Optimization of SVC device locations for stability ...

Optimization of SVC device locations for stability enhancement O. L. BEKRIa

M.K. FELLAHb

O. L. BEKRI a

b

M.K. FELLAH

M. F. BENKHORIS

R. BENABIDd

R. BENABID

LGE laboratory (Electrotechnical Engineering Laboratory ) Faculty of Technology. University Dr. Moulay Tahar, Saïda, Algeria.

ICEPS laboratory (Intelligent Control & Electrical Power Systems) Electrical Engineering Department, Faculty of Technology Sidi Bel-Abbes University, Algeria IREENA Laboratory c

d

M. F. BENKHORISc

Ecole Polytechnique de Nantes, France

Nuclear Research Center of Birine, B.P.180,17200 Ain oussera, Djelfa ,Algeria

Abstract Voltage stability has become a very importante issue of power systems analysis. One of the major causes of voltage (load) instability is the reactive power limit of the system. Improving the system’s reactive power handling capacity via Flexible AC transmission system (FACTS) devices is a remedy for prevention of voltage instability and hence voltage collapse. This paper proposes two approaches for optimal locations of static var compensator (SVC). The approaches are based on the application of continuation power flow (CPF) and particle swarm optimization (PSO) in order to assess the static voltage stability margin. While finding the optimal location for the second approach, voltage rating, firing angle and reactive power of SVC are taken as constraints. The proposed methods are validated on IEEE 6-bus test system. It has also been observed that the proposed algorithms can be applied to larger systems and do not suffer with computational difficulties. Keywords: FACTS,SVC, Optimization, CPF, PSO, stability enhancement.

1. Introduction In recent years, attention to voltage stability problem has been raised, as an abnormal phenomenon caused by voltage instability. Many major blackouts caused by power system instability have illustrated the importance of this phenomenon [1]. Historically, transient instability has been the dominant stability problem on most systems, and has been the focus of much of the industry’s attention concerning system stability. Reactive power compensation is an important issue in electrical power systems and shunt flexible ac transmission system (FACTS) devices play an important role in controlling the reactive power flow to the power network and hence the system voltage fluctuations and stability [2].Voltage collapse problems in power systems have been of permanent concern, since several major blackouts throughout the world have been directly associated with such mishaps. The collapse points are also known as maximum loadability points. Increased loading of power system, environmental restrictions, combined with a world-wide deregulation of the power industry, require more effective and efficient control means for power flow and stability control. The power flow control and static stability limits of power system can be considerably modified using the new reactive compensation equipments [1, 3]. Variable impedance devices using power electronic technology, such as Static VAR Compensators (SVCs) has the potential to increase power control and system damping.

O. L. BEKRI/ ICMS’2014

A variety of techniques are used to optimize allocations of FACTS devices in power systems. Applying Continuation Power Flow analysis (CPF), the optimal location of this controller is determined. Heuristic approaches are traditionally applied to determine the location of FACTS devices. For instance, shunt FACTS devices are usually connected to the bus with the lowest voltage. These heuristics are sufficiently accurate in a small power system; however, more scientific methods are required in larger power networks. Recently, particle swarm optimization (PSO) has shown a great promise in power system optimization problems [18]. The PSO mimics the behaviors of individuals in a swarm to maximize the survival of the species. In PSO, each individual decides based on its own experience as well as on other individual’s experiences [9, 10]. The algorithm searches a space by adjusting the trajectories of moving points in a multidimensional space. The individual particles are drawn stochastically toward the position of present velocity of each individual, their own previous best performance, and that of their neighbors [14, 15]. Several distinct models have been proposed to represent FACTS in static and dynamic analyses [2]. This paper introduces the application of CPF and PSO for the optimal locations of SVC to assess voltage stability and maximize system loadability. Rest of the paper is organized as follows: Section II introduces loading parameter and continuation power flow. Particle swarm optimization technique is presented in section III. In section IV, some interesting results are presented along with detailed discussion finally; our contributions and conclusions are summarized in Section V. 2. Loading parameter and CPF 2.1. Loading parameter The most accepted analytical tool used to investigate voltage collapse is the bifurcation theory, a general mathematical theory that is able to classify instabilities, for determining the system behavior in the neighborhood of collapse or unstable points and gives quantitative information on remedial actions to avoid critical conditions [16, 20]. In the bifurcation theory, it is assumed that system equations depend on a set of parameters together with state variables, as (1): ψ (ρ,λ)= 0

(1)

Where ∶ is power system state variable. ∶ represents loading parameter. Stability or instability properties are assessed varying “slowly” the parameters. The parameter that is used to examine system proximity to voltage collapse is called loading parameter, , which modifies load powers as follows: = 1+ = 1+ where P ∶ is the active powers at basic operating point at buses Q ∶ is the reactive powers at basic operating point at buses k and K : represent the load distribution factors.

(2)

O. L. BEKRI/ICMS’2014

In typical bifurcation diagrams, voltages are plotted as functions of λ, i.e. the measure of the system loadability, called pv or nose curves. Equations (2) are used in continuation power flow (CPF) analysis. 2.2. Continuation power flow (CPF) Continuation power flow techniques are widely recognized as a valuable tool to determine nose curves of power systems and allow estimating the maximum loading conditions and “critical” solutions (for instance, saddle-node and limit induced bifurcation points). The CPF method that is used in this paper is not affected by numerical instabilities (it is able to determine the stable and unstable fold of P–V curves) and can provide additional information, such as sensitivity factors of the current solution with respect to relevant parameters [1, 19]. From a mathematical point of view, the CPF is a homotopy technique [19] and allows exploring the stability of power system equations when varying a system parameter, which, in typical static and dynamic voltage stability studies, is the loading parameter . Generally speaking, CPF consists in a predictor step realized by the computation of the tangent vector and a corrector step that can be obtained either by means of a local parameterization or by a perpendicular intersection. A typical CPF technique that uses an iterative process involving predictor and corrector steps is shown in Fig. 1. Starting from a known initial point A, a tangent predictor step is used to estimate the solution point B for a given load direction defined by . Then the corrector step is used to determine the exact solution C using a power flow with an additional equation to find the proper value of . This process is repeated until the desired bifurcation diagram or P–V curve is obtained.

(Δ )

Predictor A

B

Corrector D

C

Voltage

Exact Solution

E

F

Maximum Loading

(Loading Factor)

Fig. 1. Continuation power flow technique illustration

3. Particle swarm optimization (PSO) technique The particle Swarm Optimization is a technical parallel optimization developed by Kennedy and Eberhart [18]. It is inspired by social behavior of individuals who tend to imitate the successful behaviors they observe around them, while making their personal variations. It uses a number of particles that constitute a swarm. Each particle traverses the search space looking for the global minimum (or maximum). In a PSO system, particles fly around in a multidimensional search space. During flight, each particle adjusts its position according to its own experience, and the experience of neighboring particles, making use of the best position encountered by itself and its neighbors. The swarm direction of a particle is defined by the set of particles neighboring the particle and its history experience. =

∗ +

+ ∗

∗ ( )∗(

( )∗( −

− )

)

(3)


=

(4)

+

Where: d: is the index of iterations, : is the current position of particle at the dth iteration, υ : is the velocity of particle at dth iteration, ω: is inertia weight factor, φ and φ are acceleration constant, rand() is a uniform random value in the range [0,1] Suitable selection of inertia weigh, provides a balance between global and local exploration. In general, is set according to (5)

=

−

×

(5)

where : is the maximum number of iterations, iter : is the number of the iterations until the current stage. Fig. 2 illustrates the search mechanism of PSO. In the above procedures, the particle velocity is limited by a maximum value, . This limit determines the local exploration of the problem space and it realistically simulates the incremental changes of human learning. If is too high, particles might fly past good solutions. If is too small, particles may not explore sufficiently beyond local solutions. In many experiences with PSO, was often set at 10–20% of the dynamic range of the variable on each dimension [11]. In [14, 15] a decaying inertia weight is proposed and tested, with the aim of favoring global search at the start of the algorithm and local search later. If the inertia weight is not reduced with time, it is suggested to select a value ∈ [0.8, … … 1.2] . Several studies propose different values for these parameters that are considered adequate for some of the usual benchmark functions in [8].

+1

+1

gBest pBest

Fig.2. Concept of modification of a search


3.1. Problem formulation 3.1.1. Objective. In this work, the goal of the optimization is to obtain the best utilization of the existing power network with the optimal placement of SVC for maximise the loading parameter denoted by (L.P). The maximization of the loading parameter can be presented as follows: (6)

Maximize (L.P)

where L.P is the loading parameter In other words, we look for increasing as much as possible the power transmitted by the network to the consumers, keeping the power system in a secure state in terms of maximizing loading level, minimizing power system total loss and flattening buses voltage. 3.1.2. SVC modelling The model considered in this paper (Fig. 3) takes into account the firing angle, assuming a balanced, fundamental frequency operation. Thus, the model can be developed with respect to a sinusoidal voltage. The differential and algebraic equations are as follows:

 M  K M V   M  TM  T     K D  K 1  M T 2 TM   x 2 - sin 2 -   2  L  x  C Q x L

  K M V   K Vref   POD   M  T2 





    V 2  b ( )V 2 SVC

α

V

K T s +1

ν

M

max

K (T s + 1) T s+K

-

1

M

2

M

D

+

V

ref

α

min

Fig. 3. Structure of an SVC

3.1.3. Equality Constraints These constraints represent typical load flow equations as follows:

(7)


−

−

−

+

−

= 0, (8)

= 1, … .

−

−

∑

−

−

−

= 0, (9)

= 1, … . where NB: is the number of buses P and Q are the generator real and reactive power, respectively. P and Q are the load real and reactive power, respectively. G and Β are the transfer conductance and susceptance between bus i and bus j, respectively. 3.1.4. Inequality constraints These constraints represent the system operating constraints as follows. 

Generation constraints. Generator voltages and reactive power outputs restricted by their lower and upper limits as follows: ≤

≤

≤

(10)

, = 1, … . . ,

≤

are

, = 1, … . . ,

(11)

where NG is number of generators. 

Transformer constraints. Transformer tap T settings are bounded as follows:

≤

≤

(12)

,

= 1, … . . , where NT is the number of transformers. SVC constraints. These include the constraints of voltage rating the reactive power as follows:

, the firing angle

and


≤

≤ (13)

= 1, . . , (

)≤ ( )≤ = 1, . . , ≤

(

)

≤

(14)

= 1, . . , where N is the number of SVCs. Only the technical benefits of the SVC are taken into account, 4. Results and discussions A 6-bus test system is used for the objective of this study (figure 4). The system has three generators at buses 1, 2 and 3 and three loads at buses 4, 5 and 6 and eleven lines. The lower voltage magnitude limits at all buses are 0.95p.u and the upper limits are 1.05 p.u for generators 1, 2 and 3. Moreover, the apparent power limits of lines are considered.

Gen# 2 bus 3

Gen# 3

L3 bus 6

bus 2

Gen# 1 bus 1 L2 L1

bus 5

bus 4

Fig. 4. 6-bus test system

4.1. Location of SVC using CPF The location of the SVC device is determined through CPF. A typical PQ model is used for the loads and the generator limits are ignored. Voltage stability analysis is performed by starting from an initial stable operating point and then increasing the loads by a factor  until singular point of power flow linearization is reached. The loads are defined as:

P P Q  Q l

0

l

0

(15)


where

P

whereas

0

and

P

l

Q

and

0

are the active and reactive base loads,

Q are the active and reactive loads at bus l for the current operating l

point as defined by  . The critical buses are identified as buses 4, 5 and 6 and their P-V curve obtained through continuation method are shown in figure. 5. Bus 4 has the weakest voltage profile and hence its profile is needed to be improved using FACTS devices. Maximum loading point or bifurcation point where the Jacobian matrix becomes singular occurs at  =11.168 pu.

1

X: 11.16 11.16 X: Y: 0.8356 0.7335 Y:

Voltages (p.u)

0.8

X: 11.16 Y: 0.536

VBus4

0.6

VBus5 VBus6

0.4

0.2

0

0

2

4

6 Lambda (p.u.)

8

10

12

Fig . 5. PV curve for 6-bus system without FACTS

When SVC is connected at bus 4 we observe from figure 6 that bus 4 has a flatter voltage profile and introducing SVC will increase the loading parameter L.P to the maximum value, (  =12.09 pu). X: 12.09 Y : 1.05

1 X: 12.09 Y: 1.003

Voltages(p.u)

0.8

VBus4

0.6

VBus5 VBus6

0.4

0.2

0 0

2

4

6 Lambda (p.u.)

8

10

12

Fig. 6. PV curve with SVC at bus 4

Voltages profiles of base case and system with SVC are illustrated in figure 7. It is obvious from this figure that SVC provides a better voltage profile at the collapse point, this is due to the reason that the SVC is installed at the weakest bus. 4.2. Location of SVC using PSO The parameters of PSO for all optimization cases are summarized in table 1.


1.5 Maximum Minimum base case with SVC

Voltages(pu)

1

0.5

0

0

1

2

3 Bus

4

5

6

Fig. 7. Voltage Profile of system with SVC

Table 1. PARAMETERS OF OPTIMIZATION TECHNIQUE Parameters Population size Weighing Factor, W Constant c1 Constant c2 N° of Iteration

PSO 20 0.9-0.4 1.4 1.4 100

To demonstrate the effectiveness of the proposed approach, two different cases have been considered as follows:  

Case 1. For comparison purpose, About SVC constraints, only the voltage rating is considered. Case 2. All constraints of SVC are considered.

Case 1. The problem was handled as a multiobjective optimization problem where both location and L.P were optimized simultaneously. The optimal solutions obtained from the PSO and their related results are provided in Tables 2 and Figures. 8. Table 2. Output of the PSO technique (CASE 1) The best location of SVC 4

Best L.P (p.u)

Best V (p.u)

12.6917

1.0262

1.5 Maximum Minimum base case with SVC

Voltages (pu)

1

0.5

0

0

1

2

3

4

5

6

Bus

Fig. 8. Voltage Profile of system with SVC (case 1)


Based on the voltage profiles (without and with SVC) results show that the voltages at the most critical buses (bus 4 and bus 5) has increased. Case 2. In this case, the complete set of problem constraints was considered, the optimal solutions obtained from the PSO and their related results are provided in Tables 3 and Figures. 9. Table 3. Output of the PSO technique (case 2) the best the best the the best the best Q location l.p (p.u) best α (deg) (p.u) of svc V (p.u) 4 12.6775 1.0003 Pi/2 Pi 1.07 2.22

The global best position for this study is bus number 4 and the corresponding size of SVC is 329 MVAR, －222 MVAR (inductive) to +107 MVAR (capacitive). Voltages profiles of base case and system with SVC are illustrated in figure 9. It is obvious from this figure that SVC provides a better voltage profile at the collapse point. This is due to the reason that the SVC is installed at the weakest bus. 1.5 Maximum Minimum base case with SVC

Voltages (pu)

1

0.5

0

0

1

2

3 Bus

4

5

6

Fig. 9. Voltage Profile of system with SVC (case 2)

5. Conclusion In this paper, two methods are presented for the optimal locations of SVC to assess the power system voltage stability. A proposed CPF analysis is implemented to determine the optimal location of FACTS devices. It is for the aim of system voltage stability margin enhancement. The second method is based on particle swarm optimizing (PSO).The algorithm is easy to implement and it is capable of finding multiple optimal solutions to the constrained multiobjective problem, giving more flexibility to make the final decision about the location of the SVC units. The system loadability, bus voltage profile improvement and size of device are employed as the measure of power system performance in optimization algorithm. For large power systems, the PSO algorithm could have a significant advantage, compared to exhaustive search and other methods, by giving better solutions with less computational effort.


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