INTRODUCTION
Purpose of optimal reactive power flow is mainly to improve the voltage profile
in the system and to minimize the real power transmission loss while satisfying
the unit and system constraints. This goal is achieved by proper adjustment
of reactive power control variables like Generator bus voltage magnitudes (Vgi)
and their reactive power (Qgi), transformer tap settings
(ai), reactive power generation of the capacitor bank (Qshi).
To solve the ORPF problem, a number of conventional optimization techniques
(Lee and Yang, 1998; Granville, 1994)
have been proposed. These include the Gradient method, non-linear programming,
quadratic programming, linear programming and Interior point method. Though
these techniques have been successfully applied for solving the reactive power
flow problem, still some difficulties are associated with them. One of the difficulties
is the multimodal characteristic of the problems to be handled. Also, due to
the non-differential, non-linearity and non-convex nature of the RPF problem,
majority of the techniques converge to a local optimum. Recently, Evolutionary
Computation techniques like Genetic Algorithm (GA) (Iba,
1994), Evolutionary Programming (Wu and Ma, 1995)
and Evolutionary Strategy (Bhagwan and Patvardhan, 2003)
have been applied to solve the optimal dispatch problem.
In this study, particle swarm optimization approach has been proposed to solve
the ORPF problem, in order to remediate the specific structure of the Algerian
Western network 220/60 kV. Because of its serious problems of tension and shortage
of the reactive power, especially for the network 60 kv which is characterized
by a great number of loading nodes which are connected radially to the principal
nodes and sometimes located far from the generators.
This approach is inspired from the behaviour of birds assembling into clouds,
fish benches under water or displaced swarms of bees. It is a stochastic optimization
technique based on a population, developed by Kennedy and
Eberhart (1995, 2001) and then Lee
and El-Sharkawi (2008). In the PSO method, the system is initialized with
a population of random solutions and it therefore seeks optima by providing
generations all like (AG). Unlike the genetic algorithms, we opted for the PSO
which does not include an operator of evolution such as crossing or mutation.
The potential solutions, called particles, move in the definite space of the
studied problem while following the existing optimal particles.
In the present study, we devote ourselves to dealing with the problem of optimal
distribution of reactive power by applying a method called Particles Swarm Optimization
(PSO), in order to remediate the specific structure of the 220/60 kV Algerian
Western network. This network has serious problems of tension and reactive power
shortage, especially for the 60 kV network, which is characterized by a great
number of loading nodes that are connected radially to the principal nodes,
sometimes located far from the generators.
PROBLEM FORMULATION
Basic concept of the PSO: According to the guiding principle of the
PSO and the simulation of the clouds of birds (Kennedy and
Eberhart, 1995, 2001; Lee and El-Sharkawi,
2008; IEEE Committee Report, 2002), the simulation
is a two-dimensional process. The position of each particle is represented by
its coordinates along the XY axes as well as by the speed, which is expressed
by Vx (the speed along the X axis) and Vy (the speed along the Y axis).
The modified position of each particle is carried out by the information of
its position and speed.
The assembly of the birds is optimized by using a certain objective function.
Each agent or particle maintains its coordinates in the space of research where
the coordinates correspond to the best-reached solution (position) at the present.
This value is called pbest. If another best value found by the PSO is the best
value, obtained up to the present by any of the nearest neighbours, this place
is called ibest. When a particle takes the entire population as its topological
neighbours, the best value is a better total and is called gbest.
This information is thus obtained from the personal experiments of each particle.
Moreover, each particle knows the best total value of the (gbest) group among
the (pbest) ones. This information is obtained from the knowledge of the way
in which the other agents are carried out Each agent tries to modify its place
based on the following information:
| • |
The current positions: (x, y) |
| • |
The current velocities (vx, vy) |
| • |
The distance between the current position and pbest |
| • |
The distance between the current position and gbest |
The concept of a particle swarm optimization changes each time the particles
take a step to change their speed (accelerating) towards the best positions
ibest of pbest.
This modification can be represented by the concept of velocity (Kennedy
and Eberhart, 2001; Lee and El-Sharkawi, 2008). Velocity
of each agent can be modified by following equation:
| Where: |
| vik+1 |
: |
Velocity of particle i at iteration k+1 |
| w |
: |
Weighting function |
| vik+1 |
: |
Weighting factor |
| rand |
: |
Random number between 1 and 0 |
| sik |
: |
Current position of particle i at iteration k |
| pbesti |
: |
Pbest of the particle i |
| gbest |
: |
gbest of the group |
The weight function, which is usually used in Eq. 2, is written
as follows (Kennedy and Eberhart, 2001; Lee
and El-Sharkawi, 2008):
| Where: |
| wmax |
: |
Initial weight |
| wmin |
: |
Final weight |
| itermax |
: |
Maximum number of iteration |
| iter |
: |
Current number of iteration |
While referring to Eq. 2, a certain speed, gradually approaching
pbest and gbest, can be calculated. The current position Sik,
which represents the research of the point within solution space, can be modified
by the following equation:
General algorithm of the PSO: General Algorithm of PSO can be described
as follows (IEEE Committee Report, 2002; Shi
and Eberhart, 1998):
| Step 1: |
Generation of initial condition of each agent. Initial searching
points (Si0) and velocities (vi0)
of each agent are usually generated randomly within the allowable range.
The current searching point is set to pbest for each agent. The best evaluated
value of pbest is set to gbest and the agent number with the best value
is stored |
| Step 2: |
Evaluation of searching point of each agent. The objective function value
is calculated for each agent. If the value is better than the current pbest
of the agent, the pbest value is replaced by the current value. If the best
value of pbest is better than the current gbest, gbest is replaced by the
best value and the agent number with the best value is stored |
| Step 3: |
Modification of each searching point. The current searching point of each
agent is changed using Eq. 1, 2 and
3 |
| Step 4: |
Checking the exit condition |
The current iteration number reaches the predetermined maximum iteration number,
then exit. Otherwise, go to step 2.
Application for active power loss minimisation: The principal objective
of the problem of the optimal flow of the reactive power is to minimize the
actives losses in the electrical supply network and to maintain the tension
within their allowed limits while satisfying constraints as a whole: equalities
and inequalities (Arif et al., 2007).
The equality constraints represent the equations of the flow of power. The
limits, applied to the tension, of the generators or the shunt of compensations
and on the ratios of the regulators in load, which constitute the constraint
inequalities. In our case, the objective function represents the active losses
in the electrical supply network and the general formulation of this problem
is then written as:
Under the constraints:
With:
Where:
| Pgi, Qgi |
: |
Active and reactive power generated in node i |
| Pli, Qli |
: |
Active and reactive power of load in node i |
| Qshi |
: |
Reactive power of shunt capacitors or facts devices shunt in node i |
| Vi, Vj |
: |
Modules of the tensions to the node i and j |
| θij= θi-θj |
: |
Angles of the tensions to the node i and j |
| Gij |
: |
Conductance between the nodes i and j |
| Bij |
: |
Susceptance between the nodes i and j |
| ng |
: |
Generator number |
| nT |
: |
Transformer number |
| nsh |
: |
Shunt condenser number |
|
| Fig. 1: |
A general flowchart of power losses minimization by meta heuristic
method |
In this study, the means of compensation considered are the groups of production,
the shunt capacities and the transformers with the regulator in load. The latter
are regarded as variables of control. The state variables are: x= [Vi,θi].
The application of the particle swarm method in optimizing the power of an
electrical supply network requires following the stages of the flow chart by
mentioning that the variables of control represent the particles or the agents
of the method and their value is found in a random way in the allowed range
that will be represented first by pbest then by gbest in another stage. The
flowchart of our method to minimize power losses by both methods used is show
in Fig. 1.
It should be noted that with the help of these values (gbest), we can calculate
the active power losses of the system studied by PF for each iteration.
| Step 1: |
For each variable of control Xi, we make a random
choice of a population within limit of function. Each item fits into a certain
position Si0 and to an initialised velocity Vi0. |
| Step 2: |
Calculate loss PL for all of the elements of each population.
If there is a value xjεXi that gives P'L<
PL, xj takes the pbest value. The best value of the
population will take the gbest value. If it is better than the previous
step, so it will be stored |
| Step 3: |
The moving or changing of the value of the items by Eq.
1, 2 and 3 |
| Step 4: |
Check stop criteria |
ILLUSTRATION
The 220/60 kV Algerian western test network studied is represented by Fig.
2. The network is made up as follows:
This system described in Table 1 and 2,
is structurally particular Fig. 2 (many load buses radially
connected to the main grid) . However, serious problems could be faced dealing
with the voltage and the reactive power shortages, especially with the 60 kV
systems.
Control parameters of the Meta heuristics methods applied: The particle
swarm optimization method contains several parameters whose values can be adjusted
so that the algorithm manages to find the optimal solution.
| Table 1: |
Main data of the Western Algerian system |
 |
| Table 2: |
The control variables limits, of and bus voltages |
 |
|
| Fig. 2: |
Algerian 220/60 KV transmission/sub-transmission system |
The same with the GA parameters, we can adjusted the values to locate the optimal
solution.
After several tests are carried out, values of the parameters adapted.
These parameters were mentioned in several references (Kennedy
and Eberhart, 2001; Lee and El-Sharkawi, 2008) and
Shi and Eberhart (1998), adding that our simulation was
made with a population of N = 150 and a maximum iteration equal to 150.
RESULTS AND DISCUSSION
We have applied this method to the studied network by using one variant u,
the represent case: u=[ Qig , ai , Qish].
The simulation results of this technique are illustrated in the tables following
before and after optimization, compared with an optimization by the algorithms
genetics (Chettih et al., 2008a, b)
and the corrections achieved with an hybrid approach by Khiat
et al. (2003a, b).
We can clearly notice that the objective function power losses is effectively
minimized by the Meta heuristics method. An analysis of the results can clearly
show that Efficiency of meta heuristics techniques to resolve the problem of
ORPF, towards the minimization of the active losses (Table 5)
as well as numbers of compensators shunt (Table 3), because
in the methods (GA, or, PSO) witch we have used so far, we note that we have
considered tow news compensators in the node 17 and 35 (Table
4). This one has proven to be successful in GA and better yet in a PSO method
(Fig. 2), controlling several variables simultaneously and
it helps improve the optimization to progress over a short period of time.
| Table 3: |
The control parameters of the Meta heuristics methods applied |
 |
| Table 4: |
The compensation of optimized device performances using the
PSO method |
 |
| Table 5: |
Total losses for all of cases |
 |
|
| Fig. 3: |
Voltages profiles in the western Algerian transmission systems
220/60 kV |
We also notice that the behaviour of the system 60 kV voltages is improving
from the GA optimization to the case where the PSO optimization method is applied;
this improvement is followed by a reduction of power losses, which can reach
16% (Table 5) with a steady control of the voltage level (Fig.
3a, b).
CONCLUSION
The Meta heuristics techniques was studied and applied to resolve the problem
of optimal reactive power flow. The model of electrical supply networks that
we test in simulations is the 220/60 kV Algerian western network. The analysis
of our results showed that this technique (meta-heuristic) gave results quantitatively
better than (Khiat et al., 2003a, b),
in terms of minimization of the loss of active power. We also noted that the
particle swarm technique is simple to implement and its execution leads quickly
and effectively to a good convergence with few parameters to be adjusted.
ACKNOWLEDGMENT
The Western dispatching Sonelgaz company Operators, acknowledge for providing
as with characteristics of the West network 220/60 kV.
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