Control variables optimized using FFA
Power quality management based environmental optimal power flow (OPF) is a vital research area for power system operation and control, this type of problem is complex with non-linear constraints where many conflicting objectives are considered like fuel cost, gaz emission, real power loss, voltage deviation, and voltage stability[1-2]. The solution of this multi objective problem becomes more complex and important when flexible ac transmission systems (FACTS) devices and renewable energy are considered. In the literature a wide variety of optimization techniques have been applied, a number of approaches have been developed for solving the standard optimal power flow problem using mathematical programming, lambda iteration method, gradient method, linear programming, quadratic programming and interior point method [3-4-5]. However all these developed techniques rely on the form of the objective function and fail to find the near global optimal solution, authors in [6-7] provide a valuable introduction and surveys the first optimization category based conventional optimization methods.
To overcome the major problem related specially to restriction on the nature of the objective function, researchers have proposed a second optimization category based evolutionary algorithms for searching near-optimum solutions considering non linear objective function characteristic and practical generating units constraints such as: Genetic algorithms (GA), simulated annealing (SA), tabu search (TS), and evolutionary programming (EP) [8-9-10], which are the forms of probabilistic heuristic algorithm, author in  gives a recent and significant review of many global optimization methods applied for solving many problems related to power system operation, planning and control.
In order to enhance the performance of these standard global optimization methods for solving complex and practical problem, many hybrid variants based evolutionary algorithm have been proposed and applied with success for solving many problems related to power system operation and control like: hybrid EP–SQP, Quantum genetic Algorithm (QGA) , artificial immune system (AIS) , Adaptive particle swarm optimization (APSO) , improved PSO (IPSO) , improved chaotic particle swarm optimization (ICPSO) , adaptive hybrid differential evolution (AHDE) , and a hybrid multi agent based particle swarm optimization (HMPSO) . These methods have a better searching ability in finding near global optimal solution compared to mathematical methods and to the standard evolutionary algorithms. Very recently a significant review of recent non-deterministic and hybrid methods applied for solving the optimal power flow problem is proposed in .
Recently a new metaheuristic optimization method called Firefly research method introduced and attracted many researchers due to its capacity and efficiency to find the global optimal solution, this method applied with success to solving many real optimization problems [20-21]. In this chapter firefly algorithm is proposed to solve the combined economic dispatch problem, and a decomposed genetic algorithm considering multi shunt FACTS proposed for solving the multi objective optimal power flow [22-23]. The remainder of this chapter is organized as follows; section 2 provides a formulation of the multi objective optimization problem, Section 3 introduces the mechanism search of the firefly algorithm. Brief description and modelling of shunt FACTS devices is developed in section 4. Simulation results based Matlab program are demonstrated in section 5, finally, the chapter is concluded in section 6.
2. Energy planning strategy: Economic issue and power quality
It is important to underline the importance of energy efficiency planning in power systems, the combined term energy planning is usually associated with power quality; how energy is produced (economic aspect), how energy is consumed at the point of end use (technical aspect), and what is the impact of the total energy produced on the environment (gaz emissions). Figure 1 shows the structure of energy planning strategy considering FACTS technology; the following points should be taken by expert engineers and researchers to assure energy efficiency.
Environmental Issue: The environment (gaz emissions) can be considered as the first step towards the improvement of energy efficiency by forcing the utilities and expert engineers to introduce the environmental constraints to the standard optimal power flow problem.
Technology Issue: Energy planning strategy becomes a complex problem and difficult to solving efficiently with the wide integration of two new technologies, flexible ac transmission systems (FACTS) and renewable energy.
Economic Issue: How to estimate economically the number, the size of FACTS Controllers coordinated with renewable sources to be installed in a practical and large electrical network?
3. Multi objective optimal power flow formulation
The real life problems involve several objectives and the decision maker would like to find solution, which gives compromise between the selected objectives. The multi objective OPF is to optimize the settings of control variables in terms of one or more objective functions while satisfying several equality and inequality constraints.
In multi objective OPF we have to optimize two or more objective functions. Figure 2 shows a general presentation of the multi objective OPF problem. The mathematical problem can be formulated as follows:
Where is the ith objective function, and is the number of objectives,and are respectively the set of equality and inequality constraints, x is the state variables and u is the vector of control variables. The control variables are generator active and reactive power outputs, bus voltages, shunt capacitors/reactors and transformers tap-setting. The state variables are voltage and angle of load buses.
3.1. Active power planning with smooth cost function
For optimal active power dispatch, the objective function is the total generation cost expressed in a simple form as follows:
where is the number of thermal units, is the active power generation at unit i and , and are the cost coefficients of the ith generator.
The equality constraints are the power flow equations.
The inequality constraints reflect the limits on physical devices in the power system as well as the limits created to ensure system security.
3.2. Emission objective function
An alternative dispatch strategy to satisfy the environmental requirement is to minimize operation cost under environmental requirement. Emission control can be included in conventional economic dispatch by adding the environmental cost to the normal dispatch. The objective function that minimizes the total emissions can be expressed as the sum of all the three pollutants (,, ) resulting from generator real power .
In this study,emission is taken as the index from the viewpoint of environment conservation. The amount of emission is given as a function of generator output (in Ton/hr), that is the sum of quadratic and exponential functions .
where , , , and are the parameters estimated on the basis of unit emissions test results.
The pollution control can be obtained by assigning a cost factor to the pollution level expressed as:
where is the emission control cost factor in $/Ton.
Fuel cost and emission are conflicting objective and can not be minimized simultaneously. However, the solutions may be obtained in which fuel cost an emission are combined in a single function with different weighting factor. This objective function is described by:
where is a weighting factor that satisfies.
In this model, when weighting factor =1, the objective function becomes a classical economic dispatch, when weighting factor =0, the problem becomes a pure minimization of the pollution control level.
3.3. Active power planning with valve-point loading effect
The valve-point loading is taken in consideration by adding a sine component to the cost of the generating units . Typically, the fuel cost function of the generating units with valve-point loadings is represented as follows:
3.4. Reactive power planning
The main role of reactive power planning (RPP) is to adjust dynamically the control variables individually or simultaneously to reduce the total power loss, transit power flow, voltages deviation, and to improve voltage stability, but still satisfying specified constraints (generators constraints and security constraints). The basic vector control structure is well presented in Fig 3.
3.4.1. Power loss objective function
The objective function here is to minimize the active power loss () in the transmission system. It is given as:
Where, is the number of transmission lines; is the conductance of branch k between buses i and j; the tap ration of transformer k; is the voltage magnitude at bus i; the voltage angle difference between buses i and j.
3.4.2. Voltage deviation objective function
One of the important indices of power system security is the bus voltage magnitude. The voltage magnitude deviation from the desired value at each load bus must be as small as possible. The deviation of voltage is given as follows:
where, is the number of load buses and is the desired or target value of the voltage magnitude at load bus k.
3.5.1. Equality constraints
The equality constraints are the real and reactive power balance equations.
Where is the number of buses, , are the active and the reactive power generation at bus i;, are the real and the reactive power demand at bus , , , the voltage magnitude at bus , , respectively; is the phase angle difference between buses and respectively, and are the real and imaginary part of the admittance ().
3.5.2. Inequality constraints
The inequality constraints reflect the generators constraints and power system security limits,
Upper and lower limits on the generator bus voltage magnitude:
The inequality constraints on security limits are given by:
Constraints on transmission lines loading
Constraints on voltage at loading buses (PQ buses)
Upper and lower limits on the tap ratio (t) of transformer.
Parameters of shunt FACTS Controllers must be restricted within their upper and lower limits.
4. Metaheuristic optimization methods
The difficulties associated with using mathematical optimization on large-scale engineering problems have contributed to the development of alternative solutions, during the last two decades; the interest in applying new metaheuristic optimization methods in power system field has grown rapidly.
In general, metaheuristic optimization methods can be classified into three large categories.
The first category such as: GA, TS, SA, ACO, PSO, DE and theirs successive developed variants.
The second category includes: Harmony search (HS), Biogeography based optimization (BBO), Artificial bee colony (ABC), Honey bee (HB), Shuffled frog leaping algorithm (SFLA), Firefly algorithm (FFA), Gravitational search algorithm (GSA), Imperialist competition algorithm (ICA) and many other variants.
The third category called hybrid optimization methods includes a combination between metaheuristics and conventional methods.
Details about their performances and the application of metaheuristic optimization techniques for solving many practical problem related to power system field particularly multi objective OPF can be found in a recent state of the art of non-deterministic optimization and hybrid methods presented in . Figure 4 shows a simplified presentation about the most popular metaheuristic methods applied by researchers for solving many complex problems.
4.1. Firefly search algorithm
This section introduces and describes a solution to the combined economic dispatch problem using a new metaheuristic nature inspired algorithm. Firefly research algorithm is one of the new Biology inspired metaheuristic algorithms which have recently introduced by Dr. Xin-She Yang at Cambridge University in 2007, as an efficient way to deal with many hard combinatorial optimization problems and non-linear optimization constrained problems [20-21]. In general, the firefly algorithm has three particular idealized rules which are based on some of the major flashing characteristics of real fireflies these are presented on the following schematic diagram (Figure 5), and described in brief as follows:
All ﬁreﬂies are unisex, and they will move towards more attractive and brighter ones regardless their sex.
The degree of attractiveness of a ﬁreﬂy is proportional to its brightness which decreases as the distance from the other ﬁreﬂy increases due to the fact that the air absorbs light. If there is not a brighter or more attractive ﬁreﬂy than a particular one, it will then move randomly.
The brightness or light intensity of a ﬁreﬂy is determined by the value of the objective function of a given problem.
The basic form of attractiveness function of a ﬁreﬂy is the following monotonically decreasing function:
Where is the distance between any two fireflies, β0 is the initial attractiveness at r=0, and γ is an absorption coefficient which controls the decrease of the light intensity.
The distance between two fireflies I and j at positions xi and xj can be defined by the following relation:
Where, is the kth component of the spatial coordinate and and d is the number of dimension.
The movement of a firefly i which is attracted by a more attractive firefly j is given by the following equation.
Where the first term is the current position of a firefly, the second term is used for considering a firefly’s attractiveness to light intensity seen by adjacent fireflies, and the third term is for the random movement of a firefly in case there are not any brighter ones.
Like many metaheuristic optimization methods, choosing the initial values of parameters is an important task which affects greatly the convergence behaviors of the algorithm. These parameters depend on the nature of the problem to be solved.
is a randomization parameter determined based on the complexity of the problem to be solved.
The attractiveness or absorption coefficient .
4.1.1. Solving combined economic emission problem based FFA
The pseudo code depicted in Figure 6 gives a brief description about the adaptation of the firefly algorithm for solving the combined economic emission problem.
At the first stage an initial solution is generated based on the following equation:
Where and are the upper range and lower range of the ith firefly (variable), respectively.
After evaluation of this initial solution, the initial solution will be updated based on the mechanism search, final solution will be achieved after a specified number of generations.
Test1: Benchmark function
In this case, the proposed algorithm is tested with many benchmark functions. All these problems are minimization problems. Due to the limited chapter length, only one function are considered for minimization problem, details can be retrieved from . Fig 7 shows the shape of the objective function, Fig 8 shows the path of fireflies during optimization's process, Fig 9 shows the Final stage corresponding to global solution.
Test 2: Combined economic-emission dispatch with smooth cost function
To demonstrate the performance of the proposed FFA, the algorithm is applied and tested to the six generating units without considering power transmission loss. The problem is solved as single and multi objective optimization. Detail data of this test system is given in [22-23]. For 20 runs, the best cost is 600.1114 ($/h), the algorithm converges at low number of iteration.
Figs 10-11 show the convergence characteristics of the proposed FFA when objective function optimized separately. Fig 12 shows the convergence characteristic for combined fuel cost and emission. Table 1 summarizes the best solutions obtained for three cases.
|Control Variables(MW)||Case 1|
4.2. Solving security OPF based parallel GA considering SVC controllers
4.2.1. Strategy of the parallel GA
Parallel Genetic Algorithms (PGAs) have been developed to reduce the large execution times that are associated with simple genetic algorithms for finding near-optimal solutions in large search spaces. They have also been used to solve larger problems and to find better solutions. PGAs can easily be implemented on networks of heterogeneous computers or on parallel mainframes. The way in which GAs can be parallelized depends upon the following elements [22-23]:
How fitness is evaluated.
How selection is applied locally or globally.
How genetic operators (crossover and mutation are used and applied)
If single or multiple subpopulations are used.
If multiple populations are used how individuals are exchanged.
How to coordinate between different subpopulations to save the proprieties of the original chromosome.
In the proposed approach the subpopulations created are dependent, efficient load flow used to test the performance of the new subpopulations generated.
The proposed algorithm decomposes the solution of such a modified OPF problem into two coordinated sub problems. The first sub problem is an active power generation planning solved by the proposed algorithm, and the second sub problem is a reactive power planning [14-15] to make fine adjustments on the optimum values obtained from the first stage. This will provide updated voltages, angles and point out generators having exceeded reactive power limits.
4.2.2. Decomposition mechanism
Problem decomposition is an important task for large-scale OPF problem, which needs answers to the following two technical questions.
How many efficient partitions needed?
Where to practice and generate the efficient inter-independent sub-systems?
The decomposition procedure decomposes a problem into several interacting sub-problems that can be solved with reduced sub-populations, and coordinate the solution processes of these sub-problems to achieve the solution of the whole problem.
4.2.3. Algorithm of the Proposed Approach
Initialization based in decomposition mechanism
In the first stage the original network was decomposed in multi sub-systems and the problem transformed to optimize the active power demand associated to each partitioned network. The main idea of the proposed approach is to optimize the active power demand for each partitioned network to minimize the total fuel cost. An initial candidate solution generated for the global N population size.
Suppose the original network under study decomposed into S sub-systems 1,2…..S with Pg1, Pg2,…..PgN, the active power control variables, for this decomposed sub-systems related to S sub-populations, 1,2……S, with population sizes, N1……..NS, and (N1+N2+……NS=N).
Each sub-population contains NP1 control variables to be optimized.
Each sub-population updated based on the GA operators.
1-For each decomposition level estimate the initial active power demand:
For NP=2 Do
Where NP the number of partition
: the active power demand for the first initial partition.
: the active power demand for the second initial partition.
: the total active power demand for the original network.
The following equilibrium equation should be verified for each decomposed level:
For level 1:
Fitness evaluation based power loss
For all sub-systems generated perform a load flow calculation to evaluate the proposed fitness function. A candidate solution formed by all sub-systems is better if its fitness is higher (low total power loss).
A global data base generated containing the best technical sub-systems.
Consequently under this concept, the final value of active power demand should satisfy the following equations.
4.2.4. Reactive power dispatch
All the sub-systems are collected to form the original network, global database generated based on the best results found from all sub-populations.
The final solution (Global) is found out after reactive power planning procedure to adjust the reactive power generation limits, and voltage deviation, the final optimal cost is modified to compensate the reactive constraints violations. Fig. 13 illustrates the mechanism search partitioning for active power planning, Fig. 14 shows an example of tree network decomposition.
5. FACTS technology
FACTS philosophy was first introduced by Hingorani  from the Electric power research institute (EPRI) in the USA. The objective of FACTS devices is to bring a system under control and to transmit power as ordered by the control centers, it also allows increasing the usable transmission capacity to its thermal limits. With FACTS devices we can control the phase angle, the voltage magnitude at chosen buses and/or line impedances. In general these FACTS devices are classified in three large categories as follows:
5.1. Shunts FACTS controllers (SVC, STATCOM)
Principally designed and integrated to adjust dynamically the voltage at specified buses. Fig 15 shows the basic principle of dynamic shunt FATCS controllers (SVC, and STATCOM).
5.2. Series FACTS controllers (TCSC, SSSC)
Principally designed and integrated to adjust dynamically the transit power at specified lines. Fig 16 shows the basic principle of dynamic series FATCS controllers (TCSC and SSSC).
5.3. Hybrid FACTS controllers (UPFC)
Principally designed and integrated to adjust dynamically and simultaneously the voltage, the active power, and the reactive power at specified buses and lines. The basic one line diagram of hybrid controller (UPFC) is well presented in Fig 17.
5.4. Shunt FACTS modelling
5.4.1. Static VAR Compensator (SVC)
The Static VAr Compensator (SVC)  is a shunt connected VAr generator or absorber whose output is adjusted dynamically to exchange capacitive or inductive current so as to maintain or control specific parameters of the electric power system, typically bus voltages. It includes separate equipment for leading and lagging VArs. The steady state model shown in Fig. 18 is used to incorporate the SVC on power flow problems. This model is based on representing the controller as variable susceptance or firing angle control.
Where, are the shunt susceptance, firing angle, inductive reactance, capacitive reactance of the SVC controller, and the bus voltage magnitude to which the SVC is connected, respectively. The reactive power exchange with the bus i can be expressed as,
6.1. Test system 1: Algerian electrical network planning
6.1.1. Active Power dispatch without SVC compensators
The proposed algorithm is developed in the Matlab programming language using 6.5 version. The proposed approach has been tested on the Algerian network (1976). It consists of 59 buses, 83 branches (lines and transformers) and 10 generators. Table I shows the technical and economic parameters of the ten generators, knowing that the generator of the bus N°=13 is not in service. Table II shows the generators emission coefficients. The values of the generator emission coefficients are given in Tables 2-3, the generators data and cost coefficients are taken from [23-28], Figs. 19-20 show the topology of the Algerian electrical network test with 59-Bus based one line diagram and in schematic representation. For the purpose of verifying the efficiency of the proposed approach, we made a comparison of our algorithm with others competing OPF algorithm. In , they presented a fuzzy controlled genetic algorithm, in , authors proposed fast successive linear programming algorithm applied to the Algerian electrical network.
To demonstrate the effectiveness and the robustness of the proposed approach, two cases have been considered with and without consideration of SVC Controllers installation:
Case 1: Minimum total operating cost (α =1).
Case 2: Minimum total emission (α =0).
|Bus Number||Pmin[MW]||Pmax[MW]||Qmin[Mvar]||Qmax[Mvar]||a [$/hr]||b [$/MWhr]||c [$/MW2hr]|
|Bus Number||Generator||α||bx1e-2||γ x1e-4||ω||μ x1e-2|
|Case1: α =1||Case 2: α =0|
|Pgi (MW)||Minimum Cost||Minimum emission|
|Power loss (MW)||16.20180||17.51580|
|Generators N°||FGA ||GA ||ACO ||FSLP ||Our Approach|
|FSLP ||Our Approach|
|Case1||Case 2||Case1||Case 2|
|Pgi (MW)||α =1||α =0||α =1||α =0|
|Power loss (MW)||19.65||19.93||16.2018||17.5158|
Table 4 shows simulation results obtained by the proposed approach for the two cases (α =1, α =0), the comparison of the results obtained by the application of the decomposed parallel GA proposed with those found by global optimization (GA, FGA, and ACO) and conventional methods are reported in the Table 5 and Table 6 The proposed approach gives better results for all cases. For example at the case corresponding to the minimum total operating cost (α =1), the fuel cost is 1765.9377 $/h, and power losses 16.2018 MW which are better compared with the results found by the global and conventional methods.
It is important to note that all results obtained by the proposed approach do not violate the physical generation capacity constraints (reactive power). The security constraints are also satisfied for voltage magnitudes (0.9<V<1.1 p.u). Fig 18 shows distribution voltage profile with and without two optimized shunt compensators installed at bus 27 and bus 36. Fig 19 shows the voltage phase profiles, it is clearly identified that all voltage phase profiles are within the constraint limit.
|Case 1||Case 2||Qmax||Qmin|
|Bus||Qg (Mvar)||Qg (p.u)||Q max||Q min|
6.1.2. Reactive power dispatch based SVC controllers
For a secure operation of the power system, it is important to maintain required level of security margin, system loadability, voltage magnitude and power loss are three important indices of power quality. In this stage, dynamic shunt compensation based SVC Controllers taken in consideration. The control variables selected for reactive power dispatch (RPD) are the generator voltages, and reactive power of the SVC compensators installed at critical buses.
6.1.3. Optimal location of shunt FACTS
Before the insertion of SVC devices, the system was pushed to its collapsing point by increasing both active and reactive load discretely using continuation load flow . In this test system according to results obtained from the continuation load flow, buses 7, 14, 17, 35, 36, 39, 44, 47, 56 are the best location points for installation and coordination between SVC Compensators and the network. Table 8 gives details of the SVC Data. Table 8, shows results of reactive power generation with SVC Compensators for the minimum cost case, the active power loss and the total cost are reduced to 15.1105 MW, 1763.5771 ($/h) respectively compared to base case considering two optimized shunt compensators. It is important to note that the system loadability improved to 1.9109 p.u compared to the base case (1.8607 p.u) with two shunt compensators, reactive power of generating units are within their admissible limits.). Fig 20 shows distribution voltage profile considering two fixed shunt compensators and multi SVC compensators. Fig 21 shows the voltage phase profiles, it is clearly identified that all voltage phase profiles are within the constraint limit.
|Bmin (p.u)||Bmax (p.u)||Binit (p.u)|
|Susceptance SVC Model||-0.5||0.5||0.025|
|Bus||BSVC(p.u)||V (p.u)with SVC||Fixed Shunt Capacitors|
|Loading Factor (p.u): without SVC||1.8607|
|Loading Factor (p.u): with 8 SVC Controllers||1.9109|
The multi objective power quality management consists in optimizing indices of power quality in coordination with FACTS devices considering fuel cost. In this chapter two techniques based metaheuristic methods have been proposed to solving the multi objective power management problem, many objective functions related to power quality have been proposed like power loss, voltage deviation, system loadability, and pollution constraint. The first proposed method called Firefly research algorithm is used to solve the combined economic emission problem, the algorithm applied to 6 generating units. The second method is a new variant based GA named parallel GA (PGA) proposed as an alternative to improve the performances of the standard GA to solving the security combined economic dispatch considering multi shunt FACTS devices. The proposed PGA applied to solving the security OPF of the Algerian power system (59-Bus) considering three indices of power quality; power loss, voltage deviation and gaz emissions. From simulation results, it is found that the two proposed approaches, FFA and PGA can converge to the near global solution and obtain competitive results in term of solution quality and convergence characteristic compared to the others recent optimization methods.
Due to these efficient properties, in the future work, author will still to apply these two methods to solving the multi objective optimal power flow based multi FACTS devices considering practical generator units (valve point effect and prohibited zones).