Interior search algorithm for optimal power flow with non-smooth cost functions

: The optimal power flow (OPF) problem considers as an optimization problem, in which the utility strives to reduce its global costs while pleasing all of its constraints. Therefore, Interior Search Algorithm (ISA) is applied to treat this problem. Where, ISA is a specific type of artificial intelligence and a mathematical programming, not a meta-heuristic algorithm. The effectiveness of this method in solving the OPF problem is evaluated on two test power systems, the IEEE 30-bus and the IEEE 57-bus test systems. For the first example, the ISA-OPF algorithm finds an answer that agrees with published results. For the 57-bus system, the ISA-OPF demonstrates its ability to transact with larger systems. Thus, the ISA-OPF algorithm is shown to be a robust tool to treat this optimization compared with other meth-ods. Moreover, the advantage of ISA is that it has only one parameter of control which makes the simplicity in the main algorithm.


Introduction
Optimization techniques are becoming the inquiry of many researchers especially that the efficiency of a particular system relies on gaining an optimal solution, which is achieved by a good optimization. The optimization is a process that discovers the best solution after evaluating the cost function which indicates the relationship between the system parameters and constraints.

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Recently, the power system has been forced to employ on its filled ability due to the urban and economic expansion that may compel strict confines on the power transfer capability. This fear recalls building novel transmission lines with their redundancy and productions units. Consequently, the planning and operation of transport networks become more complex due to the rapid growth of electricity demand, integrations of networks and movement toward the electricity markets open. Traditional optimal power flow (OPF) has provided a tool to achieve such a task and has the first treaty the cost of fuel only. Later other objectives have been incorporated in the OPF in the form of a single objective.
In the electrical power systems, the Optimal Power Flow (OPF) problem is considered as a static non-linear, which is one of the most important operational functions of the modern energy management system. It has been used as a tool to define the level of the inter-utility power exchanges. The OPF approach has gained more importance due to the increased availability of control devices and energy prices. Since its starting point, it has proven its efficiency in dealing with various issues. The OPF can represent the same objective as a scheduling problem such as minimization of the operation cost of power systems and the thermal unit cost as well. It has been considered as an optimization process to minimize or maximize a certain objective functions of the power system while satisfying system constraint as well as minimizing the fuel cost which is the main goal of the current study.
All artificial intelligent techniques that mentioned above are meta-heuristic from nature. Apart from common control parameters (like population size and maximum iteration number), several specific parameters of algorithm (like inertia weight, social and cognitive parameters in the case of PSO, crossover rate in the case of GA and DE, limit in the case of ABC, etc.) are required for proper execution of all meta-heuristic based techniques. For effective application of any metaheuristic, it is crucial to tune the algorithm-specific parameters for a specific problem. This tuning procedure requires the trial and error based simulation which need rigorous computational effort. If the number of algorithm-specific parameters increases, the tuning problem of these parameters further becomes complex and time consuming. Many times, this tuning becomes more tedious than the problem itself.
Up to 2014, a new method known as the Interior Search Algorithm (ISA) has been proposed by Gandomi (2014) is inspired by interior design and decoration. ISA has just one tuned parameter which is α, this property is mainly one of the ISA advantages. So, in this paper, an optimization approach based on ISA followed by its mathematical model is proposed to solve the OPF problem for the aim of minimizing the cost of fuel, taking into account multi-fuels options, valve-point effect and other complexities as well as to other different objectives such as voltage profile improvement.
To fulfill the task achievement, the ISA method is simulated and tested on the IEEE 30-bus and the IEEE 57-bus test systems. The obtained results are compared with other methods that have been already done and others recent works.
The organization of this paper is offered as follows: Section 2 discusses the formulation of OPF problem followed by a brief description of ISA that illustrated in Section 3. Section 4 presents the simulation results and discussion. Finally, Section 5 states the conclusion of this paper.

The formulation of OPF problem
The mathematical formulation of the OPF problem can be stated as a nonlinearly constrained optimization problem: Equations (4) and (5) give respectively the vectors of control variables "u" and state variables "x" of the problem of OPF: where P g active power generator output at PV buses except at the slack bus, V g voltages Generation bus T c transformer taps settings and Q c shunt VAR compensation.
where V L voltage profile to load buses θ argument voltages of all the buses, except the beam node (slack bus) P s active power generated to the balance bus (slack bus), Q s reactive powers generated of generators buses.

The constraints
The OPF constraints are divided into equality and inequality constraints. The equality constraints are power/reactive power equalities, the inequality constraints include bus voltage constraints, generator reactive power constraints, reproduced with reactive source reactive power capacity constraints and the transformer tap position constraints, etc. Therefore, all the above objectives functions are subjected to the below constraints:

Equality constraints
Ties constraints of the OPF reflect the physical system of electrical energy. They represent the flow equations of active and reactive power in an electric network, which is represented respectively by Equations (6) and (7): g kj , b kj elements of the admittance matrix (conductance and susceptance respectively).
(1)   A penalty function (Lai, Ma, Yokoyama, & Zhao, 1997) is added to the objective function, if the functional operating constraints violate any of the limits. The initial values of the penalty weights are considered as in (Aisac & Stott, 1974).

Interior search algorithm (ISA)
ISA is a combined optimization analysis that divides to the creative work or art relevant to Interior or internal designing (Gandomi, 2014), which consists of two stages. First one, composition stage where a number of solutions are shifted towards the optimum fitness. The second stage is reflector or mirror inspection method where the mirror is placed in the middle of every solution and best solution to yield a fancy view (Gandomi & Yang, 2012). Satisfaction of all control variables to constrained design problem using ISA is our main intention.

Algorithm description
(1) However, the position of acquired solution should be in the limitation of maximum and minimum bounds, later estimate their fitness amount (Gandomi, 2014).
(2) Evaluate the best value of a solution, the fittest solution has a maximum objective function whenever aim of an optimization problem is minimization and vice versa is always true. The solution has universally best in the jth run (iteration).
(3) Remaining solutions are collected into two categories mirror and composition elements with respect to a control parameter. Elements are categorized based on the value of the random number. (all used in this paper) are ranging between 0 and 1. Whether is less than equal to it moves to mirror category else moves towards composition category. For avoiding problems, elements must be carefully tuned.
(4) Being composition category elements, every element or solution is however transformed as described below in the limited uncertain search space.
where x j i represents ith solution in jth run, ub j , lb j upper and lower range in jth run, whereas its maximum, minimum values for all elements exists (j-1)th run and rand 2 () in ranging 0-1.
(5) For ith solution in jth run spot of mirror is described: where rand 3 () ranging 0-1 .Imaginary position of solutions is dependent on the spot where mirror is situated defined as: (6) It is auspicious for universally best to little movement in its position using uncertain walk defined: where r n vector of distributed random numbers having same dimension of x, = (0.01*(ub-lb)) scale vector, dependable on search space size.
(7) Evaluate fitness amount of new position of elements and for its virtual images. Whether its fitness value is enhanced then position should be updated for next design. For minimization optimization problem, updating are as follows: (8) If termination condition is not fulfilled, again evaluate from second step.
(A) Parameter tuning of α ISA has just one tuned parameter that is α which is almost fixed at 0.25, but the requirement is to increase its value ranging between 0 and 1 randomly as the increment in a maximum number of runs selected for a particular problem. It requires shifting search emphasised from exploration stage to exploitation optimum solution towards termination of maximum iteration.

(B) Constraint manipulation
Evolutionary edge (boundary) constraint manipulation: where r 4 and r 5 are random numbers between 0 and 1, and x gb, i is the related component of the global best solution. where nc represents number of constraints, g k (x) = kth constraint consisting problem and g maxk = maximum violation in kth constraint till yet. (Gandomi & Roke, 2014) Initialization while any stop criteria are not satisfied find the x j gb

Pseudo code of Algorithm
Check the boundaries except for decomposition elements.
end for

Application and results
The ISA algorithm is demonstrated on the two test cases the IEEE 30-bus system, and the IEEE 57-bus system that are used as a tests systems to compare results of different case studies. The detailed results are shown for all cases.
The following eleven cases are provided to demonstrate the effectiveness of the proposed approach: Cases studies on the IEEE 30-bus test system: • Case 1: Fuel cost.
• Case 7: Fuel cost with considering the prohibited zones.
• Case 8: Fuel cost with considering the prohibited zones + Valve-point effect.
Cases studies on the IEEE 57-bus test system: • Case 9: Fuel cost • Case 10: Fuel cost + Voltage profile.

IEEE-30-bus system
This system contains 6 generators, 41 transmission lines including 4 transformers and 9 compensators installed at the load bus. The total active power requires 283.4 MW while the reactive power requires 126.2 MVAR. The values of coefficients fuel costs of the six generators are presented in (Bouchekara, Chaib, Abido, & El-Sehiemy, 2016). The IEEE 30-bus test system has mainly 25 control variables.

• Case 1: Fuel cost
In this case, we are interested in solving the problem of OPF while minimizing the corresponding fuel cost production. The mathematical form of the objective function in this case is: where n total number of generators. P i real power generated by the unit i a i , b i and c i are the fuel cost coefficients of the ith generator. The obtained control variables parameters is tabulated in Table 2.
The fuel cost based objective function has achieved by 799.2776 $/hour, which is considered 11.2% lower than the normal case.
• Case 2: Fuel cost + Voltage profile In this case, two competing objectives, namely the voltage profile improvement and fuel cost are shown in the Equation (23): where η is the weight factor, it was chosen carefully. After several experiments, the weight coefficient related to the voltage profile and fuel cost is taken 1,000.
The total generation fuel cost and voltage deviations are 807.6408 $/h and 0.1273 p.u for this case compared to Case 1 which gave us 799.2776 $/h and 1.8462 p.u. we note an increase in the fuel cost by 1% but there is an improvement in the voltage profile by 93%. The voltage profile obtained for Cases 2 and 1 is shown in Figure 1 that illustrates in this case an improvement compared with Case 1, we can also note from this Figure that the voltage profile has improved and relieved.
• Case 3: Fuel cost + Multi-fuels From a practical point of view, thermal generating plants may have multi-fuel sources like coal, natural gas, and oil. Hence, the fuel cost curve can be expressed by a piecewise quadratic function as follows (Bouchekara et al., 2016): where a ik , b ik and c ik represent the cost coefficients of the ith generator for fuel type k. The generator fuel cost coefficients are given in Bouchekara et al. (2016), The results obtained for this case are shown in Table 2. The optimal cost obtained for this case is 646.2532 $/h.

• Case 4: Fuel cost + Voltage profile + Multi-fuels
We considered that the optimal solution is achieved using the proposed algorithm and presented in Table 2. This case is similar to Case 2, we can note from this table that the voltage profile has improved and relieved compared with Case 3. Because voltage deviation has been reduced from 1.6456 p.u to 0.1286 p.u. The voltage profile obtained for Case 3 and 4 is shown in Figure 1 that offers in this case enhancement compared with Case 3.
• Case 5: Fuel cost + Valve-point effect For more rational and precise modeling of fuel cost function, the generating units with multi-valve steam turbines exhibit a greater variation in the fuel-cost functions (Hardiansyah, 2013). The valve-point effects are taken into consideration in the problem by superimposing the basic quadratic fuel-cost characteristics with the rectified sinusoid component as follows: where d i and e i are the coefficients that represent the valve-point loading effects, the coefficients are given in (Bouchekara et al., 2016). The optimal settings obtained for adjusting control variables from the ISA method are given in Table 2. The minimum cost obtained from the proposed approach is 823.1379 $/h. It is increased compared with Case 1 by reason of taking the effect valve point.

• Case 6: Fuel cost + Voltage profile + Valve-point effect
This case is similar to Case 2, the objective is to optimize both the cost with considering the valve point effect and improve the voltage profile. Table 2 is given the optimal results obtained using ISA for this case. Figure 2 exposed the voltage profile obtained for Cases 5 and 6 that shown improvement compared with Case 5.
• Case 7: Fuel cost with considering the prohibited zones Resolve the problem OPF with the examination of the prohibited zones is proposed in this section. The generators have a set of allowed operating zones i.e. non-prohibited zones (POZ) and they must work in one of these zones. POZ can be included as constraints as follows:    where P min im and P max im are the minimum and maximum limits prohibited zones, limits of all generators are shown in Table 1. The obtained optimal settings of control variables for this case study are shown in Table 2, it is clear that POZ slightly contributes in increasing the generation cost compared with Case 1.
• Case 8: Fuel cost with considering the prohibited zones + Valve-point effect This Section 4.1 examines the valve point effect and prohibited zones simultaneously. The results of the present case are represented in Table 2. By comparing the results obtained here with the Case 7, it is clear that the production cost value of Case 8 is highest.

IEEE 57-bus system
In this part, we have applied ISA method to solve OPF problem in large power system and in order to prove the substance and robustness of this proposed approach, the IEEE 57-bus with 80-branch systems has been proposed, which has a 34 control variables that are given as follows: 7-generator voltage magnitudes, 17-transformer-tap settings, and 3-sbus shunt reactive compensators (Bouchekara et al., 2016). The total system demand is 12.508 p.u. for the active power while 3.364 p.u for the reactive power at 100 MVA base. The slack bus of power system has been taken in bus 1. The values of coefficients fuel costs of all generators are presented in (Sinsupan, Leeton, & Kulworawanichpong, 2010).

• Case 9: Fuel cost
In the process of ISA, the objective function of this part is given by Equation (22). The obtained results are given in Table 3. The cost obtained for this case is 41676.9466 $/h which is less when comparing with the base case.
• Case 10: Fuel cost + Voltage profile We have selected the voltage profiles as the objective function beside fuel cost in order to improve voltage profiles of the test system. The equation of the objective function is given by Equation (23). The weight coefficient related to the voltage profile and fuel cost is 120,000.
The obtained results are given in Table 3. The voltage deviation is 0.9931 p.u for this case compared to Case 9 which gave us 2.3863 p.u. It can be seen that there are differences between the proposed solutions and there is an improvement in the voltage profile by 59%. The voltage profile obtained for Cases 10 and 9 shown in Figure 1 that proves the improvement of the voltage profile in this case compared with Case 9. The effect of valve-point loading is also tested in this case. Therefore, the objective function in this case is expressed by Equation (25), the coefficients are given in Sinsupan et al. (2010). The obtained results to the aid of ISA are given in Table 3. In this case, the cost has slightly increased from 41676.9466 $/h to 41705.2941 $/h compared with Case 9.

Performance evaluation study
In order to evaluate the performance of the ISA, several optimization algorithms have been carried out including BBO, DE, HS and WDA to give equitable comparison. Control parameters of optimization algorithms are selected carefully after several trials that used in this investigation and are given in Table 4. From this table, we can see that all algorithms have more than two tuned parameters, while ISA has just one tuned parameter which is α, this property is mainly one of the ISA advantages. The results of this study are achieved after twenty different runs. Table 5 indicates that algorithm offers the minimum values of best, worst, median values of fuel cost and the standard deviation values. Also, we calculated the infeasibility rate that shows the success rate or not of the proposed algorithms. The OPF problem may become infeasible under certain operating conditions. It may not be possible to satisfy all the constraints, under these circumstances, some of the soft constraints must be relaxed to obtain a feasible solution. Detection of infeasibility is important because a solution with small violations is better than no solution at all. Infeasibility rate is calculated in this case as follows: We note from Table 5 that the difference between the minimum and the worst is very close, we can say from this table that the proposed method is robust. This is also demonstrated by the low values of the standard deviations calculated compared to other algorithms. ISA is the only algorithm that has a infeasibility rate of 0% compared with the remaining algorithms. This comparison can be seen clearly when the resolution of the OPF for larger systems has not even been able to converge with the BBO, HS and WDA as in the Cases of 9, 10 and 11.
On the other hand, the fast converge of an optimization algorithm to the optimal solution is an important issue in this domain. Figure 2 shows the trend for minimizing the generation fuel cost based objective function using the different algorithms for Cases (1, 3, 5 and 9). The algorithms were tested using same different initial solutions.
It is obvious that ISA increased the convergence speed and obtained better final results with less improvisation compared with the remaining algorithms.
For more demonstrating the convergence speed, the curve of convergence has been cut at four cut points which are 20, 40, 60 and 80% of the iterations maximum number.   • 20% represents iterations number 60; • 40% represents iterations number 120; • 60% represents iterations number 180; • 80% represents iterations number 240.
Then, we take the value of the objective function that corresponds to the number of iterations, and calculates the percentage of change value with the final value. These percentages are reported in Table 6. The final row is the average percentage for the cases of study. From this table, it can be concluded that the ISA is more robust and effective in solving the considered OPF problem. The ISA has reached more than 99% of the final objective value for each cut point. From this result, we can note that the ISA gives the best convergence speed.
For a comprehensive comparison, some published results have been traced. In Table 7, a comparison between the results is obtained by the proposed algorithm with those found in the literature,