ESSENTIAL ISSUES ON SOLVING OPTIMAL POWER FLOW PROBLEMS USING SOFT-COMPUTING

. Optimal power ﬂow (OPF) problems are important optimization problems in power systems which aim to minimize the operation cost of generators so that the load demand can be met and the loadings are within the feasible operating regions of the generators. This brief paper emphasizes two essential issues related to solving the OPF problems and which are rarely addressed in recent research into power systems: 1) the necessity to validate operational constraints on OPF, which determine the feasibility of power systems designed for the OPF problems; and 2) and the necessity to develop conventional methods for solving OPF problems which can be more eﬀective than the commonly-used heuristic methods.


1.
Introduction. For the past few decades, modern control centers of power systems have been equipped with computational tools to perform complex and extensive off-line studies in order to provide electrical power with minimum costs and minimum power interruptions, since the amount of power supply is more demanding and more stable supplies are required in the competitive environment [23]. It is necessary to maintain the power systems operating at a minimum cost, and to ensure a satisfactory power supply to all users. This can be transformed into the commonly-known optimal power flow (OPF) problem which aims to determine optimal control variables for an efficient and robust power system [2,3,7,10,24]. The OPF problems generally consist of a cost function and a set of constraint functions. The cost function aims to achieve an optimal outcome for a specific objective such as fuel cost or network loss, by setting the system control variables. The constraint functions are intended to ensure that the generated power supply is adequate for all users and to compensate for the power loses due to the transmissions, while satisfying all constraints imposed by operational and physical limitations of the power system. However, research has mostly focused on the minimization of the cost functions, but very little attention has been paid to satisfying the constraint functions which are introduced to ensure the generated power satisfactorily meets the user demands and the power losses [1,2,3,5,7,10,11,12,21,24]. This can lead to an undesirable situation, where a small generation cost can be achieved but unsustainable power is produced. Although this is an essential issue to consider when solving the OPF problems, it is rarely addressed. Also, the OPF problems are generally non-convex due to the presence of the valve-point loading effects in generators and the involvement of the flexible alternating current transmission systems [24]. This is why the heuristic algorithms such as evolutionary algorithms have commonly been used to solve the OPF problems [1,3,5,12,21,24], since such approaches can be easily applied when solving difficult optimization problems [9]. More recent research also showed that particle swarm optimization (PSO) [4] is a more robust and efficient method than genetic algorithms when solving the OPF problems [2,7,10,11]. They use heuristic operators in order to obtain the global optimal solution, since conventional gradient-based methods can find only a local optimal solution [24]. However, heuristic methods are also local search methods with no guarantee that the solution obtained will be either a global optimal solution or even a local optimal solution [17,19]. Hence, they might not be the most suitable approaches for solving the OPF problems. Conventional gradient-based methods could perform more effectively and efficiently in solving OPF problems. However, recent research on solving the OPF problems tends to focus on the development of advanced heuristic methods, and research on the development of local search methods is rarely conducted. The development of local search methods is another essential issue when solving the OPF problems. This paper aims to discuss the two essential issues that were rarely addressed in the recent research on solving the OPF problems. Section 2 provides an overview of OPF problems and one particular OPF problem, namely the economic dispatch (ED) problem, is introduced. Then, an effective local search method namely Sequential Quadratic Programming (SQP) with active set strategy [14] is applied to solve the ED problem. Section 3 presents and compares the results obtained by the applied local search method and the heuristic method. A conclusion is given in Section 4.

2.
Overview of optimal power flow problems. The OPF problem aims to optimize the performance of the steady state power system with respect to a cost function , which is the total generation cost for active and reactive power dispatch. It could represent the total generation cost or the total network loss.
Generally, an OPF problem can be formulated as [8]: x n ] is the OPF decision vector with its components being those variables such as active/reactive power of swing buses, voltage angle/magnitude of swing and load buses, tap position of LTCs. In (1a), f is the cost function which is continuously differentiable with respect to all its arguments. g i , i = 1, . . . , N E and h i , i = 1, . . . , N i are continuously differentiable with respect to all their arguments. The equality constraints (1b) are the nodal power constraints which are the operational constraints on the specified power flow conditions, such as the requirements on the load demands and system losses. The inequality constraints (1c) are the bounds of the decision variables x = [x 1 , . . . , x n ] .
As presented in [2, 3, 5, 7, 10, 12, 21, 24, 25, 26, ?], the penalty function method is used to approximate Problem (1) as formulated by the following optimization problem: where α is a sufficiently large penalty parameter, and J α (x) is known as the augmented cost function. Problem (2) is, in general, non-convex due to the presence of the valve-point loading effects in generators and the involvement of the flexible alternating current transmission systems.
2.1. Economic dispatch problem. Consider a common OPF problem, or equivalently called an economic dispatch (ED) problem, in the form of Problem (1) [8], where . . , N B and T k , k = 1, . . . , N T , represent the real and reactive power generations, bus voltage magnitudes and angles, and transformer tap-settings, where N G , N B and N T are the number of generators, buses and transformers, respectively.
The inequality constraints (1c) are specified by They can be expressed as:

KIT YAN CHAN, CHANGJUN YU, KOK LAY TEO AND SVEN NORDHOLM
The cost function (1a) is given by where C t is the generation cost; a i , b i , c i , e i and f i are the cost coefficients of the ith generator. The equality constraints (1b) are specified below by the following equality constraints which represent the power generations, power loads and power losses through transmission: where i = 1, 2, . . . , N B ; P Di (or P Gi ) and Q Di (or Q Gi ) are the real and reactive power loads (or generations) at the bus i, respectively; Y ij and θ ij are the admittance magnitude and angle between the bus i and the bus j, which are varied with respect to T k .
For the corresponding version of Problem (2), f is specified by equation (6).
The objective of the cost function (6) is to minimize the generation cost, C t , by optimizing the generated real powers, P G1 , . . . , and P GN G . The equality constraint (7a) aims to ensure that all the generated real power, P G1 , . . . and P GN G , can adequately supply all the demands, P D1 , . . ., and P DN G , and compensate for the transmission losses, while the equality constraint (7b) aims to ensure that the generated reactive powers are satisfactory. Therefore, it is necessary to find a feasible solution in order to satisfy both the equality constraints (7a) and 7b). Although some infeasible solutions may achieve small generation costs, only unsustainable power can be produced since the solution cannot satisfy the equality constraints (7a) and (7b) and is therefore infeasible.
3. Proposed local search algorithm. To solve the augmented cost function J λ (x) in (2), the conventional gradient-based methods could be used. However, the augmented cost function J λ (x) is non-differentiable due to the presence of the terms, e i |sin(f i P Gi )|, i = 1, . . . , N G in (6). Therefore, it is necessary to transform the non-differentiable terms into a differentiable estimate. Based on the approach on page 185 of [20], the terms, | sin(f i P Gi )| with i = 1, . . . , N G , can be approximated by L i,ρ (sin(f i P Gi )), where Using the approximation formulated in (8), the augmented cost function (2) can be transformed as the following corresponding approximate augmented cost function: where It is clear that J(x, α, β, ρ) is differentiable and its gradient can be readily obtained. Thus, by making the penalty parameters α and β sufficiently large, and the smoothing parameter ρ sufficiently small, this approximate augmented cost function J(x, α, β, ρ) is minimized subject to the inequality constraints specified by (5) and the equality constraints specified by (7). Problem (9) can be solved by using the sequential quadratic programming algorithm with active set strategy (namely SQP algorithm), which has attracted the interest of many mathematicians and engineers when solving real-world problems involving nonlinear constrained optimization [14,15,16]. To perform the algorithm, Problem (9) is formulated as the following quadratic programming sub-problem, namely P k by (10) as, where x k is the k−th iteration of the decision variable represented by (3); λ k = [α, β] is the associated multipliers in (9); H k is the positive-definite approximation of the Hessian of the Lagrangian function; and N I and N E are the numbers of inequality and equality constraints represented by (5) and (7) respectively. In (10), d k is the solution of Problem P k , which is solved based on the active set strategy algorithm (detailed on pages 427 to 434 in [18]). With d k , the new iterates, x k+1 , λ k+1 and H k+1 , can be determined by where σ k ∈ (0, 1] is the step-length parameter andσ k is the corresponding multipliers. Hence, P k can be constructed and be solved consequently. By repeating this process, the original constrained nonlinear optimization problem can be efficiently solved.
The following SQP algorithm is proposed to solve P k :

SQP algorithm
Step 1: Set k = 0; Choose a starting point and a positive definite matrix H 0 for the sub-problem P 0 .
Step 2: Use the active set strategy algorithm to obtain d k by solving the subproblem P j .

KIT YAN CHAN, CHANGJUN YU, KOK LAY TEO AND SVEN NORDHOLM
Step 3: If d k = 0, x k is the KKT point and go to Step 7 Step 4: Update x k+1 based on (11).
Step 7: Return x k as the optimal solution namely x opt .
The optimization procedure is carried out by the optimization toolbox in MAT-LAB. The penalty parameter and in Problem (9) are set as 10 6 . When the optimal solution of Problem (9), x opt , is obtained, we can substitute x opt into the original augmented cost function (6) to obtain the actual optimal cost (i.e. the generation cost). Also, we can substitute x opt to the equality constraints (7a) and (7b) to check whether or not x opt is a feasible solution (i.e. to check whether sustainable power can be generated).
4. Experimental results. The effectiveness of the proposed SQP algorithm was evaluated by solving the OPF problems which are involved in the design of small, medium and large scale power systems, namely WSCC 9 bus-system (with 3 generators), IEEE 30 bus-system (with 6 generators) and Poly-system (a power system with 36 generators). The numbers of decision variables in WSCC 9 bus-system, IEEE 30 bus-system and Poly-system are 26, 76 and 304 respectively. The results obtained by the SQP algorithm were compared with those obtained by an effective heuristic method, namely advanced PSO [2], which has been developed to solve OPF problems and non-convex parametrical problems.
The advanced PSO is intended to overcome the limitation of the standard PSO [6] which usually converges to a near-optimal solution. It is integrated with the genetic mutation in order to help to obtain the optimum. Similar to the standard PSO algorithm, the advanced PSO starts by randomly generating a swarm of particles, and moves the particle positions iteratively based on the operations of genetic mutation and swarm movement. [2] has demonstrated that better results can be obtained when solving the OPF problems and some non-convex parametrical problems, when comparing the standard PSO with the other evolutionary algorithms. Therefore, the advanced PSO is used in this research for comparison with the proposed SQP algorithm.
Both the proposed SQP algorithm and advanced PSO were developed using the Matlab R2011b, whereby the Matlab subroutine fmincon is used to determine the optimum of the power flow problems. The mechanism and parameters used for the advanced PSO are identical to those used in [2], and Problem (9) was used as the fitness function on the advanced PSO. For the numbers of computational evaluations, both the advanced PSO and the SQP algorithm used 10000 computational evaluations for solving the OPF on the two smaller scaled systems, WSCC 9 bus-system and the IEEE 30 bus-system. They both used 50000 computational evaluations for the larger scale system, China-system. Thirty runs were performed using both methods for each power system; the initial swarm used in the advanced PSO and the initial starting point used in the SQP algorithm were generated randomly for each run. Figures 2a, 2b and 2c illustrate the averaged results obtained by both the advanced PSO and the SQP algorithm among the 30 runs. Figure 1a illustrates the averaged results for the augmented cost J λ (x) formulated in Problem (2a). Figures  1b and 1c illustrate the generation cost f ρ (x) and the sum of equality constraint values Ng i=1 g i (x) 2 formulated in Problem (9), respectively. Figure 1a shows that the averaged augmented costs obtained by the SQP algorithm are smaller than those obtained by the advanced PSO. Figure 1b shows that the SQP algorithm outperforms the advanced PSO for which smaller generation cost among all tested systems can be obtained except IEEE 30-bus system. Also, Figure 1c shows that the SQP algorithm outperforms the advanced PSO on handling the equality constraints for all the systems where the sum of the equality constraint values obtained by the SQP algorithm are smaller.
These numerical results show the effectiveness of the SQP algorithm, where better power systems requiring small generation costs are produced. Hence, the power systems which are optimized by the SQP algorithm have lower fuel costs. Also, the equality constraints can be met by the SQP algorithm, but not by the advanced PSO. Addressing the equality constraints is important since it ensures that the generated power can adequately supply all users and compensate the power loss due to the transmissions. This can lead to an undesirable situation if those equality constraints cannot be satisfied. However, the advanced PSO performed poorly when addressing this.
Furthermore, Figures 2a, 2b and 2c illustrate the variances obtained by both the advanced PSO and the SQP algorithm for the 30 runs. The three figures illustrate the variances for the augmented cost J λ (x) in Problem (2a), the generation cost f ρ (x) in Problem (9) and the sum of equality constraint values Ng i=1 g i (x) 2 in Problem (9). They show that, in general, the variances obtained by the SQP algorithm are smaller than those obtained by the advanced PSO. Therefore, these results indicate that the SQP algorithm can produce more robust solution quality compared with the advanced PSO.
The t-test is then used to evaluate the significance of the two algorithms, Advanced PSO and SQP algorithm for the three systems in terms of the total cost, the generation cost and the constraints. Table 1