Improved Optimal Power Flow for a Power System Incorporating Wind Power Generation by Using Grey Wolf Optimizer Algorithm

In this paper, an efficient Grey Wolf Optimizer (GWO) search algorithm is presented for solving the optimal power flow problem in a power system, enhanced by wind power plant. The GWO algorithm is based on meta-heuristic method, and it has been proven to give very competitive results in different optimization problems. First, by using the proposed technique, the system independent variables such as the generators’ power outputs as well as the associated dependent variables like the bus voltage magnitudes, transformer tap setting and shunt VAR compensators values are optimized to meet the power system operation requirements. The Optimal power flow study is then performed to assess the impact of variable wind power generation on system parameters. Two standard power systems IEEE30 and IEEE57 are used to test and verify the effectiveness of the proposed GWO method. The obtained results are then compared with others given by available optimization methods in the literature. The outcome of the comparison proved the superiority of the GWO algorithm over other meta-heuristics techniques such as Modified Differential Evolution (MDE), Enhanced Genetic Algorithm (EGA), Particle Swarm Optimization (PSO), Biogeography Based Optimization (BBO), Artificial Bee Algorithm (ABC) and Tree-Seed Algorithm (TSA).


Introduction
Optimal power flow problem has been studied for many years and has become one of the most important means used for adjusting optimal settings of power systems.Therefore, it has received more attention from many researchers throughout the world [1].Several optimization techniques have been used to solve this problem, in order to find the optimal solution for operational objective functions in a power system, such as fuel cost, voltage profile and voltage stability enhancement.Some methods are based on nonlinear programming, quadratic programming, Newton techniques and interior point.These methods have many drawbacks, such as high complexity, convergence to local optimum and sensitivity to initial conditions [2].
A common drawback to meta-heuristic methods is that, in general, the optimization performance is highly dependent on fine parameter tuning.However, the proposed approach outperforms these methods in term of convergence speed to the best solution.Moreover, the use of OPF is extended to include the study of renewable energy systems like wind power, which becomes more and more useful in recent power networks, and many studies are made to integrate this natural power efficiently to a power system.Ranjit and Jadhav in [19], as well as Maskar et al. in [20], presented a study of OPF problem in a system incorporating wind power sources, using modified ABC algorithm named Gbest guided ABC algorithm; the method showed good results for fuel cost optimization case, and voltage profile enhancement, then under wind condition the total operating cost is optimized efficiently, compared to other methods.The method presented some benefits concerning reserve coefficient adjustment when considering imbalance cost of wind power.Meanwhile, Shanhe et al. [21] presented a new economic dispatch technique based on PSO-GSA algorithm for a power system including two wind power sources; the method was tested on a six generators' system connected with two stochastic wind power sources.The test yielded good results compared with other results found in the literature with different methods especially for cost and emission reduction.Panda and Tripathy [22], and Mishra and Vignesh [23] introduced another OPF algorithm based on security constrained OPF solution of wind-thermal generation system using modified bacteria foraging algorithm.The method was tested on the same system stated in [18], in which the wind power variability was modelled incorporating conventional thermal generating system.Recent works in [19], [24] and [25] presented better results and faster convergence characteristics using Grey Wolf Optimizer algorithm.Grey Wolf Optimizer (GWO) algorithm mimics the behaviour of grey wolves in nature by simulating their leadership hierarchy, through haunting, searching for, encircling, and attacking the prey [26].
The present paper aims to investigate the efficiency of GWO algorithm, as a new meta-heuristic population-based algorithm.It presents a solution to the OPF problem of a power system incorporating wind power generation.The rest of the paper is organized as follows; after the introduction, the OPF problem formulation is given in Sec 2. Subsec.2.2.deals with the OPF problem incorporating wind power.Section 3. presents the GWO algorithm and associated simulation steps for solving the OPF problem.In Sec. 4. simulation results using GWO algorithm are presented and analysed.Section 5. concludes the study.

Optimal Power Flow
The objective of conventional OPF problem is to minimize fuel cost for power generation by determining a set of control variables while satisfying system equality and inequality constraints.The OPF problem is formulated by [27]: where [ x]: is the vector of dependent variables consisting of slack bus P G 1 , load bus voltage V L , generator reactive power outputs Q G , and transmission line loading S L .This vector is expressed by: where N D, N G and N L are number of load buses, number of generators, and number of transmission lines, respectively.
[ u] is the vector of independent variables consisting of generator voltages V G , generator real power outputs P G except at the slack bus P G1 , transformer tap settings T P , shunt VAR compensation Q C .This vector is expressed by: (5) where: N G, N T , and N C are the number of thermal generators, regulating transformers, shunt compensators, respectively.

1) Fuel Cost Optimization
The function f from Eq. ( 1) concerned in the OPF study represents the total generation cost formulation and it is as: When considering valve effect, the function f ; is rewritten as: where: a i , b i , c i , d i and e i are fuel cost coefficients of i th thermal generating unit.

2) Voltage Profile Improvement
The aim of this objective function is to minimize the load bus voltage deviations from the reference value which is 1 per unit; this function is expressed by: where: VD represents the voltage deviation in (p.u); V i is the i th load bus voltage; and V ref is the reference voltage which is taken here to be 1 p.u, and thus the objective function Eq. ( 6) becomes as follows: where: w represents a weighting factor selected by the user; many works are choosing w to be 100 in order to keep the variable within the designed limits, as in [1] and [15].The OPF equality constraint such as the active power balance equation is expressed by: where: P d represents the load of the system, and P l is the total active power loss.

3) OPF Incorporating Inequality Constraints
In order to handle the inequality constraints of dependent variables, including slack bus real and reactive power, load bus voltage magnitudes and transmissions line loading; the problem is transformed into unconstrained OPF problem by penalizing these quantities using the penalty function defined as: where: h(x i ) is the penalty function of variable x i , here the x i represents dependent variables, x i min and x i max are the upper and lower limits of x i variable, respectively.
The value of the penalty function grows with a quadratic form when the constraints are violated, and equals to zero if the constraints are not violated, while the extended objective function Eq. ( 6) can be rewritten as: where: η p , η q , η v and η s are penalty factors or weights of active power generation of slack bus, reactive power output of generator buses, PQ bus magnitudes and transmission line loadings respectively.Their values are generally taken to be 100 for the same reason in Eq. ( 9) [14], [15], [16] and [17].

OPF Problem Formulation with Wind Power
The fuel cost objective in Eq. ( 6) is augmented with the cost associated with stochastic wind power, as in Eq. ( 13) [28] where; F (P wj ) is the cost for generation of wind power which is directly proportional to the wind power output and is given by: d j : is the direct cost coefficient of non-utility service, which equals to zero for the utility services.
C wj : represents the imbalance cost of investment in j th wind power source due to two components as in Eq. ( 15) [29]: where: W j,ue and W j,oe , are given by the following expressions: Pwr,j vin,j vr,j −vin,j Pwr,j vin,j vr,j −vin,j where: v 1 = v in,j + (v r,j − v in,j )P W,j /P W r,j ; k > 0, c > 0 are the shape factor and scale factor, respectively.P W r ; is the available active power for the j th wind turbine.P W r,j , is the rated wind power output, P W,j is the actual wind power output of j th wind turbine.
V in , V 0 and V r are the cut-in, cut-off and rated wind speed, respectively.
Equation ( 15) represents the stochastic nature of wind power output for which the following parameters are associated: • K p,j : penalty cost coefficient for not using all available power from j th wind turbine due to under-generation estimated from j th wind turbine, • K R,j : reserve cost coefficient due to the reserve capacity used to compensate the over-estimated wind power of j th wind turbine.
• W j,ue and W j,oe , are the expected value of j th wind turbine for over-estimated and under-estimated energy output which was calculated using Eq. ( 16) and Eq. ( 17) [2].
To deal with wind speed variations of wind turbine, the generated power from wind can be approximated with respect to particular wind speed V , as follows [2]: P w (V ) is the available wind power output, a, b, c, and d; are constants, in this study the generated wind power output is used as negative real power load connected at special bus in the test system.

1) System Equality Constraints with Wind Energy
The equality constraints for the case of wind power are expressed by [26]: The active power losses are given by the formula: where: i and j are the sending and receiving ends of particular line n.N l; is the number of lines.The equality constraints from Eq. ( 8) and Eq. ( 9) are rewritten for the wind node j as: The control variables vector is modified as: where: N W represents the number of wind generators in the power system network.

2) Wind Generators Constraints
In addition to the precedent inequality constraints, we can write; where: P W r , is the rated active power output of the i th wind turbine unit.

3) Spinning Reserve Constraints Model for OPF with Wind Energy
The spinning reserve is the reserve capacity used for sudden load increase, unpredictable fall in wind power output or forced outage of thermal generators units.
The spinning reserve has two limits which are the upper and lower limits that represent system up and system down spinning reserves USR and DSR; given by the following expressions: [2] and [30]: where; r is the influence coefficient that gives the percentage of wind power contributing to USR and DSR.
The USR can be represented with respect to the total load and total wind power by: where: U S i represents the maximum up spinning reserve limit of i th thermal unit, and s is the percentage of load contributing to USR, these constraints will be considered during the implementation of GWO algorithm.As the rate of wind power penetration increases, it becomes more difficult to predict the exact amount of power injected by all generators into the power grid.This added more uncertainty when accounting the spinning reserve requirements.

GWO Algorithm
Grey Wolf Optimizer (GWO) is a new algorithm proposed by Mirjalili et al. in 2014 [31].This algorithm mimics the leadership hierarchy and hunting technique used by grey wolves to catch their prey until stopping its movement.GWO is similar to other populationbased meta-heuristic algorithms, by simulating the natural behavior of grey wolves in their social life when searching for food; they follow hierarchy structure in the group (Fig. 1).The first level representing the leaders of the group is called (alpha), the second level in the hierarchy of grey wolves is (beta) which helps alpha to make decisions.The next levels are delta and omega; they are the lowest ranks in the group; they have to eat after all levels.In fact, these wolves are group-hunting that take three main steps; chasing, encircling and attacking.The algorithm starts with a given number of wolves whose positions are randomly generated.

Steps of GWO Algorithm
Four types of wolves groups can be used to simulate the leadership hierarchy of grey wolves.This hierarchy is represented in Fig. 1, respecting the social dominant degree, the high class is named alpha (α), mostly responsible for making decisions about hunting and order the other wolves in the pack.They can be considered as the fittest solution.The next level in the chain is called beta (β), the wolves of this level help the alpha ones in supervising other groups' actions.They can replace the alpha wolves when they die or become aged and begin to be the best candidate solution.The lowest ranking grey wolves are delta (δ) wolves and omega (ω) wolves [27] and [32].Therefore types α, β, and δ leading the optimization (hunting) process, while ω group is to track them.Kappa (κ) and lambda (λ) wolves are directed by omega in the hierarchy.
The main steps involved in the original GWO algorithm are as follows: • Initialize the search agents.
• Assign Alpha, Beta and Gamma by fitness.
• Encircling the prey: represent the circular area around the best solution (prey).This step can be represented by the following equations: where: X p is the prey's position vector.( A) and ( C), are vectors given by the following equations: where: t is the current iteration and T max , total iterations.
The parameter a decreases linearly in the range of [2, 0] for successive iterations using Eq.(30); that model wolfs behaviour approaching the prey; r 1 and r 2 are random vectors in the range [0, 1].
• Hunting step: the encircling process comes to the second step involving hunting guided by the alpha wolf group.The following equations represent this step: • Attacking the prey: Firstly, r 1 and r 2 are randomly selected for mutation (A and C), then the base vector (X) is randomly selected within the range [r 1 , r 2 ], that is to drive the algorithm to global solution and avoid local optima.The fact that "a" decreases from 2 to 0 makes the exploration more efficient, but slows down the GWO convergence characteristics.So, the final step of attacking the prey is done by decreasing linearly the value of "a" from 2 to 0 [33].
• Steps 2 to 5 are then repeated until the maximum number of iterations is reached.

Case Study and Simulation Results
In this section, the optimal power flow problem is implemented using GWO algorithm and two case studies are considered.For the first case study, the simulation is carried out on IEEE30-bus system as used in [34], by solving conventional OPF and considering quadratic model of thermal generators cost using Eq. ( 6).Then, the OPF problem is implemented considering wind power for a given wind speed and cost profiles.Later, the OPF problem is implemented considering different wind speed profiles.
In the second case study, the simulation is carried out on IEEE57-bus system.The purpose of these studies is to validate the results obtained using GWO algorithm by comparing them with the results available in the literature.

4.1.
Case Study N • 1: IEEE30 Bus Test System The objective function for this case study is given by Eq. ( 6), for all thermal generators units, the numerical data and parameters are taken from [35], the PQ bus voltages are between 0.95 and In order to assess the potential of the proposed approach, a comparison between the obtained results of fuel cost and those reported in the literature has been carried out.The results of this comparison are given in Tab. 2. It is worth mentioning that the comparison has been carried out with the same test system data.
Different OPF results of active generation powers and losses for different case studies are given in Tab. 1.The best fuel cost calculated by the proposed algorithm for this case is 801.1769$/h, which is better than Tab.2: Comparison of quadratic fuel cost case 1.1.

Methods
Fuel cost ($/h) MDE [35] 802.376ABC [3] 802.305 EGA [4] 802.060 GAMS [5] 801.519GWO 801.176 that obtained by many other algorithms as depicted in Tab. 2. The corresponding convergence graph is shown in Fig. 3.For the methods EADDE in [6], GABC in [7], EEA in [5], CSA in [8], KHA in [9], SA in [10], and ISA in [11], the PQ bus voltages are between 0.95 and 1.1 p.u, the transformers tap setting and shunt Var compensators are considered in the same case study, and the generator voltages are taken close to their high permissible limit.Table 3 shows the corresponding optimal power flow results when using the optimal settings of dependent variables.It can be observed from Tab. 4 that GWO algorithm gives better results.The system reactive generation powers for this case study are within their specified limits as in Tab. 5. Table 6 presents a comparison of optimal power flow results of the proposed algorithm with other methods found in

2) Case 1.2: OPF with Voltage Profile Improvement
Minimizing only the total fuel cost using OPF problem as in case 1.1; can result in a feasible solution, but voltage profile may not be acceptable.Thus, in this second case, the objective here is to minimize the fuel cost and improving the voltage profile at the same time by minimizing the voltage deviation of PQ buses from the unity 1.0.[36].
The results obtained using the proposed approach are compared with other methods in the literature as shown in Tab.7 where the total cost found by GWO, in this case, is better than that obtained before.
Figure 4 shows the convergence graph.Figure 5 presents the transmission load flow of the system, from this figure, we can see that the obtained transmission Tab.7: Comparison when optimizing dependent variables.

Methods
Fuel cost ($/h) BBO [13] 804.998 PSO [14] 806.380DE [1] 805.262 GWO 806.1530 loading amounts are within acceptable limits.As we can see from Fig. 6(a), the voltage magnitude is enhanced after the improvement by GWO, and all the load bus voltages are within the permissible range.

3) Case 1.3: OPF for Fuel Cost Including Valve Point Effect
Considering the same system data as in [23], the valve point effect is incorporated and the fuel cost is evaluated using the Eq. ( 7).Simulation of power flow results 0,95  of this case study is compared with other available results as in Tab. 8.

4) Case 1.4: System Analysis Under (N-1) Contingency
To investigate the efficiency of GWO under contingency, a line outage conditions are created on the test system as in [23], in which four contingency conditions are considered (lines: 12-15, 10-20, 15-23 and 6-28).For these four conditions, the voltage profile for normal and contingency conditions is shown in Fig. 7, and corresponding load flow profile is in Fig. 8.The load-bus voltages of contingency case are below their normal limit (deviated from their normal limits).To alleviate this problem we apply Eq. ( 11) and Eq. ( 12), to bring the voltage at these load buses within 0.95 and 1.05 p.u. Figure 9 shows the corrected voltage profile.

4.2.
Case 2: OPF with Wind Energy Case Study

1) Case 2.1: OPF with Stochastic Wind Power Modelling
In this section, GWO algorithm is used to solve OPF problem for system including stochastic wind power in addition to conventional thermal generators.In this case, the system has been modified by replacing conventional generators by wind farms located at buses 5, 11 and 13; each with a total capacity of 60 MW.Two case studies are considered here: in the first case, the wind power is modelled using Weibull distribution function in form of imbalance costs of wind power in the main cost objective Eq. ( 15), which is minimized subject to all given constraints.While, in the second case study, the OPF problem is solved considering different wind speeds.
The test system data given in [23] are taken for this study.The simulation convergence curve and voltage at different buses of the system are given in Fig. 10 and Fig. 11, OPF schedule is given in Tab.9; the optimal results are then compared with GABC [19] and BFA [23].The obtained results show that the GWO method performs better when compared with other methods for the same case study.The reserved power is higher than the surplus power in Tab. 9, which justifies the fact that the utility service is to purchase an important amount of reserve for covering any unavailable wind energy.As seen from Fig. 10, the total fuel cost is decreased by the integration of wind power source in the system.In this case study, we used the same configuration as in [2], by considering the nodes 1, 2, 13, 22, 23 and 27 as generator buses and total system load of 189.2 MW.First, we proceeded for optimal power flow without wind energy; the simulation results of this case are compared to those reported in [2], as shown in Tab.10.
It can be noticed that the obtained GWO cost is better comparing with the case without wind power.

Case 2.2.2: Wind Energy with Zero Cost
Two scenarios of wind power integration levels are considered in this study; 10 %, and 20 % of the system load.These levels are connected to bus 8. Using Eq. ( 25), Eq. ( 26) and Eq. ( 27), we calculated the spinning reserve under different wind speeds at the second hour, assuming the wind speed at the first hour was 3 m•s −1 .
Tab. 10: Simulation results for wind case with spinning reserve.The spinning reserve of the system was s = 15 % of the total demand, and the up-spinning reserve was set to improve the safety of the power system operation under wind intermittent conditions or uncertain wind power.Simulation results for wind case study.This was achieved by using Eq. ( 27), in which this reserve constraint of wind generation was r = 50 % of the system load.After computing the wind power using Eq. ( 25), we run the OPF program to calculate the cost associated with this wind injection then we proceeded to the calculation of different spinning reserve constraint limits, the obtained results are depicted in Tab.10.

Wind
The wind speeds values were respectively 4, 5, 6, 7, and 8 m•s −1 ; different computation results of scenarios 1 and 2 are presented in Tab. 12 and the system voltage profile is shown in Fig. 13.In this case, we assume that the wind power has the same direct cost of [19] d 1 = 1 $/h, without considering the imbalance cost.Simulation results for wind case study.
The simulation result is shown in Tab. 12, we can see that when wind speed increases, the total operation cost increases too, due to the wind direct cost impact on the total operating cost.

Case 3: OPF with Stochastic Wind Speed
In this case study and in order to check the effect of uncertain wind power on the test power system, two wind farms each with capacity of 30 MW have been connected at two separate locations; at nodes 26 and node 30 as in [19].The results obtained are then compared with the case without wind energy.
Two cases are considered here; the first one where the scale factor "c" takes the values of 3 to 30 while keeping the shape factor at k = 2, then by keeping the scale factor constant at the value 10 and varying the reserve coefficient (Krw) from its base value of 4, and with the installed wind power capacity for each wind farm of 20 MW instead of 30 MW by applying the proposed approach taking into consideration these conditions, we find the results as shown in Fig. 13(a).
For the second case study, we maintained the values of wind turbines Weibull model factors constant, , but considering the direct costs of the two wind farms d1 = d2 = 1.3 $/h.Simulation results are presented in Tab. 13, the convergence characteristics for different values of reserve coefficient "K rw " is given in Fig. 13(b).
Generally, the direct cost of wind power is less than the average cost of thermal power, and the penalty cost of not using all the available wind power is considered less than the direct cost.From Fig. 13(b), it can  be seen that, the larger the value of c the higher the value of wind speed and hence wind power penetration amount.However, the amount of wind power injected at bus 26 remains, less than that injected at bus 30, due to the thermal loading limit of the transmission line at this section.

4.4.
Case Study N • 2: IEEE57 Bus Test System This system consists of 7 thermal generators, with bus 1 is considered as slack bus; 2, 3, 6, 8, 9 and 12 as PV buses, 50 load buses and 80 lines, among which 17 lines are equipped with tap changing transformers.In addition, three shunt Var compensators are installed at buses 18, 25 and 53.The system data are taken from [37].Two cases are investigated in this case study: 1) Case 1: OPF for Quadratic Fuel Cost In this case study, the objective function to be optimized is represented by the quadratic fuel cost, related to thermal generators unit described by the Eq. ( 6).
The optimal power flow for the first case study using GWO takes the settings of the algorithm as the followings: search agent number equals to 30, and the number of runs equals to 300.These are the same system settings used for the other methods, and the obtained simulation results are shown in Tab.14.The results obtained by the proposed method were compared with others available methods, this comparison shows that the GWO algorithm gives better results when compared to many algorithms found in the literature as shown in Tab. 15.

2) Case 2: OPF with Voltage Profile Improvement of IEEE57 Test System
Bus voltage enhancement is one of the most significant safety and service qualification indices.In order to assess this case, a two-fold objective function is considered to minimize the operating fuel cost and enhancing the voltage profile at the same time by minimizing all the load bus deviations from the reference value.Voltage profile, in this case, is compared to that of the precedent one as shown in Fig. 16 and the operating cost curve is shown in Fig.It is clear that the voltage profile is enhanced efficiently compared with that in case 1.This could be achieved by the optimal tuning of the control parameters within the constraints range as given in Tab.16 using the proposed GWO technique.
It can be seen that the proposed GWO method converges to a better result than EADDE [7] method by decreasing the fuel cost from 42051.44 $/h to From the comparison of the results shown in Tab.16, it can be concluded that the solution quality of the GWO algorithm is very competitive and challenging because it converges to the best solution with less com-putational time.Figure 18 presents the convergence curve for IEEE30 bus system after 40 runs and Fig. 19 the convergence curve for IEEE57-bus system after 50 runs.

Conclusion
This paper presents an optimal power flow study using a new meta-heuristic population-based search algorithm called Grey Wolf Optimizer (GWO).Considering both wind and thermal power generators, in order to evaluate the effectiveness of the proposed technique; three case studies are considered in this work.
By adding to the normal operation condition, the N-1 contingency condition represented by lines outage and the uncertainty of wind power, which is modelled using Weibull distribution function is investigated.
Simulations results obtained by OPF analysis for two standard test systems IEEE-30, and IEEE-57 bus systems without considering wind power are compared with results of other methods available in the literature.The outcome of the comparison confirms the effectiveness and robustness of the proposed algorithm.
Similarly, the results obtained in presence of wind energy system were compared with those of other methods reported in the literature using the IEEE 30 bus system.By increasing the value of reserve coefficient, the value of the injected amount in the system can be limited by the transmission system permissible capacity of the existing network.On the other hand; when increasing the wind penetration level by increasing wind speed, the total operating cost decreases.
The method presents compromising performances measures compared to other methods found in the literature.This analysis will be extended in the future to include spinning reserve in the main optimal power flow problem.M.Sc.degree from Annaba University in 1998, his Ph.D. degree in power system from Batna University (Algeria) in 2003.His areas of interest are the application of the meta-heuristic methods in optimal power flow, FACTS control and improvement in electric power systems, Multi-Objective Optimization for power systems, and Voltage Stability and Security Analysis.He is the Editor-In-Chief of Journal of Electrical Systems (Algeria), the Co-Editor of Journal of Automation & Systems Engineering (Algeria).

1 :
OPF with Quadratic Fuel Cost

4 :
c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING Tab.Comparison when optimizing dependent variables.

Fig. 10 :
Fig. 10: Convergence graph of fuel cost for wind case.
Initialize parameters; a, A, and C Calculate the fitness of each Search_Agent; X a =the best search agent; X β =the second best search agent; X δ the third best search agent; While Iter≤ Max_Iter For j∈{search space} Sort the population of grey wolves according to their fitness Update the Update the position of the current Search Ahent using Eq.(39); endfor % search space Update a, A and C Calculate the fitness of the new search agents; Update X a , X β and X δ Iter=Iter+1; End; Return, Best solution found so far X a ; 1.05 p.u, the shunt Var Compensator are not considered in this case study, except for the two shunt capacitors banks, at nodes 10 and 24 of 19 and 4.3 Mvars respectively.The optimum control settings obtained by using GWO algorithm are presented in Tab. 1.
. 12: Simulation results for wind case study.
Tab. 13: Simulation results for wind case study.
Tab. 14: Optimal control variables settings for case 1.