Solving Multi-Contingency Transient Stability Constrained Optimal Power Flow Problems with an Improved GA K.Y. Chan, S.H. Ling, K.W. Chan, H.H.C. Iu and G.T.Y. Pong 2
Abstract— In this paper, an improved genetic algorithm has been proposed for solving multi-contingency transient stability constrained optimal power flow (MC-TSCOPF) problems. The MC-TSCOPF problem is formulated as an extended optimal power flow (OPF) with additional generator rotor angle constraints and is converted into an unconstrained optimization problem, which is suitable for genetic algorithms to deal with, using a penalty function. The improved genetic algorithm is proposed by incorporating an orthogonal design in exploring solution spaces. A case study indicates that the improved genetic algorithm outperforms the existing genetic algorithm-based method in terms of robustness of solutions and the convergence speed while the solution quality can be kept.
I. INTRODUCTION OPTIMAL power flow (OPF) [1] aims to achieve an optimal solution of a specific objective function, such as fuel cost, network loss, by setting the system control variables, while satisfying all constraints imposed by operational and physical limitations of the power system. Transient stability constraints are usually excluded in the conventional OPF. However, as the transmission system is becoming more and more stability-limited in the competitive environment and the cost of losing synchronous through a transient instability is extremely high, transient stability has now become one of the main concerns in the operations of modern power systems, and there is a pressing need for its inclusion into the OPF. Transient stability is the ability of power system to maintain synchronous operation of machines when it is subject to a set of credible contingencies. In this paper, a practical solution for the problem of multi-contingency transient stability constrained optimal power flow (MC-TSCOPF) is considered. In solving MC-TSCOPF, the difficulty mainly comes from the nonlinear differential-algebraic equations which describe the transient stability constraints of the power system. Gan et al. [2] converted the differential-algebraic equations into the numerically equivalent algebraic equations and then integrated them into the standard OPF formulation. Manuscript received March 15, 2007. K.Y. Chan is with Dept. of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Ham, Hong Kong (email: chankityan18112005@ yahoo.com) S.H. Ling and H.H.C. Iu are with School of Electrical, Electronic and Computer, University of Western Australia, WA6009, Australia (e-mail:
[email protected] and
[email protected]). K.W. Chan is with Dept. of Electrical Engineering, The Hong Kong Polytechnic University, Hung Ham, Hong Kong (email:
[email protected]). G.T.Y. Pong is with Dept. of Applied Mathematics, The Hong Kong Polytechnic University, Hung Ham, Hong Kong (email:
[email protected]).
Chen et al. [3] converted the problem into an optimization problem in the Euclidean space via a constraint transcription. Xia et al. [4] proposed to use a simple criterion to eliminate the explicit differential-algebraic constraints so as to convert the problem into a conventional OPF problem. Both nonlinear programming and interior-point method were proposed to solve the MC-TSCOPF [2-6]. Common limitations for these methods are that (1) they rely on convexity to obtain the global optimum solution and thus they are forced to simplify relationships in order to ensure convexity [7], and (2) they are heavily tied up with the power system models and hence have difficulties to deal with large scale power systems with complex non-linear models. Generally, the OPF problem is non-convex, especially when valve-point loading effects of thermal generators have to be included or FACTS devices are included on the network, and as a result many local minima may exist. For solving OPF problems, genetic algorithms have been applied successfully on the IEEE 30-bus, IEEE 3-Area RTS96 and IEEE 118-bus systems [8-15]. For solving the MC-TSCOPF problem, our recent work showed that the genetic algorithm could achieve satisfactory results [16]. Since transient stability is simulated and assessed via time-domain simulation, the problem of power system modeling is decoupled from the basic genetic algorithm and the proposed MC-TSCOPF solution approach can be applied to any practical power systems. However, the main drawback of genetic algorithms is that they often have difficulties in reaching a reasonable solution for problems with high dimensions. Slow convergence speed and incapable of reaching consistent solutions in multiple runs are other common pitfalls of genetic algorithms. In this paper, a novel orthogonal array based crossover operator (OC) [17], which is better than standard crossover on solving parametrical benchmark problems with high dimensions, is proposed to be embedded on the genetic algorithm for solving the challenging MC-TSCOPF problem. This enhanced genetic algorithm is referenced as the orthogonal array based genetic algorithm (OGA) in this paper. OGA is an efficient general-purpose algorithm capable of solving large parameter optimization problems. In OC, the chromosomes of the children are formed from the best combinations of the better genes of their parents, rather than by random combinations of genes in parents as in conventional crossovers. The better genes are chosen by a systematic reasoning approach for evaluating the contribution of individual genes based on orthogonal arrays. Thorough theoretical analysis and experimental studies for the superiority of the crossover with the use of orthogonal arrays have been done in [17].
2901 c 1-4244-1340-0/07$25.00 2007 IEEE
II. MC-TSCOPF PROBLEM FORMULATION MC-TSCOPF is mathematically defined as min f ( x , y ) s.t. g ( x, y ) = 0
(1) (2)
H ( x, y ) ≤ 0
(3)
U ( x ( t ) , y ) ≤ 0, t ∈ T
(4)
where x ( t ) is a dependent vector which includes active and reactive power of the swing bus, voltage angle and reactive power of generator buses, and voltage angle and magnitude of load buses. T = ¬ªt0 , tcl ) ∪ (tcl , te ¼º is the transient period from the occurrence of the disturbance at time t0 to the clearing time tcl and then to the ending time te . x represents the initial value of x ( t ) at t = 0 . y is a control vector which includes active power and voltage magnitude of generator buses, voltage angle and magnitude of the swing bus, and tap position of LTCs. f can be expressed as the total generation cost, total network loss, corridor transfer power, total cost of compensation, etc. g is the set of equality constraints which are usually the power flow constraints for a specified operating condition. H is inequality constraints for the steady-state security limits like bus voltage magnitude limits, generator power limits, thermal limits for transmission lines, etc. The dynamic security constraints set U is infinite in the functional space. For more details, the reader is referred to Mo et al. [14]. Since the equality constraints g are imposed implicitly by the power flow calculation incorporated within the algorithm and also the inequality constraints H is directly satisfied by the GA, the MC-TSCOPF can be formulated as a penalty function problem: (5) F ( x ) = min { f ( x, y) + β max[U ( x ( t ) , y )2 ]} Generally, transient stability constraints can be considered as hard constraints that should not be violated whilst the static constraints are soft in nature that slight violation can be tolerant. Compared with other constraint handling approaches [8, 19], penalty function offers a simple and flexible strategy to effectively deal with mixed hard and soft constraints. In addition, there is no need to have separate penalty factors for each type of constraints. In (5), any transient instability will introduce a huge angle deviation and thus produce a large violation and thus discrimination even though the same penalty factor is used for all type of violations. Typically, β = 1000 works very well in most power systems [14].
The first step of both SGA and OGA is to randomly generate strings representing the population. In (1)-(4), all generator active power PGi , generator bus voltage V j and transformer tap position Tk are all real numbers, so real-coding representation of strings is chosen in both SGA and OGA. A population of strings, represented by PGi , V j and Tk is created randomly, where i = 1,..., N G , j = 1, ..., N B and k = 1, ..., N T
.
The
string
can
be
expressed
as
ª¬ P1 , P2 , ..., PN , V1 , V2 ,..., VN , T1 , T2 ,..., TN º¼ , where N G is the G
B
T
number of generators, N B is the number of bus bars, and N T is the number of transformers. Hence the number of genes in each string is equal to N G + N B + N T . As an illustration, a representation of string is shown in Figure 1. The lower and upper bounds of the genes are based on the generation capacity constraints (3). Step 2: Fitness evaluation The fitness function is used to evaluate the fitness of all strings in the GA. It is defined as (6), which is used to evaluate the power flow solution (1) and the static violations (2)-(3). For transient stability violation evaluation (4), transient stability simulation is used to produce the generator rotor responses. The maximum rotor angle deviation from the centre of inertia, among all generators and contingencies, is then used to compute a transient stability penalty using (5). The fitness value is defined as follows: 1 (6) fitness = 1 + F ( x ) where F ( x ) is defined in (5). The objective is to maximize the fitness value (6) using the genetic algorithms. Step 3: Selection The approach of roulette-wheel is used for selecting strings in both SGA and OGA. This is one of the most common selection methods used for selecting strings to perform reproduction operations [20], in which the selection of strings is completely based on the strings’ fitness, unlike other selection approaches like rank based selection [21] or tournament selection [22] that their selective parameters need to be adjusted by the trial and error method. In the approach of roulette-wheel, the fitness fit j of the j
th
string is used to assign a value for selection th
probability prob j to the j string: III. GA-BASED MC-TSCOPF SOLUTION Basically both standard genetic algorithm SGA and OGA are identical except the crossover operator embedded on them. While the standard crossover operator discussed in Step 4 below is used in SGA, the orthogonal array based crossover operator (OC) discussed in Section 4 is used in OGA. The detailed procedures of both SGA and OGA are discussed below. Step 1: Randomly generated strings
2902
prob j =
fit j
(7)
Popsize
¦
fit j
j =1
where Popsize is the population size of the genetic algorithm. th
The larger the fit j is, the higher chance the j string can be selected. Step 4: Crossover
2007 IEEE Congress on Evolutionary Computation (CEC 2007)
In this research, two crossover operators are developed for use in SGA and OGA. The classical crossover operator, discrete crossover operator [23], is used in SGA. It can produce a new string around and between the control variables, PGi , V j , Tk , of the two selected parent strings. A child string ª¬ PGi , V j , Tk º¼ with i = 1,..., N G , j = 1, ..., N B and 3
3
3
k = 1, ..., NT is produced according to the following rule by the discrete crossover operator:
3 3 3 1 1 1 2 2 2 1 1 1 ¬ª PGi , V j , Tk ¼º = ¬ª PGi , V j , Tk ¼º + α {¬ª PGi , V j , Tk ¼º − ¬ª PGi , V j , Tk ¼º}
(8) where α is a scaling factor chosen uniformly at random over the
interval
[ −0.25, 1.25]
,
and
ª¬ PGi1 , V j1 , Tk 1 º¼ and
ª¬ PGi 2 , V j 2 , Tk 2 º¼ are the two selected parent strings. Values of control variables in a new string are the result of combining values of control variables in the parent strings according to (14) with a scaling factor α chosen for each gene. In geometric terms, intermediate crossover is capable of producing new values of control variables within a slightly larger hypercube than that defined by the parent strings but constrained by a range of the scaling factor α . On the other hand, the OC, which will be detailed in Section IV, is used in OGA. Step 5: Mutation The mutation operator of Gaussian perturbation [23] of individual parameters was used in both SGA and OGA. For example, the generator bus voltage V j is selected to be mutated. After performing mutation, its value becomes V j ' : V j ' = V j + MutMx× R j × δ
(9)
instead of the 81 ( 34 ) design parameter combinations required for a full factorial design. Therefore 81-9=72 experiments are not needed to be carried out. Due to the orthogonal nature, orthogonal arrays can separate out the effect of each parameter if strong interaction effects do not exist between the parameters. The main effect of each parameter for each level is calculated by taking the main effect of the orthogonal array for a parameter at a given level. The best combination of the parameters is obtained by considering the main effect in each parameter. The efficiency of the crossover operator has been improved in solving parametrical benchmark problems by integrating with an orthogonal array [17]. In this approach, orthogonal array is integrated into the classical crossover operator so that two parents can be used to generate a small but representative set of sampling points in the search space as children. Experimental results show that this aids the evolutionary algorithm for solving the parametrical problems with high dimensions in both convergence speed and solution quality. Since MC-TSCOPF problems are in high dimensions, the orthogonal array based crossover operator (OC) can be more effective than the standard crossover operator. In OC, the chromosomes used for solving the MC-TSCOPF are in real-coded representation. Like the classical crossover, two parents p1 and p2 are selected randomly from the population. The hypercuboid defined by the two parents is then discretized. Both p1 and p2 are divided into l genes i.e.: p1 = ( p1,1 , p1,2 ,..., p1,l ) and p 2 = ( p2 ,1 , p2 , 2 , ..., p2 ,l ) . The lower and upper levels of the orthogonal design are defined as Level (1) and Level ( Q ) respectively, where Level ( 1) = p1
where MutMx = +1 or -1 with equal probability; R j = 0.5 ×
and Level ( Q ) = p 2 . Q is the number of levels of the
the boundary of the generator bus voltage V j ; δ = a value in
orthogonal array. Each gene of Level (1) and Level ( Q ) is quantified into Q levels such that the difference between any two successive levels is the same. The i th level is denoted to
the range [0,1] for shrinking the mutation range based on Gaussian perturbation. Step 6: Termination condition The genetic algorithm goes to Step 2 to undergo the next evolutionary generation until the pre-defined number of generations is reached.
be Level ( i ) = [ ȕi ,1 , ȕi ,2 , ... ȕi ,l ] , where i=1,2,3,…,Q and ȕi, j
is denoted as:
IV. ORTHOGONAL ARRAY BASED CROSSOVER OPERATOR Orthogonal design is a systematic and efficient approach for engineers to search near optimum settings of parameters in experimental design [24-26]. A typical orthogonal array L ( 3 ) is shown in Table 1. In the orthogonal array L ( 3 ) , 4
ȕi , j
p1, j § ° = ® p + ( i −1)⋅¨ 1, j ¨ © ° ¯ p2 , j
p1, j − p2 , j · ¸ Q −1 ¸¹
4
9
4
9
for 2 ≤ i ≤ Q −1 and 1≤ j ≤ l for i = Q and 1≤ j ≤ l
(10)
9
each element in the top row represents an independent input parameter, and each element in the left-hand column represents an experimental run. The elements at the intersections indicate the level settings that apply to that input parameter for that experimental run. Using an orthogonal array L ( 3 ) means that only 9 experiments are carried out
for i =1 and 1≤ j ≤ l
After quantifying Level (1) and Level ( Q ) , the Q levels are sampled into M potential offspring based on the combinations of the M rows of factor levels in orthogonal
( ).
array L M Q
N
Specifically, we randomly generate N-1
integers k1 , k 2 ,…, k N −1 such that 1< k1 < k 2
