# Optimal Power Factor Regulation of Dispersed Wind Farms under Diverse Load and Stochastic Wind Conditions Based on Improved Firefly Algorithm

Authors: Hao Liu; Wei Wang; Jia Cui; Fen Tang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Aiming at the instability of voltage and large network loss of dispersed wind farms (DWFs) integration into distribution network (DN), an optimal power factor regulation method based on improved firefly algorithm is proposed. Firstly, the generalized load model based on the static voltage characteristics is analyzed. Then reactive power capabilities of DWFs are thoroughly discussed and the influences of DWFs with variable power factor on network loss and voltage profile are presented. In order to reduce network loss and enhance power quality, optimal power factor regulation of DWFs based on improved firefly algorithm (IFA) is proposed. Finally, results of the benchmark IEEE-33 node system show the feasibilities and validities of the proposed method and the superiorities of proposed IFA are revealed by comparison with the existing algorithms.


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## Body

## 1. Introduction

As the development of distributed energy, dispersed wind farms (DWFs) are expected to comprise a significant portion of future power generation. However, high penetration level of DWFs integration into the distribution network (DN) presents major challenges in terms of planning and operation of the power grid. According to traditional “fit and forget” practice, DWFs usually operate at maximum power point and unity power factor, which are mostly treated as negative fluctuant loads in the conventional DWF management [1, 2]. In this way, the inherent uncertainty and variability of wind power supply further deteriorate the voltage profiles of DN. To facilitate the safe and stable operation of DN, the penetration level is limited. Otherwise additional reactive power compensation equipment is needed, increasing the investment and operation cost. Actually, DWFs are connected to DN through their interfacing converters. Such power electronics based DWFs have great potentials in power regulation capabilities, which could be considered in the operation process. In this way, DWFs take some degree of responsibility for system support and their roles are changed from the passive actuators into active participators, which is an important element of changing traditional DN operation into active DN operation [3, 4].To facilitate an active DN operation, how to reestablish and resolve system optimization function with the introduction of such active participators is essential. On one hand, operation related problems mostly focus on the optimization of typical indexes under a single load type condition in the literature [5–7]. However, most of the DWFs are installed near end-users for the purpose of meeting on-site power demand. The loads are not explicitly single type such as constant power, industrial, commercial, and residential loads but rather the combination of these types in the practical applications [8, 9]. Such loads have different influences on the voltage profile of DN. Thus, load types might not be neglected in the operation of DWFs especially considering reactive power support capabilities.On the other hand, various optimization techniques have been suggested in recent decades. The solution techniques could be classified into conventional, intelligent searches, and fuzzy set applications. Compared to other intelligent algorithms, the firefly algorithm is faster and the variable parameter adjustment is simplified. But the common disadvantages of these algorithms are premature convergence of the population and slow convergence rate [10, 11].To address these issues, optimal operation of DWFs considering reactive power support capabilities and load types is proposed for network loss reduction and power quality enhancement. In this paper, the generalized load model based on the static voltage characteristics is first introduced. Afterwards, the reactive power capabilities of DWFs are thoroughly discussed. On this basis, the multiobjective optimization function is established, considering the various load and stochastic wind conditions as well as the system equality and inequality constraints. Furthermore, the improved firefly algorithm (IFA) is introduced to solve the given optimization function. Finally, results in the benchmark IEEE-33 node radial distribution system validate the effectiveness of the proposed method.


## 2. Generalized Load Model

The typical topology of DWFs in distributed network system is shown in Figure1, where DWFs are integrated at point of interconnection (POI) and near diverse end-users. The loads are not explicitly single type such as constant power (CPL), industrial (IL), commercial (CL), and residential loads (RL). Depending on the nature of area, we defined the mixed load (ML) which combines different kinds of loads. Since different loads correspond to the different characteristics of network loss and voltage profile, the loads’ characteristics need to be analyzed. Therefore, a generalized load model relating the load power to voltage should be investigated. Based on the static voltage characteristics, the generalized load model could be given as [12](1)PLi=PLiDa1UiUNα0+b1UiUNα1+c1UiUNα2+d1UiUNα3QLi=QLiDa2UiUNβ0+b2UiUNβ1+c2UiUNβ2+d2UiUNβ3where UN and Ui are, respectively, the nominal and actual node voltages, PLi, QLi, PLiD, and QLiD, are, respectively, the load active power and reactive power under nominal and actual node voltage conditions.Figure 1
Typical topology of DWFs in DN.In (1), load demands of different types are described by the exponent and weight values [13]. The typical exponent values of different load types are shown in Table 1. Obviously, the exponent values α0 and β0 are both zero for the constant power load.a1,b1,c1, andd1 are the active power weight coefficients,a2,b2,c2, andd2 are the reactive power weight coefficients, which are determined by the percentages of active and reactive power consumption or demand.Table 1
Voltage characteristic coefficients for different load types.
Season RL CL IL α 1 β 1 α 2 β 2 α 3 β 3 Spring 1.2 4.38 1.26 3.35 0.18 6 Summer 0.72 2.96 1.26 3.5 0.18 6 Autumn 0.98 3.52 0.98 3.95 0.18 6 Winter 1.04 4.18 1.5 3.15 0.18 6


## 3. Reactive Power Support Capabilities of DWFs

Doubly-Fed Induction Generator (DFIG) based wind generator is widely used in DWFs, which mainly consists of a wind turbine, a gearbox, DFIG, and back-to-back converters, as shown in Figure2.Figure 2
DFIG based wind generator.The back-to-back converters interfacing the DFIG based wind turbines to the grid play a significant role in the whole system, which are composed of grid-side converter (GSC) and rotor-side converter (RSC). The GSC is connected to the grid and is usually controlled to provide a steady DC-link voltage and meet the power quality requirements, while the RSC is connected to the rotor windings of the DFIG and control the active and reactive power injected into the grid. The total generated power is divided into the power from the stator and the power from the rotor controlled by the back-to-back converter. Therefore, their reactive power support capabilities are complicated since they could be generated by both stator and rotor sides via back-to-back converters, which are thoroughly discussed in this Section [14].By using stator voltage oriented vector control [15], due to the flux lags the voltage 90° in stator side in steady state, the grid-voltageus has no component on the d axis, which is the direct component of the stator voltage uds=0. Finally, active power Ps and reactive power Qs provided by stator side are described as(2)Ps=32uqsiqs=32LmLsusiqrQs=32uqsids=-32LmLsusidr-32us2ωsLswhere uqs, ids, iqs, idr, and iqr are, respectively, quadrature component of the stator voltage, stator current, and rotor current in the rotating frame, ωs is the air-gap magnetic field rotating angular velocity of stator, Ls and Lm are, respectively, stator winding inductance and mutual inductance between stator and rotor windings, andus is the peak value of stator voltage.Therefore, the total active powerPe injected by DFIG into the grid could be derived as(3)Pe=1+sPswhere s is the slip.The maximum and minimum reactive powerQsmax and Qsmin provided by the stator side are derived as [16]. (4)Qsmax=-3us22ωsLs+3usLmirmax2Ls2-Ps2Qsmin=-3us22ωsLs-3usLmirmax2Ls2-Ps2where irmax is the rotor current limit.Similarly, the maximum and minimum reactive powerQgmax and Qgmin provided by GSC are described as(5)Qgmax=Sc2-sPs2Qgmin=-Qgmaxwhere Sc is the apparent power of GSC.Combining (4) and (5), the total maximum and minimum reactive power Qemax and Qemin of DFIG considering both stator and rotor sides are obtained as(6)Qemax=Qsmax+QgmaxQemin=Qsmin+QgminIt can be seen from (4) that the maximum and minimum reactive power capabilities Qsmax and Qsmin are unbalanced and the lower one dominates the overall reactive power capabilities, which could be improved by installing capacitors at the grid-connected point. Consequently, the balanced reactive power capabilities Qcmax and Qcmin with proper capacitors are revised as follows:(7)Qcmax=αirmax2-Ps2+Sc2-sPs2Qcmin=-Qcmaxwhere α = 1.5usLm/Ls.Figure3 summarizes the reactive power capabilities, showing that DFIG has the great potentials in power regulation capabilities. In Figure 3, the purple solid line is the reactive power region generated by stator side and the blue dash line is the reactive power region generated by both stator and rotor sides and the shading part denoting the reactive power region of GSC. The red dash line is the reactive power region considering the added capacitor. Obviously, it has Qcmax>Qemax>Qsmax and the rotor currentirc with paralleled capacitor is much smaller thanir at the operating point (P0,Q0), which means that the reactive power capabilities could be enhanced by implementing proper control and capacitors. Furthermore, with the increasing of active power, reactive power capabilities are decreased gradually, which are mainly restricted by irmax.Figure 3
Reactive power capabilities of DFIG.


## 4. Influences on Network Loss and Voltage Profile with the Integration of DWFs

### 4.1. Influence on Network Loss

In an M-node distribution network, the total active and reactive power network lossPL andQL could be expressed as [17](8)PL=∑i=1M∑j=1MkijPiPj+QiQj+hijQiPj-PiQjQL=∑i=1M∑j=1MγijPiPj+QiQj+ξijQiPj-PiQjwhere(9)kij=rijUiUjcos⁡δijhij=rijUiUjsin⁡δijγij=xijUiUjcos⁡δijξij=rijUiUjsin⁡δijIn (8) and (9),Pi,Pj,Qi, andQj are the active and reactive power injections at the ith and jth nodes, respectively;Ui andUj are the ith and jth node voltages,δij is power angle difference between nodesi andj, andrij andxij are resistance and reactance between nodesi andj.When the DWFs are integrated into the M-node distribution network, the total active and reactive power network loss are updated as (10), where the notations of total network loss are, respectively, modified asPLDWF andQLDWF.(10)PLDWF=∑i=1M∑j=1MkijPGi+Pei-PLiPj+QjQGi+tan⁡ϕiPei-QLi+hijQGi+tan⁡ϕiPei-QLiPj-PGi+Pei-PLiQjQLDWF=∑i=1M∑j=1MγijPGi+Pei-PLiPj+QjQGi+tan⁡ϕiPei-QLi+ξijQGi+tan⁡ϕiPei-QLiPj-PGi+Pei-PLiQjIn (10),PGi andQGi are the injecting active and reactive power from grid at nodei;Pei (see (3)) andϕi are the injecting active power and power factor angle of DWF at nodei;PLi andQLi (see (1)) are load active and reactive power at nodei, respectively.Indexes of active and reactive power network loss APL and RPL could be further derived as(11)ηAPL=PLDWFPLηRPL=QLDWFQLObviously, the active power and power factor (PF) of DWFs have great influences on network loss. In this paper, negative power factor represents capacitive reactive power while positive power factor is on behalf of inductive reactive power according to the convention. For a given PF, the network loss will first decrease and then increase with the increase of the integrated power as shown in Figure4. For a given integrated power, the change trend is similar. That means that the curve which relates network loss to integrated active power and power factor presents parabolic shape. As maximum integrated power of DWF increasing, the power from upstream branch to load will be reduced, resulting in decreased active power loss. After the minimum network loss is achieved, if maximum integrated power continues to increase, the reverse power flow will be shown up, which increases the network loss as illustrated in Figure 4. It implies that the system performance could be optimized through power factor regulation of DWFs for a given distribution network.Figure 4
Impacts of maximum integrated power and PF on network loss.


### 4.2. Influence on Voltage Profile

Compared to the conventional DN, the voltage profile is quite different with the introduction of both the diverse load types and stochastic wind power. In traditional DN, the voltage profile is simple. The node voltage usually decreases as the electrical distance from the node to the main utility increases and may become lower than the minimum voltage at the end of the feeder. However, with the integration of DWFs, the voltage profile is complicated. Three DWFs were connected into IEEE 33 system, located in the points 14, 24, and 30, and the voltage curves are shown in Figure5.Figure 5
33-bus network voltage profile in the four PF scenarios.It can be concluded that the PF can make a big difference on the voltage profile. If power factor or reactive power of DWFs can be adjusted, the voltage profile may be improved. The comparative results also indicate that constant fixed power factor values are not always the preferred solutions.


## 4.1. Influence on Network Loss

In an M-node distribution network, the total active and reactive power network lossPL andQL could be expressed as [17](8)PL=∑i=1M∑j=1MkijPiPj+QiQj+hijQiPj-PiQjQL=∑i=1M∑j=1MγijPiPj+QiQj+ξijQiPj-PiQjwhere(9)kij=rijUiUjcos⁡δijhij=rijUiUjsin⁡δijγij=xijUiUjcos⁡δijξij=rijUiUjsin⁡δijIn (8) and (9),Pi,Pj,Qi, andQj are the active and reactive power injections at the ith and jth nodes, respectively;Ui andUj are the ith and jth node voltages,δij is power angle difference between nodesi andj, andrij andxij are resistance and reactance between nodesi andj.When the DWFs are integrated into the M-node distribution network, the total active and reactive power network loss are updated as (10), where the notations of total network loss are, respectively, modified asPLDWF andQLDWF.(10)PLDWF=∑i=1M∑j=1MkijPGi+Pei-PLiPj+QjQGi+tan⁡ϕiPei-QLi+hijQGi+tan⁡ϕiPei-QLiPj-PGi+Pei-PLiQjQLDWF=∑i=1M∑j=1MγijPGi+Pei-PLiPj+QjQGi+tan⁡ϕiPei-QLi+ξijQGi+tan⁡ϕiPei-QLiPj-PGi+Pei-PLiQjIn (10),PGi andQGi are the injecting active and reactive power from grid at nodei;Pei (see (3)) andϕi are the injecting active power and power factor angle of DWF at nodei;PLi andQLi (see (1)) are load active and reactive power at nodei, respectively.Indexes of active and reactive power network loss APL and RPL could be further derived as(11)ηAPL=PLDWFPLηRPL=QLDWFQLObviously, the active power and power factor (PF) of DWFs have great influences on network loss. In this paper, negative power factor represents capacitive reactive power while positive power factor is on behalf of inductive reactive power according to the convention. For a given PF, the network loss will first decrease and then increase with the increase of the integrated power as shown in Figure4. For a given integrated power, the change trend is similar. That means that the curve which relates network loss to integrated active power and power factor presents parabolic shape. As maximum integrated power of DWF increasing, the power from upstream branch to load will be reduced, resulting in decreased active power loss. After the minimum network loss is achieved, if maximum integrated power continues to increase, the reverse power flow will be shown up, which increases the network loss as illustrated in Figure 4. It implies that the system performance could be optimized through power factor regulation of DWFs for a given distribution network.Figure 4
Impacts of maximum integrated power and PF on network loss.


## 4.2. Influence on Voltage Profile

Compared to the conventional DN, the voltage profile is quite different with the introduction of both the diverse load types and stochastic wind power. In traditional DN, the voltage profile is simple. The node voltage usually decreases as the electrical distance from the node to the main utility increases and may become lower than the minimum voltage at the end of the feeder. However, with the integration of DWFs, the voltage profile is complicated. Three DWFs were connected into IEEE 33 system, located in the points 14, 24, and 30, and the voltage curves are shown in Figure5.Figure 5
33-bus network voltage profile in the four PF scenarios.It can be concluded that the PF can make a big difference on the voltage profile. If power factor or reactive power of DWFs can be adjusted, the voltage profile may be improved. The comparative results also indicate that constant fixed power factor values are not always the preferred solutions.


## 5. Optimal Power Factor Regulation of DWFs

Based on the above analysis, we can know the network loss and voltage profile can be further optimized by adjusting PF.


### 5.1. Multiobjective Optimization Function

#### 5.1.1. Multiobjective Optimization Function

In order to get the optimal results, considering various load types and stochastic wind conditions, multiobjective optimal function is constructed, whose equation is(12)MOF=min⁡k1ηAPL+k2ηRPL+k3δvwhereηAPL is the degree of active power loss,ηRPL means the degree of reactive power loss, andδv is voltage deviation, whose calculation is(13)δv=maxj=1M⁡UN-UjUNForm the above equations, it can be known that the network loss and voltage deviation are considered to construct MOF. The optimal value is the minimum MOF. Besides, all weight factors in (12) are determined according to the individual impacts and importance of the particular index. The main objective is to minimize the overall network loss of system, soηAPL gets the highest weight of 0.45 andηRPL gets the second highest weight of 0.38, while δv gets a weight of 0.17 in order to maintain good power quality and voltage profile.


### 5.2. Constraints

The constraints should be set to solve the equations, which include power flow, output power, node voltage, branch current, and climbing rate. The details are shown as follow:( 1 ) Power flow equality constraints (14)PGi+Pei-PLi-Ui∑j=1MUjGijcos⁡δij+Bijsin⁡δij=0QGi+tan⁡ϕiPei-QLi-Ui∑j=1MUjGijcos⁡δij-Bijsin⁡δij=0whereBij andGij are, respectively, susceptance and conductance between nodesi andj,PGi andQGi are, respectively, injected active and reactive power from grid,Pei andQei are the active and reactive power of DWF at nodei, Uimin and Uimax are, respectively, voltage lower and upper limits, andU1 is the first node voltage.( 2 ) Output power inequality constraints of DWFs are(15)0≤Pei≤PDWFimaxQei=tan⁡ϕiPei≤Qcmaxwhere PDWFimax is the real-time maximum power according to the real-time wind conditions and the maximum balanced reactive power Qcmax is determined according to (7), fully considering the reactive capability of the DWFs with shunt capacitor.( 3 ) Node voltage inequality constraints are(16)Uimin≤Ui≤Uimax( 4 ) Node voltage inequality constraints considering no reverse power flow at the first node are(17)Ui≤U1( 5 ) Branch current inequality constraints are(18)Ii≤IimaxHere Iimax is the ith branch current upper limit.( 6 ) Climbing rate constraints of DWFs(19)ΔPDWFimin≤Pei,t-Pei,t-1≤ΔPDWFimaxIn (19), ΔPDWFimin and ΔPDWFimax are, respectively, declining and rising rate limits of DWFs.


### 5.3. Improved Firefly Algorithm Optimization

#### 5.3.1. Firefly Algorithm Theories

The conventional Firefly algorithm (FA) is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behaviour of fireflies. FA has many advantages, such as simple operation, strong robustness, easy implementation, etc. compared with the existing genetic algorithm and particle swarm optimization algorithm, FA owns better ability to search the global optimal solution, what’s more, whose convergence speed is faster. FA has been successfully applied to the optimization of nonlinear issues [18]. The following simplified rules are considered. All fireflies are unisexual, so that any individual firefly will be attracted to all other fireflies. Attractiveness is proportional to their brightness and, for any two fireflies, the less bright one will be attracted by the brighter one. Brightness and, therefore, attractiveness decrease with the distance between the fireflies. If there are no fireflies brighter than a given firefly, it will move randomly. Mathematically, the FA can be expressed by three equations [19].( 1 ) Light intensity LI is(20)LI=LI0×e-γdijwhereLI0 is the maximum light intensity atdij = 0 anddij is the Euclidean distance between two firefliesi andj. Because of the light absorption in medium, the light intensity decreases as the mutual distance increases, which is controlled by the absorption coefficient γ.( 2 ) Attractiveness β is(21)β=β0×e-γdij2where β0 is the maximum attractiveness.( 3 ) When fireflyi is attracted by fireflyj, the update movement equation can be described as(22)xj=xi+βxi-xj+αrand-0.5wherexi andxj are the positions of firefliesi andj, α is constant and refers to the random movement factor, and rand is a random number, generated in between 0 and 1, with uniform distribution. The brightness is determined by the value of the objective function of a given problem [20].


#### 5.3.2. Improved Firefly Algorithm Principles

Based on its pattern, the chaos motion can traverse all states in a certain range without repeating and optimization search using chaos variables is better than blindly and disorderly random search. To avoid premature convergence of the population and slow convergence rate, improved FA (IFA) is introduced, where the chaos theory is integrated into optimization parameters of the firefly algorithm.Equations (20) to (22) indicate that the firefly parameters α,β, and γ play significant roles in the optimization process. If these parameters can be randomly changed in the feasible region, the global optimization ability could be improved. Thus, logistic mapping function is used to regulate these parameters, the improved equations of xj, γt, and α(t) are shown in(23)xj=xi+βxi-xj+αtrand-0.5(24)γt=u1γt-11-γt-1(25)αt=u2αt-11-αt-1Here chaos parameter[u1u2] is set to [44].After each iteration, a set of random parameter values[αβγ] are calculated by logistic mapping function. The range of three parameters is within rand [0,1] in the iteration, which effectively leads the algorithm to land at the best solution with minimum computational effort and provides a good way to move away from the local search to a search on the global scale. The modified firefly could improve the convergence performance.


### 5.4. Optimal Power Factor Regulation of Dispersed Wind Farms Based on Improved Firefly Algorithm

The comprehensive procedures for the multiobjective optimal function using IFA are given through the flowchart of Figure6.Figure 6
The flow chart of improved firefly algorithm.In Figure6, PF is the firefly position, the differences ofPF are (ΔPF), which means the attractiveness, the exponential function ofΔPF means the attractiveness,LI is MOF, and the global best MOF is the largestLI.At each iterative step, the light intensity and the attractiveness of each firefly are calculated.LI of each firefly is compared with light intensity of all other fireflies and the positions of the fireflies are updated using (22). The firefly with the largestLI is considered as the global optimal after completing an adequate number of iterations.The purpose is to obtain the optimized power factor. Therefore, the position variable in (22) is actually the power factor, whereas the second and third terms of (22) can be regarded as the correction terms for power factor.


## 5.1. Multiobjective Optimization Function

### 5.1.1. Multiobjective Optimization Function

In order to get the optimal results, considering various load types and stochastic wind conditions, multiobjective optimal function is constructed, whose equation is(12)MOF=min⁡k1ηAPL+k2ηRPL+k3δvwhereηAPL is the degree of active power loss,ηRPL means the degree of reactive power loss, andδv is voltage deviation, whose calculation is(13)δv=maxj=1M⁡UN-UjUNForm the above equations, it can be known that the network loss and voltage deviation are considered to construct MOF. The optimal value is the minimum MOF. Besides, all weight factors in (12) are determined according to the individual impacts and importance of the particular index. The main objective is to minimize the overall network loss of system, soηAPL gets the highest weight of 0.45 andηRPL gets the second highest weight of 0.38, while δv gets a weight of 0.17 in order to maintain good power quality and voltage profile.


## 5.1.1. Multiobjective Optimization Function

In order to get the optimal results, considering various load types and stochastic wind conditions, multiobjective optimal function is constructed, whose equation is(12)MOF=min⁡k1ηAPL+k2ηRPL+k3δvwhereηAPL is the degree of active power loss,ηRPL means the degree of reactive power loss, andδv is voltage deviation, whose calculation is(13)δv=maxj=1M⁡UN-UjUNForm the above equations, it can be known that the network loss and voltage deviation are considered to construct MOF. The optimal value is the minimum MOF. Besides, all weight factors in (12) are determined according to the individual impacts and importance of the particular index. The main objective is to minimize the overall network loss of system, soηAPL gets the highest weight of 0.45 andηRPL gets the second highest weight of 0.38, while δv gets a weight of 0.17 in order to maintain good power quality and voltage profile.


## 5.2. Constraints

The constraints should be set to solve the equations, which include power flow, output power, node voltage, branch current, and climbing rate. The details are shown as follow:( 1 ) Power flow equality constraints (14)PGi+Pei-PLi-Ui∑j=1MUjGijcos⁡δij+Bijsin⁡δij=0QGi+tan⁡ϕiPei-QLi-Ui∑j=1MUjGijcos⁡δij-Bijsin⁡δij=0whereBij andGij are, respectively, susceptance and conductance between nodesi andj,PGi andQGi are, respectively, injected active and reactive power from grid,Pei andQei are the active and reactive power of DWF at nodei, Uimin and Uimax are, respectively, voltage lower and upper limits, andU1 is the first node voltage.( 2 ) Output power inequality constraints of DWFs are(15)0≤Pei≤PDWFimaxQei=tan⁡ϕiPei≤Qcmaxwhere PDWFimax is the real-time maximum power according to the real-time wind conditions and the maximum balanced reactive power Qcmax is determined according to (7), fully considering the reactive capability of the DWFs with shunt capacitor.( 3 ) Node voltage inequality constraints are(16)Uimin≤Ui≤Uimax( 4 ) Node voltage inequality constraints considering no reverse power flow at the first node are(17)Ui≤U1( 5 ) Branch current inequality constraints are(18)Ii≤IimaxHere Iimax is the ith branch current upper limit.( 6 ) Climbing rate constraints of DWFs(19)ΔPDWFimin≤Pei,t-Pei,t-1≤ΔPDWFimaxIn (19), ΔPDWFimin and ΔPDWFimax are, respectively, declining and rising rate limits of DWFs.


## 5.3. Improved Firefly Algorithm Optimization

### 5.3.1. Firefly Algorithm Theories

The conventional Firefly algorithm (FA) is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behaviour of fireflies. FA has many advantages, such as simple operation, strong robustness, easy implementation, etc. compared with the existing genetic algorithm and particle swarm optimization algorithm, FA owns better ability to search the global optimal solution, what’s more, whose convergence speed is faster. FA has been successfully applied to the optimization of nonlinear issues [18]. The following simplified rules are considered. All fireflies are unisexual, so that any individual firefly will be attracted to all other fireflies. Attractiveness is proportional to their brightness and, for any two fireflies, the less bright one will be attracted by the brighter one. Brightness and, therefore, attractiveness decrease with the distance between the fireflies. If there are no fireflies brighter than a given firefly, it will move randomly. Mathematically, the FA can be expressed by three equations [19].( 1 ) Light intensity LI is(20)LI=LI0×e-γdijwhereLI0 is the maximum light intensity atdij = 0 anddij is the Euclidean distance between two firefliesi andj. Because of the light absorption in medium, the light intensity decreases as the mutual distance increases, which is controlled by the absorption coefficient γ.( 2 ) Attractiveness β is(21)β=β0×e-γdij2where β0 is the maximum attractiveness.( 3 ) When fireflyi is attracted by fireflyj, the update movement equation can be described as(22)xj=xi+βxi-xj+αrand-0.5wherexi andxj are the positions of firefliesi andj, α is constant and refers to the random movement factor, and rand is a random number, generated in between 0 and 1, with uniform distribution. The brightness is determined by the value of the objective function of a given problem [20].


### 5.3.2. Improved Firefly Algorithm Principles

Based on its pattern, the chaos motion can traverse all states in a certain range without repeating and optimization search using chaos variables is better than blindly and disorderly random search. To avoid premature convergence of the population and slow convergence rate, improved FA (IFA) is introduced, where the chaos theory is integrated into optimization parameters of the firefly algorithm.Equations (20) to (22) indicate that the firefly parameters α,β, and γ play significant roles in the optimization process. If these parameters can be randomly changed in the feasible region, the global optimization ability could be improved. Thus, logistic mapping function is used to regulate these parameters, the improved equations of xj, γt, and α(t) are shown in(23)xj=xi+βxi-xj+αtrand-0.5(24)γt=u1γt-11-γt-1(25)αt=u2αt-11-αt-1Here chaos parameter[u1u2] is set to [44].After each iteration, a set of random parameter values[αβγ] are calculated by logistic mapping function. The range of three parameters is within rand [0,1] in the iteration, which effectively leads the algorithm to land at the best solution with minimum computational effort and provides a good way to move away from the local search to a search on the global scale. The modified firefly could improve the convergence performance.


## 5.3.1. Firefly Algorithm Theories

The conventional Firefly algorithm (FA) is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behaviour of fireflies. FA has many advantages, such as simple operation, strong robustness, easy implementation, etc. compared with the existing genetic algorithm and particle swarm optimization algorithm, FA owns better ability to search the global optimal solution, what’s more, whose convergence speed is faster. FA has been successfully applied to the optimization of nonlinear issues [18]. The following simplified rules are considered. All fireflies are unisexual, so that any individual firefly will be attracted to all other fireflies. Attractiveness is proportional to their brightness and, for any two fireflies, the less bright one will be attracted by the brighter one. Brightness and, therefore, attractiveness decrease with the distance between the fireflies. If there are no fireflies brighter than a given firefly, it will move randomly. Mathematically, the FA can be expressed by three equations [19].( 1 ) Light intensity LI is(20)LI=LI0×e-γdijwhereLI0 is the maximum light intensity atdij = 0 anddij is the Euclidean distance between two firefliesi andj. Because of the light absorption in medium, the light intensity decreases as the mutual distance increases, which is controlled by the absorption coefficient γ.( 2 ) Attractiveness β is(21)β=β0×e-γdij2where β0 is the maximum attractiveness.( 3 ) When fireflyi is attracted by fireflyj, the update movement equation can be described as(22)xj=xi+βxi-xj+αrand-0.5wherexi andxj are the positions of firefliesi andj, α is constant and refers to the random movement factor, and rand is a random number, generated in between 0 and 1, with uniform distribution. The brightness is determined by the value of the objective function of a given problem [20].


## 5.3.2. Improved Firefly Algorithm Principles

Based on its pattern, the chaos motion can traverse all states in a certain range without repeating and optimization search using chaos variables is better than blindly and disorderly random search. To avoid premature convergence of the population and slow convergence rate, improved FA (IFA) is introduced, where the chaos theory is integrated into optimization parameters of the firefly algorithm.Equations (20) to (22) indicate that the firefly parameters α,β, and γ play significant roles in the optimization process. If these parameters can be randomly changed in the feasible region, the global optimization ability could be improved. Thus, logistic mapping function is used to regulate these parameters, the improved equations of xj, γt, and α(t) are shown in(23)xj=xi+βxi-xj+αtrand-0.5(24)γt=u1γt-11-γt-1(25)αt=u2αt-11-αt-1Here chaos parameter[u1u2] is set to [44].After each iteration, a set of random parameter values[αβγ] are calculated by logistic mapping function. The range of three parameters is within rand [0,1] in the iteration, which effectively leads the algorithm to land at the best solution with minimum computational effort and provides a good way to move away from the local search to a search on the global scale. The modified firefly could improve the convergence performance.


## 5.4. Optimal Power Factor Regulation of Dispersed Wind Farms Based on Improved Firefly Algorithm

The comprehensive procedures for the multiobjective optimal function using IFA are given through the flowchart of Figure6.Figure 6
The flow chart of improved firefly algorithm.In Figure6, PF is the firefly position, the differences ofPF are (ΔPF), which means the attractiveness, the exponential function ofΔPF means the attractiveness,LI is MOF, and the global best MOF is the largestLI.At each iterative step, the light intensity and the attractiveness of each firefly are calculated.LI of each firefly is compared with light intensity of all other fireflies and the positions of the fireflies are updated using (22). The firefly with the largestLI is considered as the global optimal after completing an adequate number of iterations.The purpose is to obtain the optimized power factor. Therefore, the position variable in (22) is actually the power factor, whereas the second and third terms of (22) can be regarded as the correction terms for power factor.


## 6. Verification of Control Method and Comparison

### 6.1. Verification of Optimized Control Method

The benchmark IEEE-33 node radial distribution system is used for validating the proposed optimization method as shown in Figure7. Different load types (RL, IL, CL and ML) with the same peak active and reactive power values 3.72 MW and 2.30 MVar are conducted, where mixed load (ML) consists of 40% industrial load (IL), 45% residential load (RL), and 15% commercial load (CL), respectively. Furthermore, the data with one full year from August 2014 to August 2015 in a real DWF is employed and analyzed as the basis of forecasting. The Weibull distribution is used for characterizing wind speed. Taking one-day data for test, the typical active power profiles of diverse loads and DWFs are shown in Figure 8. To find a better solution, the empirical parameter settings are listed in Table 2.Table 2
Terminologies and parameter settings of IFA and DN system with integration of DWFs.
Parameter Value Description Population size 50 Number of Fireflies/Particles N g 200 Number of generations A [ 0 ,  1 ] Randomization parameter β 0 0.1 Initial attractiveness Γ 0.2 Absorption coefficientFigure 7
IEEE-33 node radial distribution system with 3 POIs of DWFs.Figure 8
Typical active power profiles of diverse load types and DWF used.As shown in Figure8, the active power of DWF has stronger randomness compared with other kinds of loads. Generally, RL, IL, CL, and ML have similar variation characteristics in one day.In order to demonstrate the effectiveness of network loss under diverse load type conditions by using four constant PFs (0.95, unity power factor, -0.95, -0.85) and proposed optimal PF control method, respectively, both single load type and mixed load type are conducted, where three single load types (IL, RL, and CL) are with the same peak active and reactive power values 3.72 MW and 2.30 MVar, and the mixed load type consists of 40% industrial load, 45% residential load, and 15% commercial load. The resultsηAPL and ηRPL of different load types are shown in Figure 9.Figure 9
Comparative results of network loss under diverse load type conditions by using constant PFs and optimal PF regulation.
(a)
APL index (b)
RPL indexIt can be seen in Figure9 that ηAPL and ηRPL of different load types present quite differently. ηAPL and ηRPL for CL are the highest and the minimum values are achieved for RL. ηAPL and ηRPL of CL are even five times higher than those of IL, which indicates that load types might not be neglected in the optimization process. Compared to four constant PFs (0.95, unity power factor, -0.95, and -0.85), ηAPL and ηRPL are achieved by using the proposed optimal PF. The results indicate that the network loss can be decreased significantly by using the proposed optimal PF method, and the effectiveness of the proposed method is proved.To further investigate the effectiveness of the proposed method on voltage profile, one-day voltage profile is under mixed load condition by using four constant PFs (0.95, unity power factor, -0.95, and -0.85), and proposed optimal PF regulations are comparatively presented in Figure10.Figure 10
Voltage profile and power factor by using constant PFs and optimal PF regulation under mixed load condition.
(a)
Voltage profile (b)
Power factorIt can be easily known that the voltage profile is around 1 based on the proposed optimal PF method, which is to say that the voltage deviation is around 0 and the stability is the best. Compared with the positive power factor, when the values of PFs are 0.95 and 1, the DFWs need reactive power and the voltage deviation is large (>0.02). The voltage curve of Cons -0.95 is improved because the DWFs are providing reactive power and then the reactive power injected by the upstream branch will be diminished, but the difference still exists obviously. If the DWFs generate excess reactive power, the voltage profile will be also deteriorated, as demonstrated by the curve of Cons -0.85. Anyhow, as loads are real-time changed, the constant power factor could not give a good voltage profile all the time. Figure10(b) describes the real-time mutative characteristics, which shows that the PF can be adjusted adaptively to obtain the optimal voltage and lowest network loss.In summary, the varieties of loads would affect the reactive power for practical conditions and the fixed power factors cannot meet the actual distribution networks. Through the optimal PF which is calculated by the IFA algorithm, the network loss and voltage stability issues can be improved significantly.


### 6.2. Compare with Traditional Optimized Algorithms

The output power of conventional wind farm is high in the daytime and low at night, while the dispersed wind farms are different, whose power is easily affected by surrounding environment and shows the stochastic fluctuation; the regular of output power of dispersed wind farms is similar; thus any moment can be chosen as an example, and here we chose “Time 18h” as an example.Reference [21] used Particle Swarm Optimization (PSO) algorithm, FA algorithm was introduced in [22], and Artificial Bee Colony (ABC) algorithm was used in [23]; thus we compare the above algorithms and the parameters of each optimized algorithms are based on above references. To validate the effectiveness of the given IFA, the comparative convergence results and computation time under mixed load condition at 18h are compared, and the results are shown in Figures 11(a) and 11(b).Figure 11
Convergence curves of network loss and voltage deviation by using different algorithms under mixed load condition.
(a)
Network loss (b)
Voltage deviationIt can be known from Figure11, compared with PSO, ABC, and FA algorithms, that the main advantages of the IFA are the efficiency of multiobjective optimization, because it uses mainly real random numbers and it is based on the global communication among the swarming particles (the fireflies); when the iterations are around 50, the IFA method owns the minimum network loss (less than 200kW) and voltage deviation (about 0.04), while other methods need to take about 100 iterations, which means that the best results can be obtained by the proposed control method with the fewer iterations.When best results are obtained, the computation time is counted, and the comparative results are shown in Table3.Table 3
Computation time of different algorithms.
Algorithms Computation time (s) IFA 28.07 ABC 52.69 FA 59.87 PSO 65.53It can be concluded that compared with other algorithms, since IFA algorithm has less program occupation and simplified variable parameter adjustment, computation time is 28.07s, much lower than ABC (52.69s), FA (59.87s), and PSO (65.53s). Therefore, the time is much saved up by using IFA. In a word, the proposed IFA has more advantages than the existing optimal algorithms, especially in network loss, voltage deviation, and calculation time.


## 6.1. Verification of Optimized Control Method

The benchmark IEEE-33 node radial distribution system is used for validating the proposed optimization method as shown in Figure7. Different load types (RL, IL, CL and ML) with the same peak active and reactive power values 3.72 MW and 2.30 MVar are conducted, where mixed load (ML) consists of 40% industrial load (IL), 45% residential load (RL), and 15% commercial load (CL), respectively. Furthermore, the data with one full year from August 2014 to August 2015 in a real DWF is employed and analyzed as the basis of forecasting. The Weibull distribution is used for characterizing wind speed. Taking one-day data for test, the typical active power profiles of diverse loads and DWFs are shown in Figure 8. To find a better solution, the empirical parameter settings are listed in Table 2.Table 2
Terminologies and parameter settings of IFA and DN system with integration of DWFs.
Parameter Value Description Population size 50 Number of Fireflies/Particles N g 200 Number of generations A [ 0 ,  1 ] Randomization parameter β 0 0.1 Initial attractiveness Γ 0.2 Absorption coefficientFigure 7
IEEE-33 node radial distribution system with 3 POIs of DWFs.Figure 8
Typical active power profiles of diverse load types and DWF used.As shown in Figure8, the active power of DWF has stronger randomness compared with other kinds of loads. Generally, RL, IL, CL, and ML have similar variation characteristics in one day.In order to demonstrate the effectiveness of network loss under diverse load type conditions by using four constant PFs (0.95, unity power factor, -0.95, -0.85) and proposed optimal PF control method, respectively, both single load type and mixed load type are conducted, where three single load types (IL, RL, and CL) are with the same peak active and reactive power values 3.72 MW and 2.30 MVar, and the mixed load type consists of 40% industrial load, 45% residential load, and 15% commercial load. The resultsηAPL and ηRPL of different load types are shown in Figure 9.Figure 9
Comparative results of network loss under diverse load type conditions by using constant PFs and optimal PF regulation.
(a)
APL index (b)
RPL indexIt can be seen in Figure9 that ηAPL and ηRPL of different load types present quite differently. ηAPL and ηRPL for CL are the highest and the minimum values are achieved for RL. ηAPL and ηRPL of CL are even five times higher than those of IL, which indicates that load types might not be neglected in the optimization process. Compared to four constant PFs (0.95, unity power factor, -0.95, and -0.85), ηAPL and ηRPL are achieved by using the proposed optimal PF. The results indicate that the network loss can be decreased significantly by using the proposed optimal PF method, and the effectiveness of the proposed method is proved.To further investigate the effectiveness of the proposed method on voltage profile, one-day voltage profile is under mixed load condition by using four constant PFs (0.95, unity power factor, -0.95, and -0.85), and proposed optimal PF regulations are comparatively presented in Figure10.Figure 10
Voltage profile and power factor by using constant PFs and optimal PF regulation under mixed load condition.
(a)
Voltage profile (b)
Power factorIt can be easily known that the voltage profile is around 1 based on the proposed optimal PF method, which is to say that the voltage deviation is around 0 and the stability is the best. Compared with the positive power factor, when the values of PFs are 0.95 and 1, the DFWs need reactive power and the voltage deviation is large (>0.02). The voltage curve of Cons -0.95 is improved because the DWFs are providing reactive power and then the reactive power injected by the upstream branch will be diminished, but the difference still exists obviously. If the DWFs generate excess reactive power, the voltage profile will be also deteriorated, as demonstrated by the curve of Cons -0.85. Anyhow, as loads are real-time changed, the constant power factor could not give a good voltage profile all the time. Figure10(b) describes the real-time mutative characteristics, which shows that the PF can be adjusted adaptively to obtain the optimal voltage and lowest network loss.In summary, the varieties of loads would affect the reactive power for practical conditions and the fixed power factors cannot meet the actual distribution networks. Through the optimal PF which is calculated by the IFA algorithm, the network loss and voltage stability issues can be improved significantly.


## 6.2. Compare with Traditional Optimized Algorithms

The output power of conventional wind farm is high in the daytime and low at night, while the dispersed wind farms are different, whose power is easily affected by surrounding environment and shows the stochastic fluctuation; the regular of output power of dispersed wind farms is similar; thus any moment can be chosen as an example, and here we chose “Time 18h” as an example.Reference [21] used Particle Swarm Optimization (PSO) algorithm, FA algorithm was introduced in [22], and Artificial Bee Colony (ABC) algorithm was used in [23]; thus we compare the above algorithms and the parameters of each optimized algorithms are based on above references. To validate the effectiveness of the given IFA, the comparative convergence results and computation time under mixed load condition at 18h are compared, and the results are shown in Figures 11(a) and 11(b).Figure 11
Convergence curves of network loss and voltage deviation by using different algorithms under mixed load condition.
(a)
Network loss (b)
Voltage deviationIt can be known from Figure11, compared with PSO, ABC, and FA algorithms, that the main advantages of the IFA are the efficiency of multiobjective optimization, because it uses mainly real random numbers and it is based on the global communication among the swarming particles (the fireflies); when the iterations are around 50, the IFA method owns the minimum network loss (less than 200kW) and voltage deviation (about 0.04), while other methods need to take about 100 iterations, which means that the best results can be obtained by the proposed control method with the fewer iterations.When best results are obtained, the computation time is counted, and the comparative results are shown in Table3.Table 3
Computation time of different algorithms.
Algorithms Computation time (s) IFA 28.07 ABC 52.69 FA 59.87 PSO 65.53It can be concluded that compared with other algorithms, since IFA algorithm has less program occupation and simplified variable parameter adjustment, computation time is 28.07s, much lower than ABC (52.69s), FA (59.87s), and PSO (65.53s). Therefore, the time is much saved up by using IFA. In a word, the proposed IFA has more advantages than the existing optimal algorithms, especially in network loss, voltage deviation, and calculation time.


## 7. Conclusion

High penetration level of dispersed wind farms integration into distribution network presents great challenges and an optimal power factor regulation method based on improved firefly algorithm is proposed to ensure the instability of voltage and reduce network loss of dispersed wind farms. This paper proposes an optimal power factor regulation method under diverse load and stochastic wind conditions based on improved firefly algorithm and after discussion and verification, the conclusions are as follows:( 1 ) Different types of loads correspond to different characteristic of voltage profile and networks loss and the minimum networks loss and voltage deviation can be obtained based on the proposed control method.( 2 ) The chaos theory is introduced to improve the firefly algorithm effectively and the issues of local convergence and slow convergence of the original FA algorithm are well solved by IFA algorithm.( 3 ) Compared with the traditional constant PF control method, the proposed dynamic optimal PF control method can obtain the best results, which shows that the mutative PF can improve the stability of voltage and decrease the network loss and the network’s performance can be enhanced.



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*Source: 6203278-2018-09-27.xml*