A new smart approach for state estimation of distribution grids considering renewable energy sources

Energy 94 (2016) 29e37

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A new smart approach for state estimation of distribution grids considering renewable energy sources Reza Khorshidi a, Faridon Shabaninia a, *, Taher Niknam b a b

School of Electrical and Computer Engineering, Engineering Faculty No. 1, Zand St., Shiraz, Iran Electrical and Electronics Engineering Departments, Shiraz University of Technology, Shiarz, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 December 2014 Received in revised form 29 August 2015 Accepted 23 October 2015 Available online xxx

The idea of smart grids has created new opportunities in the electrical networks for monitoring the system status. One of the valuable and significant techniques for monitoring the smart grids is state estimation. In this way, this paper proposes a sufficient state estimation algorithm for inclusive monitoring of the distribution systems in the presence of RESs (renewable energy sources). The proposed method is a hybrid technique using WLS (weighted least square) method and FA (firefly algorithm) to reach more reliable and accurate state estimation of the network. FA is equipped with new optimization operators that make it possible to solve the multi-modal problems using an automatic sub-division feature. In order to improve the overall search ability of the algorithm, a new two-phase modification method is proposed. The proposed hybrid method can estimate the voltage angle using the WLS method. The simulation results show more optimal cost function value with faster response with escaping from the several local optima of the problem. © 2015 Elsevier Ltd. All rights reserved.

Keywords: SE (state estimation) RESs (smart grid, renewable energy sources) FA (firefly algorithm)

1. Introduction In the last years, a rapid progress from the conventional electrical grids toward the new smart grids has happened to deal with the increasing requirements of customers. In this regard, some of the available and significant techniques that have been considered for this evolutionary movement are fast reconfiguration [1], intellectualization of the system [2], decentralization of power system [3] and fast monitoring devices. In the first stages of creating new smart grids, monitoring of the system status is a necessary task which without it the whole idea is devastated. One of the most precious and useful strategies for monitoring the system is state estimation [4]. Power system state estimation is defined as the process of estimating the state of the electrical system from the redundant telemetry measurements located in different positions of the grid [5]. Technically, state estimation is considered as the solution of finding the voltage phasors of all buses at certain time. Generally, a direct solution is to install accurate measurement on all buses of the system to obtain the synchronized voltage phasor of

* Corresponding author. School of Electrical and Computer Engineering, Shiraz University, Namazi Square, Shiraz, Iran. Tel./fax: þ98 71 32303081. E-mail addresses: [email protected] (R. Khorshidi), [email protected] (F. Shabaninia). http://dx.doi.org/10.1016/j.energy.2015.10.096 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

these buses. However, such a solution is exposed to communication failures or measurement errors. In order to overcome this issue, redundant measurements can be employed to reduce the measurement errors and find the optimal estimation which is discussed in the state estimation problem. Some of the parameters that can be used for accurate state estimation can be named as real and reactive power injection, voltage magnitude or current. In this regard, iterative WLS (weighted least square) algorithm is a usual and widely used technique which uses a sequence of measurement samples to estimate the system and offer static or quasi-static state information [6]. The conventional state estimator function performs centrally in a control center and takes the desirable data, real time and static, to solve the TP (topology processor), SE (state estimation), and the BD (bad data) detectionidentification sequentially [7e9]. Therefore, one of the main useful outcomes of optimal state estimation is safe and efficient operation of power systems in both normal and contingency situations [10e12]. One of the new technologies that has affected the optimal state estimation problem is DG (distributed generation). The idea of DG was devised to bring generation near the consumers and thus reducing the power losses, improving the voltage of profile and providing more reliable electrical services [13]. Some the researches have also suggested to install DGs in the same building as

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R. Khorshidi et al. / Energy 94 (2016) 29e37

Nomenclature

PijBe;line

the perfect power flowing between buses ij

Be;line Pij;max

maximum value of transmission power between buses

e z e r e W

weight factor matrix

e H

PLoad;max maximum value of the jth load

functional measurement matrix

e f ðXÞ

cost function weighting factor state equation number of DGs number of loads the voltage of the jth DG voltage the jth load

Qc Is rij Iter XBest

d h ngen nload VPgj VLoadj

the measurements vector error vector

j PGen;min minimum value of active power produced by jth DGs

consumers [14e16]. Along with these benefits, the high penetration of DGs can increase the complexity of system planning, operation and communication which will affect the optimal state estimation too. In response to these complexities, new methods especially based on evolutionary algorithms have been proposed in the literature. In Ref. [17], authors proposed ant colony optimization for distribution state estimation. A hybrid method based on PSO (particle swarm optimization) was proposed in Refs. [18,19] for distribution state estimation with DGs. In Ref. [20], a new state estimation technique was devised to estimate the magnitude and phase angle of current in three phase distribution system. Here TakagieSugeno fuzzy system is used to estimate the system states in the unbalanced condition of the feeders. A two-stage approach based on WLS technique was proposed in Ref. [21] for state estimation of feeders' current. Here first the WLS sub-system is presented and then each sub-system is solved individually. In Ref. [22], state estimation of power system is done using a probabilistic load flow approach. In Ref. [23], a synchronized method was proposed for state estimation of three-phase systems. Here a new method based on HBMO (honey bee mating optimization) algorithm was suggested to increase the accuracy of state estimation in the presence of DGs. Here, the idea was to use HBMO for searching the problem space in the three-phase systems. According to the above discussions, this paper tries to address the state estimation problem in the distribution systems considering DGs. In this way, a hybrid WLS and evolutionary algorithm is used to increase the accuracy of estimation effectively. According to the high complexity of the problem and its nonlinear search space, FA (firefly algorithm) is employed. FA is a meta-heuristic optimization method that is equipped with an automatic sub-division feature to escape from the several local optima of the problem [24]. In order to increase the diversity of the firefly population, a new two-phase modification method is proposed. The feasibility and satisfying performance of the proposed method are examined on a standard distribution system. 2. State estimation in smart distribution networks Generally, the power industry tries to deliver electric power to the consumers with the highest reliability and quality and least cost and emission productions. Smart grids are new solutions for reaching these targets. In a smart grid, the producers are managed to control their production when consumers are also managed to obey a smart loading pattern with intelligent consumption pattern and cost. In order to reach these goals, the necessary of having full monitoring of the system and attaining the required information

ij j

j

D

Nequality e J ðXÞ i

a1 Xestimated

reactive power value of jth bus source intensity distance between ith and jth fireflies worst solution of the population random number laying in the range of [0,1] number of equality equality constraint constant penalty factors value estimated value

from all buses is quite obvious. Technically, the smart grid design is done in three areas of consumers, equipment and communication. As the direct result, smart grid technology is reflected in three parts of generation, transmission and distribution which can bring useful features in all parts from the electrical service point of view. The vital issue for reaching the idea of smart grids is the proper access to the required data in the network which is called shortly as state estimation. The use of appropriate state estimation will bring the below advantages: a. Peak shaving: The use of smart network can affect both the generation and consumptions sides which a direct result is peak load shaving. b. Fossil fuel reduction: By reducing the loss of energy in the feeders as well as possibility of altering the topology through some useful techniques such as reconfiguration will result in notable fossil fuel reduction. c. Interruption reduction: Reducing the outage time, the number of interruptions and unwanted shut downs. d. Reducing investment costs: The main reason for new investments in the current electric grids is the load growth which is mainly caused at peak load hours. By the use of idea of smart grid and intelligent management techniques, the load growth can be controlled which will reduce the investment costs. e. Reducing switching cost: Decreasing costs associated with the switching of remote subscribers. Fig. 1 shows a typical smart grid. In this figure, PMU (phasor measurement unit) will measure and synchronize data of the network. It is worth noting that there is less than one millisecond difference in the data acquisition of the voltage angles of buses. During the state estimation process, bad data and fake measurements are removed from the data set. It is obvious that the observability of the system should be preserved before and after data clearing process. The significant issue is that a smart grid requires adequate number of data processing devices including measuring devices and control devices. This can increase the complexity of the new smart grids greatly when the total cost of the network is increased too. Nevertheless, the use of appropriate state estimation method can help for using less hardware in the system. This paper tries to address this issue using the idea of FA and WLS technique. 2.1. WLS (weighted least square) estimator It is demonstrated in the literatures [7] that if the analog devices were synchronized for current and voltage data, then the state

R. Khorshidi et al. / Energy 94 (2016) 29e37

31

Fig. 1. A typical smart grid.

estimation equations become linear (first conditional). In this way, by the use of adequate PMU devices, new measurement function for state estimation is created that is linear in the complex plane. In this space, both states and measurements are linear and in complex plane which will result in a linear state estimation. The linear state estimation equations can be shown as below: T e e Min e r W r ;

s:t:

ee e z¼H x þe r

(1)

e is Where e z is the measurements vector, e r is the error vector, W e is the weight factor matrix, e x is the states vector of system, and H the functional measurement matrix of system connecting the measurement vector to the states. In the state estimation problem, each measurement is shown e zi ¼ zi;real þ jzi;imag , each state is shown e xi ¼ xi;real þ jxi;imag and each r i;real þ je r i;imag . In order to reach a matrix error is shown by e ri ¼ e formulation, the measurement, state and error vector are shown by       zi;real xi;real ri;real e ; e xi ¼ ; and e ri ¼ in the real plane, zi ¼ jzi;imag jxi;imag jri;imag respectively. The entry e hi;j ¼ hi;j;real þ jhi;j;imag is shown by a 2  2   e ¼ hi;j;real jhi;j;imag . Considering matrix in the real plane as H jhi;j;imag hi;j;real m measurement and n states, Eq. (1) is re-written as below: T e e Min J ¼ e r W r ; 1 0 10 1 0 1 0 e e e e e z;1 H x1 r1 1;1 / H 1;n e e r ¼@ « s:t: e z ¼ @ « A ¼ HX þ e 1 « A@ « A þ @ « A e e e e e z;m xn rm H m;1 / H m;n

(2) e is the diagonal weight matrix wherein all entries are real where W e is the 2  2 weight block for each measurement as numbers and W i follows:

Wi ¼

s2z1;real

0

0

s2z1;real

! (3)

e its partial derivative is So as to lessen J with respect to X, calculated and set to zero as: vJ ¼ 0 X ve e that includes The output of the above equation is the vector X the states of the problem. Considering normal distribution function e is calculated as follows: for the measurement error of meters, the X

1  e ¼ H T WH HT Wz X

(4)

where HT is the left pseudo inverse of matrix H which exists if m > n. It should be noted that H is full rank; i.e the number of measurements m is greater than the number of variables n. Also, the measurements are linearly independent. The necessary condition for finding the minimum of stationary point J is that the second derivative of J becomes positive semi definite. The state estimator formulation is described in the next section. 2.2. Cost function In the WLS technique, the below cost function is considered for state estimation which should be minimized:

If

m  2 X vJ e e ¼ dj zj  hj ðXÞ ¼ 0; then Min f ðXÞ e vX j¼1

8 iT h > > e e¼ X e > 1 n X > 1;VG ; X 2;Vload > > > > < iT h 1 e2 e ngen e ¼ X e ; n ¼ ngen þnload X 1VG V;Pg ; X V;Pg ;:::; X V;Pg > > > > > iT h 1 > > e2 e nload e e >X ¼ X ;X ;:::; X : 2Vload

V;load

V;load

(5)

V;load

e is the cost function, X e is the where z is the measured data, f ðXÞ vector of state variables, d is the weighting factor, h is the state equation, m is the number of measurements, ngen is the number of DGs, nload is the number of loads, VPgj is the voltage of the jth DG and VLoadj is the voltage the jth load. 2.3. Constraints and limitations The below constraints should be considered during the optimization process [24]: * Maximum capacity of DGs:

j

j

j

PGen;min  PGen  PGen;max

; j ¼ 1; 2; 3; …ngen

(6)

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R. Khorshidi et al. / Energy 94 (2016) 29e37 j

j

where PGen;min and PGen;max are the minimum and maximum value of active power produced by jth DGs, respectively. * Maximum power flow in the feeders:

   Be;line  Be;line Pij  < Pij;max

(7)

where PijBe;line is the perfect power flowing between buses ij and Be;line Pij;max is the maximum value of transmission power between buses ij. * Bus voltage limitation:

VBus;min  VBus;j  VBus;max ; j ¼ 1; 2; 3; …; nb ;

(8)

the number of buses:

where VBus,j is the voltage value of the jth bus;VBus,max and VBus,min are the maximum and minimum voltage of the jth bus, respectively. * Restriction of loads: j

j

j

PLoad;min  PLoad  PLoad;max

j ¼ 1; 2; 3; …NLoad

(9)

j j where PLoad;max and PLoad;min are the maximum and minimum values of the jth load, respectively.

* Restriction of capacitors: j

j

0  Qc  Qc;max

j ¼ 1; 2; 3; :::; nc ; number of capacitors (10)

j

j

where Qc is the reactive power value of jth bus capacitor and Qc;max is the maximum reactive power value of the jth bus capacitor. In this paper, we have considered an unbalanced three phase power flow for the above equations. The control unit will manage the performance of the VRs (voltage regulators) and capacitors, locally. Since the number of state variables is generally more than the number of measurements, the below assumptions are considered: a. The operation set point of local capacitors and voltage regulators are identified. b. The standard deviation and average output power of DGs are identified. c. Current injections and voltage at the main bus of substation are identified. d. Location of switches at any feeder is identified.

section describes original FA and the second part describes its modified version. In the recent years, nature has been a proper source for creation of a number of optimization algorithms called evolutionary algorithm. One of the most successful evolutionary algorithms is FA which mimics the behavior of fireflies in the hot summer of tropical areas. This algorithm was first introduced in 2008 to model the flashing behavior of fireflies in the eyes of each other [24].

3.1. MFA (modified firefly algorithm) The main purposes of flashing behavior are attracting the potential prey and mating partners. In order to model the behavior of fireflies for the optimization purposes, three simple assumptions should be made: 1) all fireflies are unisex, 2) a more attractive firefly is one with more brightness (a better fitness function in the optimization) and 3) when a firefly does not see any brighter firefly in its neighborhood, it will fly randomly in the air (simulating a random movement). Mathematically, the light intensity of each firefly in the distance r is as follows:

IðrÞ ¼

Is r2

(11)

where Is is the source intensity. Considering a medium whose light absorption coefficient is constant I0, the light intensity can be expressed below. Also, the Gaussian form of the light intensity is shown here:

ðgr2 Þ IðrÞ ¼ I0 e.  1 þ gr 2 IðrÞ ¼ I0

Since the attractiveness of each firefly is a function of its light intensity, it can be mathematically explained by follows: 2 bðrÞ ¼ b0 eðgr Þ

(13)

It is worth noting that rij as the distance between ith and jth fireflies can be calculated in the Cartesian distance as follows:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d  2 X  u    xi;k  xj;k rij ¼ xi  xj ¼ t

(14)

k¼1

Using the above equations, the position of each firefly Xi is updated using the below equation:

2.4. Modeling of DGs DGs can include a wide range of power sources from the fossil fuel based DGs to RESs (renewable energy sources). Nevertheless, in order to consider these devices in the state estimation problem, they should be modeled precisely. Based on the control scheme of these devices, they can be considered in two main formats of 1) autonomous three phases control and 2) concurrent three phase control. These models are shown in Fig. 2. 3. FA (firefly algorithm) This section proposes a new optimization algorithm based on FA (firefly algorithm) to solve the problem deeply. The first part of this

(12)

Fig. 2. (a) Concurrent model of DG, (b) autonomous model of DG.

R. Khorshidi et al. / Energy 94 (2016) 29e37





2 xi ¼ xi þ b0 e grij xj  xi þ a rand  1 2

(15)

=

where a is the randomization parameter and rand is a uniformly distributed generated number in the range [0,1]. As it can be seen from the equation, as the distance between two fireflies is large, the second term becomes zero and thus the firefly Xi can fly in the air randomly. FA is a meta-heuristic optimization algorithm that is equipped with many special characteristics including general usage, easy implementation, fast convergence, automatic sub-division ability and few adjusting parameters. Nevertheless, we propose a twophase modification method to increase the total search ability of this algorithm. The first part of the modification method is Levy Flight that is a slow random movement around each firefly in the air. Therefore, this modification method simulates a local search around each solution as follows [25]:

Le0 vyðuÞ  t ¼ Iteru ;

ð1 < u  3Þ

Xinew ¼ Xiold þ 41 4Le0 vyðuÞ

(16) (17)

where Iter is the iteration of algorithm and u is a constant value in the range [0,1].

33

The second part of the modification method is constructed using two mutations and three crossovers from genetic algorithm. This modification method will increase the diversity of population of fireflies and thus will improve the performance of the algorithm. In Iter and worst this way, suppose the best solution of the population XBest Iter solution of the population Xworst . For each firefly Xi, three different fireflies Xr1, Xr2, and Xr3 are chosen such that r1 s r2 s r3 s i. Now, two muted solutions are produced as follows:



XMute1 ¼ Xq1 þ D  Xq  Xq3 2  Iter Iter  XWorst XMute2 ¼ XMute1 þ D  XBest

(18)

where D is a random number laying in the range of [0,1]. The following fireflies are generated by utilizing the XMute1 and XMute2. Now by the use of crossover operator, four test fireflies are produced:

 XBest;1 ¼ xBest;1 ; xBest;2 ; :::; xBest;d  if k1  k2 xMut1;j ; xImprove1;j ¼ xBest;j ; if k1 > k2  xImprove2;j ¼

Fig. 3. Flowchart of the proposed MFA.

xMute1;j ; xj ;

if k3  k2 if k3 > k2

(19)

(20)

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R. Khorshidi et al. / Energy 94 (2016) 29e37

Fig. 4. IEEE 34 bus test system.



xBest;j ; x ;  j xMute1;j ; ¼ xMute2;j ;

xImprove3;j ¼ xImprove4;j

if k4  k3 if k4 > k3 if k5  k4 if k5 > k4

(21) (22)

where k1 ; k2 ; k3 ; k4 ; k5 are random values in the range of [0,1]. The best firefly among these five fireflies will replace the firefly Xi in the population. It is clear that the quality of fireflies is determined based on the fitness function value. The flowchart of the proposed MFA (modified firefly algorithm) is depicted in Fig. 3. 4. Implementation of MFA algorithm in state estimation problem This section tries to explain the application procedure for using MFA in solving the optimal state estimation problem. According to the problem formulation of Section 2, the state variables are voltage of DGs and load values of the buses. The measurements are currents, voltages or active powers. Therefore, the below steps are required to solve the problem: Step1: Read the input data including the network, DGs, state estimation problem and MFA. Step2: Convert the constraint problem into an unconstrained optimization problem as follows:

0 0 N Neq    ineq h  i2 X X 2



e e e e Jj X Maximum 0;gj X G X ¼ f X þa1 @ þa2 @ j¼1   e ¼ 0 ;j ¼ 1;2;3;:::;N Jj X equality   e gj X <0 ; j ¼ 1;2;3;:::;Ninequality

j¼1

(23)

Table 2 Specifications of five loads. Location

Active power (kW)

Reactive power (kVar)

SD (%)

24 29 31 32 33

133.4 19.45 19.91 19.00 27.91

106.83 15.57 15.94 15.18 22.33

20 10 15 10 15

e is the objective function of state estimation. Nequality Where GðXÞ and Ninequality are the number of equality and inequality restrictions. e and g ðXÞ e are the equality and inequality constraints. Also, a1 Ji ðXÞ i and a2 are the constant penalty factors. Here equal to 106. Step3: Generate the initial firefly population randomly. The position of each firefly Xi shows the optimal values of the state variables. Step4: Calculate the objective function value for each firefly and choose the best global solution XBest. Step5: Improved the position of fireflies in the population using (15). This equation uses the attractive function calculated for each firefly in (13). Step 6: Improve the firefly population using the proposed twophase modification method as described in Section 3.1. Step7: Check the termination criterion. If satisfied finish the algorithm otherwise return to step 5 and repeat the steps.

5. Simulation results This section uses the IEEE 34-bus test system to examine the performance of the proposed method. Fig. 4 shows the single diagram of this test system. We have considered four DGs with specifications given in Table 1. In order to have a practical analysis, five

Table 1 Specification of four distribution generations.

G1 G2 G3 G4

Mean of active power output (kW)

SD (%)

Location

Power factor

Type of RES

120 120 90 100

10 15 10 15

9 20 27 2

0.9 0.9 0.9 0.9

Photovoltaic Wind turbine Hydro power Wind turbine

R. Khorshidi et al. / Energy 94 (2016) 29e37

35

Table 3 Shows parameters of FA in the simulation. Parameter

Number of fireflies

Randomization parameter

Initial attractiveness

Maximum iterations

Value

80

0.2

0.8

500

Table 4 Specifications of each algorithm for comparing. Algorithm

Function value

MF.A. F.A. P.S.O._N.M. P.S.O. H.B.M.O. N.N. A.C.O. G.A.

Fbest

Faverage

Fworst

0.000089 0.000109 e e 0.000209 0.0024 0.000257 0.000762

0.000113 0.000173 0.000168 0.000735 0.000223 0.0024 0.000319 0.000986

0.000178 0.000196 0.000177 0.000927 0.000173 0.000348 0.000389 0.001034

MCRE Location MCAE Location

FA.

P.S.O. N.M.

P.S.O.

H.B. M.O.

N.N.

A.C.O.

G.A.

0.0183 2 0.058 2

0.02 24 0.007 29

0.6 21 1.08 32

1.98 4 6.47 32

2.77 21 3.23 14

4.25 24 6.55 32

2.43 21 4.58 32

4.89 26 4.28 32

MCRE Location MCAE Location

MCRE ¼ max

FA.

P.S.O. N.M.

P.S.O.

H.B. M.O.

N.N.

A.C.O.

G.A.

0.0507 2 0.0912 2

0.06 9 0.09 20

0.76 29 1.12 29

2.63 8 4.98 19

2.56 41 4.68 14

6 21 8.44 32

1.7 8 4.87 29

3.56 35 6.25 18

5 5 5 30 9 0 11 16

9230 11,809 20,564 45,224 36,783 ~0 39,736 56,481

* MCRE (maximum characteristic relative error)

Table 6 Results of MCRE and MCAE for estimation of DGS voltage. MFA.

Number of function evaluations

results of the best solution, worst solution and average value are shown comparatively. According to these results, the proposed MFA could reach to better results in for all scoring targets. For better comparison, the below two criteria are employed. These two criteria show the estimation error for load voltage and DG voltage using different methods.

Table 5 Results of MCRE and MCAE for estimation of load voltage. MFA.

SD (%)

jXestimated value ðjÞ  Xtrue value ðjÞj  100 jXtrue value ðjÞj

%

(24)

* MCAE (maximum characteristic absolute error)

varying loads are also considered in the test system and the data are given in Table 2. Also, two measuring devices are supposed to be installed on the buses 1 and 28. Regarding the FA, the adjusting parameters are given in Table 3. These values are found after several running of the algorithm experimentally. Table 4 shows the results of optimization of the fitness function by different algorithms. In order to have better comparisons, each algorithm has solved the problem for 10 different times and the

MCAE ¼ maxðjXestimated

value ðjÞ

 Xtrue value ðjÞjÞ

(25)

where, Xestimated value and Xtrue value are the estimated and actual values, respectively. Tables 5 and 6 show the results of MCRE and MCAE criteria using different algorithms. The superiority of the proposed MFA over the other well-known algorithm can be seen from the lower values of MCRE and MCAE in these tables. The corresponding optimal values of active and reactive power of loads and DGs calculated by MFA are shown in Tables 7 and 8.

Table 7 Results of MFA estimation for active and reactive power of loads. Bus number Active power (kW) Reactive power (kVar)

Actual Estimated Actual Estimated

2

9

17

23

28

33

85 83.7601 69 68.25

420 421.49 343 342.12

61 61.73 49 48.63

85 83.31 68 68.3

62 60.15 50 50.9

30 29.11 25 24.1

Table 8 Results of active and reactive power estimation of DGS. Bus number Active power (kW)

Actual Estimated

2

9

18

31

50 50.3917

83 82.7518

85 85.0641

85 83.7601

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R. Khorshidi et al. / Energy 94 (2016) 29e37

In order to have better insight about the estimated values, Fig. 5 shows the estimation value and real value of voltage phase (in pu) of A, B and C, comparatively. Also, Fig. 6 shows the similar results for voltage angle (in degree) for different phases of A, B and C. It is worth noting that zero value of some buses in Fig. 6 shows that these buses are single-phase (such as buses 5, 15 and 19). The accurate estimation of variables can be deduced from these figures easily. Finally, Fig. 7 displays the optimal results achieved by MFA for estimation value and real value of active and reactive of power state estimation, comparatively. Based on the above results and comparison of different methods such as MFA, FA, PSO, ACO (Ant Colony Optimization), HBMO, NM (Nelder Mead), GA, the below conclusions can be made here: Fig. 5. Show the estimation value and real value of voltage (Pu) Phase_A,B,C.

1) In comparison with other algorithms, MFA could reduce the error estimation of voltage and angle of power system network with DGs more successfully. 2) The proposed MFA could find the more optimal solution than other algorithms when preserving better stability of response. 3) MFA requires less computational effort to reach the optimal solution. 4) The response time of MFA is less than other algorithms.

6. Conclusion

Fig. 6. Shows the estimation value and real value of angle of voltage (Deg) Phase_A,B,C.

This paper proposed a new hybrid MFA and WLS technique to investigate the optimal state estimation problem in the presence of DGs. The proposed MFA makes use of a two-phase modification method to explore the entire search space of the problem deeply. The simulation results on the IEEE 34-bus test system with three DGs show the high performance and dependability of the proposed algorithm. In comparison with other well-known methods in the area, the proposed algorithm could reach to lower estimation error. According to these results, the proposed method could estimate the stat variables of the network including the voltage of loads and DGs accurately. Also, by running each algorithm for 10 trails, it was seen

Fig. 7. Shows the estimation value and real value of active and reactive of power state estimation.

R. Khorshidi et al. / Energy 94 (2016) 29e37

that the proposed algorithm has suitable stability response. The superiority of the proposed MFA over A.C.O., H.B.M.O., G.A., N.N. and P.S.O was demonstrated too.

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