Abstract―This paper approaches the issue of finding multiple optima in Optimal Power Flow (OPF) problems using a modified Artificial Immune System (AIS) algorithm. The original AIS algorithm is a methodology based on natural immune systems and intends to capture three major immunological principles: hypermutation, receptor edition and cellular memory. These characteristics enable the assessment of multiple optima using local and global search. The proposed algorithm improves the original AIS methodology by enhancing the hypermutation process (HP) and applying another immunological principle: the maturation control. The new HP uses numerical information gathered during the convergence process to reduce the number of clones, while the maturation control is responsible for eliminating redundant antibodies, reducing the initial population. Finally, to ensure optimality, the algorithm uses an approach based on the augmented Lagrangian function to find the Karush-Kuhn-Tucker (KKT) conditions. Several case results obtained with different systems illustrate the proposed AIS-based approach.

Index Terms―Artificial immune system, clustering technique, gradient-based algorithm, multiple solutions, non-linear optimization, optimal power flow.

I. INTRODUCTION

PTIMAL power flow problems appear as non-linear, non-convex and large-scale systems, involving several sets of

continuous and discrete variables. This diversity makes the optimization field to be divided, according to the solution space, convexity, and types of control variables, into several broad categories such as linear, non-linear, combinatorial, dynamic, probabilistic, and others. These fields can be further divided into two major groups: numerical and intelligence-based. Regarding numerical-based methodologies [1], [2], it is shown in [3] a comparison among three interior-point-based (IP) methods, primal-dual (PD), predictor-corrector (PC), and multiple-centrality-correction (MCC). The results show good performance for all methods, especially the MCC, although it needs accurate parametric adjustments to improve the convergence performance. These conventional methods have presented good results, though some drawbacks have appeared in actual large systems applications. For instance, it is shown in [4] that rounding off continuous variables may provide higher costs than the optimal solution. As a solution to this problem, it introduces an improvement by handling these variables through penalty functions.

Artificial intelligence-based methods are interesting alternatives for dealing with the previously discussed hurdles [5]-[7]. Several advantages can be linked to these methods: the software complexity is simple, they are able to mix integer and continuous variables, some of them are able to find more than one solution and present very appealing computational performance, especially if distributed computation is taken

into account. Even though these methodologies use a large number of individuals from a population to solve problems, they can be easily parallelized as shown in [8], therefore, decreasing the computational time. The problem with many of these methodologies is the difficulty in establishing the Karush-Kuhn-Tucker conditions at the end of the optimization process.

The AIS is based on the biological principle of bodies’ immune systems [5]. An immunological system has major characteristics that can be used in large systems optimization [6], [7]: proliferation, mutation, selection, and memory. While proliferation is the capability of generating new individuals making the optimization process dynamic, mutation is the ability of searching through the solution space for sub-optimum points. The selection is responsible for eliminating low-affinity cells, while memory is responsible for storing high-affinity cells from other solutions and using these recollections in new problems intending to reduce the optimization time. These features make AIS a powerful optimization tool, enabling the search for several local-optima and the storage of solutions that can be used in further scenarios of a given problem.

There are several variants among AIS methodologies available in the literature used to implement optimization algorithms. Reference [5] shows a very interesting approach by embedding a useful property of evolutionary algorithms, niching, which drives individuals to the most promising points in the solution space. Although this algorithm has exhibited very good results, the number of individuals used in the simulation processes is very high bearing in mind power system optimization problems. Another disadvantage of this algorithm is that it does not use memory cells, and, therefore, any modification in a base case will spend the same computational time that a new problem.

Since the purpose of this paper is also to demonstrate the effectiveness of AIS concepts in OPF problems, some modifications in the referred algorithm are being proposed by adding more relevant information to individuals. In a power flow scenario, an individual can be related to a set of control variables that defines a possible solution, which is characterized by a corresponding set of equations that describes its behavior. Therefore, when a modification in any control variable is performed, it is possible to predict the associated solution point by using the information given by the tangent/Jacobian vector (i.e. gradient) associated with these equations [2]. Thus, it is possible to lead the mutation process, which, in the original AIS algorithm was made through a completely random approach, to generate better individuals making the optimization much faster and reliable. Although

Lagrangian Method Based on PopulationApplied to Optimal Power Flow Problems

Daniele A. Barbosa, Leonardo M. Honório, Armando M. Leite da Silva Federal University of Itajubá, Brazil

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978-1-4244-5098-5/09/$26.00 ©2009 IEEE

this approach provides faster convergences, it still maintains the same number of individuals at the beginning of the search.

The AIS methodology is based on niching process and all the antigens influenced by the same attractor (a local optimum) will converge to a unique point. Thus, with the correct identification of these points, it is possible to eliminate, for each cluster of points influenced by the same attractor, all antigens except the best, reducing the necessary computation effort and speeding up the convergence process. To ensure the KKT conditions, the proposed algorithm uses a Lagrangian-based system modified from [9] as a penalty function. This approach embeds the Lagrangian characteristics into an evolutionary based solver, enhancing the computational performance and providing, at the end of the simulation, a stationary point.

This paper is organized as follows: Section II shows the theory of the Cluster Gradient-based AIS (CGbAIS); Section III presents the CGbAIS to solve the Lagrange-based method; Section IV illustrates and discusses the OPF results, obtained with IEEE standard test systems.

II. THE CLUSTER GRADIENT-BASED AIS

A. AIS Basic Concepts The AIS intends to capture some of the principles of the

previously described natural immune system within a computational framework. The main purpose is to use the successful NIS process in optimization problems. As every artificial intelligence-based method, the AIS is a search methodology that uses heuristics to explore only interesting areas in the solution space. However, unlike other similar methods, it provides tools to perform simultaneously local and global searches. These tools are based on two concepts: hypermutation and receptor edition [5]. While hypermutation is the ability to carry out small steps toward higher affinity antibodies (Ab’s) leading to local optima, receptor edition provides large random steps through the solution space, which may lead into a region where the search for a better solution is more promising.

The technical literature shows several AIS algorithms with some variants. One that has shown good results is the GbCLONALG algorithm presented in [6]. Its main statement is that progressive adaptive changes can be achieved by using numerical information of the system, instead of only computational brute force. This leads to a significant reduction in the number of clones and, thus, in computing effort. The numerical information to be used can be the entropy or just the first order derivatives or gradient, also known as the tangent vector (TV), which is presented in [2].

Under the AIS taxonomy, the set of control variables represents the Ab’s, and the applied increments yield the hypermutated clones. Under the OPF taxonomy, the TV represents the system sensitivities, given a minor disturbance, around a certain operating point. These sensitivities are assessed for each control variable, and the final result defines a vector pointing to the most likely direction to achieve the optimization objective. The number of hypermutated clones must be equal to the number of control actions, i.e. nc, since this number is sufficient to ensure that the sensitivities of all

space dimensions are being duly captured. The GbCLONALG, however, does not use two powerful

characteristics present in the NIS: maturation control and memory cells. To deal with these two concepts, this work proposes modifications in the GbCLONALG. Since the Ab’s converge to the system local optima, it is possible, after a few iterations, to generate clusters representing the points influenced by the same attractor. Once these clusters are well defined, it is possible to apply the maturation control theory and, for each cluster, eliminate all the Ab’s except the best one. This strategy can drastically reduce the initial population and improve the computational response. In order to do that, it is necessary the correct identification of these clusters. B. Clustering Data

There are several algorithms for clustering data, and in this work, the MAXMIN distance (MMD) method [10] is used. This method presents two major advantages: it automatically estimates the number of clusters, and it demands only one parameter, which can be heuristically adjusted or can be set by a simple standard deviation method. The algorithm is presented as follows:

MMD Algorithm: Step 0. Define the minimum distance parameter Dm; Step 1. Set the Ab with the highest affinity as the center of

cluster C1 and Ncluster = 1; Step 2. Assemble the VD vector, where VDi is the minimum

distance between Abi and all cluster centers Cj (j=1,Ncluster). Also, store IDXi = j, where j is the index of the cluster with the minimum distance to VDi;

Step 3. If the maximum VDi value is greater than Dm, go to Step 4, otherwise, go to Step 5;

Step 4. Set the Abi as a new cluster center, increment Ncluster, and go to Step 2;

Step 5. Finish – the clusters are provided by IDX vector. To demonstrate these concepts, Fig. 1 shows an example of

the niching and cluster process in function (1).

144 221121 ++⋅−⋅= )xsin(x)xsin(x)x,x(f Maximize πππ (1)

Figure 1a shows the initial population, and after three generations, the individuals start to group around the attractors, represented by local optima, as shown in Fig 1b. At this time, the cluster is identified using the MMD algorithm and all individuals, except the best of each cluster, were eliminated. Fig. 1c shows the final population over the function mesh. For this particular example, the population started with 60 individuals and, at the end of the simulation process, only 38 remained.

C. The CGbAIS Algorithm

Adding these features in the GbCLONALG [6] yields the proposed algorithm shown in Fig. 2, named Cluster-Gradient-based AIS (CGbAIS). Each step or block of this diagram is detailed as follows:

1. Randomly choose a population p = {Ab1,…,Abi,…,Abn}, with each individual defined as Abi = {xi,1,…,xi,j…,xi,nc}, where nc represents the number of control variables or actions. If there is a memory set present, it must be used as part of the initial population;

2. Calculate the value of the objective function for each individual; this result provides the population affinity for the optimization process;

3. For each individual Abi, a new subpopulation of hyper-mutated clones qi = {Abi,1,…,Abi,j…,Abi,nc} is generated, where Abi,j = {xi,1,…, xi,j + Δxi,j,…, xi,nc}, and nc also represents the number of hypermutated clones. The hypermutated clones are then used to evaluate the numerical information NInf utilized to evolve the population;

4. A new individual 'iAb is assessed through (2), where

iAbΔ means a value given by the NInf,

ii'i AbAbAb Δ+= ; (2)

5. Calculate the affinity of this new individual 'iAb and check

if it has a higher affinity compared to the original Abi; if it does, the hypermutated clone takes its position in the population p;

6. The bests nb individuals among the original p population are selected to stay for the next generation. The remaining individuals are replaced by randomly generated new Ab’s. This process simulates the receptor edition (re) and helps in searching for better solutions in different areas;

7. Use the MMD algorithm to cluster all individuals that converge to a single attractor. This step will generate the Cj clusters;

8. For each cluster, eliminate all the individuals but the best. If it is the end of the simulation, generate a memory of these individuals.

In the proposed algorithm, some considerations must be accounted for. First, the NInf can be provided by the TV [6], or substituted by the corresponding Jacobian vector in Step 4. This new process will further reduce the computing effort.

III. SOLVING THE LAGRANGIAN METHOD WITH CGBAIS

The OPF problem is a typical nonlinear programming problem which, in most cases, can be mathematically stated as

cscscs

maxcsmin

cs

cs

x,xx,xx,x h)x,x(hh

)x,x(g to Subject)x,xf( Minimize

≤≤≤≤

= 0 (4)

where xs is a ns×1 vector of state variables (voltage magnitudes and angles, etc.), xc is a nc×1 vector of control variables (voltage magnitudes at generation buses, power generation, shunt capacitor allocation, etc.), f(.) is a scalar function representing a power system quantity to be optimized (e.g. economic dispatch, transmission loss reduction, loadability, load shedding), g(.) is the active and reactive power balance equations with dimension ng, and hmin , and hmax are nh×1 vectors of constraints associated with the limits of some network power values, such as transmission lines flows, reactive generation, etc. For the sake of simplification, hereafter the set of variables {xs, xc} will be just treated as x.

(a) (b) (c)

Fig. 1. Niching and maturation control process.

Fig. 2. CGbAIS algorithm.

A. The Augmented Lagrangian Function The first step to solve (4), used by most commonly

numerical-based methods, is to change the original problem into an augmented1 Lagrangian function (LF) given by

∑=

−−+

+−++−++−+++=

4

1min44

max33min22

max11

.).(

).())(.())(.()(.)(),(

ii

ik

T

TT

TT

xxs

xsxxhhshsxhxgxfwxL

ρμπ

πππλ

(5)

where λ, π1, π2, π3, and π4 are the Lagrange multiplier vectors with size ng, nh, nh, ns, and ns, respectively. Vectors s1, s2, s3, and s4 store the slack variables with sizes of nh, nh, ns, and ns, respectively. The μk is a barrier parameter which tends to zero at the end of the simulation process, and ρ(.) is the penalty function (6). )ln( iii se ⋅=ρ (6) where ei = {1, 1,…, 1 } is a vector of proper dimension. For the sake of simplification, w = {λ, π1, π2, π3 π4, s1, s2, s3, s4} is a vector representing the Langrage multipliers and the slack variables. A local optimum (x*,w*) is a saddle-point over the LF that satisfies the KKT as follows

0,0,0,0,0 ≥≥==∂∂=

∂∂ ∑ ππ ss

xL

wL

ii . (7)

B. The Dual Problem Numerical methods based on the Newton-Raphson concept

do solve (7), but with some drawbacks: only one solution is found, small changes in the initial conditions may provide completely different solutions, and real-size power systems with high loadability may suffer from numerical problems. On the other hand, evolutionary algorithms [11] consider (4) as the primal objective but it also extends the problem with a dual goal, given by

0

),(≥π

πλβtoSubject

Maximize (8)

( ){ }wxLwith ,min),( =πλβ .

In fact, the primal solution x* along with w* of the dual problem satisfies the KKT condition. To solve this problem using probabilistic methods, the literature mainly shows techniques based on multi-objective or co-evolutionary optimization [8], [11]. Even though these methods provide good results, applying these techniques in power flow scenarios implies in two population sets changing information during the convergence, which can be very time consuming.

The utilization of CGbAIS in these scenarios would provide almost the same computing effort. Thus, to avoid this problem and trying to enhance the computational performance, this work proposes a small modification by mixing the primal and dual problem. This approach leads to only one objective to be accomplished and, therefore, only one population. The methodology is achieved by simply bending

1 This is not the classical Augmented Lagrangian method since it uses a

logarithmic instead of the standard quadratic penalty function.

the solution space provided by the primal problem over the KKT conditions, i.e.

∑∑==

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂=

wx n

i i

n

i i wL

xLwxwith

wxMinimize

1

2

1

2

.),(

),(

γψ

ψ (9)

where ψ(.) is the new unconstrained objective function, nx and nw are the dimension of vectors x and w, respectively, and γ is a fixed parameter responsible for enhancing the convexity in the neighborhood of the solution. In order to illustrate this concept, Fig. 4 shows a simple example where f(x) = sin(2x) and g(x) = cos(x). This would provide L(x,w) = sin(2x) + λcos(x). Fig. 4(a) shows the original Lagrangian function and Fig. 4(b) the bent augmented Lagrangian provided by (9), where the x-axis represents variations in x∈ [-3.0, 3.0], y-axis variations in λ ∈ [-5.0, 5.0], and z-axis shows the function outputs: original ⇒ [-5.0, 3.0] and augmented ⇒ [0, 40]. Fig. 5 shows the results of the CGbAIS for this example. The population started with 30 Ab’s and ended with just 2 Ab’s, representing the local optima within the analyzed range.

It is possible to visualize that the CGbAIS, applied together with the bent Lagrangian function, is able to locate multiple optima, using just one population. It is worth noting that although the bent Lagrangian may have these optima, only the ones provided by original stationary points have their values equal to zero. This feature mixes numerical-based methods with intelligent ones.

C. The Implicit Function Theorem

Before describing the results, some issues about how to apply the previous methodology in an electric power flow scenario must be discussed. Briefly, there are three major points that must be adapted to fulfill the OPF demands: how to assemble and solve (9), and define the stop criterion to be used. In the OPF domain, g(x) represents the power flow balance, which is a highly non-linear set of equations whose corresponding solution “x” is pragmatically assessed through Newton-based methods. It is a drawback in a population-based method that intends to use the conventional Lagrangian function due to the fact that it solves the equality constraints together with the optimization process. Thus, to avoid this problem, the power flow balance must be given by g(xc)=0, where xc is the vector of control variables used in the system. Therefore, the calculation of xs is assessed by the traditional Newton-Raphson method. Although this strategy solves this problem, it creates another one. Analyzing (5) and (7), it is possible to see that by enforcing limits over the states variables xs, it is necessary to evaluate the following term

43 ππ ⋅∂

∂+⋅∂

∂=∂∂

c

sT

c

sT

c xx

xx

xL . (10)

This sensitivity must be ensured by the algorithm, otherwise the direction taken by the gradient through the Lagrangian function may lead to inadequate responses. From the analytical point of view, the simplest way to assess these derivatives is through the implicit function theorem [12], i.e.

ss

cs

c

s

xP

xP

xx

∂∂

∂∂

=∂∂

or

ss

cs

c

s

xQ

xQ

xx

∂∂

∂∂

=∂∂ . (11)

In (11), P and Q represent the active and reactive power functions, respectively. The utilization of one or another depends on what control action is being considered. For instance, shunt compensation has more impact over the second equation while active power dispatch has more impact over the first one. Thus, taking xs as an output provided by a given function V(xc), and considering that the equality constraints will be calculated outside the CGbAIS, the final Lagrangian combines the objective function and the inequality constraints represented by, for instance, the voltage profile (where Vmax is the vector containing the upper limits of the states variables) as shown as follows:

∑−−++= )ln(])(.[)(),,( max sVsxVxfsxL kcT

cc μππ . (12)

Using (12) in (9) provides the final function that must be minimized. Although a stationary point over (9), i.e. ψ(.)=0, obeys the KKT conditions, it is very difficult and time consuming for a population-based method, even if it is gradient-driven as in the proposed solution, to solve high-order equality constraints. In that way, a stop criterion is needed. Thus, the strategy used in the simulations is, first, to check the primal condition and, then, the dual one. If the first one attends a pre-specified tolerance or error (ε), and the second has no improvements greater than ε after 5 generations, the process is finished. Finally, the function ψ(.), defined by (9), is used in Block 2 of Fig. 2 (CGbAIS) to substitute f(p), and the TV [20] is utilized as the NInf in Block 4 to update the new individuals, i.e. the antibodies Ab’s or {xc,w} for the OPF problem.

IV. NUMERICAL RESULTS In order to demonstrate the effectiveness and flexibility of

the proposed CGbAIS algorithm, two cases are considered: Case 1 demonstrates the cluster efficiency and convergence characteristics; Case 2 shows the ability of finding multiple solutions. All simulations were carried out in a MATLAB environment.

A. Continuous Optimization Transmission losses reduction, using a modification of the

original IEEE 14-bus test system, is firstly carried out. In order to stress the transmission network and the system voltage profile, the original load (active and reactive powers) and generation (only the active power) are multiplied by a “stressing factor” of 1.8. The optimization goal is to reduce losses installing shunt compensation at the most sensitive buses, i.e. 14, 13, 12, 11, as shown in [2]. Other considerations are that the maximum voltage limit is set as 1.03 pu, reactive power limits are duly treated during the traditional load flow, the initial loss in the system is 0.326 pu, and the maximum allowed amount of shunt per bus is 0.4 pu.

The first case (Case 1) concerns to measuring the cluster

efficiency. For that, two sets of simulations are carried out with different adjustments regarding the number of antibodies. The first set, shown in Table I, presents the results of non-cluster simulations, and the second set, shown in Table II, presents cluster-based simulations. Both sets represent the mean value obtained from 100 simulations.

TABLE I RESULTS WITHOUT CLUSTER STRATEGY – CASE 1

nAt mFLoss mGen mPerror mDerror mTime10 0.293 22.01 0.022 0.041 12.320 0.293 18.02 0.021 0.037 19.830 0.293 17.30 0.012 0.029 30.140 0.293 17.21 0.015 0.038 42.150 0.293 17.90 0.009 0.026 49.5

TABLE II

RESULTS 1 WITH CLUSTER STRATEGY – CASE 1

nAt mFLoss mGen mPerror mDerror mTime10 0.293 16.32 0.021 0.045 4.920 0.293 17.01 0.015 0.033 7.130 0.293 15.80 0.011 0.031 9.140 0.293 15.20 0.018 0.039 11.150 0.293 22.10 0.009 0.026 14.6

It can be noted that the mean loss reduction (mFLoss, in pu) is almost the same for both sets. The mean number of generations (mGen) needed to converge the simulation does not follow a fixed pattern. However, together with the number of antibodies (nAt), they have influence over the mean value of primal (mPerror, in pu) and dual (mDerror, in pu) errors. Larger populations tend to reduce the errors taking more generations in the process. Finally, if the cluster strategy is used, the mean CPU time (mTime, in seconds) has a major reduction with a speed-up around 3, without loosing accuracy. This speed-up increases if the initial population increases as well. These characteristics make the algorithm very efficient and reliable.

The Case 2 explores the ability of the proposed CGbAIS algorithm to find multiple solutions. In this simulation, the IEEE 14-bus system with a “stressing factor” of 1.2 pu (instead of 1.8 pu used in Case 1) and the maximum voltage level of 1.08 pu were adopted. In this scenario, the initial loss is 0.1343 pu and the proposed control action is to allow the installation of shunt capacitors at any PQ bus. The acceptable amount of shunt installation is limited to 0.5 pu per bus. Table III shows the result obtained with the algorithm named PDIP (i.e. Primal-Dual Interior-Point) proposed in [2]. It is important to note that the PDIP algorithm found a solution that demanded shunt installation in all original PQ buses. The total amount of shunt in the system (ShSum) was 0.3377 pu and the final loss (Floss) was 0.1245 pu. The primal and dual errors (PError) and (DError) were, respectively, 0.0001 and 0.0002.

0.3377 0.0001 0.0001 0.12454, 5, 7, 9, 10, 11, 12, 13, 14

ShSum PError DError FlossBus Number

TABLE IV

FINAL POPULATION OF A SINGLE CGBAIS SIMULATION WITH CLUSTER STRATEGY TO MINIMIZE LOSSES – CASE 2

0.3063 0.0001 0.0020 0.12410.3129 0.0001 0.0030 0.12410.3442 0.0003 0.0060 0.12430.3557 0.0001 0.0020 0.12450.3379 0.0002 0.0020 0.1246

13, 1410, 11, 12, 13, 14

4, 5, 11, 12, 13, 144, 5, 13, 14

4, 5, 7, 9, 10, 11, 12, 13, 14

Bus Number ShSum PError DError Floss

The CGbAIS algorithm, used in this scenario (Case 2), started with 30 individuals and finished with 5, and the corresponding results are shown in Table IV. By comparing these results with those shown in Table III, it can be noted that the proposed CGbAIS was able to achieve five local-optimum solutions, including the one found by the PDIP algorithm. The errors presented in Table IV indicate that all solutions reached a stationary point over the Lagrangian function, meaning that all of them found the KKT conditions. The best solution among all five local-optima indicated only two buses (13 and 14) to install shunt capacitors, and the final amount (0.3063 pu) was almost 10% lower than the one found by the PDIP. This case demonstrated that a more restricted solution space, i.e., the one formed by all possible solutions through installing shunt compensation only at buses 13 and 14, may offer a better result than the one when all PQ buses are available for shunt reinforcements.

Cases 1 and 2 have shown the efficiency of the clustering process proposed by the CGbAIS and its capability of achieving multiple local-optima.

V. CONCLUSIONS

In this paper, an Artificial Immune System (AIS) based approach has been proposed. It combines AIS with cluster techniques and numerical information in order to improve both computing effort and search robustness. The numerical information, brought by the gradient vector, leads to a more efficient hypermutation process, and consequently, to a faster approaching to local optima. Different from most evolution techniques, the proposed algorithm reduces the population during evolution. Cluster techniques are used to identify individuals that might be driven to the same local optima for further elimination of all except the best of each group.

To induce the population to a feasible operational point, instead of just eliminating unfeasible individuals, the proposed Cluster-Gradient-based AIS (CGbAIS) algorithm uses a bent augmented Lagrangian. The utilization of this concept also provides crisp stop criteria, which avoid unnecessary computational effort. Another advantage is that the proposed CGbAIS algorithm generates as output a set of interesting solutions instead of just the best one. This can be helpful due to the fact that there are several constraints that may be difficult or even impossible to be modeled under certain

operation condition. In that way, this feature provides a powerful analysis tool.

The proposed algorithm has been successfully implemented to minimize stressed systems and locating and tracking multiple optima in time-variant environments. Moreover, very good results were achieved in nonlinear optimization problems. The proposed CGbAIS algorithm has been compared with traditional interior-point-based methods. The results showed that it is an interesting alternative to these well established approaches, even being a population-based technique, and it should be definitely considered in practical power systems applications.

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[2] L.M. Honório, D.A. Barbosa, A.C.Z. Souza, and C.V. Lopes, “Intelligent optimal power flow system development using aspect-oriented modeling,” IEEE Trans. on Power Systems, vol. 22, no. 4, pp. 1826-1834, Nov. 2007.

[3] G.L Torres, and V.H. Quintana, “On a nonlinear multiple-centrality-corrections interior-point method for optimal power flow,” IEEE Trans. on Power Systems, vol. 16, no. 2, pp. 222-228, May 2001.

[4] M. Liu, S.K. Tso, and Y. Cheng, “An extended nonlinear primal-dual interior-point algorithm for reactive-power optimization of large-scale power systems with discrete control variables,” IEEE Trans. on Power Systems, vol. 17, no. 4, pp. 982-991, Nov. 2002.

[5] L.N. Castro, and F.J.V. Zubben, “Learning and optimization using the clonal selection principle,” IEEE Trans. on Evolutionary Computation, vol. 6, no. 3, pp. 239-251, Jun. 2002.

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[7] E. Carpaneto, C. Cavallero, F. Freschi, and M. Repetto, “Immune procedure for optimal scheduling of complex energy systems,” Lecture Notes in Computer Science, Springer-Verlag, vol. 4163, pp. 309-320, Sept. 2006.

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TABLE III

PRIMAL-DUAL INTERIOR-POINT (PDIP) RESULTS – CASE 2