**4.2 Active power dispatch for conventional source**

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

$$J\_{-1} = \sum\_{l=1}^{\text{MC}} f\_l \tag{30}$$

$$P d\mathbf{1} = \sum\_{i=1}^{\text{NC}} P \mathbf{g}\_i \tag{31}$$

Where;

42 Electrical Generation and Distribution Systems and Power Quality Disturbances

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 differential evolution mechanism search is presented based on the following steps

**Step 1.** Initialize the initial population of individuals: Initialize the generation's counter, *G=1*,

according to a uniform probability distribution in the n-dimensional space.

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

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

> ( ) () () ( ) ( 1) 3 21 *G G GG*

**Step 3.** Evaluate the fitness function value: for each individual, evaluate its fitness

**Step 4.** following the mutation operation, the crossover operator creates the trial vectors,

using a uniform distribution, and a trail vector, ( 1) ( 1) ( 1) ( 1)

avoid search stagnation and it is usually taken from the range [0,1].

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

which are used in the selection process. A trial vector is a combination of a mutant vector and a parent vector which is formed based on probability distributions.

( ) [ ] ( ) ( ) ( 1)

in the next generation. These vectors are selected from the current population and the trial population. Each individual of the trial population is compared with its

*v if rand CR or j rnbr i*

and also initialize a population of individuals, *x(G)* with random values generated

( ) ( ) () ( ) ( ) [0,1] *<sup>G</sup> L U <sup>L</sup> X x rand x x i ij* =+ ∗ − *ij ij* (27)

*i r mr r v x fx x* <sup>+</sup> = +∗ − (28)

*<sup>i</sup> v* <sup>+</sup> , an index *rnbr i n* ( )∈{1,2,..., } is randomly chosen

12 2 , ,..., *<sup>T</sup>*

(29)

+ ++ + <sup>=</sup> is

*G GG G u uu u i ii i*

*<sup>r</sup> x* are randomly selected from the population and the vector

*ij x* are lower and upper boundaries of the parameters *ij x* respectively for

the next generation.

(Gonzalez et al., 2008):

Where:

( ) *L ij x* and ( ) *<sup>U</sup>*

*j n* = 1,2,..., .

Two vectors ( )

2 *G <sup>r</sup> x* and ( ) 1 *G*

(objective function) value.

For each mutate vector, ( 1) *<sup>G</sup>*

generated according to equation:

( 1)

*u*

*G ij ij G ij*

counterpart in the current population.

( )

+

0,1 *<sup>G</sup>*

<sup>+</sup> ≤ = <sup>=</sup>

**Step 5.** the selection operator chooses the vectors that are going to compose the population

**Step 6.** Verification of the stopping criterion: Loop to step 3 until a stopping criterion is

*x otherwise*

satisfied, usually a maximum number of iterations, *G*max .

**4.1 Differential evolution mechanism search** 

*G* : is the generation or iteration

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

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

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

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

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

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

$$J\_{\
u\_2} = \text{Min } P\_{\text{loss}} \tag{32}$$

$$P\_{\rm loss} = \sum\_{k=1}^{N\_l} \mathcal{g}\_k \left[ \left( t\_k V\_i \right)^2 + V\_j^2 - 2t\_k V\_i V\_j \cos \delta\_{ij} \right] \tag{33}$$

The equality constraints to be satisfied are given as follows:

$$P d\_2 = \sum\_{i=1}^{\text{MV}} P w\_i \tag{34}$$

Optimal Location and Control of

**5.1 Test system 1** 

**1** 

**2** 

**3 4** 

**13 12** 

**14** 

**29** 

**11** 

**15** 

**27** 

**16** 

**5** 

**10** 

**18 <sup>19</sup>**

**23 <sup>24</sup>**

**20** 

**22** 

**21** 

**9** 

Fig. 13. Single line diagram for the modified IEEE 30-Bus test system (with FACTS devices)

**17** 

**26 25** 

**6** 

**7** 

Multi Hybrid Model Based Wind-Shunt FACTS to Enhance Power Quality 45

IEEE 14-Bus, IEEE 30-Bus, and to large power system size). After a number of careful experimentation, following optimum values of DE parameters have been settled for this test case: population size = 30, mutation factor =0.8, crossover rate = 0.7, maximum generation =100. Due to the limited chapter length, details results related to values of control variables (active power generation, voltage magnitudes, reactive power compensation ( *QSTATCOM* ) for other

The first test system has 6 generating units; 41 branch system, the system data taken from (). It has a total of 24 control variables as follows: five units active power outputs, six generator-bus voltage magnitudes, four transformer-tap settings, nine bus shunt FACTS controllers (STATCOM). The modified IEEE 30-Bus electrical network is shown in Fig 13.

practical network test will be given in the next contribution.

$$P d\mathbf{1} + P d\mathbf{2} - \sum\_{i=1}^{NG} P g\_i - \sum\_{i=1}^{NW} P w\_i = P\_{\text{loss}} \tag{35}$$

$$Pd\mathbf{1} + Pd\mathbf{2} = \text{PD} \tag{36}$$

Where, *<sup>l</sup> <sup>N</sup>* is the number of transmission lines; *<sup>k</sup> g* is the conductance of branch *k* between buses *i* and *j*; *<sup>k</sup> t* the tap ration of transformer *k*; *Vi* is the voltage magnitude at bus *i; ij* δ the voltage angle difference between buses *i* and *j.* 

Fig. 11. Vector control structure: conventional source

The inequality constraints to be satisfied are all the security constraints related to the state varaibles and the control variables mentioned in section 2.1.

Fig. 12. Coordinated vector control

Fig. 12 shows the two coordinated vectors control structure related to this stage, the individual of the combined vector control denoted by *X P PQ Q pq w wN STC STCN* = [ 1 1 ,..., , ,..., ] Where [*P P w wN* <sup>1</sup> ,..., ] indicate active power outputs of all units based wind source, [*Q Q STC STCN* <sup>1</sup> ,..., ] represent reactive power magnitude settings of all STATCOM controllers exchanged with the network.

#### **5. Simulation results**

The proposed algorithm is developed in the Matlab programming language using 6.5 version. The proposed approach has been tested on many practical electrical test systems (small size: IEEE 14-Bus, IEEE 30-Bus, and to large power system size). After a number of careful experimentation, following optimum values of DE parameters have been settled for this test case: population size = 30, mutation factor =0.8, crossover rate = 0.7, maximum generation =100. Due to the limited chapter length, details results related to values of control variables (active power generation, voltage magnitudes, reactive power compensation ( *QSTATCOM* ) for other practical network test will be given in the next contribution.
