Fractional PSOGSA Algorithm Approach to Solve Optimal Reactive Power Dispatch Problems With Uncertainty of Renewable Energy Resources

The optimal reactive power dispatch (ORPD) is a major tool, and it plays a vital role for enhancement of the power system performance. ORPD is one multimodal, non-convex, and non-linear problem. Many elegant benefits can be obtained by using the renewable energy resources (RERs), but many technical issues related to the RERs including the stochastic characteristics of these resources due to continuous variations of solar irradiance and the wind speed lead to increasing the uncertainties of system. Thus, solving the ORPD problem with RERs is a crucial task. The contribution of the paper includes application a modified hybrid algorithm for solving the ORPD considering the uncertainties of the RERs and the load demand. The proposed algorithm is based on Fractional Calculus with Particle Swarm Optimization Gravitational Search Algorithm (FPSOGSA) which aims to enhance the searching capabilities of the conventional PSOGSA algorithm and overcome its tendency to stagnation. The proposed algorithm is tested on IEEE 30-bus system for reducing power losses and voltage deviation as well as enhancing voltage stability. The scenario-based method is employed to produce a set of scenarios from the uncertainties of load, wind speed and solar irradiance. The simulation results verify the effectiveness of the proposed algorithm for solving the ORPD problem with and without considering the uncertainties in the system. Furthermore, the proposed algorithm is superior compared with the state-of-the-art techniques in terms of the reduction of power losses and voltage deviations as well as the stability enhancement.


I. INTRODUCTION
In this section, the different techniques related to the optimal reactive power dispatch (ORPD) discussed with and without integration of renewable energy resources (RERs) as well as research gap and paper contribution are given in detail.
ORPD is taking as an important consideration in the secure operation of electric systems that aims to get the best profile of the voltage and reduction of power losses by adjusting a set of control variable values including the voltages of generator, The associate editor coordinating the review of this manuscript and approving it for publication was Ying Xu . shunt VAR reactive compensators and the tap changing of the transformers. Meanwhile, optimization constraints generator reactive power capabilities, voltages of load bus, power balance and line capacities must be satisfied. Therefore, the reactive power flow is taken as the important consideration in the electrical network and cannot be avoided. Most of the loads in the electrical systems are inductive, such as transmission lines and transformers but it should be sufficient to supply VAR consumers within the limits. Otherwise, it will cuase unwanted voltage and heat loss. An objective will be set to minimize the power losses, voltage deviation and voltage stability index while the desired objective can be VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ achieved by settings of the control variables. Nowadays, the contribution of RERs in electric power system is intensively considered [1]- [8].
In the past few decades, numerous optimization techniques have been studied to use conventional thermal unit for solving ORPD. These optimization techniques include gradient-based approach, linear programing, interior point, quadratic programming and non-linear programming [9]- [13]. However, they have some of difficulties to solve the intricated problem of ORPD such as trapping into the local minima, premature convergence and the algorithmic complexity. To resolve these problems, the development of the new meta-heuristic optimization techniques like differential evolution technique, whale optimization algorithm, sine cosine algorithm, moth-flame optimization, ant lion optimizer algorithm, cuckoo search algorithm, plant propagation algorithm, grey wolf algorithm and particle swarm optimization [14]- [21] are developed to overcomes these issues.
The new hybrid techniques are used for the conventional thermal units to solve ORPD. Singh and Srivastava [22] used the newly Hybrid Multi-Swarm Particle Swarm Optimization (HMPSO) algorithm to minimize the real power loss and improving the voltage stability. In this technique, a swarm is split into the sub-swarms then the sub-swarm is applied as search engine. Jirwadee et al [23] proposed an improved pseudo-gradient search PSO (IPG-PSO) using a linearly decrease chaotic weight factor led by pseudo gradient search technique. A novel hybrid population based on PSOGSA algorithm was proposed by Lenin et al [24] to minimize real power loss and voltage deviations using IEEE30 standard. Khan et al. [25] proposed the fractional PSOGSA algorithm to alleviate power losses and voltage deviations. The fractional approach was incorporated into the PSOGSA algorithm for enhancing the convergence properties of the algorithm.
However, the new developments and the generations are introduced to the electric power system due to increasing demand of electric power supply to the grid. These generations are the renewable energy resources (RERs) which are cost-effective, pollution-free but have the uncertainty in the generations due to variation in windspeed and solar irradiance. There are few studies related to integration of RERs into ORPD. Biswas [26], presented success-history differential evolution (SHADE-EC) algorithm for solving the single and multi-objective stochastic ORPD with the integration of the RERs using IEEE30 and IEEE57 standards. The appropriate PDF functions are taken into consideration to model stochastic power generation and load from RERs at different scenarios. The implementation of feasible solution search with PSO algorithm is discussed by Marcela Martinez-Rojas [27]. The ORPD on PCC is considered to minimize the power losses in a windfarm. A lightning attachment procedure optimization was introduced by Ramadan [28]. The paper explained the uncertainty of the wind speed and the solar irradiance which are modeled using Weibull PDF and Lognormal PDF to reduce the power losses of IEEE30 standard at different 25 scenarios. Mohseni-Bonab [29], [30] formulated the multi-objective stochastic problem related to ORPD considering the uncertainty of load, and it was tested on IEEE14 and IEEE118 standards.
The traditional PSO algorithm is typically facing with sub-optimal problem and caught into the local minima. While, traditional GSA has required a lengthy computational time for solving some optimization issues to find the global solution. PSO has a propensity of quick convergence in a multi variable optimization problem while the global exploration performance of GSA is mostly conspicuous. Hence, PSO and GSA algorithms have their individual perspectives and encourage us to develop an effectual hybridization technique of PSOGSA algorithm to overcome the weakness of the existed traditional PSO and GSA algorithms [24].
The fractional calculus (FC) is a strong mathematical tool that attracts the attentions of the most researchers and implemented in the field of science and technology, for example, electromagnetism, robotics, electronics, physics, telecommunication and control systems [31]- [34] but it is not yet much explored in the field of ORPD with the integration of RERs. By using the FC concept, the fractional properties are applied to update the velocity of traditional PSOGSA algorithm to obtain the fractional FPSOGSA that aims to enhance the convergence performance of the algorithm and memory effects of all past events [25].
The stochastic ORPD problem is taken as a single or multi objective problem with integration of uncertainty load demand, solar irradiance and wind speed. The research is using the single objective approach and ORPD problem with uncertainty is solved in load demand, solar irradiance and windspeed by using the scenario-based approach in order to optimize the power losses, voltage stability index and voltage deviation.
The objective of this research is to implement the novel FPSOGSA algorithm for ORPD with integration of RERs for reducing the power losses, voltage deviation as well as improving the voltage stability index.
From the comprehensive literature review, the few research gaps are observed and discussed as follows.
• ORPD without RERs is simply solved by FPSOGSA algorithm and the uncertainty of solar irradiance, wind speed and load demand at different scenarios is not considered.
• The voltage stability index was not discussed in [26], [28] for the scenario-based approach. The salient features of the research can be summarized as below: • The novel FPSOGSA approach was implemented to the ORPD with RERs according to IEEE30 standard using 13 and 19 control variables.
• The mathematical model of ORPD was built under the uncertainties of load demand, solar and wind power on IEEE 30-bus system.
• Solving ORPD for reducing power losses and voltage deviations as well as improving voltage stability by applying FPSOGSA algorithm with and without integration of RERs.
• The outcomes of FPSOGSA was analyzed and compared with those of the different meta-heuristic techniques for ORPD. In organizing rest of the article, section II describes the problem formulation. Section III presents the mathematical model with the uncertainty in load demand, solar PV unit and wind power for ORPD. Section IV explains the mathematical models of PSO, GSA, PSOGSA, FC with graphical abstract. Section V explains the result and discussion while Section VI is the conclusion section.

II. PROBLEM FORMULATION
The first objective of solving ORPD is to find the best control parameter settings to diminish the power losses. ORPD is assigned as follows.
where, h n and g q are the inequality and the equality constraints while r and s are the control and state variables.
where, V g denotes the generator voltage, Q C denotes reactive power of the compensator, T p is the transformer tap, P 1 is the slack bus power, V is the load bus voltage, Q g is the reactive power of the generator, S t is apparent power flow in the transmission line. While, R, NPV, NTr and NPQ represent the numbers of generators, transformer taps, load buses and transmission lines.

A. OBJECTIVE FUNCTIONS
There are three minimization objectives of ORPD and related details are discussed in below sub-sections.

1) POWER LOESSES
where, Ploss denotes the reactive power loss and G ij denotes the conductance of the transmission line. While, the inequality constraints are given for generators, transformers, shunt VAR compensators and security. Where, max and min are superscripts of the maximum and minimum limit of the control and dependent variables. The equality constraint is representing as follows: While, the penalty factor is representing as the given mathematical expression.

2) VOLTAGE DEVIATION
It sets another objective to improve the voltage profile in the electrical grid, the following expression is used to measure the voltage deviation (VD). 3

) VOLTAGE STABILITY
In the electric power networks, the instability of the voltages is considered as the most critical phenomena which leads to the voltage collapse steadily or immediately. In the improvement of voltage stability, the minimization of voltage stability factor is denoted as L-index of each bus. It can be improved by alleviating the L-index values at one bus and formulated as follows.
Here, L i denotes the stability index value of bus i, while y ij represents the mutual admittance between bus i and j.

III. UNCERTAINTY MODEL OF RERS A. MODEL OF LOAD UNCERTAINTY
The load uncertainty can be represented by the normal PDF function that is formulated by the following expression [30].
where, L d represents normal probability distribution function (PDF), ω d and ρ d are the mean and standard deviations with the values 10 and 70, respectively [28].

B. MODEL OF WIND POWER
The wind speed is varying continuously and stochastically which causes the uncertainty in power system. This means that the solution cannot be presented as a certain point of the wind speed and the uncertain characteristics of the wind speed should be considered. Commonly, the Weibull PDF (11) is used to cultivate the wind power.
where, W (v ν ) represents the PDF of wind speed v ν , α and β refer to the scale and shape parameters of Weibull function, P Wt represents the rated power generated by the wind turbine while v ν,i , v ν,r and v ν,o denote the cut-in, rated and cut-out wind speed, and their parameters with limits are given in Table 1 [25]. The active and reactive generated powers of the wind farm are depicted as follows.
where, P Wt and Q Wf represent the active and the reactive power generated by the windfarm while N Wt represents the number of the wind turbines connected in a windfarm.

C. MODEL OF SOLAR POWER
The output of the solar PV units also fluctuates due to variation of the solar irradiance daily and seasonally which leads to the change in power system. For optimal planning, the uncertain characteristics of the solar irradiance should be considered. Commonly, lognormal PDF function is used to describe the solar irradiance (W/m 2 ). The mathematical expression is given as follows.
where, ρ(g s ) represents the probability density of the solar irradiance in (W/m 2 ), σ s and µ s denote the standard and mean deviation, P Sr is the generated power by PV, X c is the irradiance point while g std is the solar environment solar irradiance and their limit values are given in Table 2 [28]. The wind and solar powers are mainly affecting the dispatch solution of ORPD. The integrated mathematical expression of ORPD containing windfarm and PV is given as (16), shown at the bottom of the page. It should be highlighted here that Eq. (16) represents the balanced power flow in system considering the renewable energy resources where the generated active or reactive powers from the conventional power sources (P g,i , Q g,i ) add the generated powers from the solar PV (P s , Q s ) and the windfarm (P Wf , Q Wf ) on condition that the load demands (P d,i , Q d,i ) and the power losses in the system must be covered.

D. EXPECTED SUM OF OBJECTIVE FUNCTIONS
In case of the uncertainity in the power system, several scenarios will be taken into account. Thus, the losses, the voltage deviations and the stability index should be assigned in these scenarios which are known as the expected values. To find the expected sum of the power losses (E Ploss ), the given expression is used as follows.
where, Nsc denotes the total number of the generated scenarios, sc is the probability of the given scenarios while P losses,sc represents the power losses of each scenario. Similarly, to find the expected sum of voltage deviation, the expression is used as follows.
where, E VD represents the sum of the expected voltage deviation in the given scenarios while VD sc represents the voltage deviation of each scenario. Likewise, the expected sum of voltage stability index is formulated by the following expression.

E VSI =
Nsc sc=1 sc × VSI sc (19) where, E VSI is the expected sum of voltage stability index while VSI sc is the voltage stability index of each scenario.

IV. METHODOLOGY A. PARTICLE SWARM OPTIMIZATION (PSO)
The traditional PSO algorithm was proposed by Kennedy and Eberhart in 1995 [35] where the solution is considered as a particle. The representation of the position and velocity are given as follows.
where, v i,t+1 denotes the velocity of the i th particle at given iteration (t + 1) while x i,t+1 denotes its position, w represents the inertia weight, r1 and r2 denote random values in range [0,1], c1 and c2 are the coefficient for P BST and G BST positions, and w is the inertia parameter that achieves better stability and it is given as follows.
where, w max GSA is presented by Rashedi in 2009 [36]. The algorithm is conceptualized from the Newton's Law where the collection of agents having masses accommodate to the fitness objective value. The initial number of agents are expressed as follows.
where, x dm i is the position of the i th agent while the worst and best for each agent at each iteration is expressed as follows.
where, G const at t th iteration is given in Eq. (25), T represents the total iteration while G e and α values are set to 1 and 23, respectively.
The gravitational and the inertial masses are calculated as follows.
In the search space, overall acting forces on the agent is computed as follows.
where, F dm i signifies the gravitational force applied from j th to i th agent at the explicit time computed as follows.
The acceleration of an agent at the d th dimension is computed as follows.
The velocity and position of the traditional GSA are calculated by the following equations.

C. PARTICLE SWARM OPTIMIZATION AND GRAVITATIONAL SEARCH ALGORITHM (PSOGSA)
The novel hybrid PSOGSA algorithm was firstly introduced by Mirjalili [25], the hybridization of PSO and GSA algorithms aims to enhance the seaching capabilities of these algorithms. Both algorithms are hybridized at low-level co-evolutionary heterogenous.
The position of particles are updated as follows.
x t+1 The FC concept plays an important role in the fields of mathematics, science, and technologies. Past few years, the researchers are using this mathematical tool for enhancing the performance of the algorithms applied in different fields, such as controllability, edge detection, stability, filtering, pattern recognition, identification and observability.
In the literature, there are numerous different interpretations of FC, e.g. Grünwald-Letnikov interpretation signal x(t) is expressed as follows [34].
represents the Euler gamma function.
Generally, a simple integer order only involves a finite series while Grünwald-Letnikov interpretation using fractional derivatives needs a number of infinite series. The fractional derivatives of Grünwald-Letnikov has the oblique memory effect of entire past events that will be reduced over the time. While, the discrete time interpolation of D α (x [t]) is given using (38).
where, T is the sampling time, α is fractional order, is Euler gamma function, q is the index which represents number of terms in power series expansion, and r is the truncation order. The velocity in (39) is modified to amend the velocity as follows.
where, r 1 and r 2 are the random numbers between [0,1], ρ 1 and ρ 2 are the local and global coefficients, n is particle index crossponding velocity v, while P BEST ,i and G BEST are the local and global positions. The velocity order (α) could be a real number in [0,1] that learns the fractional optimization behavior of this novel mechanism, the fractional testing are between a=0 and α = 1 with incrementation of steps α =0.1. Thus, using r = 4, the updated velocity is represented as follows [42].
While the velocity of FPSOGSA is updated, Eq. (34) and (42) will be used as follows.
v n k+1 = α v n k + The steps of FPSOGSA for solving the ORPD including RERs is depicted in Fig. 1.

V. RESULTS AND DISCUSSION
In this section, FPSOGSA is applied for solving ORPD with and without integration of RERs in order to optimize three objectives including power losses and voltage deviations reduction and improvement of voltage stability. The proposed algorithm is test on IEEE30 standard bus system. The IEEE30 standard bus system contains 41 branches, 30 buses, 6 generators, 4 transformers with 9 shunt VAR reactive compensators [51]. 19 control variables are considered for 9 shunt VAR compensators while 13 control variables are considered for 3 shunt VAR compensators based on IEEE 30-bus standard [25]. The simulations are carried on Core I5 PC and the FPSOGSA code for ORPD was programmed. For studying ORPD with RERs, the solar and wind powers resources have been considered as depicted in Fig. 2 where a wind farm is connected to the bus 5 while the solar power plant is injected to bus 8. The rated power of the wind farm is 75 MW where it consists of 25 wind turbines, each wind turbine capacity is about 3 MW. The selected parameters of a windfarm are given in Table 1 [26], [28]. Table 2 lists the parameters of the solar power PV plants which are connected to the generator bus 8 and its rated power is 50 MW. The parameters of FPSOGSA including number of particles, number of iterations, velocity bounds, size of population, fractional coefficient, cognitive/social  acceleration, and inertia weight are given in Table 3. The selection of fractional order depends upon the information of the optimization problem, experimentations with experience and extensive care [25], [39], [41]. The learning convergence curves of FPSOGSA algorithm are obtained at fractional order α = 0.6, population size 50, iteration 100 for all given objectives with and without integration of RERs while, the control parameter values for the generator voltages, shunt VAR reactive compensators and transformer tap are taken as given in Table 4.

A. ORPD PROBLEM WITHOUT RERs
The FPSOGSA algorithm is successfully applied on IEEE30 bus standard to minimize three objectives without integration of RERs including the power losses and voltage deviations reduction and voltage stability enhancement. The FPSOGSA simulation outcomes are given for IEEE30 with 13 and 19 control variable settings. Tables 5 and 7 shows the best control values for the Ploss minimization, VD minimization and VSI enhancement for IEEE30 with 13 and 19 control variables, respectively.    Table 9 and Table 10 show the obtained results by application of different optimization algorithms, respectively. In case of minimizing the power losses for 13 control variables, the Ploss by using FPSOGSA is reduced from 5.811 MW (base case) to 4.5308 MW. Judging from Table 9, the percentage reduction of losses is reported as: MFO is 19.01%, VOLUME 8, 2020         demonstrated the best convergence characteristics achieved by the proposed algorithm for minimization of VSI with 13 and 19 control variables which is 0.0953 p.u. and 0.0878 p.u., respectively. Figs. 3, 4 and 5 illustrates the trend of the objective functions versusiterations.
According to Figs. 3, 4 and 5, FPSOGSA has the stable convergence capacity for given objectives without considering RERs. While, the estimation time of simulation are given in Tables 5 and 7. In addition, the dependent variables of ORPD i.e., reactive power outputs of non-considering RERs and the voltages of the load buses are given in Table 6 and 8  for IEEE30 standard with both 13 and 19 control variables. The result shows that the dependent variables are remained within their permissible limits and there is no violation.

B. ORPD PROBLEM WITH UNCERTAINTY OF RERS
In this section, ORPD is solved using FPSOGSA with RERs. With the uncertainties of the load, solar irradiance and the wind speed, voltage deviations and the expected stability index according to (17), (18) and (19). Table 11 shows the percentage load demands for each scenario and the output powers from the 25 scenarios can be obtained. The objective function is minimizing the expected power loss, the expected wind farm and the PV plant as well as the corresponding probabilities. The total expected values of the power loss, the voltage deviations and stability index without incorporating RERs are reported as 4.7985 MW, 0.7229 p.u., 0.1114 p.u., respectively, for IEEE30 with 13 control variables. Table 12 shows the obtained results with RERs including the Ploss, VD and VSI for each scenario in case of 13 and 19 control variables.
As ORPD with 13 control variables is resolved, E Ploss is reduced from 4.7985 MW to 2.2462 MW (53.19 %), E VD is reduced from 0.7229 p.u to 0.1126 p.u (84.42 %), and E VSI is reduced from 0.1114 p.u to 0.0952 p.u. (14.54%). When ORPD with 19 control variables is resolved, the E Ploss is reduced from 4.7985 MW to 2.1782 MW (54.61 E VD is reduced from 0.7229 p.u to 0.0840 p.u (88.38 %) and E VSI is reduced from 0.1114 p.u to 0.0870 p.u. (21.90 %). The simulation time (sec) for IEEE30 with 13 and 19 control variables considering RERs are given in Table 12.
From the aforementioned results, inclusion of RERs can enhance the performance of the system considerably. Referring to Table 12, values of the Ploss, VD and VSI are changed according to variations of load demand, solar irradiance and the wind speed. the highest values of the Ploss, VD and VSI are obtained in the 1 st and 7 th scenarios where in the 1 st scenario, the percentage load is high (105.784 %) and there is no output power from the wind farm, while in the 7 th scenario, the output power from the wind farm and PV plant are small and the load is high.

VI. CONCLUSION
In the article, a novel heuristic approach of Fractional PSOGSA algorithm has been successfully implemented and applied to ORPD with and without RERs (Wind + PV) in the electrical systems. The uncertainties in RERs is considered by varying the windspeed, solar irradiance, the load demand by applying the Weibull and thee Lognormal probability distribution functions.
• The proposed algorithm is successfully applied to IEEE30 bus standard with 13 and 19 control variables to minimize the three objective functions, such as power losses, voltage deviation and voltage stability index.
• The overall results of FPSOGSA algorithm shows the better performance for ORPD problem with and without RERs as compared to the other heuristic approaches.
• The integration of FC into PSOGSA enhances the overall convergence strength with memory effect of the algorithm.
In future, FC will be added to other heuristic approaches for improvement.