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Reactive power control of wind farm made up with doubly fed induction generators in distribution system Jingjing Zhao a,∗ , Xin Li a , Jutao Hao b , Jiping Lu a a b

State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, 400044 Chongqing, China School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, 200093 Shanghai, China

a r t i c l e

i n f o

Article history: Received 7 April 2009 Received in revised form 8 August 2009 Accepted 31 October 2009 Available online 6 December 2009 Keywords: DFIG wind turbine Network reconﬁguration Particle swarm optimization Reactive power control Wind farm

a b s t r a c t In recent years, the number of small size wind farm made up with doubly fed induction generators (DFIG) located within the distribution system is rapidly increasing. DFIG can be utilized as the continuous reactive power source to support system voltage control by taking advantage of their reactive power control capability. In this paper, considering both reactive power control and distribution network reconﬁguration can be used to reduce power losses and improve voltage proﬁle, a joint optimization algorithm of combining reactive power control of wind farm and network reconﬁguration is proposed to obtain the optimal reactive power output of wind farm and network structure simultaneously. The proposed algorithm has been successfully implemented on the 16 bus distribution network and the results obtained demonstrate the efﬁciency of the algorithm. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Wind energy is one of the most important and promising renewable energy resources in the world. The penetration of the wind energy in electrical system is rapidly increasing. Currently, a growing number of small size wind farms used as DG sources are located within the distribution system. Installing wind farm in the distribution system can defer the investments for the distribution system expansion, but the intermittent and volatile nature of wind power generation may impact distribution system voltages, frequency and generation adequacy, so the electrical parameters of the distribution network have to be maintained [1–4]. When wind energy penetration is high, voltage control in the distribution system becomes particularly important. As the consequences, in many countries, the new established grid codes demand that wind farm made up with doubly fed induction generators should actively participate in improving voltage control in the distribution system [5]. The variable-speed wind turbine equipped with DFIG is the most popularly employed generator for the recently built wind farm. The variable-speed wind turbine has the ability to obtain the maximum active power from wind speed and control the reactive power independently [6,7]. Utilizing DFIG reactive power control capability, wind farm composed of DFIG can be used as the continuous reactive

∗ Corresponding author. E-mail addresses: [email protected], jjzhao [email protected] (J. Zhao). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.10.036

power source to support system voltage control with fewer costs on the reactive power compensation device. Wind farm reactive power control can reduce power losses and improve the voltage proﬁle at the user terminal by providing reactive power compensation in distribution systems. Wind farm reactive power output is controlled by the system optimal operation condition and the reactive power control capability of each DFIG wind turbine. There are many previous works on wind farm reactive power control. Ref. [8] proposes a detailed mathematical model of the DFIG and two alternative simulation models for the analysis of both the active and reactive power performances associated with a wind farm constituted exclusively by DFIG. Ref. [9] proposes an optimized dispatch control strategy for active and reactive powers delivered by a doubly fed induction generator in a wind park. Ref. [10] presents a control strategy developed for the reactive power regulation of wind farms made up with DFIG, in order to contribute to the voltage regulation of the electrical grid to which farms are connected. Ref. [11] describes the relation between active and reactive power in order to keep each DFIG operating inside the maximum stator and rotor currents and the steady state stability limit. Ref. [12] describes a PI-based control algorithm to govern the net reactive power ﬂowing between wind farms composed of doubly fed induction generators and the grid. Network reconﬁguration is one of the most signiﬁcant control schemes in the distribution system, which alters the topological structure of distribution feeders by changing open/closed status of sectionalizing and tie switches. The purpose of the optimal distribution network reconﬁguration problem is to identify an

J. Zhao et al. / Electric Power Systems Research 80 (2010) 698–706

optimal radial operating structure that reduces real power losses or improves voltage proﬁle while satisfying operating constraints. Most of the methods used for the network reconﬁguration in the literature are heuristic methods [13–15]. The other class of approaches applied to network reconﬁguration problem is based on artiﬁcial intelligence searching algorithms, such as genetic algorithm, simulated annealing algorithm, tabu search algorithm, etc. [16–18]. However, the previous studies perform wind farm reactive power control with no consideration of network reconﬁguration, or perform network reconﬁguration with no consideration of wind farm reactive power control, which cannot ﬁnd the optimal network structure and wind farm reactive power output at the same time for system optimal operation condition. In this paper, a joint optimization algorithm of combining reactive power control of wind farm and network reconﬁguration is proposed to obtain the optimal reactive power output of wind farm and the optimal network structure simultaneity. To ﬁnd the optimal reactive power output of wind farm, an improved hybrid particle swarm optimization with wavelet mutation (HPSOWM) algorithm is utilized, meanwhile a binary particle swarm optimization (BPSO) algorithm is developed to ﬁnd the optimal network structure for each particle updating instance at each iteration of wind farm reactive power output optimization algorithm.

699

Fig. 2. Power curve for a typical 1500 kW DFIG wind turbine.

2. System model and control 2.1. DFIG wind turbine model Fig. 3. DFIG equivalent circuit.

Fig. 1 shows the model of DFIG wind turbine consisting of a pitch controlled wind turbine and an induction generator [19]. The stator of the DFIG is directly connected to the grid, while the rotor is connected to a converter consisting of two back-to-back PWM inverters, which allows direct control of the rotor currents. Direct control of the rotor currents allows for variable-speed operation and reactive power control thus DFIG can operate at a higher efﬁciency over a wide range of wind speeds and help provide voltage support for the grid. These characteristics make the DFIG ideal for use as a wind generator. Generally, the reference value of the active power that a DFIG should generate is established through optimum power curves, which provide the active power as a function of the generator rotational speed [19]. Fig. 2 shows a power curve for a typical 1500 kW DFIG wind turbine. Such curves are derived as a result of analysis of the wind turbine aerodynamics, and by deﬁning the maximum mechanical power the DFIG can extract from the wind at any angular speed [8,12]. 2.2. DFIG capability limits curve Fig. 3 shows the single-phase equivalent circuit of the DFIG, where US is the stator voltage, UR is the rotor voltage, IS is the stator

current, IR is the rotor current, RS is the stator resistance, RR is the rotor resistance, XS is the stator reactance, XR is the rotor reactance, XM is the mutual reactance, and s is the slip. The doubly fed asynchronous generator converts the wind turbine mechanical power into electrical power that is fed into the grid through the stator and the rotor by means of a frequency converter consisting of two back-to-back inverters. The total active power of the DFIG fed into the grid is the sum of stator and rotor active power. PT = PS + PR

(1)

Taking into account that PR = −sPS

(2)

PT = (1 − s)PS

(3)

where PT is the total active power of the DFIG fed into the grid, PS is the stator active power, and PR is the rotor active power. In opposition to active power, total reactive power fed into the grid is not the addition of stator and rotor reactive power because rotor reactive power cannot ﬂow through the frequency converter. The grid side inverter of the frequency converter has its own reactive power capability, so total reactive power fed into the grid is the sum of the stator and the grid side inverter reactive power. Usually, in the commercial systems, this inverter works with unity power factor, being total reactive power, in such case, equal to stator reactive power. QT = QS

(4)

The stator active and reactive power can be expressed as a function of stator and rotor maximum allowable current [11]: PS2 + QS2 = (3US IS )2

Fig. 1. DFIG wind turbine.

PS2 +

(5)

2 2

QS + 3

US

XS

=

X M 3

XS

US IR

2 (6)

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2.3. Wind farm model In this paper, the “generator by generator” wind farm model is considered, which treats each of the wind farm generators separately [8]. It consists of N identical DFIG wind turbines whose stators are connected in parallel to an inﬁnite bus bar supplying a US peak voltage, being N is the number of DFIG wind turbines in the wind farm. This model allows considering different wind conditions for each generator. Consequently, the total active and reactive powers output of the wind farm equal to the sum of the active and reactive power generated by each DFIG wind turbine in the wind farm: PWF =

n

PTi

(9)

i=1

QWF =

n

QTi

(10)

i=1

Fig. 4. 1500 kW DFIG capability limits curve.

In the PQ plane, (5) represents a circumference centered at the origin with radius equal to the stator rated apparent power. Eq. (6) represents a circumference centered at [−3US2 /XS , 0] and radius equal to 3US IR XM /XS . Introducing (3) and (4) into (5) and (6):

P 2 T 1−s

P 2 T 1−s

+ QT2 = (3US IS )2

+

QT + 3

US2 XS

(7)

2 =

X M 3

XS

US IR

2 (8)

According to (7) and (8), the DFIG capability limits can be obtained by considering the stator and rotor maximum allowable currents ISmax and IRmax . Fig. 4 shows the capability limits curves of 1500 kW DFIG, which is obtained by taking into account the maximum stator and rotor currents and the steady state stability limit of the DFIG [11]. The electric parameters of 1500 kW DFIG are given in Table 1. In this ﬁgure, the solid and dashed curves represent the maximum reactive power that DFIG can generate or absorb corresponding to the stator and rotor maximum allowable currents for terminal voltage US = 1.00 p.u., respectively. The vertical dotted line at [−3US2 /XS , 0] coordinate represents the stability limit of the DFIG. It means that the generator becomes unstable if reactive power absorption is higher than 3US2 /XS . The shadow part is the area where the operation of the DFIG can be considered as feasible. One can observe that when the available active power is close to its maximum limit, the reactive power operation range decreases. The active power that a DFIG should generate is established through optimum generation curves, which provide the active power as a function of the generator rotational speed. When the active power that a DFIG should generate is given, the maximum reactive power operation range can be obtained.

where PWF represents the active power output of the wind farm, QWF represents the reactive power output of the wind farm, PTi represents the generated active power of each i DFIG and QTi represents the generated or absorbed reactive power of each i DFIG. When the reactive power output reference for the wind farm is obtained, the reactive power reference for each DFIG can be calculated applying the proportional distribution algorithm: QTiset =

Q

WFeref QTi max

where QTiset is the reactive power output set point calculated for each DFIG, QTimax is the maximum reactive power that each DFIG can generate or absorb and QWFeref is the reactive power reference for the wind farm. 2.4. Load ﬂow including wind farm In this paper, the node integrating wind farm is treated as PQ nodes in a load ﬂow analysis. In situations where the wind speed at wind farm is speciﬁed and the loads at buses are known, the real power output of DFIG can be calculated by means of the power curve. The reactive power of the wind farm is obtained from the optimization algorithm proposed in this paper. Then a backward–forward load ﬂow algorithm is utilized to determine the real and reactive current injection at all the buses. Using these currents and a backward–forward sweep scheme the branch currents are found and voltages at all the buses are updated for this iteration. 3. Problem formulation In this section, wind farm reactive power control and network reconﬁguration joint optimization have been modeled as a multiobjective, non-differentiable optimization problem. The objective is to minimize the system real power losses and the deviation of the bus voltage, subject to operating constraints. In this paper, the constant load models are considered in all time periods. The objective function is expressed as follows: ¯ = 1 min f1 (X)

Table 1 DFIG electric parameters.

Nl Pi2 + Qi2

Ri

i=1

Parameter

Value

RS , stator resistance per phase XS , stator leakage reactance per phase XM , mutual reactance RR , rotor resistance per phase XR , rotor leakage reactance per phase

0.001692 0.03692 1.4568 0.002423 0.03759

(11)

QTi max

|Vi |2

+ 2 max|Vi − Vrat |

(12)

X¯ = [Q¯ WF , S¯ w ] S w = [S w1 , S w2 , . . . , S wNs ] where X¯ is the state variables vector. Q WF is wind farm reactive power output vector, S w is the switches status vector, which

J. Zhao et al. / Electric Power Systems Research 80 (2010) 698–706

represents the status of switches speciﬁed in terms of on/off status, taking 0 or 1 as its value. Ri , Pi and Qi are the resistance, real power, and reactive power of branch i, respectively. Nl is the total number of branches. Ns is the number of switches. Vi and Vrat are the real and rated voltage on bus i. 1 and 2 represent weighting factors. 1 + 2 = 1. Owing to the DFIG operational requirements, the minimization of the objective function is subjected to the following constraints: (1) Distribution power ﬂow equations: Pi + PWFi = PDi + Vi

Nb

Vj (Gij cos ıij + Bij sin ij )

(13)

j=1

Qi + QWFi = QDi + Vi

Nb

701

4. Particle swarm optimization 4.1. The standard particle swarm optimization (PSO) The PSO is a population-based optimization method ﬁrst proposed by Kennedy and Eberhart [20]. The PSO algorithm is initialized with the population of individuals being randomly placed in the search space and search for an optimal solution by updating individual generations. At each iteration, the velocity and the position of each particle are updated according to its previous best position (Pbesti ) and the best position found by informants (Gbest). Each particle’s velocity and position are adjusted by the following formula:

vki (t) = ω · vki (t) + c1 · r1 (Pbestik (t − 1) − xik (t − 1)) Vj (Gij sin ıij − Bij cos ij )

+ c2 · r2 (Gbest k (t − 1) − xik (t − 1))

(14)

(20)

j=1

where Pi and Qi are the substation injected active and reactive power at the ith bus. PWFi and QWFi are the wind farm injected active and reactive power at the ith bus. PDi and QDi are the active and reactive load power at the ith bus. Vi and Vj are the amplitude of voltage at the ith and jth bus, respectively. Gij and Bij are the conductance and the susceptance between the ith and jth nodes. ıij and ij are the phase angle difference between the ith and jth nodes. (2) DFIG active capacity limits: PTimin ≤ PTi ≤ PTi max

(15)

where PTi , PTimin and PTimax are scheduled, minimum and maximum active power output of each i DFIG, respectively. (3) DFIG reactive capacity limits:

QTi ≥ − QTi ≤

3

XM US IR XS

XM 3 US IR XS

2

2

−

−

P 2 Ti 1−s

P 2 Ti 1−s

−3

−3

US2 XS

(16)

US2 XS

(17)

where Vi is the voltage magnitude of node i, Vmin and Vmax are low and upper bound of nodal voltage, respectively. (5) Distribution line limits: line |Pijline | < Pijmax

|Pijline |

(18) line Pijmax

where and are absolute power ﬂowing over distribution lines and maximum transmission power between nodes i and j, respectively. (6) Radial structure of the network. In this paper, the inequality and equality constraints are included into the objective function by using penalty function method. Therefore, the objective function in the joint optimization algorithm is written as F(x) = f (x) + k1

Nu j=1

Uj (x) + k2

Ne

Ej (x)

(21)

where i is the number of the particle in the swarm, k is the number of element in the particle xi (t), and t is the iteration number. vki (t)

and xik (t) are the velocity and the position of kth element of the ith particle at the tth iteration, respectively. r1 and r2 are the random numbers uniformly distributed between 0 and 1.The constants c1 and c2 are the weighting factors of the stochastic acceleration terms and ω is the positive inertia weight. The suitable selection of inertia weight ω in (20) provides a balance between global and local explorations [21]. The inertia weight ω can be dynamically set with the following equation: ωt+1 = ωmax −

ωmax − ωmin ×t tmax

(22)

where tmax is the maximum number of iteration, and t is the current iteration number. ωmax and ωmin are the upper and lower limits of the inertia weight. 4.2. Binary particle swarm optimization (BPSO)

where QTi is reactive power output of each i DFIG wind turbine. (4) Node voltage magnitude limits: Vmin ≤ Vi ≤ Vmax

xik (t) = xik (t) + vki (t)

(19)

j=i

where f(x) is the objective function values of optimization problem. Nu and Ne are the number of inequality and equality constraints, respectively. Uj (x) and Ej (x) are the inequality and equality constraints. k1 and k2 are the penalty factors, respectively.

The BPSO algorithm was introduced by Kennedy and Eberhart to allow the PSO algorithm to operate in binary problem spaces [22]. It uses the concept of velocity as a probability that a bit takes on one or zero. In the BPSO, (20) for updating the velocity remains unchanged, but (21) for updating the position is re-deﬁned by the rule:

xik (t) = 1,

r < S(vki (t − 1))

xik (t) = 0,

r ≥ S(vki (t − 1))

(23)

where S(vki ) is the sigmoid function for transforming the velocity to the probability as the following expression: S(x) =

1 1 + e−x

(24)

4.3. Hybrid particle swarm optimization with wavelet mutation (HPSOWM) The PSO performs well in the early iterations, but it presents problems reaching the near optimal solution. The behavior of the PSO presents some problems related with the velocity update. If a particle’s current position coincides with the global best position, the particle will only move away from this point if its inertia weight and previous velocity are different from zero. If their previous velocities are very close to zero, then all the particles will stop moving once they catch up with the global best particle, which may

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lead to premature convergence of the algorithm. This phenomenon is known as stagnation. To avoid this problem, a hybrid particle swarm optimization with wavelet mutation (HPSOWM) is proposed, which incorporates a genetic algorithm’s evolutionary operations of mutations [23]. In the HPSOWM, a mutation with a dynamic mutating space by incorporating a wavelet function is proposed. The mutating space is dynamically varying along the search based on the properties of the wavelet function. The resulting mutation operation aids the HPSOWM to perform more efﬁciently and provides faster convergence. The details of the operation are as follows. Every particle element of the swarm will have a chance to mutate what is governed by a probability of mutation pm ∈ [0, 1], which is deﬁned by the user. For each particle element, a random number between 0 and 1 will be generated such that if it is less than or equal to pm , a mutation will take place on that element. For instance, if xi = [xi1 , xi2 , . . . , xik ] is the selected ith particle, and the element of particle xik is randomly selected for the mutation, the

resulting particle is given by xi = [xi1 , xi2 , . . . , xik ], i.e.

x¯ ik (t)

=

xik (t) + ı × (parakmax − xik (t))

if ı > 0

xik (t) + ı × (xik (t) − parakmin )

if ı ≤ 0

(25)

where k ∈ 1, 2, . . ., ; denotes the dimension of the particle and ı is the mother wavelet. There are many kinds of wavelets which can be used as a mother wavelet, such as the Harr wavelet, Meyer wavelet, Coiﬂet wavelet, Daubechies wavelet, Morlet wavelet and so on. These wavelets have different speciﬁcities. In this paper, the Morlet wavelet is chosen as the mother wavelet because the Morlet wavelet offers the best performance. Its mathematical form is shown as follow: 2 1 ı(ϕ) = √ e−(ϕ/a) /2 cos a

ω0

ϕ a

(26)

where ω0 is the central frequency of wavelet. 5. Joint optimization algorithm 5.1. Joint optimization algorithm In this paper, a joint optimization algorithm of combining wind farm reactive power output and network reconﬁguration to minimize the real power losses of the system and the deviation of the bus voltage is proposed. The state variable vector is X¯ = [Q¯ WF , S¯ w ]. Q¯ WF is continuous variable, which represents the wind farm reactive power output. S¯ w is discrete variable, which represents the status of switches. In proposed algorithm, HPSOWM is utilized to optimize wind farm reactive power output, and BPSO is developed to ﬁnd the optimal network structure for each particle updating instance at the each iteration of wind farm reactive power output optimization. The stopping criteria of the algorithm is the maximum number of iteration is reached. The ﬂow of the joint optimization algorithm is illustrated in Fig. 5. 5.2. Distribution network reconﬁguration using BPSO In this paper, we use BPSO-based algorithm for distribution network reconﬁguration. The tie and sectionalizing switches status of all feeders are chosen as a set of control variables. With such a variable expression, each element of the solution vector represents one feeder with a switch. The value 0 or 1 of one element in the solution vector denotes that the status of corresponding switch in the feeder is open or closed, respectively. It was found that such a variable expression is often not efﬁcient because the extremely large number of unfeasible non-radial solutions appearing at each

Fig. 5. Flowchart of the proposed joint optimization algorithm.

generation will lead to a long computing time before reaching an optimal solution. Therefore, in a radial distribution network, when a tie switch is closed, a loop is formed and a sectionalizing switch in the loop should be opened to retain the radial structure of the system. In this paper, the coding method that recognizes the positions of the tie switches described in Ref. [24] is utilized. The total number of open switches is equal to the total number of loops. To ensure that no feeder section can be left out of service, a switch not included in any loop must be closed. 5.3. Wind farm reactive power output optimization using HPSOWM In this paper, the HPSOWM optimization algorithm described in the above section is used to optimize reactive power output of wind farm. In situations where the wind speed at each DFIG wind turbine is speciﬁed, the active power generated by each DFIG wind turbine can be calculated by means of the power curve. According to (7) and (8), the maximum reactive power that each DFIG can generate or absorb can be obtained. The total active power output and maximum reactive power output of the wind farm are obtained by (9) and (10). The steps followed for the implementation of the algorithm are described as follows:

J. Zhao et al. / Electric Power Systems Research 80 (2010) 698–706

Step 1: The reactive power output of the wind farm is used as the control variable. Initialize a population of particles with random position and velocities within the reactive power capability limits. Step 2: Evaluate the objective function values of all particles according to (19) using the result of distribution load ﬂow. Step 3: Set Pbest of each particle and its objective value equal to its current position and objective value, and set Gbest and its objective value equal to the position and objective value of the best particle. Step 4: Select the ith particle. Step 5: Update the velocity and position of the ith particle according to (20) and (21). Step 6: Perform wavelet mutation on the selected element of particle according to (25) to create new particle. Step 7: Evaluate the objective function value of the new created particle, and compare its current objective value with the objective value of its Pbest. If current value is better, then update Pbest and its objective value with the current position and objective value. Step 8: If all individuals are selected, go to the next step, otherwise i = i + 1 and returns to Step 4. Step 9: Determine the best particle of current whole population with the best objective value. If the objective value is better than the objective value of Gbest, then update Gbest and its objective value with the position and objective value of the current best particle. Step 10: If the maximum number of iteration is reached, the search procedure is stopped, otherwise returns to Step 4.

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Table 2 DFIG performance parameters. Parameter

Value

Rated capacity Cut-in wind speed Cut-out wind speed Rated wind speed Rated voltage

1500 kW 4 m/s 25 m/s 12 m/s 0.69 kV

hourly wind speed curves of each wind turbines, obtained from a wind speed forecasting for consecutive 24 h of a day. The actual active power outputs of each DFIG in each period are shown in Fig. 8, calculated by means of the power curve of each DFIG. Considering the DFIG capability limits described in Section 2, the maximum reactive power outputs limits of each DFIG wind turbines are shown Fig. 9. In Fig. 9, the undermost curve represents the stability limit of the DFIG. From Figs. 8 and 9, it can be observed that wind farm made up of DFIG wind turbine can generate high quantities of reactive power when the available active power is far from its maximum. For example, when the average wind speed at wind farm is 4.5 m/s, the maximum active and reactive power wind farm can generate are 0.3563 MW and 3.6657 MVAR, respectively. But the maximum reactive power wind farm can generate become very low when the available active power is near to its rated power. For

6. Simulation results In this paper, the 16 bus distribution network given in Ref. [13] is used to verify the validity and performance of the proposed joint optimization algorithm (see Fig. 6). Test system has 13 sectionalizing branches and 3 tie branches, S15, S21 and S26 are three tie switches. The tie switches and sectionalizing switches are normally open and closed, respectively. A small wind farm comprising 4 DFIG wind turbines of 1500 kW, with a power installed of 6 MW is connected at node 12 through a rated 23/0.69 kV transformer. The performance parameters of 1500 kW DFIG are given in Table 2. The system loads are 28.7 MW and 17.3 MVAR. Voltage limits are assumed to be within the range 0.95–1.05 p.u. 6.1. Available active and reactive power in wind farm Fig. 7. Curve of wind speed.

In this paper, each of wind turbines in wind farm was considered to have different instantaneous wind speed. Fig. 7 shows the mean

Fig. 6. 16 Node test feeder.

Fig. 8. Active power of each DFIG wind turbine.

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Table 3 Optimization results of different cases. Case

PWF (MW)

QWF (MVAR)

Tie switch set

Objective function value

Losses (MW)

Minimum nodal voltage (p.u.)

Original system 1 2 3 4 5

3.34 3.34 3.34 3.34 3.34 3.34

0.678 0.678 2.73 2.14 2.73 2.18

15, 21, 26 17, 19, 26 15, 21, 26 17, 19, 26 17, 19, 26 17, 19, 26

0.1860 0.1676 0.1699 0.1555 0.1569 0.1554

0.338 0.310 0.347 0.315 0.320 0.315

0.979 0.983 0.984 0.988 0.988 0.988

example, in the period 10, the average wind speed is 14 m/s, the active power that wind farm generated is rated power 5.9813 MW, and the maximum reactive power DFIG can generate reduce to 1.8220 MVAR. 6.2. Joint optimization In this paper, the parameters of the joint optimization algorithm are as follows: the number of particles is set as 20 and the maximum iteration number of the algorithm is set as 100. c1 = c2 = 2, ωmax = 0.9 and ωmin = 0.4.The upper and lower bounds of mutation probability are set as 0.05 and 0.3, respectively. The Morlet wavelet is chosen as the mutation wavelet parameter. It is assumed that the network reconﬁguration and reactive power control operate once per hour and the load is the constant during all time periods. Choosing the data during period 3 from Figs. 7–9 as the example, in this period, the wind speed is 8.1 m/s, the total active power wind farm generated is 3.3375 MW, and the maximum available reactive power in wind farm is 2.7253 MVAR. To demonstrate the performance of the proposed joint optimization algorithm, the following ﬁve cases are studied. Table 3 provides the simulation results of these ﬁve cases, using MATLAB software carried out on a P4 1.6 GHz/1 GB RAM computer system. Case 1: Only perform network reconﬁguration, power factor of the DFIG is 0.98. Case 2: Only perform wind farm reactive power optimization. Case 3: Perform network reconﬁguration ﬁrst, and then optimize wind farm reactive power. Case 4: Perform wind farm reactive power optimization ﬁrst, and then carry out network reconﬁguration. Case 5: Perform the proposed joint optimization algorithm. As shown in Table 3, the objective function value in Case 5 is lowest in the ﬁve cases. Compared with original system, real power losses are reduced about 7%, from 0.338 MW to 0.315 MW, and the minimum node voltage is improved from 0.979 p.u. to 0.988 p.u.

Fig. 10. Convergence performance of PSO for the best solution.

Table 4 Maximum active power available in each wind turbine (MW). Period

Turbine

1 2 8 12

1

2

3

4

Total

0.1688 0.5250 1.2188 1.5000

0.0938 0.4688 1.2000 1.5000

0.0562 0.5625 1.1438 1.5000

0.0375 0.4312 1.1438 1.4813

0.3563 1.9875 4.7064 5.9813

The computation time used in the proposed joint optimization algorithm is 232 s. It means that the proposed joint optimization has proven to be an effective algorithm to ﬁnd the optimal network structure and wind farm reactive power output to minimize losses and improve voltage proﬁles. The optimization results of Cases 3 and 4 are better than those of Cases 1 and 2. It demonstrates that performing wind farm reactive power control followed by network reconﬁguration, or vice verse, can do a better job than only performing wind farm reactive power control or network reconﬁguration alone. Fig. 10 illustrates the convergence performance of the joint optimization algorithm for the best solutions. It can be evidently seen from this ﬁgure that the algorithm converged to a good solution well before the maximum iterations number 100 was reached. Table 5 Maximum reactive power available in each wind turbine (MVAR). Period

Fig. 9. Maximum reactive power limit of each DFIG wind turbine.

1 2 8 12

Turbine 1

2

3

4

Total

0.9063 0.7962 0.4624 0.4555

0.9174 0.8144 0.4788 0.4555

0.9205 0.7835 0.5250 0.4555

0.9215 0.8156 0.5250 0.4555

3.6657 3.2097 1.9912 1.8220

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Table 6 Results of joint optimization of four chosen periods of time. Period

PWF (MW)

QWFmax (MVAR)

QWF (MVAR)

Tie switch set

Objective function value

Losses (MW)

Minimum nodal voltage (p.u.)

1 2 8 12

0.356 1.99 4.71 5.98

3.66 3.21 1.99 1.82

3.66 3.21 1.16 1.07

17, 19, 26 17, 19, 26 17, 19, 26 17, 19, 26

0.2119 0.1766 0.1409 0.1315

0.465 0.379 0.267 0.236

0.984 0.988 0.989 0.992

Table 7 Reactive power output of each DFIG (MVAR). Period

1 2 8 12

Turbine 1

2

3

4

Total

0.9049 0.7938 0.2694 0.2675

0.9160 0.8119 0.2789 0.2675

0.9191 0.7811 0.3058 0.2675

0.9201 0.8131 0.3058 0.2675

3.66 3.21 1.16 1.07

Tables 4 and 5 show the maximum available active and reactive power values of each DFIG during four different periods of time, chosen from Figs. 7 and 8. The ﬁrst column shows the number of the time period, and the other columns show the power available at each DFIG in each period. According to joint optimization algorithm proposed in this paper, the joint optimization results of four chosen periods of time are shown in Table 6. The optimization results show that when wind farm active power output are 0.356 MW and 1.99 MW, the optimal reactive power output results equal to the 3.66 MVAR and 3.21 MVAR, which are the maximum reactive power that wind farm can generate. This shows that the wind farm needs to generate maximum reactive power to improve the voltage proﬁle when wind farm active power output is small. When wind farm active power output increases to 4.71 MW and 5.98 MW, the wind farm optimal reactive power output becomes very low, the results are 1.16 MVAR and 1.07 MVAR. From Table 6, it can be found that the real power losses are reduced and the minimum nodal voltage is improved compared with original system after joint optimization. After the wind farm optimal reactive power output is obtained, the reactive power reference for each DFIG is calculated applying the proportional distribution algorithm described in (11). The reactive power distribution results are shown in Table 7. 7. Conclusions In this paper, a joint optimization algorithm of combining reactive power control of wind farm and network reconﬁguration is proposed. In the proposed joint optimization algorithm, reactive power output of wind farm and status of switches are utilized as the control variable for losses minimization and voltage proﬁle improvement. The optimal reactive power output of wind farm and the optimal network structure are efﬁciently obtained by taken into account DFIG reactive capability limits in the simulation. From the results obtained in the simulations, it can be concluded that wind farm made up of DFIG can constitute an important continuous reactive power source to support system voltage control. The simulation results also show that the joint optimization gets better solution results than using the wind farm reactive power control optimization algorithm or the network reconﬁguration algorithm alone. References [1] R. Piwko, N. Miller, J. Sanchez-Gasca, Integrating large wind farms into weak power grids with long transmission lines, IEEE/PES Transmission and Distribution Conference & Exhibition: Asia and Paciﬁc, Dalian, China, 2005, pp. 1–7.

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Xin Li is a professor at the College of Electrical Engineering of Chongqing University, China. Her research interests include power system operation, power system dynamics and control. Jutao Hao was born in 1976. He received the Ph.D. degree in computer science from Shanghai Jiao Tong University, Shanghai, China, in 2008. He is currently a lecture of computer science engineering at University of Shanghai for Science and Technol-

ogy. His research interests include soft computing methodologies applied to power system analysis and planning. Jiping Lu is a professor at the College of Electrical Engineering of Chongqing University, China. His research interests include power system automation, relay protection and probability application in power systems.