 (2012) 150–155
3.3.1. Selection The selection operator chooses the vectors that are going to compose the population in the next generation. This operator compares the fitness of the trial vector and the corresponding target vector and selects the one that provides the best solution. The fitter of the two vectors is then allowed to enter into the next generation. ðGþ1Þ Xi
¼
8 < U ðGÞ
ðGÞ ðGÞ if f ðU i Þ 6 f X i
:
otherwise
i
ðGÞ Xi
ð10Þ
The DEA optimization process is repeated across generations to improve the fitness of individuals. 4. Overview of particle swarm optimization It is a population based algorithm and has since proven to be a powerful tool for optimization problem. The algorithm is also very simple. The PSO model consists of a number of particles moving around in the search space, each representing a possible solution to a numerical problem. Each particle has a position vector (xi) and a velocity vector (vei), each particle keeps track of its coordinates in the problem space, which are associated with the best solution (fitness) it has achieved so far. This value is called Pbest. Another best value that is tracked by the global version of the particle swarm optimizer is the overall best value, and its location, obtained so far by any particle in the population. This location is called Gbest. The PSO concept consists of, at each time, changing the velocity of each particle flying towards its Pbest and Gbest. Acceleration is weighted by random numbers. The velocity vector is given by
Ve ¼ w v e þ ðc1 a ðGbest ppÞ þ c2 b ðPbest ppÞÞ
ð11Þ
The position of each particle is updated by using this velocity vector by
x ¼ x þ ve
ð12Þ
where a and b are random numbers varying between (0 and 1), c1 and c2 are constants usually chosen to be 2, pp is present particle and w is weighting factor In HPSO, breeding property of genetic algorithm is included into PSO [7,8]. The particle produces worst result (child) is replaced from the best particles (parent). This algorithm is limited for small scale applications (limited to 14 bus only). 5. DEPSO DE algorithm has some advantages, such as its ability to maintain the diversity of population, and to explore local search, but it has no mechanism to memory the previous process and use the global information about the search space, so it results in a waste of computing power and may get trapped in local optima. The differential information can be helpful for the search ability, but it also leads to instability of some solutions. Although PSO converges quickly, easily gets stuck in local optima because of loss of diversity of swarm [18–21]. To get the advantages of both, DEPSO algorithm is developed. Here PSO algorithm is incorporated into DE algorithm. 5.1. Proposed DEPSO algorithm Initial population set is the parent vector for both DE and PSO. After mutation and crossover of DE, a target vector is created. Comparing the fitness values, a candidate solution set is obtained either from parent or target vector. PSO’s Gbest and Pbest are selected from the parent vector or from the target vector. Using PSO algorithm the position and velocity of particles in parent vector are updated
to produce another set of candidate solution. Comparing the candidates of DE and PSO, the candidate that produces lesser fitness value, be selected as the member of the parent vector for next iteration. 5.2. DEPSO algorithm for maximum loadability problem The DEPSO algorithm explained in section the previous section is applied to determine the maximum loadability limit (1). The methodology is explained step by step as follows: 1. Initialize a set of random values for k Xi = [kji]; i = 1 to Np; j e NPQ. Np – population size; NPQ – load buses. 2. Set iteration count G = 1. 3. Using Eq. (1) calculate the fitness 1 f ðxGi Þ. Run NR power flow for five iterations and check for voltage limit in all PQ buses. If any one of the bus voltages is violating the limit then f ðxGi Þ = 0. 4. DE operator: (i) Perform mutation and cross over. Generate a new set of 0 k: X G. (ii) Substituting this in Eq. (1), calculate the fitness 2 f ðx0G i Þ. Run NR power flow for five iterations and check for the voltage limit. 5. PSO operator (i) Choose XG as present particle (pp). (ii) Generate Pbest. For ith particle, if fitness 1 is greater than fitness 2, then Pbesti is selected from initial parent vector, else it is selected from DE mutant vector. (iii) Generate Gbesti from overall best. (iv) Using (8) and (9) update the velocity and position of XG and get the new loading parameters x00G . (v) The new k calculated in step (iv) is used to calculate the fitness 3 f ðx00G i Þ using Eq. (1) by considering the voltage limits. 6. By comparing fitness 1, 2 and 3, select the best particle for next iteration;
X Gþ1 ¼ xGi i ¼ ¼
x0G i x00G i
if fitness 1 is greater if fitness 2 is greater if fitness 3is greater:
7. Increase the iteration count G = G + 1. 8. Check for maximum number of iterations. If not go to step 3 and repeat the steps.
6. Case studies The proposed algorithm is tested on the sample Matpower 30 bus and IEEE 118 bus systems. The simulation studies are performed in MATLAB in the Pentium Dual core, 1800 MHz system. For all the cases DE and PSO parameters are set as follows. PSO constants: c1 = 1.5; c2 = 2; w = 0.9 to 0.7; as the iteration increases the inertia weight decreases. DE constants: cross over constant = 0.8; mutation constant = 1.2; Literature survey shows that MAHPSO is superior to other evolutionary algorithms like HPSO, MAPSO. Hence, here the results are compared with the results obtained by MAHPSO and DE. 6.1. Matpower case30 system [22] DE, and DEPSO algorithms are tested on 30 bus system which consists of six generators. The initial load and total generations for the system are given as follows:
153
K. Gnanambal, C.K. Babulal / Electrical Power and Energy Systems 43 (2012) 150–155 Table 1 P Comparison of maximum loadability limits PDmax in p.u.
2.8
MAHPSO [9]
DE
DEPSO
30 Bus 118 Bus
2.608080 56.450
2.6709 56.5430
2.6974 57.0156
Table 2 The maximum loading (PDmax) at various buses for 30 bus system. Bus no.
Real power demand (p.u)
Bus no.
Real power demand (p.u)
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.21700000000000 0.15002339659131 0.11997460498977 0.11751924369242 0.06587178824539 0.22809925810236 0.30425668684641 0.00077140893639 0.06190634999534 0.09056018005652 0.19774738106643 0 0.06202540795097 0.20869352775483
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.04066493281219 0.09000857624661 0.04596108709000 0.09508361535178 0.02242039776266 0.23815651652000 0 0.03200000000000 0.10462883728880 0.00576385147577 0.05189323840793 0 0.00443430392603 0.02407280722437 0.11782858547791
Table 3 Bus voltages at the maximum loading point for 30 bus system. Bus no.
Bus voltage (p.u)
Bus no.
Bus voltage (p.u)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.0 1.0 0.97261200426218 0.96960030098020 0.97364262373538 0.96268240115439 0.95735744566324 0.94999121014668 0.97558075651480 0.98314939774109 0.9753859020558 0.98295812113907 1.0 0.97331444260981 0.97380333288614
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.97518578879304 0.97535305691873 0.96299530427879 0.96118427918268 0.96573997868401 0.99266378186036 1.0 1.0 0.98683946965025 0.98780971231111 0.96520768309565 1.0 0.96444776142341 0.97830594316959 0.96531057991667
X X
Pload ¼ 1:892 p:u:; PG ¼ 1:9164 p:u:;
X X
Q load ¼ 1:072 p:u:;
Q G ¼ 1:0041 p:u;
The maximum loadability limit is determined for this system using DE and proposed DEPSO algorithms [9] shows that MAHPSO is superior to PSO, HPSO and MAPSO algorithms hence the results are compared with MAHPSO algorithm. The population size taken for this case is 30. The comparison of maximum loadability limit obtained by MAPSO, DE and DEPSO is given in Table 1. The maximum allowable loads at various buses are depicted in Table 2. The maximum loadability obtained by DEPSO method is 2.6974 p.u. Bus voltages obtained at this maximum loading condition are illustrated in Table 3. The simulation results are taken after
2.7 2.6 2.5
total demand
System
2.4 2.3 2.2 DEPSO
2.1
DE
2
1.9 1.8
5
10
15
20
25
30
iteration Fig. 1. Convergence characteristics for 30 bus system.
50 iterations. The statistical measures like mean, best, and standard deviation are calculated from 20 independent trials. Statistical performances are given in Table 4. Best and mean values prove the accuracy and consistency of the proposed method. The convergence characteristics curves are shown in Fig. 1. The best solution is considered for the characteristic curves. The solutions are taken for 50 iterations. There is no change in the fitness values after 30 iterations. Characteristic curves show that, DEPSO converges quickly than DE. The total time taken by DEPSO method is 13.512 s.
6.2. IEEE 118 Bus system To check the ability of the proposed method to solve higher order systems, it is tested on IEEE 118 system. The initial load and total generations for the system are given as follows:
X X
Pload ¼ 42:42 p:u:; PG ¼ 43:7486 p:u:;
X
Q load ¼ 14:39 p:u:; X Q G ¼ 79:568 p:u;
ð13Þ
The population size is increased to 50. The performances are carried out for 20 trials and for each trial 50 iterations are considered. The maximum loadabilty obtained by DEPSO algorithm is 57.0156 p.u. It converges with the total time of 22.8240 s. Statistical performances of the evolutionary algorithms along with the time per iteration are given in Table 5. Best and mean values of DEPSO algorithm show the superiority of this over DE. The characteristic curves are given in Fig. 2. Even though, DEPSO takes more time for single iteration, it converges in lesser number of iterations and so the total computational time is lesser. DE takes more time to converge. The PSO and DE parameters are varied and the above said algorithm is tested. For a set of parameter, the best value is taken by running 20 independent trials. The best values are given in Table 6. From the table, it is evident that the following parameters are giving the best results.
Table 4 Statistical performance for 30 bus system. Method
Best
Worst
Mean
Standard deviation
Time per iteration (s)
No. of iteration
DE DEPSO
2.6709 2.6974
2.5813 2.642
2.6334 2.6759
0.0234 0.02149
1.03 1.126
15 12
154
K. Gnanambal, C.K. Babulal / Electrical Power and Energy Systems 43 (2012) 150–155
Table 5 Statistical performance for IEEE 118 bus system. Method
Best
Worst
Mean
Standard deviation
Time per iteration (s)
No. of iterations
DE DEPSO
56.6212 57.0156
54.4993 56.2189
56.2432 56.4966
0.6999 0.2700
1.3012 1.5216
18 15
58
presented here. The DEPSO algorithm gives more accurate results compared to other evolutionary algorithms. The characteristic curves show the effectiveness of the proposed DEPSO method for higher order systems. It gives higher accuracy with lesser number of iterations. Even though time per iteration for the proposed DEPSO algorithm is greater, the total computational time is less as it converges with lesser number of iterations. Statistical measures indicate the consistency of this DEPSO algorithm. Compared to MAHPSO and DE, the proposed DEPSO algorithm gives better results for the maximum loadability problem.
56
total demand
54 52
DEPSO DE
50 48 46
References 44 42
0
5
10
15
20
25
30
35
40
45
50
iteration Fig. 2. Convergence characteristics for IEEE 118 system.
Table 6 Comparison of best values with various DE and PSO parameters. DE parameter
PSO parameter
Mutation constant
Cross over constant
c1
c2
1.0 1.0 1.1 1.2 1.3 1.4 1.2 1.2 1.2 1.2 1.2
0.8 0.7 0.8 0.8 0.8 0.8 0.9 1 0.8 0.8 0.8
1 1 1 1 1 1 1 1 1.5 1.5 2
1 1 1 1 1 1 1 1 1.5 2 2
Best values for 118 bus system
57.0132 56.9874 57.0132 57.0132 57.0131 57.0131 57.0125 56.0122 56.8765 57.0156 57.0156
PSO parameters: c1 = 1.5; c2 = 2; DE parameters: cross over constant = 0.8; mutation constant = 1.2. 6.3. Indian 181 bus system [23] The proposed DEPSO algorithm is tested on the Indian practical system [23]. The total base load is 6017.86 MW. Even in the base case loading, some buses experience very low voltage. Voltage limits are violating without any increase in load. Now the objective function is slightly modified that the k is included in the generator. That is the generation at the PV buses are increased until the voltage is not violating the limits. 7. Conclusion A new DEPSO algorithm which combines the advantages of DE and PSO to determine the maximum loading point has been
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Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Maximum loadability limit of power system using hybrid differential evolution with particle swarm optimization K. Gnanambal a,⇑, C.K. Babulal b a b
Department of Electrical and Electronics Engineering, K.L.N. College of Engineering, Pottapalayam 630 611, Tamilnadu, India Department of Electrical and Electronics Engineering, Thiagarajar College of Engineering, Madurai 625 015, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 15 May 2010 Received in revised form 27 April 2012 Accepted 29 April 2012 Available online 18 June 2012 Keywords: Maximum loadability limit Particle swarm optimization Differential evolution
a b s t r a c t Differential evolution (DE), a simple evolutionary algorithm which shows superior performance in global optimization. Since it utilizes the differential information to get the new candidate solution, sometimes it results in instability of performance. Particle swarm optimization (PSO) is widely used to solve the optimization problems as it can converge quickly. But PSO easily gets stuck in local optima. Hybridization of DE and PSO (DEPSO) eliminates the disadvantages of both. This paper presents the application of DEPSO algorithm to determine the maximum loadability limit of power system. It is tested on Matpower 30 bus and IEEE 118 bus systems. To compare the performance of this DEPSO algorithm with other evolutionary algorithms like DE and Multi Agent Hybrid PSO, statistical measures like best, mean, standard deviation of results and average computation time over 20 independent trials are considered here. The results show the better performance of DEPSO algorithm to solve the maximum loadability problem. DEPSO algorithm provides high maximum loading point in reduced time. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The power systems are heavily stressed due to the increased loading and this leads to the voltage stability problem in power systems. Recently, various studies for detecting loadability limits of power systems have been proposed in relation to voltage stability and security monitoring. The maximum loadability limit is the margin between the operating point of the system and the maximum loading point. Various methods have been proposed to determine maximum loadability limit [1–4]. Among these methods, the continuation power flow (CPF) technique has been widely used. This method fails to give the accurate result if the step length is more. Even though Interior Point method (IP) is efficient to solve the maximum loading problem, this method has the limitation of starting and terminating condition. The Sequential Quadratic Programming (SQP) algorithm includes the differentiation of the constraints. This method is very slow as it involves many matrices during the iteration process. Fuzzy logic has been used to find the loadability limit in [5], this algorithm does not give global optima. Evolutionary algorithms have been applied to solve this problem. PSO is a computational intelligence-based technique that is not largely affected by the size and nonlinearity of the problem, ⇑ Corresponding author. E-mail addresses: [email protected] (K. Gnanambal), [email protected] (C.K. Babulal). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.033
and can converge to the optimal solution in many problems [6]. PSO performance is improved by incorporating breeding of genetic algorithm into PSO [7]. This Hybrid PSO has been used to solve the maximum loadability problem [8]. This paper does not consider the voltage limit and this algorithm is not suitable for large scale system (limited to 14 bus system). Multi Agent Hybrid PSO (MAHPSO) has been developed and applied to determine the maximum loadability limit in [9]. This MAHPSO has the advantages of both HPSO and MAPSO. The optimal allocation of generators at this maximum loading point is determined in [10]. In this algorithm, the load is uniformly increased in all the load buses until the voltage limits are violated. Maximum loadability limit is identified using genetic algorithm in [11] and using CAPSO in [12]. Here the maximum limit is the voltage stability limit where the voltage magnitude is much lower than the lower limit. Power system loadability discussed in [13] considers the steady state and dynamic security constraints. One of the most powerful algorithms of evolutionary computation is differential evolution (DE) because of its excellent convergence characteristics and a few control parameters. The computational algorithm of DE is simple to understand and implement. Only a few parameters are required to be set by the users [10–12]. Even though DE is shown to be precise, fast as well as robust, the faster convergence yields in a higher probability searching towards a local optimum or getting premature convergence. Recently hybrid algorithms that combine the advantages
K. Gnanambal, C.K. Babulal / Electrical Power and Energy Systems 43 (2012) 150–155
of DE and PSO were developed [13–17]. But there is no published work on hybrid DE to solve the maximum loadability problem. This paper proposes a DEPSO algorithm to determine the maximum loadability limit. In this approach, the powerful DE algorithm is combined with PSO to get the advantages of both. The main operator is DE and its search capability is improved by PSO. The proposed algorithm is successfully tested on Matpower 30 bus and IEEE 118 bus systems. The simulation results are compared with the results obtained by MAHPSO and DE. The performance of DEPSO is compared with DE on statistical measures like best, mean, standard deviation of results and average computation time, over 20 independent trials. Results show the superiority of DEPSO over other evolutionary algorithms on accuracy and consistency. The organization of the paper is as follows: Section 2 presents the formulation of maximum loadability problem as an optimization problem. Sections 3 and 4 explain the overview of DE and PSO. In Section 5 the algorithm of DEPSO in solving the maximum loadability problem is presented. To check the validity of the DEPSO algorithm, it is tested on different systems and the results are discussed in Section 6. Finally conclusions are drawn in the last section. 2. Formulation of maximum loadability limit Maximum loadability can be formulated as an optimization problem. The objective of this problem is to determine the maximum load increase in a power system without violating the voltage limits. This can be mathematically formulated as follows:
Maximize f ¼
X k
ð1Þ
i
ieN PQ NPQ – load buses Subjected to the constraints: 1. Equality constraints:
0 ¼ PGi PDi V i
X
0 ¼ Q Gi Q Di þ V i
V j Y ij cosðhij þ dj di Þ;
X
V j Y ij sinðhij þ dj di Þ;
ieN 0
ð2Þ
ieNPQ
ð3Þ
N0 ¼ NPQ þ N PV For i
e NPQ ; PDi ¼ PD0 þ k
ð4Þ
j 2 nb
Real and reactive power limits on PV buses
Pgi min 6 Pgi 6 Pgi max
QGi – reactive power generation at bus i. QDi – reactive power demand at bus i. The real power demand is randomly increased in all the load buses for the given generation. For this loading condition, the power flow Eqs. (2) and (3) are solved using Newton Raphson method, if the voltages are within the limits, the increase in load is considered otherwise the load increase is 0. 3. Overview of differential evolution 3.1. Initialization This is the first step in DE. Typically, each decision parameter in every vector of the initial population is assigned a randomly chosen value from within its corresponding feasible bounds. ðG¼0Þ
xj;i
¼ xmin þ randj ½0; 1 ðxmax xmin Þ j j j
where i = 1, . . . , NP and j = 1, . . . , D. xj,i(G = 0) is the initial value (G = 0) of the jth parameter of the ith individual vector. xmin and j xmax are the lower and upper bounds of the jth decision parameter, j respectively. Once every vector of the population has been initialized, its corresponding fitness value is calculated and stored for future reference. 3.2. Mutation The DEA optimization process is carried out by applying the following three basic genetic operations; mutation, recombination (also known as crossover) and selection. After the population is initialized, the operators of mutation, crossover and selection create the population of the next generation P(G+1) by using the current population P(G). At every generation G, each vector in the populaðGÞ tion has to serve once as a target vector X i , the parameter vector has index i, and is compared with a mutant vector. The mutation ðGÞ operator generates mutant vectors (V i ) by perturbing a randomly selected vector (Xr1) with the difference of two other randomly selected vectors (Xr2 and Xr3).
V Gi ¼ X Gr1 þ FðX Gr2 X Gr3 Þ; i ¼ 1; 2; . . . ; Np
3.3. Crossover
ð6Þ
In this step, crossover operation is applied in DEA because it helps to increase the diversity among the mutant parameter vectors. At the generation G, the crossover operation creates trial vectors (Ui) by mixing the parameters of the mutant vectors (Vi) with the target vectors (Xi) according to a selected probability distribution. ðGÞ Ui
¼
ðGÞ uj;i
¼
8 < v ðGÞ
if randj ð0; 1Þ 6 CR or j ¼ s
: xðGÞ
otherwise
j;i
j;i
NPV – PV buses. Vi – magnitude of voltage at bus i. di – voltage angle at bus i. Yij – magnitude of Yij element in bus admittance matrix. hij – angle of Yij element in bus admittance matrix. PGi – real power generation at bus i. PDi – real power demand at bus i.
ð8Þ
Vector indices r1, r2 and r3 are randomly chosen, which r1, r2 and r3 {1, . . . , NP} and r1 – r2 – r3 – i. Xr1, Xr2 and Xr3 are selected anew for each parent vector. F is scaling mutation factor.
Q gi min 6 Q gi 6 Q gi max : Voltage limits on PQ buses & Q limits on PV buses.
ð7Þ
ð5Þ
2. Inequality constraints Voltage limit on all PQ and PV buses
V min 6 V max ; j j
151
ð9Þ
The crossover constant CR is usually selected from within the range [0, 1]. The crossover constant controls the diversity of the population and aids the algorithm to escape from local optima. randj is a uniformly distributed random number within the range (0, 1) generated anew for each value of j. s is the trial parameter with randomly chosen index e {1, . . . ,D}, which ensures that the trial vector gets at least one parameter from the mutant vector.
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3.3.1. Selection The selection operator chooses the vectors that are going to compose the population in the next generation. This operator compares the fitness of the trial vector and the corresponding target vector and selects the one that provides the best solution. The fitter of the two vectors is then allowed to enter into the next generation. ðGþ1Þ Xi
¼
8 < U ðGÞ
ðGÞ ðGÞ if f ðU i Þ 6 f X i
:
otherwise
i
ðGÞ Xi
ð10Þ
The DEA optimization process is repeated across generations to improve the fitness of individuals. 4. Overview of particle swarm optimization It is a population based algorithm and has since proven to be a powerful tool for optimization problem. The algorithm is also very simple. The PSO model consists of a number of particles moving around in the search space, each representing a possible solution to a numerical problem. Each particle has a position vector (xi) and a velocity vector (vei), each particle keeps track of its coordinates in the problem space, which are associated with the best solution (fitness) it has achieved so far. This value is called Pbest. Another best value that is tracked by the global version of the particle swarm optimizer is the overall best value, and its location, obtained so far by any particle in the population. This location is called Gbest. The PSO concept consists of, at each time, changing the velocity of each particle flying towards its Pbest and Gbest. Acceleration is weighted by random numbers. The velocity vector is given by
Ve ¼ w v e þ ðc1 a ðGbest ppÞ þ c2 b ðPbest ppÞÞ
ð11Þ
The position of each particle is updated by using this velocity vector by
x ¼ x þ ve
ð12Þ
where a and b are random numbers varying between (0 and 1), c1 and c2 are constants usually chosen to be 2, pp is present particle and w is weighting factor In HPSO, breeding property of genetic algorithm is included into PSO [7,8]. The particle produces worst result (child) is replaced from the best particles (parent). This algorithm is limited for small scale applications (limited to 14 bus only). 5. DEPSO DE algorithm has some advantages, such as its ability to maintain the diversity of population, and to explore local search, but it has no mechanism to memory the previous process and use the global information about the search space, so it results in a waste of computing power and may get trapped in local optima. The differential information can be helpful for the search ability, but it also leads to instability of some solutions. Although PSO converges quickly, easily gets stuck in local optima because of loss of diversity of swarm [18–21]. To get the advantages of both, DEPSO algorithm is developed. Here PSO algorithm is incorporated into DE algorithm. 5.1. Proposed DEPSO algorithm Initial population set is the parent vector for both DE and PSO. After mutation and crossover of DE, a target vector is created. Comparing the fitness values, a candidate solution set is obtained either from parent or target vector. PSO’s Gbest and Pbest are selected from the parent vector or from the target vector. Using PSO algorithm the position and velocity of particles in parent vector are updated
to produce another set of candidate solution. Comparing the candidates of DE and PSO, the candidate that produces lesser fitness value, be selected as the member of the parent vector for next iteration. 5.2. DEPSO algorithm for maximum loadability problem The DEPSO algorithm explained in section the previous section is applied to determine the maximum loadability limit (1). The methodology is explained step by step as follows: 1. Initialize a set of random values for k Xi = [kji]; i = 1 to Np; j e NPQ. Np – population size; NPQ – load buses. 2. Set iteration count G = 1. 3. Using Eq. (1) calculate the fitness 1 f ðxGi Þ. Run NR power flow for five iterations and check for voltage limit in all PQ buses. If any one of the bus voltages is violating the limit then f ðxGi Þ = 0. 4. DE operator: (i) Perform mutation and cross over. Generate a new set of 0 k: X G. (ii) Substituting this in Eq. (1), calculate the fitness 2 f ðx0G i Þ. Run NR power flow for five iterations and check for the voltage limit. 5. PSO operator (i) Choose XG as present particle (pp). (ii) Generate Pbest. For ith particle, if fitness 1 is greater than fitness 2, then Pbesti is selected from initial parent vector, else it is selected from DE mutant vector. (iii) Generate Gbesti from overall best. (iv) Using (8) and (9) update the velocity and position of XG and get the new loading parameters x00G . (v) The new k calculated in step (iv) is used to calculate the fitness 3 f ðx00G i Þ using Eq. (1) by considering the voltage limits. 6. By comparing fitness 1, 2 and 3, select the best particle for next iteration;
X Gþ1 ¼ xGi i ¼ ¼
x0G i x00G i
if fitness 1 is greater if fitness 2 is greater if fitness 3is greater:
7. Increase the iteration count G = G + 1. 8. Check for maximum number of iterations. If not go to step 3 and repeat the steps.
6. Case studies The proposed algorithm is tested on the sample Matpower 30 bus and IEEE 118 bus systems. The simulation studies are performed in MATLAB in the Pentium Dual core, 1800 MHz system. For all the cases DE and PSO parameters are set as follows. PSO constants: c1 = 1.5; c2 = 2; w = 0.9 to 0.7; as the iteration increases the inertia weight decreases. DE constants: cross over constant = 0.8; mutation constant = 1.2; Literature survey shows that MAHPSO is superior to other evolutionary algorithms like HPSO, MAPSO. Hence, here the results are compared with the results obtained by MAHPSO and DE. 6.1. Matpower case30 system [22] DE, and DEPSO algorithms are tested on 30 bus system which consists of six generators. The initial load and total generations for the system are given as follows:
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2.8
MAHPSO [9]
DE
DEPSO
30 Bus 118 Bus
2.608080 56.450
2.6709 56.5430
2.6974 57.0156
Table 2 The maximum loading (PDmax) at various buses for 30 bus system. Bus no.
Real power demand (p.u)
Bus no.
Real power demand (p.u)
2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.21700000000000 0.15002339659131 0.11997460498977 0.11751924369242 0.06587178824539 0.22809925810236 0.30425668684641 0.00077140893639 0.06190634999534 0.09056018005652 0.19774738106643 0 0.06202540795097 0.20869352775483
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.04066493281219 0.09000857624661 0.04596108709000 0.09508361535178 0.02242039776266 0.23815651652000 0 0.03200000000000 0.10462883728880 0.00576385147577 0.05189323840793 0 0.00443430392603 0.02407280722437 0.11782858547791
Table 3 Bus voltages at the maximum loading point for 30 bus system. Bus no.
Bus voltage (p.u)
Bus no.
Bus voltage (p.u)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.0 1.0 0.97261200426218 0.96960030098020 0.97364262373538 0.96268240115439 0.95735744566324 0.94999121014668 0.97558075651480 0.98314939774109 0.9753859020558 0.98295812113907 1.0 0.97331444260981 0.97380333288614
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.97518578879304 0.97535305691873 0.96299530427879 0.96118427918268 0.96573997868401 0.99266378186036 1.0 1.0 0.98683946965025 0.98780971231111 0.96520768309565 1.0 0.96444776142341 0.97830594316959 0.96531057991667
X X
Pload ¼ 1:892 p:u:; PG ¼ 1:9164 p:u:;
X X
Q load ¼ 1:072 p:u:;
Q G ¼ 1:0041 p:u;
The maximum loadability limit is determined for this system using DE and proposed DEPSO algorithms [9] shows that MAHPSO is superior to PSO, HPSO and MAPSO algorithms hence the results are compared with MAHPSO algorithm. The population size taken for this case is 30. The comparison of maximum loadability limit obtained by MAPSO, DE and DEPSO is given in Table 1. The maximum allowable loads at various buses are depicted in Table 2. The maximum loadability obtained by DEPSO method is 2.6974 p.u. Bus voltages obtained at this maximum loading condition are illustrated in Table 3. The simulation results are taken after
2.7 2.6 2.5
total demand
System
2.4 2.3 2.2 DEPSO
2.1
DE
2
1.9 1.8
5
10
15
20
25
30
iteration Fig. 1. Convergence characteristics for 30 bus system.
50 iterations. The statistical measures like mean, best, and standard deviation are calculated from 20 independent trials. Statistical performances are given in Table 4. Best and mean values prove the accuracy and consistency of the proposed method. The convergence characteristics curves are shown in Fig. 1. The best solution is considered for the characteristic curves. The solutions are taken for 50 iterations. There is no change in the fitness values after 30 iterations. Characteristic curves show that, DEPSO converges quickly than DE. The total time taken by DEPSO method is 13.512 s.
6.2. IEEE 118 Bus system To check the ability of the proposed method to solve higher order systems, it is tested on IEEE 118 system. The initial load and total generations for the system are given as follows:
X X
Pload ¼ 42:42 p:u:; PG ¼ 43:7486 p:u:;
X
Q load ¼ 14:39 p:u:; X Q G ¼ 79:568 p:u;
ð13Þ
The population size is increased to 50. The performances are carried out for 20 trials and for each trial 50 iterations are considered. The maximum loadabilty obtained by DEPSO algorithm is 57.0156 p.u. It converges with the total time of 22.8240 s. Statistical performances of the evolutionary algorithms along with the time per iteration are given in Table 5. Best and mean values of DEPSO algorithm show the superiority of this over DE. The characteristic curves are given in Fig. 2. Even though, DEPSO takes more time for single iteration, it converges in lesser number of iterations and so the total computational time is lesser. DE takes more time to converge. The PSO and DE parameters are varied and the above said algorithm is tested. For a set of parameter, the best value is taken by running 20 independent trials. The best values are given in Table 6. From the table, it is evident that the following parameters are giving the best results.
Table 4 Statistical performance for 30 bus system. Method
Best
Worst
Mean
Standard deviation
Time per iteration (s)
No. of iteration
DE DEPSO
2.6709 2.6974
2.5813 2.642
2.6334 2.6759
0.0234 0.02149
1.03 1.126
15 12
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K. Gnanambal, C.K. Babulal / Electrical Power and Energy Systems 43 (2012) 150–155
Table 5 Statistical performance for IEEE 118 bus system. Method
Best
Worst
Mean
Standard deviation
Time per iteration (s)
No. of iterations
DE DEPSO
56.6212 57.0156
54.4993 56.2189
56.2432 56.4966
0.6999 0.2700
1.3012 1.5216
18 15
58
presented here. The DEPSO algorithm gives more accurate results compared to other evolutionary algorithms. The characteristic curves show the effectiveness of the proposed DEPSO method for higher order systems. It gives higher accuracy with lesser number of iterations. Even though time per iteration for the proposed DEPSO algorithm is greater, the total computational time is less as it converges with lesser number of iterations. Statistical measures indicate the consistency of this DEPSO algorithm. Compared to MAHPSO and DE, the proposed DEPSO algorithm gives better results for the maximum loadability problem.
56
total demand
54 52
DEPSO DE
50 48 46
References 44 42
0
5
10
15
20
25
30
35
40
45
50
iteration Fig. 2. Convergence characteristics for IEEE 118 system.
Table 6 Comparison of best values with various DE and PSO parameters. DE parameter
PSO parameter
Mutation constant
Cross over constant
c1
c2
1.0 1.0 1.1 1.2 1.3 1.4 1.2 1.2 1.2 1.2 1.2
0.8 0.7 0.8 0.8 0.8 0.8 0.9 1 0.8 0.8 0.8
1 1 1 1 1 1 1 1 1.5 1.5 2
1 1 1 1 1 1 1 1 1.5 2 2
Best values for 118 bus system
57.0132 56.9874 57.0132 57.0132 57.0131 57.0131 57.0125 56.0122 56.8765 57.0156 57.0156
PSO parameters: c1 = 1.5; c2 = 2; DE parameters: cross over constant = 0.8; mutation constant = 1.2. 6.3. Indian 181 bus system [23] The proposed DEPSO algorithm is tested on the Indian practical system [23]. The total base load is 6017.86 MW. Even in the base case loading, some buses experience very low voltage. Voltage limits are violating without any increase in load. Now the objective function is slightly modified that the k is included in the generator. That is the generation at the PV buses are increased until the voltage is not violating the limits. 7. Conclusion A new DEPSO algorithm which combines the advantages of DE and PSO to determine the maximum loading point has been
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