A practical algorithm for distribution state estimation ...

Renewable Energy 34 (2009) 2309–2316

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A practical algorithm for distribution state estimation including renewable energy sources Taher Niknam a, *, Bahman Bahmani Firouzi b a b

Electronic and Electrical Department, Shiraz University of Technology, Modares Blvd., P.O. 71555-313, Shiraz, Iran Islamic Azad University Marvdasht Branch, Marvdasht, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 September 2008 Accepted 7 March 2009 Available online 27 March 2009

Renewable energy is energy that is in continuous supply over time. These kinds of energy sources are divided into five principal renewable sources of energy: the sun, the wind, flowing water, biomass and heat from within the earth. According to some studies carried out by the research institutes, about 25% of the new generation will be generated by Renewable Energy Sources (RESs) in the near future. Therefore, it is necessary to study the impact of RESs on the power systems, especially on the distribution networks. This paper presents a practical Distribution State Estimation (DSE) including RESs and some practical consideration. The proposed algorithm is based on the combination of Nelder–Mead simplex search and Particle Swarm Optimization (PSO) algorithms, called PSO-NM. The proposed algorithm can estimate load and RES output values by Weighted Least-Square (WLS) approach. Some practical considerations are var compensators, Voltage Regulators (VRs), Under Load Tap Changer (ULTC) transformer modeling, which usually have nonlinear and discrete characteristics, and unbalanced three-phase power flow equations. The comparison results with other evolutionary optimization algorithms such as original PSO, Honey Bee Mating Optimization (HBMO), Neural Networks (NNs), Ant Colony Optimization (ACO), and Genetic Algorithm (GA) for a test system demonstrate that PSO-NM is extremely effective and efficient for the DSE problems.  2009 Elsevier Ltd. All rights reserved.

Keywords: Renewable Energy Sources (RESs) State estimation Hybrid particle swarm optimization

1. Introduction Traditional energy sources, which use oil, coal, and natural gas as the fuel, are damaging economic progress, environment and human life. These energy sources are facing growing pressure on a host of environmental fronts, with perhaps the most serious challenge confronting the future use of coal being the Kyoto Protocol greenhouse gas reduction targets. Renewable Energy Sources (RESs) currently supply somewhere between 15 percent and 20 percent of world’s total energy demand [1]. RESs usually refer to those energies that do not pollute environment and could be recycled in nature [1–5]. RESs are now categorized to traditional and new renewable energy. The traditional RESs mainly include giant hydropower and biomass burnt directly; the new RESs mainly refer to small hydropower, solar energy, wind energy, biomass energy, geothermal energy, ocean energy, etc. [4].

* Corresponding author. Tel.: þ98 711 7264121; fax: þ98 711 7353502. E-mail addresses: [email protected], [email protected] (T. Niknam). 0960-1481/$ – see front matter  2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2009.03.005

The most of RESs are connected to distribution networks and it is predicted they play an increasing role in the distribution systems of present and future. RESs reduce the amount of energy loss in distributing electricity because the electricity is generated very near where it is used, perhaps even in the same building. The use of RESs also reduces the size and number of power lines that must be constructed [1–5]. Connection of RESs to distribution grids cannot effectively be made, unless the some especial control, monitoring and optimization tools are available and utilizable. State estimation in these kinds of networks is the preliminary and essential tool to fulfill this requirement. In fact, state estimation is a mixed integer nonlinear optimization problem, whose objective function is to minimize the difference between the calculated and measured values of variables, such as voltage of nodes, active/reactive powers and current in the branches. DSE can provide effective information support for on-line functions, such as optimal schedule, security assessment, preventive control and reconfiguration. It provides the system state to other advanced power system analysis tools such as Distribution Management Systems (DMS) to track the real-time state of a distribution network. Because of the limited real-time measurements in the distribution systems, the DSE cannot estimate state

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variables correctly; therefore, pseudo-measurements are necessary for a distribution system state estimator. A load modeling procedure can be utilized as the pseudo-measurements for DSE. A number of efforts have been done as an advance to formulate and solve the DSE problem [6–21]. For instance, Naka et al. have proposed a hybrid particle swarm optimization (PSO) for DSE including Distributed Generators (DGs) [6]. The nonlinear characteristics of the practical equipment and actual limited measurements in distribution systems have been considered in the proposed method. Lu et al. have presented an algorithm for harmonic estimation [7]. The algorithm utilizes the particle swarm optimizer with passive congregation (PSOPC) to estimate the phases of the harmonics and a least-square (LS) method that is used to estimate the amplitudes. Niknam et al. have presented an approach based on ant colony optimization for DSE including DGs [8]. In that approach, the values of loads and DGs are considered as state variables. A regressionbased prediction approach to address the load estimation problem has been presented in [4]. In [9], a general framework for self-adapting dynamic estimator has been proposed to improve the forecasting and filtering models for power system dynamic state estimation. Sun et al. have proposed a mathematical-based of state estimation method for power systems according to information theory [11]. In [12] and [17] the authors have presented a branch currentbased three-phase state estimation algorithm for distribution systems whose state variables are chosen the magnitude and phase angle of the branch current. A methodology for estimating load curves at low-voltage substations has been described in [13]. The system is constructed by the aggregation of a fuzzy inference system of the Takagi–Sugeno type. Load estimation problem in unbalanced, radial power distribution networks has been modeled as a weighted HN estimation problem with equality and inequality constraints in [13]. In [14], a branch-based state estimation method which is an estimation technique for radial distribution systems that can support most kinds of real-time measurements has been proposed. In that algorithm, WLS problem of a whole network was decomposed into a series of WLS sub-problems and each sub-problem deals with only single branch state estimation. In [15], the authors have presented an approach to DSE using a probabilistic extension of the radial load flow algorithm. A method considering uncertainty of network data, load distribution information and telemetric data for voltage measurement has been proposed in [16]. In the presented method, data involved in the calculation are calculated by using fuzzy numbers. A three-phase power flow model and state estimation for distribution systems have been formulated in [18]. The proposed model is based on fully asymmetric modeling of the power distribution system. The state estimation utilizes the model in conjunction with synchronized measurements. A method for state distribution of electric power distribution in quasi real-time conditions has been presented in [19]. A three-phase state estimation method has been developed to increase the accuracy of load data in [20]. Niknam has presented an approach based on HBMO for DSE including DGs [21]. Due to the existence of RESs, as well as var compensators, VRs, ULTC transformer with discrete operation, we need to utilize algorithms that reach global minima and have short convergence time. On the contrary traditional algorithms required continuous and differentiable objective functions and constraints, the evolutionary algorithms and expert systems such as neural networks, genetic algorithms, can be utilized. PSO is one of the modern

heuristic algorithms and can be applied to nonlinear and noncontinuous optimization problems with continuous variables such as DSE. However, PSO based algorithms are of little practical use in solving constraint optimization problems due to slow convergence rates. The Nelder–Mead simplex algorithm is very powerful local algorithm, making no use of the objective function derivatives. In this paper a hybrid evolutionary optimization algorithm based on combination of Nelder–Mead simplex search and PSO (PSO-NM) is presented so that can generate a high-quality solution within short calculation time and apply to practical problems associated with DSE. In the following section, the DSE problem is formulated. In Section 3, a suitable RES modeling is offered. Sections 4–6 briefly review the fundamentals of PSO, NM, and the proposed hybrid PSONM, respectively. Application of the proposed optimization algorithm in the DSE problem is investigated in Section 7. Finally, in Section 8, the performance of the proposed algorithm is shown and compared with algorithms based on original PSO, HBMO, NNs, ACO, and GA for a test system. 2. Distribution state estimation including renewable energy sources From a mathematical point of view, the DSE problem is an optimization problem with equality and inequality constraints. The objective function is the summation of difference between the measured and calculated values. DSE including RESs can be expressed as follows. 2.1. Objective function

Pm 2 min f ðXÞ ¼ i ¼ 1 ui ðzi  hi ðXÞÞ   X ¼ PG ; PLoad 1n i h PG ¼ PG1 ; PG2 ; .; PGNg i h NL 1 2 PL ¼ PLoad ; PLoad ; .; PLoad n ¼ Ng þ NL

(1)

where X is the state variables vector including the loads’ and RESs’ outputs. zi is the measured value. ui is the weighting factor of the ith measured variable. hi is the state equation of the ith measured variable. m is the number of measurements. Ng is the number of RESs with variable outputs. NL is the number of loads with variable i is the active outputs. PGi is the active power of the ith RES. PLoad power of the ith load. n is the number of state variables. 2.2. Constraints Constraints are defined as follows:  Active power constraints of RESs: i i PG;min  PGi  PG;max ; i PG;max

i ¼ 1; 2; 3; .; Ng

(2)

i PG;min

and are the maximum and minimum active power of the ith RES, respectively.  Distribution line limits:

   Line  Line Pij  < Pij;max

(3)

Line are the absolute power flowing over distribujPijLine j and Pij;max tion lines and the maximum transmission power between the nodes i and j, respectively.

T. Niknam, B.B. Firouzi / Renewable Energy 34 (2009) 2309–2316

 Tap of transformers:

Tapmin < Tapi < Tapmax ; i i Tapmin , i

i ¼ 1; 2; .; Nt

(4)

Tapmin i

and Tapi are the minimum, maximum and current tap positions of the ith transformer, respectively. Nt is the number of transformers and VRs installed along feeder.  Bus voltage magnitude

Vmin  Vi  Vmax ;

i ¼ 1; 2; 3; .; Nb

(5)

Vi, Vmax and Vmin are the actual voltage magnitude of the ith bus, the maximum and the minimum values of voltage magnitudes, respectively. Nb is the number of buses.  Active power constraints of loads: i i i PLoad;min  PLoad  PLoad;max ;

i ¼ 1; 2; 3; .; NL

(6)

i i PLoad;max and PLoad;min are the maximum and minimum active power of the ith load, respectively.  Reactive power constraint of capacitors

i 0  Qci  Qc;max ;

i ¼ 1; 2; 3; .; Nc

(7)

i Qci and Qc;max are the reactive power and maximum reactive power of the ith capacitor, respectively. Nc is the number of capacitors installed along feeder.  Unbalanced three-phase power flow equations.

It is assumed, in this paper, that capacitors and VRs, which change stepwise and are installed along feeders, are locally controlled. During the search procedure, change of state variables (loads’ and RESs’ outputs) may cause change of tap positions and capacitor banks, which consequently make the objective function change non-continuously. The number of measurements in distribution systems is usually less than that of the state variables. In order to have a unique solution, these assumptions should be made:  Status of distribution lines and switches is known.  A contracted load and RES values are known at each node.  Voltage and current at the substation bus (main bus) are known.  If outputs of RESs and loads are fixed, the outputs and power factors will be available.

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 If outputs of RESs and loads are variable, the average outputs, the standard deviations and the power factors can be obtained.  Set points of VRs and local capacitors are known. In this paper average outputs and the standard deviations of RESs and loads, which are variable, are considered as pseudo instrument devices. The value of ui for real instrument devices should be considered high and for pseudo instrument devices should be considered low. In this paper these values are 100 and 0.1, respectively.

3. Renewable Energy Sources modeling Renewable Energy Sources (RESs) are energy sources that are continually regenerated. RESs get energy from water, wind, the sun, geothermal sources, and biomass sources such as energy crops. RESs are the electric energy resources which are usually connected to the distribution network. Since RESs are connected close to the loads, they can increase the power quality and reliability from the customers’ perspective and help the utilities to face the load growth delaying the upgrade of distribution/transmission lines [1–5]. Generally, RESs in distribution networks can be modeled as a PV or a PQ model. Since distribution networks are unbalanced threephase systems, RESs can be controlled and operated in two forms:  Simultaneous three-phases control.  Independent three-phases control or single phase control. Therefore, with regard to the control methods and RESs models, four models can be defined for the simulation of these generators:    

PQ model with simultaneous three-phase control (Fig. 1a). PV model with simultaneous three-phase control (Fig. 1c). PQ model with independent three-phase control (Fig. 1b). PV model with independent three-phase control (Fig. 1d).

It must be taken into account that when RESs are considered as the PV models, they have to be able to generate reactive power to maintain their voltage magnitudes. Many researchers have presented several procedures to model RESs as the PV buses [22–24]. Fig. 1 shows model of RESs based on the type of their control. In this paper, RESs are modeled as the PQ model with simultaneous threephase control (Fig. 1a).

Fig. 1. Models of RESs. (a) PQ Model with simultaneous three-phase control. (b) PQ Model with independent three-phase control. (c) PV Model with simultaneous three-phase control. (d) PV Model with independent three-phase control.

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Read input data

Generate initial population and initial velocity

Calculate the objective function for each individual

Select the global position based on the objective function

Search around the global solution by NM

Is the new solution better than the global best solution?

No

Yes Replace the new solution with Gbest

Select the ith individual

Select the local position for the ith individual

Calculate the modified velocity for each individual Fig. 3. Single line diagram of IEEE 70 bus.

i=i+1

share individual knowledge when such swarms flock, migrate, or hunt. In fact PSO simulates a case that a member of a swarm recognizes a favorite path to go and the other members follow it soon. For each particle i, the velocity and position of particles can be updated by the following equations [25–28]:

Calculate the next position for each individual

Check the new position with its limits

No

All individuals are selected? Yes Convergence condition is satisfied?

¼ uVi

ðtþ1Þ

¼ Xi

Xi No

  ðtÞ þ c1 rand1 ð+Þ Pbesti  Xi   ðtÞ þc2 rand2 ð+Þ Gbest  Xi

ðtþ1Þ

Vi

ðtÞ

ðtÞ

ðtþ1Þ

þ Vi

uðtþ1Þ ¼ umax 

umax  umin tmax

t (8)

i ¼ 1; 2; .; NSwarm Yes Stop and print the results. Fig. 2. Flowchart of PSO–NM.

Table 1 Characteristic of generators. Average of active power output (kW)

Standard deviation (%)

Location Power factor

Type of RES

G1 300 G2 450

10 15

8 14

1 1

G3 500 G4 350 G5 650

10 15 15

21 29 35

1 1 1

G6 500

10

41

1

G7 200 G8 300

15 20

62 58

1 1

Photovoltaic Wind turbine Hydropower Solar Wind turbine Wind turbine Fuel cell Photovoltaic

4. Original PSO One of the population-based random search algorithms is called PSO. This algorithm was represented by Kennedy and Eberhart for the first time and since then, it has been widely used to solve a broad range of optimization problems. This algorithm is represented as a simulation of animals’ social activities, e.g. insects, birds, etc., which simulates natural procedure of group communication to

T. Niknam, B.B. Firouzi / Renewable Energy 34 (2009) 2309–2316 Table 2 Characteristic of variable loads. Active power (kW)

Reactive power (KVar)

Standard deviation (%)

4 14 26 21 34 42 53 64

100 320 210 150 260 170 230 400

30 230 134 86 134 93 134 183

20 15 15 10 20 10 15 20

5. Nelder–Mead method The Nelder–Mead (NM) method is a generally used nonlinear optimization algorithm. NM is a numerical method to minimize an objective function in a multi-dimensional space [29–31]. The operations of the method are to rescale the simplex based on the local behavior of the function by using four basic procedures: reflection, expansion, contraction and shrinkage [29–31]. The original NM simplex procedure is described as below: Step 1: initialization Generate N þ 1 individuals to form an initial N-dimensional simplex, randomly. Calculate the functional value at each vertex point of the simplex. The individuals sort ascending based on the objective function values as below:

Xlow 6« 6 4 Xhighs Xhigh

Xrefl

1 N

(10)

where a is the reflection coefficient (a > 0). Nelder and Mead suggested that a ¼ 1. If flow < frefl < fhighs, accept the reflection by replacing Xhigh with Xrefl, and Step 2 is repeated again for a new iteration. If frefl < flow, go to Step 3. If fhigh > frefl > fhighs, Xrefl replaces Xhigh and go to Step 4. If fhigh < frefl, go to Step 4 without the replacement of Xhigh by Xrefl.

where, i is the index of each particle, t is the current iteration number, rand1 ð+Þ and rand2 ð+Þ are random numbers between 0 and 1. Pbesti is the best previous experience of the ith particle that is recorded. Gbest is the best particle among the entire population. NSwarm is the number of the swarms. Constants c1 and c2 are the weighting factors of the stochastic acceleration terms, which pull each particle towards the Pbesti and Gbest positions. u is the inertia weight. umax and umin are the maximum and minimum of the inertia weights, respectively.

2

PNþ1 Xj j ¼ 1 Xj sXhigh ¼ ð1 þ aÞ  Xcent  a  Xhigh

Xcent ¼

Location

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Step 3: expansion If the reflection produce a function value smaller than flow (i.e., frefl < flow), the reflection is expanded in order to extend the search space in the same direction and the expansion point is calculated by the following equation:

Xexp ¼ g  Xrefl þ ð1  gÞ  Xcent

where g is the expansion coefficient (g > 1). Nelder and Mead suggested g ¼ 2. If fexp < flow, the expansion is accepted by replacing Xhigh with Xexp; otherwise, Xexp replaces Xhigh. The algorithm continues with a new iteration in Step 2. Step 4: contraction The contraction vertex is calculated by the following equation:

Xcont ¼ g  Xhigh þ ð1  bÞ  Xcent

(9)

where, Xlow, Xhigh, and Xhighs are the vertices with the lowest, the highest and the second highest function values, respectively. flow, fhigh and fhighs represent the corresponding observed function values, respectively. n is the number of state variables. Step 2: reflection Find Xcent, the center of the simplex without Xhigh in the minimization problem. Generate a new vertex Xrefl by reflecting the worst point according to the following equation:

(12)

where b is the contraction coefficient (0 < b < 1). Nelder and Mead suggested b ¼ 0.5. If fcont < flow, the contraction is accepted by replacing Xhigh with Xcont and then a new iteration begins with Step 2. If fcont > fhigh then go to Step 5. Step 5: shrinkage In this step, shrink the entire simplex except Xlow by

Xi ¼ g  Xlow þ ð1  dÞ  Xlow ;

3 flow 7 « 7 fhighs 5 fhigh ðNþ1Þðnþ1Þ

(11)

0