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Stochastic reactive power dispatch in hybrid power system with intermittent wind power generation
Energy xxx (2015) 1e8
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Energy journal homepage: www.elsevier.com/locate/energy
Stochastic reactive power dispatch in hybrid power system with intermittent wind power generation Reza Taghavi, Ali Reza Seifi, Haidar Samet* School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 May 2014 Received in revised form 3 June 2015 Accepted 6 June 2015 Available online xxx
Environmental concerns besides fuel costs are the predominant reasons for unprecedented escalating integration of wind turbine on power systems. Operation and planning of power systems are affected by this type of energy due to the intermittent nature of wind speed inputs with high uncertainty in the optimization output variables. Consequently, in order to model this high inherent uncertainty, a PRPO (probabilistic reactive power optimization) framework should be devised. Although MC (Monte-Carlo) techniques can solve the PRPO with high precision, PEMs (point estimate methods) can preserve the accuracy to attain reasonable results when diminishing the computational effort. Also, this paper introduces a methodology for optimally dispatching the reactive power in the transmission system, while minimizing the active power losses. The optimization problem is formulated as a LFP (linear fuzzy programing). The core of the problem lay on generation of 2m þ 1 point estimates for solving PRPO, where n is the number of input stochastic variables. The proposed methodology is investigated using the IEEE-14 bus test system equipped with HVDC (high voltage direct current), UPFC (unified power flow controller) and DFIG (doubly fed induction generator) devices. The accuracy of the method is demonstrated in the case study. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Linear programming Fuzzy optimization AC/DC system Probabilistic load flow Point estimate method DFIG
1. Introduction Progressively ascending fuel prices accompanied by environmental concerns have induced countries to expand their power system infrastructure to incorporate further renewable energy, notably wind generation [1,2]. Annual statistics show that wind capacity increased by 16% from 2013 to 2014 with 51 GW of new capacity added [3,4]. The unmanageable nature of this category of energy necessitates probabilistic analysis of the power system operation. MC (Mont-Carlo) technique has been well proven as a trustworthy means to deal with stochastic variables and to attain distribution of the resultant variables with high precision [5e7]. In the view of aforementioned characteristics, MC methods are perceived as the most precise, reliable and robust SAM (stochastic analysis method), which is commonly used as a reference method to inspect the correctness of other probabilistic methods [8e10]. However,
* Corresponding author. E-mail address: [email protected] (H. Samet).
the enormous computational effort which is required for this method persuades researchers to use alternative approaches which are capable of quickly acquiring the probabilistic output attributes of the system. Abundant studies have confirmed that PEMs (point estimate methods) are capable of producing convincing stochastic results while effectively reducing the computational efforts [11e13]. However, the primary version of this method requires the input random variables’ PDF (probability density function) to evaluate the 2m þ 1 central moments. The aforementioned constraint within the PEM algorithm grew more apparent when stochastic input variables follow no common PDF [13]. This is most assuredly the case when the wind generation sources are integrated into the power grid. Although the stochastic behavior of wind speed is commonly modeled by a Weibull or Rayleigh PDF, the nonlinear relationship between the wind speed and the wind power makes it difficult to match the wind power to any generally known PDF. In order to cope with this issue, a discrete point estimate method is utilized. The core of the discrete point estimate method is that it employs sufficient sample measurement to approximate the (2m1)th central moment of the input
http://dx.doi.org/10.1016/j.energy.2015.06.018 0360-5442/© 2015 Elsevier Ltd. All rights reserved.
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Nomenclature V dc I dc Rc a P dc Q dc j jdc Vcr Vvr qcr qvr Pmk Qmk ~j tij Vg Qz Pst Qst Iro xsout m
xout
dc terminal voltage dc terminal current commutation resistance transformer tap connected to DC terminals active power at DC terminals reactive power flowing from the AC to DC the voltage angle of AC bus connected to DC the angle of AC current injected into the DC series voltage magnitude of UPFC parallel voltage magnitude of UPFC angle of series voltage source angle of parallel voltage source transferred active power between bus m and K transferred reactive power between bus m, k probabilistic jacobian matrix transformer tab between buses i and j voltage magnitude of generator bus (PV buses) compensators reactive power DFIG's stator active power DFIG's stator reactive power DFIG's rotor current the relative error of the standard deviation of the output random variable the mean error indices of a set of out put
s
xout the standard deviation error indices of a set of out put z ¼ f ðXÞ the PRPO objective function X set of n random variable XM Mth sample point of x KT matrix which links objective functions to variables S sensitivity matrix b dependent parameters matrix bmax , bmin upper and lower boundary of dependent variables xmax maximum limit of linearized control parameters xmin minimum limit of linearized control parameters fuzzy less than or equal to e D fuzzy decision G, C fuzzysubsets for objective functions and constraints
variables, where m refers to the number of concentration points in PEM approach. In the other hand, HVDC (high voltage direct current) transmission lines along with FACTS (flexible AC transmission systems) offer an excellent opportunity to support and improve the power supply of sustainable, efficient and reliable future grids [14,15]. FACTS are well known for the fast-response devices that can control the active and reactive power as well as the bus voltage. One of the more comprehensive FACTS devices is UPFC (Unified Power Flow Controller) that helps power systems to share the excessive loads from the lines and leads to the loss decline and high stable operations. To the best of authors’ knowledge, only few papers [16e20] have considered PRPO (probabilistic reactive power optimization) in the presence of HVDC and UPFC along with the uncertainties injected by the wind generation source. Therefore, this paper intended to propose a comprehensive model which consists of HVDC, UPFC and DFIG (double fed induction generator) as one of the most widely used generators in the wind turbine. This model takes the stochasticity under consideration in order to fill this gap. The main idea of this work is to attain
mD ðXÞ membership function of D mG ðXÞ; mC ðXÞ membership function of G, C l the degree of satisfaction for fuzzy objective function and fuzzy constraints mfi ðXÞ fuzzy membership function of fi ðXÞ [i threshold of constraint ðrÞ PL active power loss in iteration r Pro DFIG's rotor active power Ptot DFIG's total active power Qro DFIG's rotor reactive power Vst DFIG's stator voltage Ist DFIG's stator current Qtot DFIG's total reactive power mrj rth sample central moment ofjth random variable mxj average value of the M observation of jth variable the standard deviation of xj sxj rj;k kth concentration points correspond to jth input random xj;k kth standard location for jth input variable wj;k kth weighting factor for jth input variable xmout the relative error of the mean of the output random variable Abbreviation Rec rectifier Inv inverter MC Monte Carlo PDF probability density function PEM point estimate method MW Mega Watt MVAr Mega VAr p.u, Per Unit PRPO probabilistic reactive power optimization RPO reactive power optimization LFP linear fuzzy programming HVDC high voltage direct current FACTS flexible ac transmission systems UPFC unified power flow controller
the optimal control variables that are appropriate for stochastic situation, meanwhile minimizing the active power loss. Contrary to deterministic RPO (reactive power optimization), the probabilistic nature of input variables results in outputs from the optimization process become probabilistic. In addition, this paper presents a method that takes the uncertainty in optimization process into accounts, while conducting the proposed method in hybrid power system to consider a comprehensive model for investigation the proposed methodology. The remainder of this paper is arranged as follows: description of system model consisting of DFIG wind turbine, HVDC and UPFC models are introduced in Section 2. Details of discrete PEM are discussed in Section 3. A probabilistic reactive power optimization algorithm is proposed that aims to minimize active power loss in Section 3. The problem is formulated as linear fuzzy programing [21] considering the stochastic effects of the wind generation and loads. In Section 4 the performance of the proposed method is investigated with modified IEEE-14 bus test system and the correctness of the method is evaluated through comparing obtained results with those provided by MC technique.
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or power at the scheduled value. The firing angle must be small in order to draw in a little reactive power. However, to maintain an effective commutation, it must be set at its minimum value. 2.2. UPFC model In this study, UPFC is described by two ideal voltage sources, both working in conjunction with each other. One source is connected to the system in parallel and the other source comes in series. The equivalent circuit of this model is depicted in Fig. 3. Equations of series and parallel voltages are given by Ref. [23]:
Vcr ¼ Vcr ðcosðqcr Þ þ j sinðqcr ÞÞ
(7)
2. Description of model
Vvr ¼ Vvr ðcosðqvr Þ þ j sinðqvr ÞÞ
(8)
2.1. Converter model
min V V max Þ and q ð0 q 2pÞ are voltage where Vcr ðVcr cr cr cr cr magnitude and adjustable angle of series voltage source. The voltage magnitude and corresponding angle ðVvr ; qvr Þ indicate the parallel voltage source parameters as shown in Fig. 2.
Fig. 1. Equivalent representation of a monopolar station.
A basic analogous circuit of a typical monopolar converter is illustrated in Fig. 1. According to [21,22], the equations for each terminal can be formulated as:
V dc ¼ aV ac cosðaÞ Rc I dc P
dc
dc dc
¼V I
2.3. Wind turbine model
(1) (2)
where Rc , a and a are commutation resistance, transformer tab and firing angle. By neglecting the loss of converter and its attached transformer and counting the balance between power of AC and DC parts, the power factor can be formulated as:
V dc ¼ aV ac cosðj zÞ
(3)
The reactive power which streams from AC side to the converter is as follows:
Q dc ¼ P ac tanðj jdc Þ
(4)
where j and z are voltage and current angle of AC bus which are connected to DC converter. The relation between rec and inv voltages can be stated as follows: dc dc Vrec ¼ Vinv þ Rdc I dc
2 Pst þ Qst2 ¼ ðVst Ist Þ2
(6)
The practical control rule for a DC system is to oblige one terminal's determined voltage while the other terminals force current
Fig. 2. Unified power flow controller equivalent circuit.
1. Stator and rotor current limit [25]: The stator and rotor currents which are responsible of armature and field coil's heating due to joules losses are represented as follows:
(9)
(5)
An equation for DC current flowing between the two converters is given by:
ac ac I dc ¼ arec Vrec cosðarec Þ ainv Vinv cosðainv Þ Rdc
One of the most widely used generators in the wind turbine is DFIG. This adjustable speed generator comprises of a wound rotor induction generator that fed the grid by both the rotor and stator. The stator is directly connected to the grid while the rotor is linked to the system through back-to-back four-quadrant converter [24]. In Comparison with single fed induction generator that can only consume reactive power, DFIGs have the ability to control active and reactive power independently. When PRPO is concerned, the mathematical representation of DFIGs’ reactive (active) power constraints should be acquired. By considering the stator and rotor rating current as well as steady state stability limit, DFIG capability limits are obtained [25]:
Pst ¼
Xm Vst Irt sinðdÞ Xst
(10)
Qst ¼
Xm V2 Vst Iro cosðdÞ st Xst Xst
(11)
Fig. 3. Total DFIG's reactive power capability curve.
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2 Pst þ
Qst þ
Vst2 Xst
!2
¼
Xm Vst Iro Xst
2 (12)
Equations (9) and (12) represent two circles with the center [0, " # 0] and
Vst2 Xst ;
0 and radiuses amount to Vst Ist and
Xm Xst
Vst Iro in the
P Q plane respectively. 2. Steadyestate stability limit: according to [25] the maximum stable operation of power system is acquired when this limit is as follows:
calculated and scheduled values in the Jacobean matrix becomes less than a certain stopping criterion. The schedule values of Jacobean matrix in ac part include: sch sch sch Pical for i ¼ 2:::n; Qical for i ¼ m þ 2:::n; Pmk ; Qmk ; Pbb
In DC part: Vdc is a voltage by which the converter performs the voltage control, while Idc and Pdc are the current and power that the converter requires for this object. The under study system is a twoterminal monopolar system in which the rectifier controls the voltage, and inverters adjust the injected current.
3.2. Discrete point estimated method
V2 Qst ¼ st Xst
(13)
3. Total capability limit: The DFIG's total capability limit is attained by adding rotor reactive power to the stator ones. This limit is expressed as follows:
In comparison with PEMs that needs distribution information, this method employs the input sample data to ascertain the input 2m1 central moment [27]. Consider z as a function of n random input variables X:
z ¼ f ðXÞ Pro ¼ sPst
(14)
Ptot ¼ Pst þ Pro ¼ ð1 sÞPst
(15)
Qtot ¼ Qst þ Qro ¼ Qst
(16)
Qtotmax
Qtotmin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ AV 2st P 2tot BVst2 V2 ¼ st Xst
where A ¼
Xm Xst Iro
where Xmn ¼ ½x1 …xn T , let X1 …XM stand for M sample point of x. The unbiased estimates of the expectation and the central moments of the random vector X are stated as follows:
mxj ¼ (17)
(18) 2
M 1 X x M i¼1 ij
3. Probabilistic load flow The essence of the PEMs centered on approximation of the statistical information of the output random variables. The theory, in the first place, is proposed by Hong in 1998 [13] that presumed input random variables have continues distribution function. The PEMs’ procedure can be summarized as follows: The 2m1 central moments of input random variables are obtained from their distribution function (regularly m is set as 3). Then concentration points and the corresponding weights for each random variable are acquired through matching 2m equations. Finally, the statistical properties of output random variables are acquired based on aforementioned concentration points and their weight. 3.1. AC/DC power flow solution method Based on Rbus Gauss-Seidel method [26], the DC voltage and current for all terminals can be obtained. After substituting the values of Vdc , Idc , Rc and a in Eq. (1), the firing angle is computed. The active and reactive power conveyed from AC side to the DC terminals are also acquired by Eqs. (2)e(4). This set of power values is established as loads connected to the DC part. The first iteration of load flow solution delivers the deviation of the values of bus angles and bus voltages as well as the change of firing angle of converters. Thereby, the new active and reactive power of DC terminals is gained. This trend continues until the error between the
j ¼ 1:::n
(20)
where mxj denote the average value of the M observation of jth variable The rth sample central moment of jth random variable is determined as:
, B ¼ 1=Xst
Fig. 3 demonstrates these operational limits. The shaded area in Fig. 3 stands for the possible area of operation for the DFIG wind turbine.
(19)
M 1 X xk mxj M
mrj ¼
j ¼ 1…n
(21)
k¼1
In accordance with PEM procedure, concentration points correspond to each input random variable are defined as follows:
rj;k ¼ mxj þ xj;k sxj k ¼ 1; 2; 3
(22)
where xj;k denote the standard location and mxj , sxj are the mean and the standard deviation of xj . The standard locations are achieved by:
xj;k ¼
lxj ;3 2
þ ð 1Þ
3k
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 lxj ;4 l2xj ;3 k ¼ 1; 2 4
xj;3 ¼ 0
(23) (24)
where lxj ;3 and lxj ;4 are, the third and fourth standardized central moments of xj , also recognized as skewness and kurtosis. A weighting factor wj;k is defined corresponding to each random variable:
wj;k ¼
ð 1Þ3k k ¼ 1; 2 xj;k xj;1 xj;2
(25)
wj;3 ¼
1 1 m lxj ;4 l2x ;3
(26)
j
The row moments of the output random variable z are obtained through evaluation of the output at each of the concentration points zði; kÞ while considering the weighing factors wj;k [13].
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In view of PRPO, the objective function can be considered as a function of stochastic input random variables. Thus, its row moments are determined using PEM algorithm. The procedure of the proposed methodology is depicted in Fig. 4. 3.3. Description of fuzzy optimization model Linear programming, which is used for reactive power control aims at minimizing the total power loss of the system for all load conditions as follows:
parameters. S is the sensitivity matrix that stands for the relationship between the control and dependent parameters. b consists of the linearized dependent parameters. bmax and bmin are the up and down limits of the linear dependent variables. x consists of linearized control parameters. xmax and xmin are the maximum and minimum limits of linearized control parameters. In real-life situations, the objective function(s) and/or constraints in a fuzzy form have to be considered, particularly where planners are concerned with future plans. LP problem with indefinite constraints, this form of general LP is
minimize minimize subjected to and
T
DPL ¼ K x bmin b ¼ Sx bmax xmin x xmax
(27)
where:
vPL vPL vPL vPL vPL vPL vPL vPL ; ; ; ; ; ; ; vVg vQz vPmk vQmk vVk vVdc vIdc va
DX ¼ DVg ; DQz ; Dtij ; DPmk ; DQmk ; DVk ; DVdc ; DIdc ; Da KT ¼
(28)
In Eq. (27) K T is the linearized matrix which establishes the relationship between the objective function and the control
5
K T x f subjected tob min b e 0 e
¼ Sx bmax andxmin x xmax e e e
(29)
where is ‘fuzzy less than or equal to ‘and shows that variables or e parameters that take a fuzzy form. Also, fuzzy objective function(s) and constraints are characterized by membership functions. Here, K T , S and bmax have been changed to cope with multi-load levels. The optimal solution (fuzzy decision D) is the one which is able to meet (optimize) the fuzzy objective function(s) and the fuzzy constraints simultaneously. If G and C are defined as the fuzzy subsets of variables x in set X which satisfy the fuzzy objective(s)
Fig. 4. The proposed PEM reactive power optimization algorithm.
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and fuzzy constraints, respectively, then the membership function of fuzzy decision mD , is defined [28,29] as:
mD ðXÞ ¼ minfmG ðXÞ; mC ðXÞg
(30)
where the membership function of objective function is
(31)
mC ðXÞ ¼ min mCj ðXÞ j ¼ 1; 2; …; m
(32)
An optimal solution x* 2 X must satisfy both the objective function and constraints which can be obtained by taking the maximum of Eq. (30) Such as,
(33)
where l is the degree of satisfaction for fuzzy objective function and fuzzy constraints. While the optimal solution x* 2 X satisfies all constraints and objective functions for different load levels. In this study the l formulation is used to solve our problem as follows:
maximize subjected to
l l mfi ðXÞ
0 l1
(34)
The membership function mfi ðXÞ is arbitrary and is assumed to have a linear form. So the constraint fi ðXÞ di will have a fuzzy membership function in the fuzzy environment as can be observed below:
8 1 > > > > < ðd þ [ Þ f ðXÞ i i i mfi ðXÞ ¼ > [i > > > : 0
(37)
where:
mD ðXÞ ¼ minfmG ðXÞ; mC ðXÞg
l ¼ max mD X * ¼ max½minfmG ðXÞ; mC ðXÞg
maximize lðrþ1Þ subjected to S0 DX þ [lðrþ1Þ b0ðrÞ þ [ 0 lðrþ1Þ 1 X ðrþ1Þ ¼ X ðrÞ þ DX ðrþ1Þ
T
_ 0 S S ¼ K ¼ S b S S "
ðrÞ # _ P 0 bmax b ¼ b ðrÞ ¼ _ðrÞL bmin b
(38)
ðrÞ
where PL is the active power loss in iteration r, and [ is the threshold of objective function and constraints. Since S is constant in each iteration, we will encounter only ordinary linear programming. 4. Case studies In this section, the effectiveness of the proposed PEM PRPO method is investigated on the modified IEEE_14 bus test system as depicted in Fig. 5 [21]. It is noteworthy to mention that the load level is 20% increased when comparing with the original value for this system [31]. The wind farm equipped with DFIG is located at bus 6, having the total capacity of 621 MW. The wind speed is presumed to be two-parameter Weibull with quantities a and b equal to 9 and 2.025 respectively. Probabilistic distribution corresponding to the wind farm output power is illustrated in Fig. 6. The
fi ðXÞ di di fi ðXÞ di þ [i
(35)
fi ðXÞ di þ [i
where [i is the threshold of constraint fi ðXÞ di . From Eqs. (34) and (35), for the constraints we can write:
l mfi ðXÞ l
ðdi þ [i Þ fi ðXÞ [i
(36)
fi ðXÞ þ l[i di þ [i It is the planner who settles the membership function of the optimization process. Some researchers [28,29] maintain that the membership and objective functions are subjective and should be established based on the planners’ preferences and past experience. Nevertheless, it can be seen that by choosing an appropriate membership function for objective function(s), computation time can be considerably reduced [30]. In the present article, the abovementioned method has been improved and a methodology has been introduced which can significantly reduce the computation time. In our optimization, because the minimization of active power loss in the system is the main objective function, the ideal case is achieved when the loss equals zero. Therefore, the membership function of one will be considered for it. Because our objective is to reduce the system loss from the base case of the load flow, it is assumed that the membership function is zero. The membership function between these two points is assumed to be linear. Eq. (27) has been modeled as can be seen below,
Fig. 5. Modified IEEE 14 bus test system with DG connected to bus 6.
Fig. 6. Output power histogram in p.u. for wind turbine.
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Table 1 Trapezoidal possibility of variables for the IEEE 14 bus system. Control
Max
[max
Min
[min
Dependent
Max
[max
Min
[min
V1 V2 V3 V6 V8 9 Qsvc 14 Vupfc t47 t49 t56 Pmk Qmk Vdc Idc arec ainv
1.18 1.15 1.15 1.15 1.18 5 1.15 1.1 1.1 1.1 40 22 1.02 0.5 1.1 1.1
0.0 0.0 0.0 0.0 0.0 5 0.0 0.05 0.05 0.05 0.0 0 0.0 0.2 0.05 0.05
1 1 1 1.05 1 0.0 1 1 1 1 30 7 0.9 0.2 0.98 1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05 0.05 0.05 5 2 0.05 0.1 0.0 0.0
Q1 Q2 Q3 Q6 Q8 V5 V7 V8 V13 V10 V11 V12 V13 arad inv arad rec
100 50 40 24 24 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 0.4 0.4
10 10 10 10 10 0.05 0.05 0.05 0.05 0.05 0.05 0 0.05 0 0
100 40 0 6 6 1 1 1 1 1 1 1 1 0.17 0.17
10 10 10 10 10 0.03 0.03 0.03 0.03 0.03 0.03 0 0.03 0.17 0.17
Table 2 Statistical characteristic of the power system variables comparing MC results to that of PEM method for the IEEE 14-bus test system (active and reactive power in MW and MVAr respectively with Voltages in p.u.). Stochastic variables
V1 V2 Q1 Ploss Pmk
Expectation (m)
Second-order moment (m2 )
Standard deviation (s)
Relative error (%)
MC RPO
PEM RPO
MC RPO
PEM RPO
MC RPO
PEM RPO
xmout
xsout
1.132 1.032 47.53 19.54 27.01
1.131 1.031 47.26 19.47 26.79
1.281 1.065 2270 383.9 732.6
1.279 1.063 2241 382.5 719.8
0.0056 0.0043 3.43 1.45 1.76 (MW)
0.0068 0.0054 2.89 1.85 1.47
0.051 0.078 0.57 0.36 0.78
21.43 25.58 15.74 27.59 16.48
Table 3 The mean and maximum error indices of the categorized outputs for the IEEE 14-bus test system. Stochastic variables type m
xout Max ðxmout Þ s xout Max ðxsout Þ
qi
Vi
Pi
Qi
0.078 0.141 0.1754 0.2343
0.00062 0.0043 0.2018 0.2102
0.00041 0.00047 0.2387 0.2856
0.0065 0.0087 0.1453 0.1878
active power absorbed by busses 13 and 14 are considered to have normal distribution with means equal to the values given in Ref. [21], and the standard deviation equals to 70% of the mean value. The reactive power consuming is controlled to keep power factor constant. The optimization process is formulated as linear fuzzy programing considering active power loss minimization as corresponding goal. Its fuzzy domain is shown in Table 1. The membership function used for the optimization variables are linear. The allowed maximum value of variables has a membership function of zero and for the ideal case where variables are within the satisfied region, a membership function of unity is assigned. In order to authenticate the efficiency and precision of the proposed method, results of both PEM PRPO and those provided by Monte-Carlo method with a sample size of 5000 are compared. The statistical characteristics of active power loss after performing optimization with two stochastic methods are shown in Table 2. The results are obtained with the proposed PEM method are compared to those provided by MC. It is shown that the proposed method can obtain the statistical characteristic of the output random variables such as the expectation, second-order central moment and variance in satisfactory relative error and short time comparing to MC method.
The MC method with sample size of 5000, taking 6.5 h of CPU times while PEM RPO computes the mean and standard deviation of outputs in 262 s. For the sake of obtaining numerical measurement of the proposed method's accuracy, the relative error of the mean and standard deviation of the output random variable is used as follows:
mout mout r p xmout ¼ mout r
(39)
sout sout r p xsout ¼ sout r
(40)
where mout and sout denote the value of the mean and standard r r deviation of a certain output correspond to MC method. mout and p sout p are the obtained values provided by the proposed method. The mean and maximum error indices of the categorized statistical characteristic of outputs are predominantly used for additional evaluation of the methodology [9]. m
xout ¼
n 1X xm n i¼1 outi
o n max xmout ¼ max xmouti i ¼ 1…n s
xout ¼
n 1X xs n i¼1 outi
o n max xsout ¼ max xsouti i ¼ 1…n
(41)
(42)
(43)
(44)
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The term out stands for a specific category of output variables such as a set of voltage magnitude V, or voltage phase angle d. Table 3 demonstrates aforementioned indices. 5. Conclusion This paper proposed a PRPO (probabilistic reactive power optimization) method that considers the effect of wind farm stochastic outputs. Comparing to deterministic RPO, the output results of the PRPO has stochastic nature that can be depicted with distribution functions. A PLF simulation based on point-estimated method was conducted while the input random variables corresponding to wind farm power output follow no common PDF. In order to tackle this issue, discrete point estimate method is investigated which utilizes wind power sample data to obtain statistical properties of input random variables when the PDF of wind farm power output is unknown. The computational burden of the proposed method is remarkably lower than MC method while retaining accuracy in the satisfied degree. References [1] Hatziargyrigu N, Zervos A. Wind power development in Europe. Proc IEEE 2001;89(12):1765e82. [2] Sun Xiaojing, Huang Diangui, Wu Guoqing. The current state of offshore wind energy technology development. Energy 2012;41(1):298e312. [3] Pullen A. Global wind report 2014 global. Wind energy council-renewable energy house (GWEC). April 2014. Available from: http://www.gwec.net/ global-figures/graphs. [4] Bjeli c Ilija Batas, Rajakovi c Nikola, Cosi c Boris, Dui c Neven. Increasing wind power penetration into the existing Serbian energy system. Energy 2013;57: 30e7. [5] Leite da silva AM, Arrienti VL. Probabilistic load flow by a multilinear simulation algorithm. Proc Inst Elect Eng Gener Treansmiss Disturb 1990;137(4): 276e82. [6] Rubinstein RY. Simulation and Monte Carlo method. John Wiley and Sons; 1981. [7] Leite da Silva AM, Arienti VL, Allan RN. Probabilistic load flow considering dependence between input nodal powers. IEEE Trans Power Appar Syst 1984;103(6):1524e30. [8] Fan M, Vittal V, Heydr GT, Ayannar R. Probabilistic power flow studies for transmission systems with photovoltaic generation using cumulants. IEEE Trans 2012;27(4):2251e61. [9] Usaola J. Probabilistic load flow in systems with wind generation. IET Gener Transm Distrib 2009;3(12):1031e41. rio GJ, Lujano-Rojas JM, Matias JCO, Catal~ [10] Oso ao JPS. A probabilistic approach to solve the economic dispatch problem with intermittent renewable energy sources. Energy 2015;82:949e59.
[11] Morales JM, Bringo L, Conejo AJ, Minguez R. Probabilistic power flow with correlated wind source. IET Gener Transm Distrib 2010;4(5):641e51. [12] Morales JM, Perez-Ruiz J. Point estimates schemes to solve the probabilistic power flow. IEEE Trans Power Syst 2007;22(4):1594e601. [13] Hong HP. An efficient point estimate method for probabilistic analysis. Rel Eng Syst Saf 1998;59(3):261e7. [14] Siemens fact sheet high-voltage direct current transmission (HVDC). 2014. available on: http://www.energy.siemens.com/hq/en/power-transmission/ hvdc. lez, Oriol Gomis-Bellmunt, Andreas [15] Mikel De-Prada-Gil, Francisco Díaz-Gonza Sumper. DFIG-based offshore wind power plant connected to a single VSCHVDC operated at variable frequency: Energy yield assessment. In Press, Corrected Proof. Available online 16.05.15. [16] Saunders Christopher Scott. Point estimate method addressing correlated wind power for probabilistic optimal power flow. IEEE Trans Power Syst 2013;29(3):1045e54. [17] Rabiee Abbas, Soroudi Alireza. Stochastic multiperiod OPF model of power systems with HVDC-connected intermittent wind power generation. IEEE Trans Power Deliv 2014;29(1):336e44. [18] Atwa YM, El-Saadany EF. Probabilistic approach for optimal allocation of wind-based distributed generation in distributed systems. IET Renew Power Gener 2009;5(1):79e88. [19] Jabr RA, Pal BC. Intermittent wind generation in optimal power flow dispatching. IET Gener Transm Distrib 2008;3(1):66e74. [20] Zare Mohsen, Niknam Taher, Azizipanah-Abarghooee Rasoul, Amiri Babak. Multi-objective probabilistic reactive power and voltage control with wind site correlations. Energy 2014;66:810e22. [21] Taghavi R, Seifi AR, Pourahmadi-Nakhli M. Fuzzy reactive power optimization in hybrid power systems. Elect Power Eng Syst 2012;42(1):75e383. [22] Jenkins Thukaram DL, Visaka K. Optimum allocation of reactive power for voltage stability improvement in ACeDC power systems. IEEE Proc 2006;153(2):237e46. [23] Fuerte CR, Acha Esquivel E. Unified power flow controller: a critical comparison of Newton-Raphson UPFC algorithms in power flow studies. IEEE Proc 1997;144:437e44. [24] Muller S, Deicke M, De Donker RW. Doubly fed induction generator systems for wind turbine. IEEE Ind Appl Mag 2002;8(3):26e33. [25] De Santos-Martin, Arnaltes S, Rodriguez-Amenedo JL. Reactive power capability of doubly fed of doubly fed asynchronous generators. Elect Power Syst Res 2008;78:1837e40. [26] Fudeh H, Ong CM. A simple and efficient acedc load-flow method for multiterminal dc systems. IEEE Trans Power Appar Syst November 1981;PAS100(11). [27] Miller AC, Rice TR. Discrete approximations of probability distributions. Manage 1983;29(3):352e62. [28] Bellman RE, Zadeh LA. Decision-making in a fuzzy environment. Manage Sci 1970;17(4):141e64. [29] Zimmermann HJ. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1978;1(1):45e55. [30] Tomsovic K. A fuzzy linear programming approach to the reactive power/ voltage control problem. IEEE Trans Power Syst 1992;7(1):287e93. [31] Power system test case archive. Available on: http://www.ee.washington.edu/ research/pstca, [accessed February 2012].
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Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Stochastic reactive power dispatch in hybrid power system with intermittent wind power generation Reza Taghavi, Ali Reza Seifi, Haidar Samet* School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 May 2014 Received in revised form 3 June 2015 Accepted 6 June 2015 Available online xxx
Environmental concerns besides fuel costs are the predominant reasons for unprecedented escalating integration of wind turbine on power systems. Operation and planning of power systems are affected by this type of energy due to the intermittent nature of wind speed inputs with high uncertainty in the optimization output variables. Consequently, in order to model this high inherent uncertainty, a PRPO (probabilistic reactive power optimization) framework should be devised. Although MC (Monte-Carlo) techniques can solve the PRPO with high precision, PEMs (point estimate methods) can preserve the accuracy to attain reasonable results when diminishing the computational effort. Also, this paper introduces a methodology for optimally dispatching the reactive power in the transmission system, while minimizing the active power losses. The optimization problem is formulated as a LFP (linear fuzzy programing). The core of the problem lay on generation of 2m þ 1 point estimates for solving PRPO, where n is the number of input stochastic variables. The proposed methodology is investigated using the IEEE-14 bus test system equipped with HVDC (high voltage direct current), UPFC (unified power flow controller) and DFIG (doubly fed induction generator) devices. The accuracy of the method is demonstrated in the case study. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Linear programming Fuzzy optimization AC/DC system Probabilistic load flow Point estimate method DFIG
1. Introduction Progressively ascending fuel prices accompanied by environmental concerns have induced countries to expand their power system infrastructure to incorporate further renewable energy, notably wind generation [1,2]. Annual statistics show that wind capacity increased by 16% from 2013 to 2014 with 51 GW of new capacity added [3,4]. The unmanageable nature of this category of energy necessitates probabilistic analysis of the power system operation. MC (Mont-Carlo) technique has been well proven as a trustworthy means to deal with stochastic variables and to attain distribution of the resultant variables with high precision [5e7]. In the view of aforementioned characteristics, MC methods are perceived as the most precise, reliable and robust SAM (stochastic analysis method), which is commonly used as a reference method to inspect the correctness of other probabilistic methods [8e10]. However,
* Corresponding author. E-mail address: [email protected] (H. Samet).
the enormous computational effort which is required for this method persuades researchers to use alternative approaches which are capable of quickly acquiring the probabilistic output attributes of the system. Abundant studies have confirmed that PEMs (point estimate methods) are capable of producing convincing stochastic results while effectively reducing the computational efforts [11e13]. However, the primary version of this method requires the input random variables’ PDF (probability density function) to evaluate the 2m þ 1 central moments. The aforementioned constraint within the PEM algorithm grew more apparent when stochastic input variables follow no common PDF [13]. This is most assuredly the case when the wind generation sources are integrated into the power grid. Although the stochastic behavior of wind speed is commonly modeled by a Weibull or Rayleigh PDF, the nonlinear relationship between the wind speed and the wind power makes it difficult to match the wind power to any generally known PDF. In order to cope with this issue, a discrete point estimate method is utilized. The core of the discrete point estimate method is that it employs sufficient sample measurement to approximate the (2m1)th central moment of the input
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Nomenclature V dc I dc Rc a P dc Q dc j jdc Vcr Vvr qcr qvr Pmk Qmk ~j tij Vg Qz Pst Qst Iro xsout m
xout
dc terminal voltage dc terminal current commutation resistance transformer tap connected to DC terminals active power at DC terminals reactive power flowing from the AC to DC the voltage angle of AC bus connected to DC the angle of AC current injected into the DC series voltage magnitude of UPFC parallel voltage magnitude of UPFC angle of series voltage source angle of parallel voltage source transferred active power between bus m and K transferred reactive power between bus m, k probabilistic jacobian matrix transformer tab between buses i and j voltage magnitude of generator bus (PV buses) compensators reactive power DFIG's stator active power DFIG's stator reactive power DFIG's rotor current the relative error of the standard deviation of the output random variable the mean error indices of a set of out put
s
xout the standard deviation error indices of a set of out put z ¼ f ðXÞ the PRPO objective function X set of n random variable XM Mth sample point of x KT matrix which links objective functions to variables S sensitivity matrix b dependent parameters matrix bmax , bmin upper and lower boundary of dependent variables xmax maximum limit of linearized control parameters xmin minimum limit of linearized control parameters fuzzy less than or equal to e D fuzzy decision G, C fuzzysubsets for objective functions and constraints
variables, where m refers to the number of concentration points in PEM approach. In the other hand, HVDC (high voltage direct current) transmission lines along with FACTS (flexible AC transmission systems) offer an excellent opportunity to support and improve the power supply of sustainable, efficient and reliable future grids [14,15]. FACTS are well known for the fast-response devices that can control the active and reactive power as well as the bus voltage. One of the more comprehensive FACTS devices is UPFC (Unified Power Flow Controller) that helps power systems to share the excessive loads from the lines and leads to the loss decline and high stable operations. To the best of authors’ knowledge, only few papers [16e20] have considered PRPO (probabilistic reactive power optimization) in the presence of HVDC and UPFC along with the uncertainties injected by the wind generation source. Therefore, this paper intended to propose a comprehensive model which consists of HVDC, UPFC and DFIG (double fed induction generator) as one of the most widely used generators in the wind turbine. This model takes the stochasticity under consideration in order to fill this gap. The main idea of this work is to attain
mD ðXÞ membership function of D mG ðXÞ; mC ðXÞ membership function of G, C l the degree of satisfaction for fuzzy objective function and fuzzy constraints mfi ðXÞ fuzzy membership function of fi ðXÞ [i threshold of constraint ðrÞ PL active power loss in iteration r Pro DFIG's rotor active power Ptot DFIG's total active power Qro DFIG's rotor reactive power Vst DFIG's stator voltage Ist DFIG's stator current Qtot DFIG's total reactive power mrj rth sample central moment ofjth random variable mxj average value of the M observation of jth variable the standard deviation of xj sxj rj;k kth concentration points correspond to jth input random xj;k kth standard location for jth input variable wj;k kth weighting factor for jth input variable xmout the relative error of the mean of the output random variable Abbreviation Rec rectifier Inv inverter MC Monte Carlo PDF probability density function PEM point estimate method MW Mega Watt MVAr Mega VAr p.u, Per Unit PRPO probabilistic reactive power optimization RPO reactive power optimization LFP linear fuzzy programming HVDC high voltage direct current FACTS flexible ac transmission systems UPFC unified power flow controller
the optimal control variables that are appropriate for stochastic situation, meanwhile minimizing the active power loss. Contrary to deterministic RPO (reactive power optimization), the probabilistic nature of input variables results in outputs from the optimization process become probabilistic. In addition, this paper presents a method that takes the uncertainty in optimization process into accounts, while conducting the proposed method in hybrid power system to consider a comprehensive model for investigation the proposed methodology. The remainder of this paper is arranged as follows: description of system model consisting of DFIG wind turbine, HVDC and UPFC models are introduced in Section 2. Details of discrete PEM are discussed in Section 3. A probabilistic reactive power optimization algorithm is proposed that aims to minimize active power loss in Section 3. The problem is formulated as linear fuzzy programing [21] considering the stochastic effects of the wind generation and loads. In Section 4 the performance of the proposed method is investigated with modified IEEE-14 bus test system and the correctness of the method is evaluated through comparing obtained results with those provided by MC technique.
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3
or power at the scheduled value. The firing angle must be small in order to draw in a little reactive power. However, to maintain an effective commutation, it must be set at its minimum value. 2.2. UPFC model In this study, UPFC is described by two ideal voltage sources, both working in conjunction with each other. One source is connected to the system in parallel and the other source comes in series. The equivalent circuit of this model is depicted in Fig. 3. Equations of series and parallel voltages are given by Ref. [23]:
Vcr ¼ Vcr ðcosðqcr Þ þ j sinðqcr ÞÞ
(7)
2. Description of model
Vvr ¼ Vvr ðcosðqvr Þ þ j sinðqvr ÞÞ
(8)
2.1. Converter model
min V V max Þ and q ð0 q 2pÞ are voltage where Vcr ðVcr cr cr cr cr magnitude and adjustable angle of series voltage source. The voltage magnitude and corresponding angle ðVvr ; qvr Þ indicate the parallel voltage source parameters as shown in Fig. 2.
Fig. 1. Equivalent representation of a monopolar station.
A basic analogous circuit of a typical monopolar converter is illustrated in Fig. 1. According to [21,22], the equations for each terminal can be formulated as:
V dc ¼ aV ac cosðaÞ Rc I dc P
dc
dc dc
¼V I
2.3. Wind turbine model
(1) (2)
where Rc , a and a are commutation resistance, transformer tab and firing angle. By neglecting the loss of converter and its attached transformer and counting the balance between power of AC and DC parts, the power factor can be formulated as:
V dc ¼ aV ac cosðj zÞ
(3)
The reactive power which streams from AC side to the converter is as follows:
Q dc ¼ P ac tanðj jdc Þ
(4)
where j and z are voltage and current angle of AC bus which are connected to DC converter. The relation between rec and inv voltages can be stated as follows: dc dc Vrec ¼ Vinv þ Rdc I dc
2 Pst þ Qst2 ¼ ðVst Ist Þ2
(6)
The practical control rule for a DC system is to oblige one terminal's determined voltage while the other terminals force current
Fig. 2. Unified power flow controller equivalent circuit.
1. Stator and rotor current limit [25]: The stator and rotor currents which are responsible of armature and field coil's heating due to joules losses are represented as follows:
(9)
(5)
An equation for DC current flowing between the two converters is given by:
ac ac I dc ¼ arec Vrec cosðarec Þ ainv Vinv cosðainv Þ Rdc
One of the most widely used generators in the wind turbine is DFIG. This adjustable speed generator comprises of a wound rotor induction generator that fed the grid by both the rotor and stator. The stator is directly connected to the grid while the rotor is linked to the system through back-to-back four-quadrant converter [24]. In Comparison with single fed induction generator that can only consume reactive power, DFIGs have the ability to control active and reactive power independently. When PRPO is concerned, the mathematical representation of DFIGs’ reactive (active) power constraints should be acquired. By considering the stator and rotor rating current as well as steady state stability limit, DFIG capability limits are obtained [25]:
Pst ¼
Xm Vst Irt sinðdÞ Xst
(10)
Qst ¼
Xm V2 Vst Iro cosðdÞ st Xst Xst
(11)
Fig. 3. Total DFIG's reactive power capability curve.
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2 Pst þ
Qst þ
Vst2 Xst
!2
¼
Xm Vst Iro Xst
2 (12)
Equations (9) and (12) represent two circles with the center [0, " # 0] and
Vst2 Xst ;
0 and radiuses amount to Vst Ist and
Xm Xst
Vst Iro in the
P Q plane respectively. 2. Steadyestate stability limit: according to [25] the maximum stable operation of power system is acquired when this limit is as follows:
calculated and scheduled values in the Jacobean matrix becomes less than a certain stopping criterion. The schedule values of Jacobean matrix in ac part include: sch sch sch Pical for i ¼ 2:::n; Qical for i ¼ m þ 2:::n; Pmk ; Qmk ; Pbb
In DC part: Vdc is a voltage by which the converter performs the voltage control, while Idc and Pdc are the current and power that the converter requires for this object. The under study system is a twoterminal monopolar system in which the rectifier controls the voltage, and inverters adjust the injected current.
3.2. Discrete point estimated method
V2 Qst ¼ st Xst
(13)
3. Total capability limit: The DFIG's total capability limit is attained by adding rotor reactive power to the stator ones. This limit is expressed as follows:
In comparison with PEMs that needs distribution information, this method employs the input sample data to ascertain the input 2m1 central moment [27]. Consider z as a function of n random input variables X:
z ¼ f ðXÞ Pro ¼ sPst
(14)
Ptot ¼ Pst þ Pro ¼ ð1 sÞPst
(15)
Qtot ¼ Qst þ Qro ¼ Qst
(16)
Qtotmax
Qtotmin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ AV 2st P 2tot BVst2 V2 ¼ st Xst
where A ¼
Xm Xst Iro
where Xmn ¼ ½x1 …xn T , let X1 …XM stand for M sample point of x. The unbiased estimates of the expectation and the central moments of the random vector X are stated as follows:
mxj ¼ (17)
(18) 2
M 1 X x M i¼1 ij
3. Probabilistic load flow The essence of the PEMs centered on approximation of the statistical information of the output random variables. The theory, in the first place, is proposed by Hong in 1998 [13] that presumed input random variables have continues distribution function. The PEMs’ procedure can be summarized as follows: The 2m1 central moments of input random variables are obtained from their distribution function (regularly m is set as 3). Then concentration points and the corresponding weights for each random variable are acquired through matching 2m equations. Finally, the statistical properties of output random variables are acquired based on aforementioned concentration points and their weight. 3.1. AC/DC power flow solution method Based on Rbus Gauss-Seidel method [26], the DC voltage and current for all terminals can be obtained. After substituting the values of Vdc , Idc , Rc and a in Eq. (1), the firing angle is computed. The active and reactive power conveyed from AC side to the DC terminals are also acquired by Eqs. (2)e(4). This set of power values is established as loads connected to the DC part. The first iteration of load flow solution delivers the deviation of the values of bus angles and bus voltages as well as the change of firing angle of converters. Thereby, the new active and reactive power of DC terminals is gained. This trend continues until the error between the
j ¼ 1:::n
(20)
where mxj denote the average value of the M observation of jth variable The rth sample central moment of jth random variable is determined as:
, B ¼ 1=Xst
Fig. 3 demonstrates these operational limits. The shaded area in Fig. 3 stands for the possible area of operation for the DFIG wind turbine.
(19)
M 1 X xk mxj M
mrj ¼
j ¼ 1…n
(21)
k¼1
In accordance with PEM procedure, concentration points correspond to each input random variable are defined as follows:
rj;k ¼ mxj þ xj;k sxj k ¼ 1; 2; 3
(22)
where xj;k denote the standard location and mxj , sxj are the mean and the standard deviation of xj . The standard locations are achieved by:
xj;k ¼
lxj ;3 2
þ ð 1Þ
3k
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 lxj ;4 l2xj ;3 k ¼ 1; 2 4
xj;3 ¼ 0
(23) (24)
where lxj ;3 and lxj ;4 are, the third and fourth standardized central moments of xj , also recognized as skewness and kurtosis. A weighting factor wj;k is defined corresponding to each random variable:
wj;k ¼
ð 1Þ3k k ¼ 1; 2 xj;k xj;1 xj;2
(25)
wj;3 ¼
1 1 m lxj ;4 l2x ;3
(26)
j
The row moments of the output random variable z are obtained through evaluation of the output at each of the concentration points zði; kÞ while considering the weighing factors wj;k [13].
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In view of PRPO, the objective function can be considered as a function of stochastic input random variables. Thus, its row moments are determined using PEM algorithm. The procedure of the proposed methodology is depicted in Fig. 4. 3.3. Description of fuzzy optimization model Linear programming, which is used for reactive power control aims at minimizing the total power loss of the system for all load conditions as follows:
parameters. S is the sensitivity matrix that stands for the relationship between the control and dependent parameters. b consists of the linearized dependent parameters. bmax and bmin are the up and down limits of the linear dependent variables. x consists of linearized control parameters. xmax and xmin are the maximum and minimum limits of linearized control parameters. In real-life situations, the objective function(s) and/or constraints in a fuzzy form have to be considered, particularly where planners are concerned with future plans. LP problem with indefinite constraints, this form of general LP is
minimize minimize subjected to and
T
DPL ¼ K x bmin b ¼ Sx bmax xmin x xmax
(27)
where:
vPL vPL vPL vPL vPL vPL vPL vPL ; ; ; ; ; ; ; vVg vQz vPmk vQmk vVk vVdc vIdc va
DX ¼ DVg ; DQz ; Dtij ; DPmk ; DQmk ; DVk ; DVdc ; DIdc ; Da KT ¼
(28)
In Eq. (27) K T is the linearized matrix which establishes the relationship between the objective function and the control
5
K T x f subjected tob min b e 0 e
¼ Sx bmax andxmin x xmax e e e
(29)
where is ‘fuzzy less than or equal to ‘and shows that variables or e parameters that take a fuzzy form. Also, fuzzy objective function(s) and constraints are characterized by membership functions. Here, K T , S and bmax have been changed to cope with multi-load levels. The optimal solution (fuzzy decision D) is the one which is able to meet (optimize) the fuzzy objective function(s) and the fuzzy constraints simultaneously. If G and C are defined as the fuzzy subsets of variables x in set X which satisfy the fuzzy objective(s)
Fig. 4. The proposed PEM reactive power optimization algorithm.
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R. Taghavi et al. / Energy xxx (2015) 1e8
and fuzzy constraints, respectively, then the membership function of fuzzy decision mD , is defined [28,29] as:
mD ðXÞ ¼ minfmG ðXÞ; mC ðXÞg
(30)
where the membership function of objective function is
(31)
mC ðXÞ ¼ min mCj ðXÞ j ¼ 1; 2; …; m
(32)
An optimal solution x* 2 X must satisfy both the objective function and constraints which can be obtained by taking the maximum of Eq. (30) Such as,
(33)
where l is the degree of satisfaction for fuzzy objective function and fuzzy constraints. While the optimal solution x* 2 X satisfies all constraints and objective functions for different load levels. In this study the l formulation is used to solve our problem as follows:
maximize subjected to
l l mfi ðXÞ
0 l1
(34)
The membership function mfi ðXÞ is arbitrary and is assumed to have a linear form. So the constraint fi ðXÞ di will have a fuzzy membership function in the fuzzy environment as can be observed below:
8 1 > > > > < ðd þ [ Þ f ðXÞ i i i mfi ðXÞ ¼ > [i > > > : 0
(37)
where:
mD ðXÞ ¼ minfmG ðXÞ; mC ðXÞg
l ¼ max mD X * ¼ max½minfmG ðXÞ; mC ðXÞg
maximize lðrþ1Þ subjected to S0 DX þ [lðrþ1Þ b0ðrÞ þ [ 0 lðrþ1Þ 1 X ðrþ1Þ ¼ X ðrÞ þ DX ðrþ1Þ
T
_ 0 S S ¼ K ¼ S b S S "
ðrÞ # _ P 0 bmax b ¼ b ðrÞ ¼ _ðrÞL bmin b
(38)
ðrÞ
where PL is the active power loss in iteration r, and [ is the threshold of objective function and constraints. Since S is constant in each iteration, we will encounter only ordinary linear programming. 4. Case studies In this section, the effectiveness of the proposed PEM PRPO method is investigated on the modified IEEE_14 bus test system as depicted in Fig. 5 [21]. It is noteworthy to mention that the load level is 20% increased when comparing with the original value for this system [31]. The wind farm equipped with DFIG is located at bus 6, having the total capacity of 621 MW. The wind speed is presumed to be two-parameter Weibull with quantities a and b equal to 9 and 2.025 respectively. Probabilistic distribution corresponding to the wind farm output power is illustrated in Fig. 6. The
fi ðXÞ di di fi ðXÞ di þ [i
(35)
fi ðXÞ di þ [i
where [i is the threshold of constraint fi ðXÞ di . From Eqs. (34) and (35), for the constraints we can write:
l mfi ðXÞ l
ðdi þ [i Þ fi ðXÞ [i
(36)
fi ðXÞ þ l[i di þ [i It is the planner who settles the membership function of the optimization process. Some researchers [28,29] maintain that the membership and objective functions are subjective and should be established based on the planners’ preferences and past experience. Nevertheless, it can be seen that by choosing an appropriate membership function for objective function(s), computation time can be considerably reduced [30]. In the present article, the abovementioned method has been improved and a methodology has been introduced which can significantly reduce the computation time. In our optimization, because the minimization of active power loss in the system is the main objective function, the ideal case is achieved when the loss equals zero. Therefore, the membership function of one will be considered for it. Because our objective is to reduce the system loss from the base case of the load flow, it is assumed that the membership function is zero. The membership function between these two points is assumed to be linear. Eq. (27) has been modeled as can be seen below,
Fig. 5. Modified IEEE 14 bus test system with DG connected to bus 6.
Fig. 6. Output power histogram in p.u. for wind turbine.
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Table 1 Trapezoidal possibility of variables for the IEEE 14 bus system. Control
Max
[max
Min
[min
Dependent
Max
[max
Min
[min
V1 V2 V3 V6 V8 9 Qsvc 14 Vupfc t47 t49 t56 Pmk Qmk Vdc Idc arec ainv
1.18 1.15 1.15 1.15 1.18 5 1.15 1.1 1.1 1.1 40 22 1.02 0.5 1.1 1.1
0.0 0.0 0.0 0.0 0.0 5 0.0 0.05 0.05 0.05 0.0 0 0.0 0.2 0.05 0.05
1 1 1 1.05 1 0.0 1 1 1 1 30 7 0.9 0.2 0.98 1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05 0.05 0.05 5 2 0.05 0.1 0.0 0.0
Q1 Q2 Q3 Q6 Q8 V5 V7 V8 V13 V10 V11 V12 V13 arad inv arad rec
100 50 40 24 24 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 0.4 0.4
10 10 10 10 10 0.05 0.05 0.05 0.05 0.05 0.05 0 0.05 0 0
100 40 0 6 6 1 1 1 1 1 1 1 1 0.17 0.17
10 10 10 10 10 0.03 0.03 0.03 0.03 0.03 0.03 0 0.03 0.17 0.17
Table 2 Statistical characteristic of the power system variables comparing MC results to that of PEM method for the IEEE 14-bus test system (active and reactive power in MW and MVAr respectively with Voltages in p.u.). Stochastic variables
V1 V2 Q1 Ploss Pmk
Expectation (m)
Second-order moment (m2 )
Standard deviation (s)
Relative error (%)
MC RPO
PEM RPO
MC RPO
PEM RPO
MC RPO
PEM RPO
xmout
xsout
1.132 1.032 47.53 19.54 27.01
1.131 1.031 47.26 19.47 26.79
1.281 1.065 2270 383.9 732.6
1.279 1.063 2241 382.5 719.8
0.0056 0.0043 3.43 1.45 1.76 (MW)
0.0068 0.0054 2.89 1.85 1.47
0.051 0.078 0.57 0.36 0.78
21.43 25.58 15.74 27.59 16.48
Table 3 The mean and maximum error indices of the categorized outputs for the IEEE 14-bus test system. Stochastic variables type m
xout Max ðxmout Þ s xout Max ðxsout Þ
qi
Vi
Pi
Qi
0.078 0.141 0.1754 0.2343
0.00062 0.0043 0.2018 0.2102
0.00041 0.00047 0.2387 0.2856
0.0065 0.0087 0.1453 0.1878
active power absorbed by busses 13 and 14 are considered to have normal distribution with means equal to the values given in Ref. [21], and the standard deviation equals to 70% of the mean value. The reactive power consuming is controlled to keep power factor constant. The optimization process is formulated as linear fuzzy programing considering active power loss minimization as corresponding goal. Its fuzzy domain is shown in Table 1. The membership function used for the optimization variables are linear. The allowed maximum value of variables has a membership function of zero and for the ideal case where variables are within the satisfied region, a membership function of unity is assigned. In order to authenticate the efficiency and precision of the proposed method, results of both PEM PRPO and those provided by Monte-Carlo method with a sample size of 5000 are compared. The statistical characteristics of active power loss after performing optimization with two stochastic methods are shown in Table 2. The results are obtained with the proposed PEM method are compared to those provided by MC. It is shown that the proposed method can obtain the statistical characteristic of the output random variables such as the expectation, second-order central moment and variance in satisfactory relative error and short time comparing to MC method.
The MC method with sample size of 5000, taking 6.5 h of CPU times while PEM RPO computes the mean and standard deviation of outputs in 262 s. For the sake of obtaining numerical measurement of the proposed method's accuracy, the relative error of the mean and standard deviation of the output random variable is used as follows:
mout mout r p xmout ¼ mout r
(39)
sout sout r p xsout ¼ sout r
(40)
where mout and sout denote the value of the mean and standard r r deviation of a certain output correspond to MC method. mout and p sout p are the obtained values provided by the proposed method. The mean and maximum error indices of the categorized statistical characteristic of outputs are predominantly used for additional evaluation of the methodology [9]. m
xout ¼
n 1X xm n i¼1 outi
o n max xmout ¼ max xmouti i ¼ 1…n s
xout ¼
n 1X xs n i¼1 outi
o n max xsout ¼ max xsouti i ¼ 1…n
(41)
(42)
(43)
(44)
Please cite this article in press as: Taghavi R, et al., Stochastic reactive power dispatch in hybrid power system with intermittent wind power generation, Energy (2015), http://dx.doi.org/10.1016/j.energy.2015.06.018
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Please cite this article in press as: Taghavi R, et al., Stochastic reactive power dispatch in hybrid power system with intermittent wind power generation, Energy (2015), http://dx.doi.org/10.1016/j.energy.2015.06.018