**3.2 Steady state model of Static Compensator (STATCOM)**

The first SVC with voltage source converter called STATCOM commissioned and installed in 1999 (Hingorani, N.G., 1999). STATCOM is build with Thyristors with turn-off capability like GTO or today IGCT or with more and more IGBTs. The steady state circuit for power flow is shown in Fig. 7, the V-I characteristic presented in Fig. 8.

Fig. 7. STATCOM steady state circuit representation

Fig. 8. V-I Characteristic of the STATCOM

#### **3.2.1 Advantages of STATCOM**

The advantages of a STATCOM compared to SVC Compensators summarized as follows (Hingorani, N.G., 1999):


#### **3.2.2 STATCOM modelling based power flow**

In the literature many STATCOM models have been developed and integrated within the load flow program based modified Newton-Raphson, the model proposed by (Acha, et al, 2004), is one of the based and efficient models largely used by researchers. Fig. 9 shows the equivalent circuit of STATCOM, the STATCOM has the ability to exchange dynamically reactive power (absorbed or generated) with the network.

Fig. 9. STATCOM equivalent circuit

Based on the simplified equivalent circuit presented in Fig. 9, the following equation can be formulated as follows:

$$I\_s = Y\_s \left(V\_s - V\_k\right);\tag{21}$$

Where,

40 Electrical Generation and Distribution Systems and Power Quality Disturbances

The first SVC with voltage source converter called STATCOM commissioned and installed in 1999 (Hingorani, N.G., 1999). STATCOM is build with Thyristors with turn-off capability like GTO or today IGCT or with more and more IGBTs. The steady state circuit for power

α max

*Power Flow* 

αmin

a*) Firing angle Model* 

*V* 

IL max IC max

*IL IC* 

The advantages of a STATCOM compared to SVC Compensators summarized as follows

• The reactive power to be exchanged is independent from the actual voltage on the connection point. This can be seen in the diagram (Fig. 9) for the maximum currents being independent of the voltage in comparison to the SVC. This means, that even

during most severe loading conditions, the STATCOM keeps its full capability. • Reduced size is another advantage of the STATCOM as compared to the SVC Controller, sometimes even to less than 50%, and also the potential cost reduction achieved from the elimination of many passive components required by SVC, like

Vref

**3.2 Steady state model of Static Compensator (STATCOM)** 

flow is shown in Fig. 7, the V-I characteristic presented in Fig. 8.

*Vr*

α min

Fig. 7. STATCOM steady state circuit representation

αmax

Fig. 8. V-I Characteristic of the STATCOM

**3.2.1 Advantages of STATCOM** 

capacitor and reactor banks.

(Hingorani, N.G., 1999):

XC

*Y G jB ss s* = + ; is the equivalent admittance of the STATCOM;

The active and reactive power exchanged with the network at a specified bus expressed as follows:

$$S\_s = V\_s I\_s^" = V\_s Y\_s^" . \left(V\_s^" - V\_k^"\right);\tag{22}$$

After performing complex transformations; the following equations are deduced:

$$P\_s = \left| V\_s \right|^2 \mathcal{G}\_s - \left| V\_s \right| \left| V\_k \right| \left[ \mathcal{G}\_s \cos \left( \theta\_s - \theta\_k \right) + B\_s \sin \left( \theta\_s - \theta\_k \right) \right] \tag{23}$$

$$Q\_s = -\left|V\_s\right|^2 B\_s - \left|V\_s\right| \left|V\_k\right| \left\{ G\_s \sin\left(\theta\_s - \theta\_k\right) - B\_s \sin\left(\theta\_s - \theta\_k\right) \right\} \tag{24}$$

The modified power flow equations with consideration of STATCOM at bus k are expressed as follows:

$$P\_k = P\_s + \sum\_{i=1}^{N} \left| Y\_{ki} \right| . \left| V\_k \right| . \left| V\_i \right| \cos \left( \theta\_k - \theta\_i - \theta\_{ki} \right) \tag{25}$$

$$Q\_k = Q\_s + \sum\_{i=1}^{N} \left| Y\_{ki} \right| . \left| V\_k \right| . \left| V\_i \right| \sin \left( \theta\_k - \theta\_i - \theta\_{ki} \right) \tag{26}$$

### **4. Overview of Differential Evolution technique**

Differential Evolution (DE) is a new branch of EA proposed by (Storn and Price, 1995). DE has proven to be promising candidate to solve real and practical optimization problem. The strategy of DE is based on stochastic searches, in which function parameters are encoded as floating point variables. The key idea behind differential evolution approach is a new mechanism introduced for generating trial parameter vectors. In each step DE mutates

Optimal Location and Control of

Where;

**4.2 Active power dispatch for conventional source** 

stage the fuel cost objective <sup>1</sup> *J* is considered as:

*<sup>i</sup> f* : is the fuel cost of the *ith* generating unit.

*Active power dispatch Conventional source* 

( *Ploss* ) in the transmission system. It is given as:

Fig. 10. Three phase strategy based differential evolution (DE)

**4.3 Combined active and reactive power planning based hybrid model** 

1

=

*k*

The equality constraints to be satisfied are given as follows:

*Nl*

*Pd*1 : the new active power associated to the conventional units; *Pd*2 : the new active power associated to the wind source;

Multi Hybrid Model Based Wind-Shunt FACTS to Enhance Power Quality 43

The main objective of this first stage is to optimize the active power generation for conventional units (>=80% of the total power demand) to minimize the total cost, Fig 9 shows the three phase strategy based deferential evolution (DE). Fig. 10 shows the structure of the control variables related to active power dispatch for conventional source. In this

> 1 1 *NG i i J f* =

> > 1 *NG*

*Solution Strategy* 

Reactive Power Planning Shunt FACTS

The main objective of this second stage is to optimize the active power generation for wind source (<=20% of the total power demand) in coordination with the STATCOM installed at the same specified buses, the objective function here is to minimize the active power loss

( )<sup>2</sup> <sup>2</sup>

*P g t V V t VV*

*loss k k i j k i j ij*

1

*i Pd Pg* =

*i*

<sup>=</sup> (30)

<sup>=</sup> (31)

*Active power dispatch Wind source* 

<sup>2</sup> *loss J Min P* = (32)

δ

= +− (33)

2 cos

vectors by adding weighted, random vector differentials to them. If the fitness function of the trial vector is better than that of the target, the target vector is replaced by trial vector in the next generation.

#### **4.1 Differential evolution mechanism search**

The differential evolution mechanism search is presented based on the following steps (Gonzalez et al., 2008):

**Step 1.** Initialize the initial population of individuals: Initialize the generation's counter, *G=1*, and also initialize a population of individuals, *x(G)* with random values generated according to a uniform probability distribution in the n-dimensional space.

$$X\_{\boldsymbol{l}}^{(G)} = \boldsymbol{x}\_{\boldsymbol{\eta}}^{(\boldsymbol{l})} + rand[\mathbf{0}, \mathbf{1}] \* \left(\boldsymbol{x}\_{\boldsymbol{\eta}}^{(\boldsymbol{l})} - \boldsymbol{x}\_{\boldsymbol{\eta}}^{(\boldsymbol{l})}\right) \tag{27}$$

Where:

*G* : is the generation or iteration

*rand*[0,1] : denotes a uniformly distributed random value within [0, 1].

( ) *L ij x* and ( ) *<sup>U</sup> ij x* are lower and upper boundaries of the parameters *ij x* respectively for *j n* = 1,2,..., .

**Step 2.** the main role of mutation operation (or differential operation) is to introduce new parameters into the population according to the following equation:

$$\mathbf{x}\_{l}^{(G\ast 1)} = \mathbf{x}\_{r3}^{(G)} + f\_{m} \ast \left(\mathbf{x}\_{r2}^{(G)} - \mathbf{x}\_{r1}^{(G)}\right) \tag{28}$$

Two vectors ( ) 2 *G <sup>r</sup> x* and ( ) 1 *G <sup>r</sup> x* are randomly selected from the population and the vector difference between them is established. 0 *mf* is a real parameter, called mutation factor, which the amplification of the difference between two individuals so as to avoid search stagnation and it is usually taken from the range [0,1].


For each mutate vector, ( 1) *<sup>G</sup> <sup>i</sup> v* <sup>+</sup> , an index *rnbr i n* ( )∈{1,2,..., } is randomly chosen using a uniform distribution, and a trail vector, ( 1) ( 1) ( 1) ( 1) 12 2 , ,..., *<sup>T</sup> G GG G u uu u i ii i* + ++ + <sup>=</sup> is generated according to equation:

$$\mu\_{ij}^{(G+1)} = \begin{cases} \upsilon\_{ij}^{(G+1)} \text{ if } \left( \text{rand}\left[0, 1\right] \le \text{CR} \right) \text{ or } \left( j = rnbr\left(i\right) \right) \\ \upsilon\_{ij}^{(G)} \text{ otherwise} \end{cases} \tag{29}$$

