A Constraint-Handling Genetic Algorithm to Power Economic Dispatch

A Constraint-Handling Genetic Algorithm toPower Economic Dispatch

Felix Calderon, Claudio R. Fuerte-Esquivel, Juan J. Flores and Juan C. Silva

Universidad Michoacana de San Nicolas de HidalgoDivision de Estudios de Posgrado. Facultad de Ingenierıa ElectricaSantiago Tapia 403 Centro. Morelia, Michoacan, Mexico. CP 58000

[email protected] [email protected]

[email protected] [email protected]

Abstract. This paper presents a new constraint-handling genetic ap-proach for solving the economic dispatch problem in electric power sys-tems. A real code genetic algorithm is implemented to minimize theactive power generation cost while satisfying power balance (energy con-servation) and generation limit constraints simultaneously during theoptimization process. This is achieved by introducing a novel strategyfor searching the solution on the energy conservative space, producingonly individuals that fulfill the energy conservation constraint, and re-ducing the search space in one dimension. Computer simulations on threebenchmark electrical systems show the prowess of the proposed approachwhose results are very close to those reported by other authors using dif-ferent methods.

1 Introduction

An electric power systems can be seen as the interconnection of generators andloads through a transmission network. The function of the transmission networkis always to transport the electric energy from power plants to load centers.In this context, the Economic Dispatch (ED) problem is defined as the processof providing the required active power load demand and transmission losses byallocating generation among power plant units such that the total generation costis minimized [1]. This goal is achieved if generator units are dispatching based ontheir function cost and generation limits while satisfying the total active powerbalance in the network. Hence, the ED can be stated as a constrained nonlinearoptimization problem. Over the years, many classical optimization techniqueshave been used to solve this problem by generating a finite succession of testsolutions, feasible or not, with search directions based on the gradient or superiororder operators that produce solutions that converge to local optima (see [2]).

Evolutionary optimization techniques constitute an efficient and robust al-ternative to treat the ED problem. Among these techniques, Genetic Algorithms(GAs) basic principles were proposed in the 70’s [3–5], and they have been ap-plied for the ED problem for the last two decades. These algorithms are based

on the natural evolution of species (Natural Selection)and offer many outstand-ing features over traditional optimization methods. For instance, their abilityto search non-convex solution spaces with multiple and isolated maxima, globalconvergence, and inherent parallel search capability, among others characteristics[3].

In general, GAs applied to solve the ED problem use a set of random solu-tions associated to generator’s active power. Each set constitutes a chromosomewhich is substituted in the objective cost function to compute the total activepower generation cost. This process is repeated for each chromosome to assessthe lowest production cost. Owing to the fact that GAs are unconstrained searchtechniques, methods based on penalty functions have been used to consider prob-lem constraints during the optimization process. However, these methods requiresetting additional search parameters affecting the convergence performance. An-other option is to preserve the feasibility of the solution by generating chromo-somes that satisfy problem constraints. This can be achieved if it is possible toobtain the feasible solution region.

In this paper a constraint-handling GA approach is proposed based on pre-serving the feasibility of the ED solution. The proposed approach computes thefeasible solution region by obtaining a vectorial search space based on the prob-lem constraints. Results and analysis are presented to show the computationalperformance of the algorithm.

2 Economic Dispatch Problem Formulation

The ED problem is defined in terms of minimizing total active generation costsubject to satisfying the total active power demand and transmission losses, whilemaintaining the active power generation within limits [1]. For a system with Ngeneration units, the cost c(Pi, αi), for each generator is defined as a polynomialfunction given by Equation (1) where Pi is the active power for each generationunit, αi is a set of polynomial coefficients, αi,j is the j−th polynomial coefficientassociated to the i− th generation unit and Nc is the polynomial order.

c(Pi, αi) = αi,0 + αi,1Pi + αi,2P2i + ... =

Nc∑

j=0

αi,jPji (1)

The total cost C is the sum of costs for each generation unit and the problemof minimizing the total cost can be formulated as an optimization problem (2)with restrictions (3) and (4).

Minimize C =N∑

i=1

c(Pi, αi) (2)

s.t

N∑

i=1

Pi = PT (3)

Pmini ≤ Pi ≤ Pmax

i (4)

where PT is the total active power demanded by loads and transmission system,and Pmin

i and Pmaxi the generation limits for each unit. Constraint given by

Equation (3) corresponds to the total active power balance that must exists inthe network, and it is referred to as Energy Conservation Constraint (ECC). Onthe other hand, constraints given by Equation (4) correspond to active powergeneration limits, referred to as Capability Constraints (CC).

For a quadratic cost, (Nc = 2), a unique minimum exists and the way tominimize it is by using Lagrange Multipliers [6]. Using Lagrange multipliers anew function, given by Equation (5), which includes the ECC constraint andlagrange multiplier λ, must be minimized. The most common method used inPower Engineering to solve (5) is the Newton approach.

L =N∑

i=1

[c(Pi, αi) + λ

(N∑

i=1

Pi − PT

)](5)

In the case of a non-quadratic system, the Newton algorithm does not guar-antee to reach the global minimum and some authors as Wood and Wollenberg[7] apply an iterative strategy for computing the minimum, but this proceduredoes not guarantee to reach the global minimum; the solution depends on initialconditions. Other authors as Sheble, Orero, and Song [8–10] began to use search-ing strategies based on Simulated Annealing (SA) [11] and GA [4], which allowsto find the global minimum for quadratic and non quadratic cost functions.

3 Economic Power Dispatch Using Genetic Algorithms

GAs are population based techniques which sample the solution space randomlyby using several candidate solutions. These solutions are evaluated and thosethat perform better are selected to compose and generate the population in thenext generation. After several generations, these solutions improve the objectivefunction value as they explore the solution space for an optimal value.

In the context of ED problem, GAs optimize the cost function starting froma random population of possible solutions. Each individual solution is stored intoa N -dimensional vector, whose i − th floating point element or gene representsthe active power output of the i− th dispatchable generator. This data structurecontaining the active power outputs is known as chromosome. The fitness of achromosome relates to the evaluation of the objective function and is a measureof goodness of this solution with respect to other solutions in the population.

Once the fitness of the initial population has been computed, a search foran optimal solution is carried out by updating the population according to rulesof selection and stochastic operations called crossover and mutation. After per-forming a certain number of stochastic operations, some of the solutions in theold population are replaced by new solutions; this concludes one generation ofthe algorithm. Generations are repeated to minimize the total production costuntil a stopping criteria is satisfied. In our case, the convergence criterion is givenby the number of generation units.

Several authors have proposed GAs to tackle the ED problem. Sheble andBrittig [8] and Song and Chou [10] solve ED using GA. Both proposals considera chromosome as a binary string representing all Pi’s. The ECC (3) is handled asan error term and a measure of this error is calculated during the optimizationprocess.

Orero and Irving [9] solve ED with prohibited operating zones using GA.They proposed to solve a non-continuous objective cost function given by Equa-tion (6),

L =N∑

i=1

c(Pi) + Ψ

(N∑

i=1

Pi − PT

)+ Φ (P ) (6)

where c is a cost for the i − th generation unit, Ψ and Φ are penalty functionsfor ECC and CC; respectively. This approach is not clear on how to select theΨ and Φ functions; additionally chromosomes have binary representation.

Some Simulated Annealing (SA) based solution to the ED problem are pre-sented by Wong and Fung [12], as well as Ongsakul and Ruangpayoongsak [13].In [13], the purpose of this combination is to provide one initial solution forGA. It is used a quadratic cost function and 16 bits for chromosomes’ binaryrepresentation.

A drawback of GAs using a chromosome’s binary representation is that ac-curacy suffers of radical changes when the GA does not use the correct bitrepresentation. For some cases, the chromosome length would have more bitsthan those needed by a floating point number representation in a computer (32bits).

To avoid this problem, the proposed GA uses a real representation in order toincrease the accuracy. Furthermore, instead of including the ECC into the costfunction, a feasible solution region is generated by a linear combination of basevectors, such that feasible solutions are generated randomly within a so calledvectorial search space to fulfill both ECC and CC.

4 Constraint-Handling Genetic algorithm

The proposed approach to solve the ED optimization problem is described inthis section. First, a reduction on the search space is accomplished through a setof vectors that define the intersection between a hypercube that represents CC,and the hyperplane that represents ECC. Next the GA solution based on thosevectors is described.

4.1 Vectorial Search Space

Let A = {a1, a2, a3, . . .} be a field of real number, whose elements will be calledscalars. A vector space over the field A is a set V together two binary operation,vector addition and scalar multiplication. Two axiom are interesting for us: V isclosed under vector addition (if u, v ∈ V then u+v ∈ V ) and scalar multiplication

(if a ∈ R , v ∈ V, then av ∈ V ); see [14]. Hence, any power vector P (a) =[P1(a), P2(a), . . . , PN (a)]T in the vector space (P (a) ∈ V ) can be expressed as alinear combination of the vectors vk making up the vector space, as given by (7).From (7) it is clear that the elements of a vector vk corresponds to the on-linedispatchable active power generation.

Pi(a) =M∑

k=1

akvk,i i = 1, 2, . . . , N (7)

In the ED problem, the vector space needs to fulfill the ECC given by Equa-tion (3). So it is necessary to compute the scalars ak such that the

∑Ni=1 Pi(a) =

PT and∑N

i=1 vk,i = PT ∀k. Applying this condition to Equation (7) we obtainEquation (8).

N∑

i=1

M∑

k=1

akvk,i =M∑

k=1

ak

N∑

i=1

vk,i =N∑

i=1

Pi(a) (8)

M∑

k=1

akPT = PT ⇔M∑

k=1

ak = 1

From Equation (8) one can easily conclude that if the sum of the scalarsis one, the linear combination of the vector will be a vector that fulfills ECC.Furthermore, if the active power generation representing by the elements ofthese vectors are within limits, the set of CC are also satisfied. Hence, the wayof computing the vector collection vk determines that both constraints couldbe fulfill simultaneously during the iterative process. The proposed approach toachieve this goal is described below.

InRN , the set of CCs are a hypercube whose corners are defined by the activepower generation limits, and the set of ECCs constraint are a hyperplane. Themain idea is to compute a vector set over the intersection between the hypercubeand the hyperplane; the vector set can be created by a vector collection vk, whereeach vk corresponds to a vertex of the polygon that defines the intersection ofCC and ECC. This condition reduces the search space in one dimension. Eachvk lies at the intersection between an edge of the hypercube and the hyperplane;for this reason vk fulfills ECC – a desirable condition.

Given two vertexes, Pm = [Pm,1, . . . , Pm,N ]T and Pl = [Pl,1, . . . , Pl,N ]T in thehypercube CC, a parametric representation for edge Em,l(t) is given by Equation(9) with t ∈ [0, 1].

Em,l(t) = Pm + t (Pl − Pm) (9)

There exists a value t∗, for the intersection between the hyperplane and theedge; such

∑Ni=1 Em,l,i(t∗) = PT , so if we add the components of edge Em,l,i(t),

at position t∗, we can obtain an expression for t∗ given by (10).

N∑

i=1

Em,l,i(t∗) =N∑

i=1

Pm,i + t∗N∑

i=1

(Pl,i − Pm,i) = PT

t∗ =PT −

∑Ni=1 Pm,i∑N

i=1 (Pl,i − Pm,i)(10)

If t∗ ∈ [0, 1] exists an intersection between edge Em,l(t) and the hyperplaneECC. So the corresponding vector vk = Em,l(t∗) is accepted in the collectionvector V . The procedure to compute the vector collection V is presented in Algo-rithm 1. The complexity for Algorithm 1 is lineal with respect the edge number,but the edge number is exponential with respect the dimension (N × 2N−1) andnot necessarily exits a vector for each edge. Figure 1 shows the intersection ofECC (solid polygon) and CC (pipeline cube) in a three-dimension example. Thedoted lines correspond to vectors vk from Wood network data presented in Table2. Note that the search space instead of being a cube will be a plane in threedimensions, one less dimension. In this case the Generation units are N = 3, thenumber of vectors are M = 4 and the edges number are 8.

300400

500600

700800 200

400600

8001000

1200200

300

400

500

600

700

800

900

1000

1100

CC

ECC

Fig. 1. Intersection between the CC and ECC in 3D

4.2 GA algorithm

Haupt [4] describes the steps to minimize a continuous cost function using GA.In our case, the chromosome instead of having a power representation, has arepresentation given by Equation (11), where each gene ak is a floating-pointrandom number in [0, 1] and will represent the scalar field to computing Pi(a)by eq. (8). In order to fulfill the restriction imposed by Equation (8), the chro-mosome a is normalized using Equation (12). Given a chromosome a, the powervector P (a) is computed by Equation (7) and the fitness function is given byEquation (13), which combines Equations (1) and (2).

Algorithm 1 Vector searching SpaceGiven P min

i , P maxi and PT

- set k=1 and V = {∅}- Compute de coordinates Qm for each vertex of the Hypercube- Compute the edges Em,l(t) of the hypercube- For all edges Em,l(t)

* Compute t∗ by Equation (10)* if 0 ≤ t∗ ≤ 1

compute vk = Em,l(t∗) by Equation (9) and V ← V ∩ {vk}

k ← k + 1

a = [a1, a2, · · · , ak, · · · , aM ] (11)

ak ← ak∑Mj=1 aj

(12)

C =N∑

i=0

Nc∑

j=0

αi,jPji (a)

(13)

The initial population with size Npop, after applying a sorting algorithm, issplit in two halves. The best half is selected for mating (with Ngood = Npop/2members); the other half will be discarded and replaced by the offsprings of thebest half. Instead of using a random selection, a cost weighing function is used;the paring probability is computed using a normalized cost. The normalizedcost for each chromosome is computed subtracting the best cost of the discardedchromosomes to the cost of all the chromosomes in the mating pool Ci = Ci −CNgood+1, i ∈ [1, 2, . . . , Npop]. The mating probability for each chromosome πi

is computed using Equation (14), so the best individual has the highest matingprobability.

πi =Ci∑Ngood

j=1 Cj

i = 1, 2, . . . , Ngood (14)

Mating is carried out from a set of random couples. Given a couple of chro-mosomes p and m, a crossover point j is computed as a random integer numberin [1, M ]; the parameters around gene j are exchanged as shown by Equation(15). The genes at the j − th position are combined using Equation (16), whereβ is a floating-point random number in [0, 1].

O1 = [p1, · · · , o1,j , · · · ,mM ] O2 = [m1, · · · , o2,j , · · · , pM ] (15)o1,j = mj − β (mj − pj) o2,j = pj + β (mj − pj) (16)

Mutation is applied only to a percentage of the population, taking care of notmutating the best individual. So a random integer k is generated in [2, Ngood]

and then a random integer j is generated in the interval [1,M ]. The gene ak,j ,corresponding to the k − th member, is replaced by a random floating-pointnumber in [0, 1]. The new chromosomes generated by mating or mutation arenormalized applying Equation (12). The procedure proposed in our approach(called GA-V) is shown in Algorithm 2.

Algorithm 2 Genetic Algorithm GA-V- Compute the vector collection V using algorithm 1- Given Npop and the mutation percentage, set Ngood = Npop/4- Compute a initial population

for each population memberCreate a chromosome a = [0, 0, ..., ] similar to (11)Change randomly only M ′ genes with M ′ ∈ [1, M ]Normalize the chromosome using (12)Evaluate the Power vector P (a) by (7)Given P (a) and coefficients αi evaluate the fitness functionby eq. (13) and sort by fitnessReduce the population size Npop ← Npop/2

- Repeat Ngen

* Compute the Mating probability using (14)* Create random couples using the Mating probability and thebest population half* For each couple compute the offsprings by (15, 16)* Replace the worst population half by the offsprings* Mutate randomly the k − th chromosome at the j − th gene* Normalize all the chromosomes by (12)* Evaluate the fitness function (13) and sort by fitness

- For the best population member compute the power vector using (7)

5 Study cases

The total power demanded for each network is 300.5576 KW, 2500 KW and1443.4 KW, respectively; other data for each network are presented in Table1. The first network contains five generators with a production cost given bya quadratic function cost (Nc = 2) whose coefficients are presented in column2. The maximum and minimum limits for active power generation are given incolumns 3 and 4, respectively. On the other hand, three generators are connectedto the second and to the third network and the production cost is representedby a cubic cost function (Nc = 3). Data for generators of each network are alsogiven in Table 1 in the same manner as indicated for the first network. Theproposed GA approach (GA-V) was implemented using Java. The results of itsapplication were obtained considering a population size of Npop = 1000, number

Table 1. System data

Network Polynomial coefficientsGen. limitsPmax Pmin

IEEE 14

α1 = [0.02e− 3, 0.003, 0.01]T 80.0 10.0α2 = [0.02e− 3, 0.003, 0.01]T 60.0 10.0α3 = [0.02e− 3, 0.003, 0.01]T 60.0 10.0α4 = [0.02e− 3, 0.003, 0.01]T 60.0 10.0α5 = [0.02e− 3, 0.003, 0.01]T 80.0 10.0

Woodα1 = [749.55, 6.95, 9.68e− 4, 1.270e− 7]T 800 320

α2 = [1285.0, 7.051, 7.375e− 4, 6.453e− 8]T 1200 300α3 = [1531.0, 6.531, 1.04e− 3, 9.98e− 8]T 1100 275

Wongα1 = [11.200, 5.10238,−2.64290e− 3, 3.33333e− 6]T 500 100

α2 = [−632.000, 13.01000,−3.05714e− 2, 3.33330e− 5]T 500 100α3 = [147.144, 4.28997, 3.08450e− 4,−1.76770e− 7]T 1000 200

Table 2. Searching space Vector computed by algorithm 1

Network Active Power Vectors in KW

IEEE 14

v1 = [40.5576, 60.0000, 60.0000, 60.0000, 80.0000]T

v2 = [80.0000, 20.5576, 60.0000, 60.0000, 80.0000]T

v3 = [80.0000, 60.0000, 20.5576, 60.0000, 80.0000]T

v4 = [80.0000, 60.0000, 60.0000, 20.5576, 80.0000]T

v5 = [80.0000, 60.0000, 60.0000, 60.0000, 40.5576]T

Wood

v1 = [320.0, 1080.0, 1100.0]T

v2 = [800.0, 600.0, 1100.0]T

v3 = [320.0, 1200.0, 980.0]T

v4 = [800.0, 1200.0, 500.0]T

Wong

v1 = [343.40, 100.00, 1000.00]T

v2 = [100.00, 343.40, 1000.00]T

v3 = [100.00, 500.00, 843.40]T

v4 = [500.00, 500.00, 443.40]T

v5 = [500.00, 100.00, 843.40]T

Table 3. Comparative Results.

Network Algorithm Demand Generation Time Active PowerCost (seg.) Vector Solution

IEEE 14[13] 300.5576 181.5724 -

[60.2788 60.0000 . . .60.0000 60.0000 60.2788]

GA-V 300.5576 181.5724 4.276[60.2789 59.9999 . . .

59.9999 59.9997 60.2789]

Wood[7] 2500.1000 22730.21669 - [726.9000 912.8000 860.4000]

GA-V 2500.0000 22729.32458 2.814 [725.0078 910.1251 864.8670]

Wong[12] 1462.4480 6639.50400 - [376.1226 100.0521 986.2728]

GA-V 1443.4000 6552.23790 2.842 [343.3980 100.0415 999.9604]

of generations Ngen = 1000, and mutation probability of 10%. These parameterswere chosen in order to have a good performance for GA in the three cases.

The computed vectors that define the search space for these networks arepresented in Table 2. It must be observed that each vector has N−1 active powergenerations at one of their limits, whilst the remaining is within limits. By wayof example, for vector v1 of the IEEE 14 system containing five generators, P2,P3, P4 and P5 generate at their maximum limit. On the other hand, generator P1

generates 40.5576 MWs. The same observation applies to each vector vk givenin this table. Lastly, each vector satisfies

∑Nk=1 vk = PT , such that it is in RN ;

however in general the number of vectors vk differs of the number or Generationunits, as it is observed in table 2.

5.1 Numerical Results

A comparison of the results obtained by the proposed approach GA-V, withrespect to those reported by other authors are shown in Table 3. This table showsthe algorithm used, the power demand, the computed generation Cost, the timerequired to compute the solution using GA-V (Algorithm 1 and 2) and the finalgenerator active power vector. We select the IEEE14 and Wong networks becausetheir solution are obtained by an algorithm based on Simulated Annealing in[13] and [12]. Results obtained for the IEEE14 system are identical to thosereported by Ongsakul [13]. For the second and third networks, GA-V yieldsbetter solutions than those reported in [12] and [7], respectively. Additionally tothe improvement of the generation cost, it must be pointed out that the solutionsreported by Wood and Wong do not fulfill the energy conservation equations,exactly. This is a necessary condition to a feasible steady state operation of anyelectric power system.

A comparison in computational time is not possible because it was not re-ported in [7], [12] and [13]. However, computational times for the networks solu-tion based on the proposed approach is also reported in Table 3. This informationprovides an idea of the algorithm complexity.

6 Conclusion

In this work, a new strategy called vector search space has been proposed forstochastic search of the active power dispatch that produce the lowest generationcost. This strategy guarantees that generation dispatch fulfill the restrictions im-posed by the ED problem, reducing the search space in one dimension, thereforereducing the convergence time.

One drawback of GA-V is that it needs to compute the vectors that intersectthe edges of the hypercube with the hyperplane. In RN the number of edges isN × 2N−1, so its complexity is exponential. In future works we will try to devisea strategy to compute the vector collection in polynomial time.

References

1. Stevenson, W.D.: Elements of Power System Analysis. Mc Graw Hill (1982)2. Momoh, J.A.: Electric Power System Applications of Optimization. CRC Press

(2005)3. Pohlheim, H.: Evolutionary Algorithms. Springer-Verlag (2003)4. Haupt, R.L., Haupt, S.E.: Practical Genetic Algorithms. second edn. John Wiley

and Son, Inc, New York, N.Y (2004)5. Goldberg, D.E.: Genetic Algorithms. Addison-Wesley, New York, N.Y (1989)6. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Verlag, New York,

N.Y (1999)7. Wood, A., Wollenberg, B.: Power Generation, Operation, and Control. John Wiley

and Sons Inc (1984)8. Sheble, G.B., Brittig, K.: Refined genetic algorithm-economic dispatch example.

Power Systems, IEEE Transactions on 10, (1995) 117–1249. Orero, S.O., Irving, M.R.: Economic dispatch of generators with prohibited op-

erating zones : A genetic algorithm approach. In: IEE proceedings. Generation,transmission and distribution. Volume 143., Institution of Electrical Engineers,(1996) 529–534

10. Song, Y., Chou, C.: Advanced engineered-conditioning genetic approach to powereconomic dispatch. In: Generation, Transmission and Distribution, IEE Proceed-ings. Volume 144. (1997) 285–292

11. Aarts, E., Korst, J.: Simulated Annealing and Boltzmann Machines: A StochasticApproach to Combinatorial Optimization and Neural Computing. John Wiley(1991)

12. Wong, K., Fung, C.: Simulated annealing based economic dispatch algorithm.In: Generation, Transmission and Distribution, IEE Proceedings C. Volume 140.(1993) 509 – 515

13. Ongsakul, W., Ruangpayoongsak, N.: Constrained dynamic economic dispatchby simulatedannealing/genetic algorithms. In: Power Industry Computer Applica-tions, 2001. PICA 2001. Innovative Computing for Power - Electric Energy Meetsthe Market. 22nd IEEE Power Engineering Society International Conference on.(2001) 207 – 212

14. Strang, G.: Linear Algebra and Its Applications. Third edn. Thomson Learning,USA (1988)