Application of Grey Wolf Optimizer Algorithm for Optimal Power Flow of Two-Terminal HVDC Transmission System

This paper applies a relatively new optimization method, the Grey Wolf Optimizer (GWO) algorithm for Optimal Power Flow (OPF) of twoterminal High Voltage Direct Current (HVDC) electrical power system. The OPF problem of pure AC power systems considers the minimization of total costs under equality and inequality constraints. Hence, the OPF problem of integrated AC-DC power systems is extended to incorporate HVDC links, while taking into consideration the power transfer control characteristics using a GWO algorithm. This algorithm is inspired by the hunting behavior and social leadership of grey wolves in nature. The proposed algorithm is applied to two different case-studies: the modified 5-bus and WSCC 9-bus test systems. The validity of the proposed algorithm is demonstrated by comparing the obtained results with those reported in literature using other optimization techniques. Analysis of the obtained results show that the proposed GWO algorithm is able to achieve shorter CPU time, as well as minimized total cost when compared with already existing optimization techniques. This conclusion proves the efficiency of the GWO algorithm.


Introduction
In a High Voltage Direct Current (HVDC) transmission system, an inverter station converts the AC electrical power into DC.After transmission, a rectifier converts the DC electrical power back to AC.These converters can be located in one place as a back-toback HVDC system, or electrical power can be transmitted from one converter station to another over long distance via an overhead transmission line or an underground cable [1].HVDC systems serve as ideal supplements to existing AC power networks.The advantages of using HVDC systems include providing economical and more efficient transmission of electrical power over long distances, solving synchronism-related problems by connecting asynchronous networks or networks which operate at different frequencies, providing controlled power supply in either direction and offering access for onshore and offshore power generation from renewable energy sources [2].
As reported in literature, the first commercial application of HVDC transmission took place between the Swedish mainland and the island of Gotland in 1954, using mercury-arc valves.The first 320 MW, thyristorbased HVDC system was commissioned in 1972 between Canadian provinces of New-Brunswick and Quebec [3].The converters used for HVDC systems are grouped into these categories: line-commutated converters and voltage-source converters or current-source converters.In AC power systems, the Optimal Power Flow (OPF) problem is defined by nonlinear, nonconvex equations.
Feasibility studies are required to determine preliminary parameters of the planned system modifications.To incorporate results of power flow and other parame-ters, more detailed studies are needed.Finally, operating studies are necessary to successfully integrate the HVDC facility into the power system [4].In HVDC systems, where no reactive power is involved, the OPF problem is less complex but it still retains its nonlinear characteristic when voltage control and optimal storage operation are included in the formulation.There are several different methods solving the resulting nonlinear equations [5].
More recently, some authors have proposed other methods to solve this problem.Authors in [6] and [7] present the model of a voltage-source converter suitable for OPF solution of HVDC using Newton Raphson Algorithm (NRA) and a sequential method was introduced [8].A new approach for load flow analysis of integrated HVDC power systems using sequential modified Gauss-Seidel method was reported [9].In [10], a multi-terminal HVDC power flow with a conventional AC power flow has been proposed.In [11], a steadystate multi-terminal HVDC model for power flow has been developed and it includes converter limits, as well as different converter topologies.Other authors have solved this problem by applying new techniques, such as Artificial Bee Colony (ABC) algorithm [12], Genetic Algorithm (GA) [13] and Backtracking Search Algorithm (BSA) [14].Authors in [15] proposed an OPF in order to minimize the losses in a multi-terminal HVDC grid.Application of transient stability constraints for OPF, to a transmission system including an HVDC, was proposed [16].Authors in [17] applied an information gap decision theory to the OPF model for the optimal operation of AC-DC systems with offshore wind farms.
In the last two years, Grey Wolf Optimizer (GWO) algorithm has been applied in power systems for solving combined economic emission dispatch problems [18], studying the blackout risk prevention in a smart grid based flexible optimal strategy [19] and estimating the parameters of the Proportional Integral (PI) controller for automatic generation control of two-area power system [20].Furthermore, it has been used for optimizing wide-area power system stabilizer design [21], solving OPF problem [22] and [23], load frequency control of interconnected power system [24] and economic dispatch problems [25].It has been also used for solving the optimizing PID controller for automatic generation control of a multi-area thermal power system [26] and the design of Static Synchronous Series Compensator (SSSC) based stabilizer to damp inter-area oscillations [27].
In this paper, the GWO algorithm is used to achieve OPF of the two-terminal HVDC system.The proposed algorithm is applied to two different case-studies which are: the modified 5-bus and WSCC 9-bus test systems.The validity and efficiency of the proposed algorithm are evaluated by comparing the obtained re-sults with those obtained when applying other methods reported in literature such as Backtracking Search Algorithm (BSA) [14], Artificial Bee Colony (ABC) algorithm [12], Genetic Algorithm (GA) [13] and Newton-Raphson Method (NRM) [7].

Two-Terminal HVDC Modeling
A basic schematic diagram of a two-terminal HVDC transmission link is given in Fig. 1.In Fig. 1, v r and v i are the AC voltages (rms) at the converter transformer primary, i r and i i are the currents at the AC sides of the rectifier and inverter.δ r and δ i are the bus voltage phase angles, ξ r and ξ i are the current angles and r dc is DC link resistance.p r and p i are the active powers at the rectifier and inverter sides, q r and q i are the reactive powers at the rectifier and inverter sides and i d is the direct current of HVDC link.The basic converter equations between AC and DC sides for the rectifier terminal are expressed as follows [7], [12], [13] and [14]: where the constant k is equal to 3 √ 2/π, v dor is the open circuit DC voltage at the rectifier side; r cr is the equivalent commutation resistance at the rectifier side and equal to √ 3x cr /π (x cr is the equivalent commutation reactance at the rectifier side), and φ r = δ r − ξ r is the phase angle between the AC voltage and the fundamental AC current at the rectifier side.
The basic converter equations between AC and DC sides for the inverter terminal are also expressed as follows [7], [12], [13] and [14]: where v doi is the open circuit dc voltage at the inverter side; r ci is the equivalent commutation resistance at the inverter side, and Φ i = δ i − ξ i is the phase angle between the AC voltage and the fundamental AC current at the inverter side.An equivalent circuit of a two-terminal HVDC link is shown in Fig. 2. In Fig. 2, α is the ignition delay angle, γ is the extinction advance angle, and v dr and v di are the DC link voltages at the rectifier and inverter terminals.The relationship between the rectifier and inverter terminal voltages of the DC link can be expressed by considering DC link resistance as follows:

Problem Formulation and Constraints
The OPF is an optimization problem whose mathematical equations are expressed as follows: Subject to: The objective function f (x, u) considers the production cost of the entire power system and the equality constraints g(x, u) consider the power flow equations related to the entire power system.The inequality constraints h(x, u) consider the limits of the variables related to the entire power system.The variables x = (x 1 , . . ., x n ) and u = (u 1 , . . ., u n ) of these functions are the state and control vectors, respectively.

Control Variables
The control variables should be the same as those of the problem to be optimized.The AC and DC system state variables in per unit are selected as follows [12], [13] and [14]: x AC = [p gslack , q g1 , . . ., q gNg , v L1 , . . ., v LNl ], where p gslack is the slack bus active power output, q gi is the reactive power outputs, v Li is the load bus voltage magnitudes, and N l is the number of load buses.The AC and DC system control variables in per unit are also selected as follows [12], [13] and [14]: where p gi (except for the slack bus p gslack ) is the generator active power outputs, v gi is the generator voltage magnitudes, N g is the number of the generator buses, t i is the transformer tap ratios, N T is the number of transformers, p r and p i are the active powers at the rectifier and inverter sides, q r and q i are the reactive powers at the rectifier and inverter sides, and i d is the direct current of HVDC link.

Objective Function
The problem is minimization of the total production cost (F cost ) in a power system.In other words, the aim is minimization of objective function which is power loss in an energy system.At the same time, the objective function of whole system is minimized under equality and inequality constraints.Therefore, the objective function (f ) can be as follows: where F cost represents the total production cost; a i , b i and c i represent the production cost coefficients of the i th generator.

1) AC System Equality Constraints
A representation of the AC bus connected to the DC transmission link is shown in Fig. 3.The equalities related to the k th bus are given by: q gk + q sk − q lk − q dk − q k = 0.
In Fig. 3, p, q, v and δ represent the active power, reactive power, bus voltage magnitudes, and bus voltage angles, respectively.The subscripts g, l, s and d represent generator, load, shunt reactive compensator, and DC link, respectively.The active and reactive powers transferred from the k th bus to the AC line are also expressed as: where v j and v k are the voltage magnitudes of the j th and k th buses; G kj and B kj are the transfer conductance and susceptance between buses k and j of the bus admittance matrix Y bus .δ kj is the voltage phase angle difference between buses k and j and N is the number of buses in the power system.

2) DC System Equality Constraints
By neglecting the converter and transformer losses in the power system, the power of the rectifier bus becomes equal to that of the inverter bus.Hence, the equations that represent the equality constraints of the DC system are:

Generation Capacity Constraints
For stable operation, the values of the generator active and reactive power outputs, bus voltage magnitudes, transformer tap ratios and shunt VAR compensation are restricted by their lower and upper limits as follows [7], [12], [13] and [14]: where N c is the number of the compensation devices.

DC Transmission Link Constraints
These constraints are represented by the upper and lower limits of the corresponding variables as follows [7], [12], [13] and [14]:

Grey Wolf Optimizer (GWO) Algorithm
The Grey Wolf Optimizer (GWO) algorithm mimics the leadership hierarchy and hunting mechanism of grey wolves in nature proposed by Mirjalili et al. in 2014 [28] and [29].The hunting technique and the social hierarchy of grey wolves are mathematically modeled in order to design GWO and perform optimization.
The mathematical model of the encircling behavior is represented by the following equations [28]: where iter is the current iteration and X P(iter) represents the position vector of the victim.The A and C are coefficient vectors which are given by: where a is linearly decreased from 2 to 0 over the course of iterations, r 1 and r 2 are random vectors in the range of 0 to 1.
In GWO, the first three best solutions obtained are stored so far and push the other search agents to update their positions due to the position of the best search agents.In order to formulate the social hierarchy of wolves when designing GWO algorithm, the population is split into four groups: alpha (α), beta (β), delta (δ) and omega (ω).
Over the course of iterations, the first three best solutions are called α and δ, respectively.Figure 4 shows how to update the locations of α, β and δ, respectively, in a two-dimensional space.The rest of the candidate solutions are denoted as ω.In this algorithm, the hunting/optimization is guided by α, β, δ and ω.The wolves are required to encircle α, β and δ to find better solutions [28], [29] and [30].

D delta D alpha D beta Move
Save the first three best solutions obtained so far and oblige the other search agents (including the omegas) to update their positions according to the position of the best search agent.The following formulas are proposed in this regard.
With these equations, a search agent updates its position according to α, β, δ, in a dimensional search space.
In these formulas, vectors A and C are obliging the GWO algorithm to explore and exploit the search space.With decreasing A, half of the iterations is devoted to exploration (| A ≥ 1|) and the other half is dedicated to exploitation (| A < 1|).
The range of C is 2 ≤ C ≤ 0 and the vector C also improves exploration when C > 1. Exploitation is emphasized when C < 1; A is decreased linearly over the course of the iterations [29] and [30].In contrast, C is generated randomly to emphasize the exploration/exploitation at any stage and to avoid local optima.

Simulation Results
To show the applicability and efficiency of the proposed GWO optimization algorithm in solving the OPF problem of a two-terminal HVDC system, we tested it on the two test systems.The parameters used for the proposed GWO algorithm are given by: the population size is 80, the coefficient a is between [0, 2], the random vectors r 1 and r 2 belong to interval [0, 1] and the stopping criterion of the algorithm is set as 100 iterations for each of the two test systems.The developed program using MATLAB is run on a computer with processor Intel-core i3-5010 U, CPU 2.1 GHz, 4 GB RAM.
Figure 5 shows the flow chart of the sequential power flow algorithm now combined in a sequential AC/DC [11] and [31].Applied the traditional Newton-Raphson method is used for the sequential AC power flow algorithm is the first step and a linear current-balancing method is used for the sequential DC power flow algorithm.Fig. 5: Flow chart of the sequential VSC AC/DC power flow algorithm [11] and [31].

First Case-Study: 5-Bus Test System
The system shown in Fig. 6 [7] has five buses and two generators and it is extended with a two-terminal HVDC link.The AC network and HVDC converters are assumed to work under three-phase balanced conditions.The experiment is performed for two different scenarios, according to the power and current of the DC link, which are: • Scenario A: The current of the DC link is considered to be 0.10 p.u.
• Scenario B: The current of the DC link is considered to be 0.15 p.u.
G.2 The convergence characteristics of the GWO optimization algorithm for the two scenarios of the first case-study are presented in Fig. 7.A lower DC current, as given in Scenario A, encounters less iterations than for Scenario B. Table 1 and Tab. 2 represent the simulation results obtained when applying BSA [14], ABC [12], GA [13], NRM [7], and GWO algorithms to the two scenarios of the first case-study.

Second-Case Study: WSCC 9-Bus Test System
The WSCC 9-bus system shown in Fig. 8 [13] consists of three generators, six transmission line, three power transformers, and three loads connected at buses 5, 6, The convergence characteristics of the GWO optimization algorithm for the second case-study are presented in Fig. 9. Table 3 represents the simulation results obtained when applying BSA [14], GA [13], and GWO algorithms to the second case-study.

Comparative Study
Table 4 and Tab. 5 compare the total cost and CPU time for all of the tested techniques in the first and second case-study, respectively.
When using various optimization algorithms including GWO, the comparative costs and CPU times of the two case-studies are represented in Fig. 10, and Fig. 11, respectively.
As shown in Fig. 10, the cost encountered by applying the proposed GWO algorithm is lower than that obtained when applying the BSA, ABC, GA, or NRM algorithm for the first case-study.This observation is still valid for the second case-study when comparing the cost of applying the GWO algorithm with that of applying BSA or GA algorithm.
According to Fig. 11, applying the proposed GWO algorithm attracts the lowest CPU time when compared with other optimization algorithms for the two cases studied in this paper.

Conclusion
In this paper, the OPF problem of a two-terminal HVDC transmission power system was addressed using the proposed GWO algorithm.The algorithm was applied to two HVDC test systems, namely the 5-bus and the WSCC 9-bus test systems.
The total costs and CPU times encountered when applying the GWO algorithm showed to be lower than those obtained when using other optimization algorithms, such as BSA, ABC, GA, and NRM, with a faster rate of convergence.
The GWO algorithm demonstrated several advantages such as fast convergence, adaptability and reliability to the optimal solution with a performance that was not sensitive to the initial conditions.Therefore, the scope of the future work is to apply the GWO algorithm to solve the OPF problems of large power systems equipped with FACTS devices and renewable energy sources.Hassan was selected by reputable universities in India as an External Ph.D. Examiner and as a Keynote Speaker in several international conferences.She was appointed by the Omani MoHE as a Reviewer of newly submitted academic programs.Dr. Hassan is a Senior IEEE Member (SMIEEE), an IET Member (MIET), an Associate Fellow of the Higher Education Academy-UK (AFHEA) and a Certified Associate Academic Trainer by the International Board of Certified Trainers (IBCT).She is the Chief Editor of two international referred journals in the field.She is also serving as an Editorial Board Member and a Reviewer for many international journals and conferences in power engineering.Dr. Hassan research interests include electrical power systems stability and control, FACTS modeling, optimal and robust adaptive control and quality assurance of higher education.

Fig. 2 :
Fig. 2: An equivalent circuit of a two-terminal HVDC link.

Fig. 3 :Fig. 3 :
Fig. 3: A representation of the AC bus connected to the DC transmission link.

Fig. 7 :
Fig. 7: Convergence curve of the GWO algorithm for the two scenarios of the first case-study.

Fig. 9 :
Fig. 9: Convergence curve of the GWO algorithm for the second case-study.
Mohamed ZELLAGUI was born in Constantine, Algeria, in 1984.He received the engineering degree (Honors with first class) and M.Sc.degree in Electrical Engineering (Power System) from the Department of Electrical Engineering, University of Constantine, Algeria in 2007 and 2010, respectively.He received Doctor Degree in Power Systems from the Department of Electrical Engineering, University of Batna, Algeria in 2014.In 2012 obtained the national c 2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING award for the best Ph.D. student in science and technology.He has membership at International Association of Engineers (IAENG), Institute of Electrical and Electronics Engineers (IEEE), Power and Energy Society (PES), Smart Grid Community (SGC) and The Institute of Engineering and Technology (IET).He is a Senior Member of the Universal Association of Computer and Electronics Engineers (UACEE), International Scientific Academy of Engineering & Technology (ISAET) and International Association of Computer Science and Information Technology (IACSIT).His research interests include power systems protection, power electronics, renewable energy, FACTS devices and optimization techniques.
Tab. 1: The first case-study (Scenario A): Simulation results obtained for BSA, ABC, GA, NRM and GWO algorithms., of 315 MW active loads and 115 MVAR reactive loads.It is extended with a two-terminal HVDC link which replaces the AC line between buses 4 and 5 in the original WSCC 9-bus test system.
Tab. 3: The second case-study: Simulation results obtained for BSA, GA and GWO algorithms.
Tab. 5: Total costs and CPU times of the second case-study.
Dispatch with Cogeneration Systems.International Journal of Electrical Power & Energy Systems. 2016, vol.74, iss. 1, pp. 252-264.ISSN 0142-0615.DOI: 10.1016/j.ijepes.2015.07.031.Heba Ahmed HASSAN received her B.Sc. and M.Sc.with Distinction First Honors degree from Electrical Power and Machines Department, Faculty of Engineering, Cairo University, Egypt, in 1995 and 1999, respectively.She obtained her Ph.D. degree in Electrical Engineering from the University of Ulster, UK, in 2004 when she was selected to present her Ph.D. work at the House of Commons, Parliament House, Westminster, London, UK.Dr. Hassan is a full-time faculty in Electrical Power and Machines Department, Cairo University, currently on leave.She joined Dhofar University, Sultanate of Oman in 2008 where she was promoted to several senior leadership positions.She was the Dean and Assistant Dean of College of Engineering, Dhofar University.She joined Oman Academic Accreditation Authority (OAAA) as Senior Quality Assurance Expert in October 2015.She was an Academic Visitor at the Imperial College, London, UK (1998), a Teaching and Research Assistant at the University of Ulster, UK (2001-2005), and a part-time faculty at many respectable private engineering universities in Egypt (2005-2008).During that period, she worked as a quality auditor for the Quality Assurance and Accreditation Project (QAAP) and a consultant for several Egyptian Ministry of Higher Education (MoHE) development projects financed by IBRD.She co-supervised master Students in Faculty of Engineering, Cairo University (2005-2012).Dr.