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Journal of Intelligent & Fuzzy Systems 16 (2005) 119–131 119IOS Press

A new approach for distribution stateestimation based on ant colony algorithm withregard to distributed generation

T. Niknama,∗, A.M. Ranjbara and A.R. ShiranibaSharif University of Technology, Tehran, IranbNiroo Research Institute, Tehran, Iran

Abstract. Technology enhancement of Distributed Generations, as well as deregulation and privatization in power system industry,shows a new perspective for power systems and subsystems. When a substantial portion of generation is in the form of dispersedand small units, a new connection pattern emerges whereby the dispersed units are embedded in reticulation infrastructure. Now,the flow of power is no longer the same as in the conventional systems, since the dispersed generating plants contribute withgeneration also at the distribution grids level. Connection of generation to distribution grids cannot effectively be made, unlessthe some especial control and monitoring tools are available and utilizable. State estimation in these kinds of networks, oftencalled mixtribution, is the preliminary and essential tool to fulfill this requirement and also is the subject of this article. Actually,state estimation is an optimization problem including discrete and continuous variables, whose objective function is to minimizethe difference between calculated and measured values of variables, i.e. voltage of nodes, and active/reactive powers in thebranches. In this paper, a new approach based on Ant Colony Optimization (ACO) is proposed to solve this optimization problem.The feasibility of the proposed approach is demonstrated and compared with methods based on neural networks and geneticalgorithms for two test systems.

Keywords: Distributed generation, state estimation, Ant Colony Optimization (ACO)

1. Introduction

Return back to 20 years ago, when vertical structurewas the dominant structure for power system industryaround the world, large-scale power plants were thebest choice for system expansion due to their numer-ous advantages, such as low investment, operation andmaintenance cost. As the capacities of these kinds ofpower plants were high, they were usually connected totransmission systems, and consequently to load centers,where distribution networks were responsible for sup-plying the loads. Now, the situation has been changed;the vertical structure is going to be eliminated all around

∗Corresponding author. Tel.: +98 21 8079395; Fax: +98 218590174; E-mails: [email protected]; taher [email protected].

the world by means of unbundling of generation, trans-mission and distribution systems.

Privatization and deregulation create this opportunitythat the share of private sector is going to be increasedin power system investment, and so give this possibilityto the customers to be responsible for their own con-sumption with new types of generations, which havesmall size and user friendly performance. Utilizationof these kinds of generations in power systems, again,changed the philosophy of classification of power sys-tem sub-sectors; now, the flow of power is no longer thesame as in the conventional systems, since distributionsystems have some kinds of generations by themselves.On the other hand, refer to huge investment in thesekinds of generations, it is estimated that about 25% oftotal electricity generation will be produced by thesekinds of generations till 2020 [14]. Therefore, devel-opment of new tools and algorithms to guarantee the

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120 T. Niknam et al. / A new approach for distribution state estimation based on ant colony algorithm

performance of the whole system will be an essentialrequirement, which must be taken into considerationfirmly.

In this paper, state estimation in a distribution sys-tem incorporating DGs has been reviewed. It is clearthat while a huge number of small size generations areconnected into distribution systems, appropriate mon-itoring and controlling algorithms are required to con-trol the state variables of the system and minimize theoperational cost of the whole network and at the sametime the general security of the system should be takeninto consideration.

Distribution systems usually have a large number ofnodes and connections, which need to be monitoredand controlled. For this purpose, two methods can beapplied. The first one is based on installation of mon-itoring and controlling devices in each node while thesecond one is based on installation of these equipmentsin selected nodes.

The first method incorporates the measuring devices,which are equipped with some types of modem tosend and receive the data. These measuring devicesare going to be used in most of the new distributionsystems to facilitate locally data logging and genera-tions/consumptions management. The second method,due to lower number of monitoring and control devices,is much cheaper than the first one, and it can be imple-mented in both new and old distribution systems; how-ever, an estimation algorithm is needed to extend theresults to the whole system. It is clear that, because ofneed to a mathematical estimation in this method, theaccuracy is less than the first one. This paper presentsan algorithm for state estimation of distribution sys-tems, where limited measuring devices are available.

State estimators have been developed to provide thenecessary data for real time control of transmissionnetworks. They are typically based on the weighted– least – squares approach, which solves the systemstates by minimizing the mean-square-error of an over-determined system of equations. A number of Distri-bution State Estimation (DSE) methods have been de-veloped in distribution systems, which are divided intotwo main categories [1,2,5,7,8,11,12,19,20]:

– Statistical methods, which usually use an iterativeconvergence method.

– Load adjustment state estimation, which usuallyutilize sensitivity analysis.

In the conventional methods belonging to both cat-egories, it is assumed that the objective functions andconstraints should be continuous and differentiable.

Existence of distributed generations, as well as SVCsand transformer tap changers with discrete perfor-mances, causes that these methods could not be easilyused when a lot of discrete variables are incorporated.

It has been revealed that, when there are some opti-mization problems where ordinary mathematical meth-ods cannot be used easily, evolutionary methods andexpert systems, such as neural networks and geneticalgorithms would be good alternatives.

Recently, a new evolutionary global optimizationtechnique known as Ant Colony Optimization (ACO)has become a candidate for many optimization appli-cations. The ant colony optimization has been usedto solve several combinatorial optimization problemssuch as the traditional ones like Traveling Sales-man Problem (TSP), Quadratic Assignment Problem(QAP), Job Shop Scheduling Problem (JSP), SingleMachine Total Tardiness Problem (SMTTP), or someapplication of power system problems like unit com-mitment, economic dispatch, hydroelectric generationscheduling, reactive power pricing in deregulated sys-tem, voltage and var control in distribution systems [3,4,6,9,10,13,15–18].

In this paper, a new approach based on ant colonyalgorithm for a practical distribution state estimationincluding DGs, SVCs and Voltage Regulators (VRs)is presented. In this method, DGs and loads, whichdo not have constant outputs, are considered as statevariables in which the difference between measured andcalculated values is assumed as the objective function.

Following this section, in Section 2, distribution stateestimation problem is formulated. Unbalanced three-phase power flow is described in Section 3. Modelingof distributed generation is proposed in Section 4. InSections 5 and 6, ant colony system mechanism and theproposed ant colony algorithm are presented, respec-tively. Implementing the proposed approach to distri-bution state estimation is shown in Section 7. Finally,in Section 8, the feasibility of the proposed approachis demonstrated and compared with methods based onneural networks and genetic algorithms for two testsystems.

2. Distribution state estimation with regard todistributed generation

From a mathematical point of view, the state estima-tion problem is an optimization problem with equalityand inequality constraints. The objective function isthe summation of difference between the measured and

T. Niknam et al. / A new approach for distribution state estimation based on ant colony algorithm 121

calculated values. Distribution state estimation consid-ering DGs can be expressed as follows:

Min f(X) =n∑i

= ωi(zi − hi(X))2

X = [PG, PLoad]

PG = [P 1G, P 2

G, . . . , PNgG ]

PL = [P 1Load, P

2Load, . . . , P

NLLoad]

s.t

P iG,min P i

G P iG,maxi = 1, 2, 3, . . . , Ng (1)

P iLoad,min P i

Load P iLoad,max

i = 1, 2, 3, . . . , NL∣∣∣P ijLine

∣∣∣ < P ijLine,max

Tapimin

Tapi Tapimax

i = 1, 2, 3, . . . , Nt

0 Qic Qi

c,maxi = 2, 2, 3, . . . , Nc

where:X: state variables vector including loads’ and DGs’

outputs.zi: measured values.ωi: weighting factor of the ith measured variable.hi: state equation of the ith measured variable.m: number of measurements.Ng: number of DGs with variable outputs.NL: number of loads with variable outputs.Nc: number of capacitors installed along the feeder.Nt: number of VRs installed along the feeder.Qi

c, Qic,max: reactive power and maximum reactive

power of the ith capacitor, respectively.P i

G, P iG,max and P i

G,min: active power, maximumand minimum active power of the ith DG, respectively.

P iLoad, P i

Load,max and P iLoad,min: active power, max-

imum and minimum active power of the i th load, re-spectively.

P ijLine and P ij

Line,max: transmission line flow andmaximum transmission line, respectively.

Tapi: tap position of VRsTapi

min and Tapimax: minimum and maximum tap,

respectively.In this paper, it is assumed that capacitors and VRs,

which change stepwise and are installed along feeders,are controlled locally. During the search procedure,change of state variables (loads’ and DGs’ outputs)may cause change of tap positions and capacitor banks,which in consequence cause that the objective functionchanges non-continuously.

The number of measurements in distribution systemsis usually less than state variables. In order to have aunique answer, these assumptions should be made:

– Transmission lines and switches status are known.– A contracted load and distributed generation val-

ues are known at each node.– Voltage and current at the substation bus (main

bus) are known.– If outputs of DGs and loads are fixed, outputs and

power factors will be available.– If outputs of DGs and loads are variable, the aver-

age outputs, standard deviations and power factorscan be obtained.

– Set points of VRs and local capacitors are known.

3. Unbalanced three phase power flow

In unbalanced three-phase power flow, the followingcomponents are modeled by their equivalent circuitsin terms of inductance, capacitance, resistance and in-jected current.

a) Distributed generations: DGs are modeled as PQnodes.

b) Transformers: transformers are modeled asequivalent circuit with fictitious current injec-tions.

c) Capacitors: capacitors are represented by theirequivalent injected currents.

d) Demands or loads: system loads are basicallyconsidered asymmetrical because of single-phaseloads and unequal three phase loads.

In this paper, a network-topology-based on three-phase distribution power flow algorithm is used. Twomatrices are used to obtain the power flow solution,which are the Bus Injection to Branch Current (BIBC)and the Branch Current to Bus Voltage (BCBV) matri-ces [18].

4. Distributed generation modeling

Depending on the contract and control status of agenerator, it may be operated in one of the followingmodes:

– In “parallel operation” with the feeder, i.e., thegenerator is located near the loads and designed tosupply a large load with a fixed real and reactivepower output.


– Constant output power at a specified power factor.– Constant output power at a specified terminal volt-

age.

The generation nodes in the first two cases can bewell represented as PQ nodes. The generation nodesin the third case must be modeled as PV nodes. T.Niknam and A.M Ranjbar have presented an approachbased on compensation method to model the generatoras a PV node [18]. In this paper, DGs are consideredas PQ nodes.

5. Ant colony system mechanism

Ants are insects, which live together. Since they areblind animals, they find the shortest path from nest tofood with the aid of pheromone. The pheromone is thechemical material deposited by the ants, which servesas a critical communication media among ants, therebyguiding the determination of the next movement. Onthe other hand, ants find the shortest path based on theintensity of pheromone deposited on different paths.For better understanding,assume that ants want to movefrom A to B and vice versa, to obtain food (Fig. 1).

At first, if there is no obstacle, all of them will movealong the straight path (Fig. 1a). At the next stage,assume that there is an obstacle; in this case, ants willnot be able to follow the original trial in their move-ment. Therefore, randomly, they turn to the left (ACB)and to the right (ADB) (Fig. 1b). Since ADB path isshorter than ACB path, the intensity of pheromone de-posited on ADB path is more than the other path. Soants will be increasingly guided to move on the shorterpath (Fig. 1c). This behavior forms the fundamentalparadigm of ant colony system.

As it was indicated in Fig. 1, the intensity of de-posited pheromone is one of the most important factorsfor ants to find the shortest path. Therefore, this factorshould be used to simulate the behavior of ants. Gen-erally, the following factors are used to simulate antsystems:

– Intensity of pheromone– Length of path

To select the next path, the state transition probabilityis defined as follows:

Pij =(τij)γ1(1/Lij)γ2∑(τij)γ1(1/Lij)γ2

(2)

After selecting the next path, trail intensity ofpheromone is updated as:

τij(k + 1) = ρτij(k) + ∆τij (3)

Where:τij : intensity of pheromone between nodes i and j,Lij : length of path between nodes i and j,ρ: a coefficient such that (1-ρ ) represents the evap-

oration of trail between time k and k + 1.γ1 and γ2: control parameters for determining

weights of trail intensity and length of path.

6. The proposed ant colony algorithm

Up to now, a number of studies based on ant colonyalgorithm have been carried out in order to solve someoptimization problems such as TSP, unit commitmentand etc; however, the search domains in these stud-ies are in discrete form. In addition to that, a numberof methods based on ant colony have been presentedto solve the problem in both continuous and discretesearch domains. The main problem associated withthese methods is that the time of convergence increasesdrastically when the number of variables is increased.In this section, a new approach based on ant colonyalgorithm will be presented to solve optimization prob-lems in continuous and discrete domains simultane-ously.

In order to follow this goal, a number of N colonies,each of which contains M individuals and a Masterare considered. The ant, which has the best position(minimum value of objective function) in each colony,is known as the Master of the colony. Now, to choose amovement direction, each of colonies needs to find thebest local and global positions as follows:

6.1. Finding the best local position

Suppose the ith colony wants to change its position.At first, the transition probabilities between the Masterand the rest of the ants of colony are calculated asindicated in Eq. (4):

[PLn]i = [PLi1, PLi2, . . . , PLiM ]1∗M(4)

PLij =(τLij)γ1(1/Lij)γ2

M∑j=1

(τLij)γ1(1/Lij)γ2

where: Pij is the transition probability between the ith

Master and the jth individual in the ith colony.Cumulative probabilities are calculated as:


Fig. 1. An example of finding the shortest path by ants.

[CLn]i = [CL1, CL2, . . . , CLM ]1∗M

where

CL1 = PLi1

CL2 = CL1 + PLi2

CL3 = CL2 + PLi3

. . . (5)

CLj = CLj−1 + PLij. . .

CLM = CLM−1 + PLiM

In the above equation, CLj is the cumulative proba-bility for the jth individual.

The roulette wheel is used for stochastic selection ofthe best local position as follows:

A number between 0 and 1 is randomly generated andcompared to the calculated cumulative probabilities.The first term of cumulative probabilities (CLj), whichis bigger than the generated number, is selected andthe associated position is considered as the best localposition.

6.2. Finding the best global position

Finding the best global position is similar to find-ing the best local position. The main difference be-tween them is that in this case, Masters are consideredto calculate transition probabilities. In other words, tochoose the best global position, it is necessary to cal-culate the transition probability between each pair ofMasters.

6.3. Determination of the next position

The movement direction for each ant is obtainedfrom a linear combination of the best global and local

positions. All ants at each colony should be movedtogether along the specified direction.

Figure 2 shows the above-mentioned process graph-ically.

7. Implementing the proposed ACO to distributionstate estimation

This section presents the implementation of the pro-posed algorithm to solve distribution state estimationproblem. It should be noted that the state variablesare loads’ and DGs’ values whose outputs are variable,rather than voltage or current as used by conventionalstate estimations.

Step 1: Generating the initial population and trailintensity for Masters

An initial population of Masters of colonies, X i,which must meet constraints, is generated randomly.At initialization phase, it is assumed that trail intensitiesbetween each pair of Masters are the same.

Master Population = [X1 , X2 , . . . , XN ]

Xi = [PG, PLoad]1∗n

PG = [P 1G, P 2

G, , PNgG ]

PL = [P 1Load, P

2Load, . . . , P

NLLoad] (6)

Xi min(j) ≤ Xi(j) ≤ Xi max(j)

Xi(j) = rand() ∗ (Xi max(j) − Xi min(j))

−Xi min(j) j = 1, 2, 3, ..., n

Master Trail Intensity = [τGij ]N∗N

where N is the number of the colonies and n is the num-ber of variables. rand() is a uniform random generatorfunction.


* * … * *

* * … * *

* * … * *

* * … * *

* * … * *

* * … * *

* * … * *

* * … * *

N: Number of colonies M: Number of ants in each colony : Master ant

*: Ant in colony Global: The best global position Local: The best local position Movement: Movement direction of colony

Local Global

Movement

1 2 3

M

1

i

j

N

Fig. 2. Determination of the movement direction for ant colony.

Step 2: Generating the initial population and trailintensity for ants in each colony

In this step, an initial population is generated ran-domly for each ant colony. Also local trail intensitiesbetween each pair of ants in each colony are generated.

Local Population = [Y 1, Y2, . . . , YM ]

Xi(j) − δ ≤ Yi(j) ≤ Xi(j) + δ

Yi = [PG, PLoad]1∗n

PG = [P 1G, P 2

G, . . . , PNgG ]

PL = [P 1Load, P

2Load, . . . , P

NLLoad] (7)

Yi(j) = rand() ∗ (Yi max(j) − Yi min(j))

−Yi min(j) j = 1, 2, 3, . . . , n

Local Trial Intensity = [τLij ]M∗M

In this equation, M is the number of the ants in eachcolony, δ is the radius of local search area and n isthe number of variables. rand() is a uniform randomgenerator function.

After generating the local population, the objectivefunction is calculated for these individuals and the cor-

responding Master. The position, which has the min-imum objective function, is selected as the Master ofthe colony.

Step 3: Determination of the next positionAssume that the ith colony wants to determine its

next position. As mentioned before, the movementdirection of each ant is a linear combination of the bestglobal and local positions, which can be selected as:

A). Selection of the best global positionAs stated before, the Masters are representatives of

the colonies; therefore, the best global position is foundbased on them. Since Lij is not known in state estima-tion optimization problem, we can define its inverse asfollows:

1/Lij = φGij = f(X i) − f(Xj); j = i (8)

f(Xi) and f(X j) are the objective function valuesof the state estimation problem for the ith and the jth

Masters.Transition probabilities between the ith and the rest

of the Masters are defined as:


Read data include transmission line status, DGs and loads status, average and standard

deviation of loads and DGs

Read real and pseudo measured values

Generate an initial population and trail intensities for Masters based on DGs and load values

Generate an initial population and trail intensities for each ant of one colony based on DGs and Load values

Calculate local transition probabilities based on trail

intensities and different costs between each pair of

ants in one colony

Update trail intensities

Select the local position based on roulette wheel

Calculate values of measured points based on DGs and Loads values

Calculate and determine the next position based on the best local and global positions

Is convergence condition satisfied?

Stop and print results.

Yes

Calculate global transition probabilities based on trail intensity and different costs

between each pair of Masters

Update trail intensities

Select the global position based on roulette wheel

No

Fig. 3. Flowchart of the proposed algorithm.

PGij =(φGij)γ1(τGij)γ2⎛

⎜⎝ N∑j=1j =i

(φGij)γ1(τGij)γ2

⎞⎟⎠

(9)j = 1, 2, . . . , N ; i = j

The cumulative probabilities for Masters are calcu-lated based on calculated transition probabilities. The

best global position is selected by roulette wheel. Afterselecting the global position, global trail intensities areupdated as follows:

∆τGij = PGij(10)

τGij(k + 1) = ρτGij(k) + ∆τGij

B). Finding the best local positionFinding the best local position is similar to finding

the best global one and is as follows:


Fig. 4. Single Line Diagram of IEEE 34 bus.

0

0.05

0.1

0.15

0.2

1 3 5 7 9 11 13 15 17 19

0.0000

0.0100

0.0200

0.0300

1 21 41 61 81 101

Objective function value

Iteration

ACO


Iteration

GA

Fig. 5. Convergence characteristic of GA and ACO for the bestsolution.

At first, transition probabilities between the ith Mas-ter and the local ants are calculated as:

φLij = f(Xi) − f(Y j);

PLij =(φLij)γ1(τLij)γ2(M∑

j=1

(φLij)γ1(τLij)γ2

) (11)

j = 1, 2, . . . , M ;

f(Xi) and f(Y j) are the objective function values ofthe state estimation problem for the ith Master and thejth ant in the ith colony.

The cumulative probabilities are calculated and thebest local position is selected by roulette wheel.

Table 1Characteristic of generators

G1 G2 G3

Average of output (kW) 65 75 95Standard deviation (%) 0 10 10Power factor 0.8 0.8 0.8

After selecting the best local position, local trail in-tensities are updated as follows:

∆τLij = PLij(12)

τLij(k + 1) = ρτLij(k) + ∆τLij

C). Determination of the next positionAfter selecting the best local and global position, the

next position is determined as follows:

Xi(k + 1)=Xi(k)+rand() ∗ (Y Local−Xi(k))

+rand() ∗ (XGlobal − X i(k)) (13)

In above equation, Y Local and XGlobal are the bestlocal and global positions, respectively. rand() is auniform random generator function. It must be notedthat, in the new position, the constraint restriction mustbe completely satisfied.

Step 4: Check of convergenceAfter all of the ant colonies found their next posi-

tions, the convergence is checked by:√√√√ N∑i=1

∣∣∣Xk+1

i − Xk

i

∣∣∣2 < ε (14)

If the convergence condition is satisfied, the task iscomplete and if not, the process must be repeated fromstep 3. Figure 3 shows the complete flowchart of theprocess.

The variations range of M and N are between 4 to 10and 10 to 100, respectively and depend on the numberof unknown variables (dimension of vectorX).


Table 2Characteristic of variable loads

L1 L2

Average of output (kW) 80 90Standard deviation (%) 10 10Power factor 0.75 0.75Location 25 34

0.00E+00

1.00E-07

2.00E-07

3.00E-07

4.00E-07

5.00E-07

6.00E-07

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0000

0.0020

0.0040

0.0060

0.0080

0.0100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Execution

GA Min=1.18*10e(-5) Max=877*10e(-5)


Execution

ACOMin=5.87*10e(-8) Max=48*10e(-8)

Fig. 6. Comparison of objective function values for different execu-tions.

8. Simulation

In this section, the proposed method is applied tothe distribution state estimation problem on two distri-bution test systems. It is assumed that the followinginformation is available.

– Value of output for each of the constant loads andDGs.

– Average value and standard deviation for each ofthe variable DGs and loads.

– Values of the measured points– Power factors of Loads and DGs– Set points of VRs and local capacitors

In following, results for two cases are presented.Case 1: IEEE 34 bus radial test feedersFigure 4 shows the IEEE 34 bus radial distribution

test feeders whose associated specifications are pre-sented in [22].

For this system, it is assumed that there are three DGs(two variable and one constant) connected at buses 9,23

Table 3Comparison of measured and estimated values

Method Actual Calculated valuevalue Best Worst

solution solution

ACO 70 69.95 70.2G2 NN 70 69.4 69.4

GA 70 69.78 72.234ACO 80 80.02 79.5

G3 NN 80 80.3 80.3GA 80 79.98 78.64ACO 90 89.84 89.2

L1 NN 90 89.2 89.2GA 90 90.5 92.378ACO 75 75.213 74.3

L2 NN 75 74 74GA 75 76.4 77.63

Table 4Comparison of the execution time and objective function values

Method Objective Executionfunction time(s)

NN Best solution 6.84* 10(−5) ∼ 0

Worst solution 6.84* 10(−5) ∼ 0

GA Best solution 1.18* 10(−5) 40Worst solution 887* 10(−5) 120

ACO Best solution 5.87* 10(−8) 4Worst solution 48*10(−8) 14

Table 5Comparison of average and standard deviation for different execu-tions

Method Average Standard deviation

ACO 32*10(−8) 14*10(−8)

GA 294* 10(−5) 311* 10(−5)

NN 6.84* 10(−5) 0

and 27, respectively whose specifications are presentedin Table 1. There are also two variable loads whosespecifications are demonstrated in Table 2.

It is assumed that there are three measurement de-vices installed on buses 1, 15 and 25. These devicesare Ammeters and Wattmeters.

Tables 3, 4 and 5 and Figs 5 and 6 show the com-parison between the results of the proposed method,Genetic Algorithm and Neural Networks, while theirsimulation parameters are presented in appendix.

Case 2. A realistic 23 bus 20 Kv networkThe proposed method is also applied to a rural net-

work as shown in Fig. 7. This system is used to supplypower demand in the village located in the north of Iran.Line and load characteristics are shown in Tables 6 and7, respectively. Line Impedance Matrix is presentedin Eq. (15). As there is no DG in this network cur-rently, two typical DGs have been considered on buses13 and 21 whose specifications have been presented


Fig. 7. Single Line Diagram of the rural network.

Table 6Line characteristics

No From To Length (m)

1 1 2 402 2 3 2803 3 4 1404 4 5 1205 5 6 3306 6 7 7257 7 8 2108 8 9 2109 9 10 55

10 10 11 6011 11 12 100012 12 13 102013 13 14 87014 14 15 86515 15 16 86516 10 17 140017 17 18 170018 17 19 7019 19 20 7020 18 21 106021 21 22 150022 22 23 520

in Table 8. In this system, there are 2 variable loadswhose characteristics are shown in Table 9.

ZLine(Ω/m) = (1e − 4)⎡⎣ 7 + j7 0.2 + j.15 0.2 + j.15

0.2 + j.15 7 + j7 0.2 + j.150.2 + j.15 0.2 + j.15 7 + j7

⎤⎦ (15)

It is assumed that there are three measuring devicesinstalled on buses 1, 11 and 19. These devices areAmmeters and Wattmeters.

A comparison between the proposed algorithm(ACO), Genetic Algorithm and Neural Network isshown in Tables 10 and 11.

Input Layer

Hidden Layer

Output Layer

Fig. A1. The structure of neural network.

9. Discussion

As shown in Tables 3, 4, 5, 6 and 7 and Figs 5 and6, the proposed method can be used to estimate statevariables in distribution networks. The results, pre-sented in these Tables and Figures, can be summarizedas follows:

– The execution time of the proposed method is sig-nificantly short with respect to GA and gives ageneral idea that the method can be implementedwithout any restriction in practical networks.

– The execution time of Neural Network is less thanthe proposed method, however, it is more feasiblethan neural networks, which needs a lot of trainingpatterns, as the proposed algorithm requires onlynetwork data and measured values.

– Presented method is very precise. In other words,not only does this method reach to better optimalsolution with respect to other methods, but also thestandard deviation for different trials (with regardto GA) is very small (Fig. 6 and Table 5).


Table 7Load characteristics

No Pa(Kw) Qa(Kvar) Pb(Kw) Qb(Kvar) Pc(Kw) Qc(Kvar)

1 0.00 0.00 0.00 0.00 0.00 0.002 105.00 78.75 114.45 85.84 95.55 71.663 83.33 62.50 90.83 68.13 75.83 56.884 83.33 62.50 90.83 68.13 75.83 56.885 83.33 62.50 90.83 68.13 75.83 56.886 83.33 62.50 90.83 68.13 75.83 56.887 83.33 62.50 90.83 68.13 75.83 56.888 83.33 62.50 90.83 68.13 75.83 56.889 83.33 62.50 90.83 68.13 75.83 56.88

10 83.33 62.50 90.83 68.13 75.83 56.8811 105.00 78.75 114.45 85.84 95.55 71.6612 105.00 78.75 114.45 85.84 95.55 71.6613 83.33 62.50 90.83 68.13 75.83 56.8814 83.33 62.50 90.83 68.13 75.83 56.8815 21.00 15.75 22.89 17.17 19.11 14.3316 333.33 250.00 363.33 272.50 303.33 227.5017 133.33 100.00 145.33 109.00 121.33 91.0018 83.33 62.50 90.83 68.13 75.83 56.8819 105.00 78.75 114.45 85.84 95.55 71.6620 105.00 78.75 114.45 85.84 95.55 71.6621 50.00 37.50 54.50 40.88 45.50 34.1322 0.00 0.00 0.00 0.00 0.00 0.0023 105.00 78.75 114.45 85.84 95.55 71.66

– The proposed method converges fast (Fig. 5).– The simulation results show that estimation errors

are in acceptable levels (Tables 4 and 11).– The method can estimate the appropriate target

system condition even with measuring devices er-rors (Tables 2 and 5).

– The proposed algorithm can estimate appropriateloads’ and DGs’ output values at each node withlimited measurement points in distribution net-works.

– The method can be applied to a wide variety of sim-ilar optimization problems with non-differentialand non-continuous objective functions and con-straints.

10. Conclusion

As the number of DGs grows, their impacts on powersystem have to be studied more. One of the most impor-tant issues in distribution systems is distribution man-agement system (DMS), which can be affected by DGs.State estimation in DMS plays a key role in estimatingthe system real-time state. An efficient approach toestimate distribution state variables in the presence ofDGs is presented in this paper. The results of simula-tions indicate that the method can estimate target sys-tem conditions accurately. Also, the proposed methodcould be applied to a wide variety of similar problems.

Table 8Characteristic of generators

G1 G2

Average of output (kW) 850 950Standard deviation (%) 10 10Power factor 0.8 0.8

Table 9Characteristic of variable loads

L1 L2

Average of output (kW) 300 250Standard deviation (%) 10 10Power factor 0.75 0.75Location 16 19

The execution time of proposed method is remarkablyshort and gives a general idea that the method can beimplemented without any restriction in practical net-works.

The following subjects can be considered in futureworks:

– Estimation of DGs’ outputs when they are modeledand controlled as PV nodes.

– Estimation of tap position of VRs.– Estimation of DGs’ outputs when they are con-

trolled and modeled separately. In this situation,voltage magnitude and reactive power for eachphase of DGs can be controlled separately.


Table 10Comparison of measured and estimated values

Method Actual value Calculated valueBest solution Worst solution

ACO 800 799.9 801.2G1 NN 800 798.1 798.1

GA 800 796.1 810ACO 920 920.3 918.4

G2 NN 920 921.123 921.123GA 920 926.42 928.78ACO 310 309.71 311.58

L1 NN 310 312.4 312.4GA 310 311.56 317.75ACO 240 239.56 238.89

L2 NN 240 246.2 246.2GA 240 235.1 247.63

Table 11Comparison of the execution time and objective function values

Method Objective Executionfunction time(s)

NN Best solution 45.56*10(−5) ∼ 0

Worst solution 45.56*10(−5) ∼ 0

GA Best solution 488.1*10(−5) 35Worst solution 986.45*10(−5) 70

ACO Best solution 2.523*10(−8) 3Worst solution 284*10(−8) 12

Appendix

This appendix describes processes of the used ACO,GA and NN as well as illustrates their simulation con-ditions introduced in Section 8.

A1. Ant Colony Algorithm

N = 15; M = 4; γ2 = 2; γ1 = 4; ρ = 0.9;

A2. Genetic Algorithm

In this paper, Integer strings instead of binary codingare used to represent value of variables, and includethese processes:

– Representation and initialization– Fitness function– Reproduction operation– Crossover operation– Mutation operation

Simulation conditions are:Initial population = 1000;Selected Population = 100;Mutation = 4 Percent;Cross Over Probability = 0.2 to 0.3;

A3. Neural Network

In this paper, a MLP neural network with BP learningtechnique is ‘used whose structure is shown in Fig. A1.

Simulation conditions are:Number of total patterns = 2000;Number of training patterns = 1500;Test data = 500;Number of hidden layers = 1;Number of neurons in hidden layers = 4;Number of neurons in input layers = Number of

measurement devicesNumber of neurons in output layers = Number of

DGs + Number of Loads

Acknowledgments

The authors gratefully acknowledge the helps fromDr. Mahmoud Fotohi, Mehran Mirjafari, Babak Moza-fari and Sohrab Amini.

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