Economic/Environmental power dispatch for power systems including wind farms
Imen BEN JAOUED^{1*}, Tawfik GUESMI^{2} , Yosra WELHAZI^{3},
National Engineering School of Sfax, Sfax University, B.P.W. 3038 Sfax, Tunisia.
Emails: imenbenjaoued@live.com, tawfik.guesmi@istmt.rnu.tn
yosrawelhazi@yahoo.fr, hsan.haj@enis.rnu.tn
Abstract
This paper presents the problem of the Economic/Environmental power Dispatching (EED) of hybrid power system including wind energies. The power flow model for a stall regulated fixed speed wind generator (SRFSWG) system is discussed to assess the steadystate condition of power systems with wind farms. Modified NewtonRaphson algorithm including SRFSWG is used to solve the load flow equations in which the state variables of the wind generators are combined with the nodal voltage magnitudes and angles of the entire network. The EED problem is a nonlinear constrained multiobjective optimization problem, two competing fuel cost and pollutant emission objectives should be minimized simultaneously while satisfying certain system constraints. In this paper, the resolution is done by the algorithm multiobjective particle swarm optimization (MOPSO). The effectiveness of the proposed method has been verified on IEEE 6generator 30bus test system and using MATLAB software package.
Keywords
Economic/Environmental power Dispatching (EED); Wind farm; Stall Regulated Fixed Speed Wind Generator (SRFSWG); Power flow; NewtonRaphson algorithm; MultiObjective Particle Swarm Optimization (MOPSO)
Introduction
The main objective of the Environmental Economic power Dispatch (EED) consists in the schedule of the power generator units outputs with load demand at minimum operating cost, emissions and pollution while satisfying operational system constraints. A lot of different strategies have been reported in the literature pertaining to the reduction of the atmospheric emissions in power plants [1,2]. These include the use of alternative fuels with a low emission potential, replacement of the existing technologies with energyefficient ones and emission dispatching [3,4] which is an attractive shortterm alternative. In recent years, the environmental and economic concerns have lead to the use of renewable energy resources such as wind power and solar radiation. The use of wind energy conversion systems (WECS) has been considered the most growing renewable energy source [5]. However, the integration of wind generation into the electric power network requires more attention while planning and operating an electrical power system. In the last few decades, different Power Flow (PF) solution techniques such as GaussSeidel, NewtonRaphson and Fast decoupled load flow [6] have been developed in order to operate and control the power system. The NewtonRaphson technique is a fundamentally approach for modeling the wind energy systems. This method simultaneously combines the state variables corresponding to the wind generators and the network in a single frameofreference.
In literature, several techniques [3,4,7] have been reported in order to handle the EED problem. In recent directions, both fuel cost and emission are considered simultaneously competing objectives. Stochastic search and Fuzzybased multiobjective optimization techniques have been proposed for the EED problem [7,8]. However, these algorithms are unable to provide a systematic framework for directing the search towards Paretooptimal front and the extension of these approaches to include more objectives is a very involved question. The EED problem can be also solved by using genetic algorithm based multiobjective techniques [9].
In recent years, multiobjective evolutionary algorithms [10] such as niched Pareto genetic algorithm (NPGA) and strength Pareto evolutionary algorithm (SPEA) algorithms have been used for the EED problem optimization in order to find the optimal solution. Recently, modern metaheuristic algorithms are used for nonlinear optimization problems. The multiobjective particle swarm optimization (MOPSO) [11] is a typical populationbased optimization method. Unlike other heuristic techniques such as genetic algorithm (GA), MOPSO has a flexible mechanism to carry out both global and local search in each iteration process within a short calculation time.
In this paper, MOPSO has proposed to solve the EED problem of hybrid power system including wind energies. In addition, a fuzzybased mechanism was used in order to extract the best compromise solution. Modified NewtonRaphson algorithm including SRFSWG was used to solve the load flow equations. To illustrate the effectiveness and potential of the proposed approach to solve the multiobjective EED problem, several runs have been carried out on the IEEE 6generators 30bus test system and the results are compared to the recently reported methods. The results show that the proposed approach is efficiently used to solve the EED problem including wind energies and it is superior to other multiobjective methods.
Materials and Method
Modelling of wind generator
At this moment, different types of wind turbine generating units were installed and they can be classified into three categories, namely fixed, semivariable and variable speed types. This paper addresses the mathematical representation of directly gridconnected wind generators such as SRFSWG. The idea of this machine is based on an asynchronous squirrelcage motor generator shown in Figure 1, which is driven by a wind turbine with the stator directly connected to the grid through a power transformer. In this SRFSWG a fixed shunt capacitor is used to control reactive power compensation.
Figure 1. Induction machine equivalent circuit
The power output of this SRFSWG depends on the turbine and generator characteristics, wind speed, rotor speed and the terminal voltage.
From the equivalent circuit shown in Figure 1, the power converted from mechanical to electrical form P_{g} can be represented by (1).
_{} 
(1) 
where R_{r} is the rotor resistance, s is the slip of the induction generator and I_{r} is the rotor current given by the following equation :
_{} 
(2) 
The active and reactive powers, determined by equations (3) and (4), are dependent on the machine’s slip s and the terminal voltage_{ }V.
_{} 
(3) 
_{} 
(4) 
where the variables are given in [12].
The wind turbine mechanical power output P_{m} [W] extracted from the wind by this generator [13] can be written as
_{} 
(5) 
were ρA [g/m^{3}] is the density of air, Vw [m/s] is the wind speed, [m^{2}] is the area swept by the rotor and Cp(λ,β) is the power coefficient. The Cp given by (6) is a nonlinear function of the tip speed ratio λ and the pitch angle β:
_{} 
(6) 
where, depends on the wind speed Vw and the radius of the rotor R [m] as given in (7).
_{} 
(7) 
where W_{r} [rad/s] is the angular speed of the turbine and μ is given in (8):
_{} 
(8) 
μ is represented by (8), [degrees] is the pitch angle and the constants C1 to C9 are the parameters of design of the wind turbine
Power flow model
The objective of this section is to give a power flow model for a power system without and with wind farm device.
Power flow analysis without wind farm
The injected real and reactive power flow at bus_{}, for power system with N buses, can be written as in [14]:
_{} 
(9) 
_{} 
(10) 
where V_{i} and α_{i} are respectively, modulus and argument of the complex voltage at bus I_{.} Y_{ij} and θ_{ij} are respectively, modulus and argument of the ij th element of the nodal admittance matrix Y. The resolution of the problem of power flow uses the NewtonRaphson method [14]. The nonlinear system is represented by the linearized Jacobian equation given by the following equation:
_{} 
(11) 
Power flow analysis with wind farm
When the SRFSWG is connected at terminal bus f of the system, the set of mismatch power flow equations is
_{} 
(12) 
_{} 
(13) 
where P_{lf} and Q_{lf} represent the active and reactive powers drawn by the load at bus f
_{} 
(14) 
_{} 
(15) 
P_{f}^{inj} and Q_{f}^{inj} are active and reactive power injections at bus f, G_{f} and B_{f} are transfer conductance and susceptance between buses f and i respectively.
The power balance inside the induction machine is represented by (16).
_{} 
(16) 
Finally, the modified power flow equations can be solved with the NewtonRaphson method by using equation (17).
_{} 
(17) 
Problem formulation
The OPF is a mathematical optimization problem set up to minimize a multiobjective function subject to equality and inequality constraints.
Objective Functions
The EED problem is to minimize two competing objective functions, fuel cost and emission, while satisfying several equality and inequality constraints. It can be considered as a nonlinear multiobjective problem (MOP). The objectives functions are [9,15].
· Fuel Cost Function
_{} 
(18) 
where a_{i}, b_{i} and c_{i} are the cost coefficients of the ith generator and N_{g} is the number of generators committed to the operating system. is the output power of the ith generator.
· Emission Function
_{} 
(19) 
where α_{i}, β_{i}, γ_{i}, ξ_{i} and λ_{i} are the emission coefficients of ith generator
Problem constraints
In this manuscript, the equality and inequality constraints of the problem are as follows.
· Production capacity constraints
The generated real power of each generator at the bus i is restricted by lower limit p_{g}^{max} and upper limit p_{g}^{max} :
_{} 
(20) 
· Active power loss constraint:
Active power losses p of the transmission and transport lines, are positives:
_{} 
(21) 
· Load flow constraints
_{} 
(22) 
_{} 
(23) 
where P_{Gi} and Q_{Gi} are generated real and reactive powers at bus i, respectively. P_{Di} and Q_{Di} are respectively, real and reactive power loads at bus i. P_{i} and Q_{i} are respectively the injected real and reactive power flow at bus_{ }i
· Line flow constraints :
This constrains can be described as:
_{} 
(24) 
where P_{l} the real power flow of line_{}. P_{l}^{max} is the power flow up limit of line l and N_{L} is the number of transmission lines.
The MOPSO technique
This approach is populationbased, it uses an external memory, called repository, and a geographicallybased approach to maintain diversity. MOPSO is based on the idea of having a global repository in which every particle will deposit its flight experiences after each flight cycle. The general algorithm of MOPSO can be described as follows [11]:
Step 1: Initialize an array of particles with random positions POP and their associated velocities VEL.
Step 2: Evaluate the fitness function of each particle.
Step 3: Store the positions of the particles that represent nondominated vectors in the repository REP.
Step 4: Generate hypercubes of the search space explored so far, and locate the particles using these hypercubes as a coordinate system.
Step 5: Initialize the memory of each particle.
Step 6: Compute the speed of each particle using the following expression:
_{} 
(25) 
where φ_{1} and φ_{2} are weights affecting the cognitive and social factors, respectively. r_{1} and r_{2} are random numbers in the range [01]. χ is the constriction factor that ensures convergence which is calculated as in (26)
_{} 
(26) 
where 0<k<1 and ϕ= φ1 +φ2 
(27) 
PBEST(i) is the best position that the particle i has had; REP(h) is a value that is taken from the repository; the index h is selected by applying roulettewheel selection
Step 7: Update the position for each particle
POP(i)=POP(i)+REP(i) 
(28) 
Step 8: Maintain the particles within the search.
Step 9: Evaluate each of the particles in POP.
Step 10: Update the contents of REP together with the geographical representation of the particles within the hypercubes.
Step 11: Update the particle’s position using Pareto dominance.
Step 12: Repeat Step 611 until a stopping criterion is satisfied or the maximum number of iterations is reached.
Results and Discussion
The effectiveness of the proposed algorithms is tested using IEEE 30 bus system including wind farms comprising ten wind generators. Data and results of system are based on 100 MVA. Bus 30 is the slack bus. The test system data can be found in [16].
The values of fuel cost and emission coefficients corresponding to the generators, G_{i} are shown in [17]. The bounds of generated powers are: P_{gi}^{min}=0.05 p.u. and P_{gi}^{max}=1.5 p.u.
The initial value for the slip of the induction generator to execute simulations is given by s(0)=s_{nom}/2, where s_{nom}=0.005 The value of fixed capacitors installed at each wind generator is 30% of rated power. The induction generator circuit parameters are given in [12].
Power flow of base case
Table 1 shows the voltage magnitudes and angles given by the power flow program for the system without and with wind farm. However, slip, active and reactive powers given by ten SR_FSWG is also the outputs of power flow program of the system with wind farm.
The results assuming that wind speed is V_{w}=10 m/s at all wind farms. The active power requested (PD) is _{}.
The convergence characteristic, of the power flow program without and with wind farm is given in Figure 2.
Table 1. Solution of the power flow program for the base case
Bus No 
Without wind farm 
With wind farm 

_{}[pu] 
_{}[Degree] 
_{}[pu] 
_{}[Degree] 

1 
0.9568 
18.4720 
0.9569 
11.5578 
2 
0.9697 
17.5551 
0.9698 
10.6411 
3 
1.0067 
11.9744 
1.0105 
5.7416 
4 
0.9878 
16.1597 
0.9880 
9.2461 
5 
0.9608 
17.1391 
0.9617 
9.7909 
6 
0.9792 
16.6855 
0.9801 
9.3381 
7 
0.9796 
17.0775 
0.9822 
9.0301 
8 
0.9920 
17.1170 
0.9955 
8.5782 
9 
0.9935 
16.7448 
0.9959 
8.6576 
10 
0.9930 
16.7642 
0.9954 
8.6738 
11 
1.0028 
17.1434 
1.0057 
8.8268 
12 
0.9992 
17.7798 
1.0022 
9.2322 
13 
1.0002 
17.6750 
1.0033 
9.0559 
14 
1.0047 
16.4141 
1.0072 
8.2195 
15 
1.0133 
16.2660 
1.0171 
7.5764 
16 
1.0078 
16.8697 
1.0121 
7.9755 
17 
1.0133 
16.7887 
1.0189 
7.7447 
18 
1.0293 
15.8452 
1.0351 
6.6557 
19 
1.0064 
16.2977 
1.0087 
8.2003 
20 
1.0264 
14.5852 
1.0290 
6.4553 
21 
1.0025 
13.1126 
1.0055 
7.1430 
22 
1.0113 
11.3614 
1.0162 
5.3686 
23 
1.0169 
9.6984 
1.0251 
4.5893 
24 
1.0245 
8.0293 
1.0318 
3.7798 
25 
1.0710 
15.8452 
1.0710 
4.1010 
26 
1.0820 
14.5852 
1.0820 
4.5152 
27 
1.0100 
12.0944 
1.0100 
5.5252 
28 
1.0100 
14.3647 
1.0100 
8.5163 
29 
1.0450 
5.5222 
1.0450 
2.3737 
30 
1.0600 
0 
1.0600 
0 
s 
 
0.0029 

10.P_{w}_{ }[MW] 
 
6.3291 

10.Q_{w}_{ }[MVAR] 
 
1.5165 
Figure 2. Convergence criterion of the power flow algorithm
Optimal solutions
In order to demonstrate the effectiveness of the MOPSO to solve the EED problem a compromise with two multiobjective evolutionary algorithms (MOEA) such as NSGA and SPEA [10] has been done in this study. Two cases without and with wind farm have been considered. The convergence of objective functions and Pareto optimal fronts are given respectively in Figure 3 and 4.
· Without wind farm
Table 2.The best solution without wind farm

without wind farm 

Best cost 
Best Emission 

NPGA 
SPEA 
MOPSO 
NPGA 
SPEA 
MOPSO 

cost [$/h] 
620.46 
619.60 
607.52 
657.59 
651.71 
644.33 
Emission[ton/h] 
0.2243 
0.2244 
0.2198 
0.2017 
0.2019 
0.1942 
Pg1 [pu] 
0.1127 
0.1319 
0.1117 
0.4753 
0.4419 
0.4110 
Pg2 [pu] 
0.3747 
0.3654 
0.3097 
0.5162 
0.4598 
0.4583 
Pg3 [pu] 
0.8057 
0.7791 
0.5954 
0.6513 
0.6944 
0.5438 
Pg4 [pu] 
0.9031 
0.9282 
0.9778 
0.4363 
0.4616 
0.3933 
Pg5 [pu] 
0.1347 
0.1308 
0.5227 
0.1896 
0.1952 
0.5502 
Pg6 [pu] 
0.5331 
0.5292 
0.3486 
0.5988 
0.6131 
0.5072 
Table 3 gives the best compromise solution extracted using membership functions [10]. It is clear that the MOPSO has the best results compared to NPGA and SPEA.
Table 3. Best compromise solutions without wind farm

without wind farm 

NPGA 
SPEA 
MOPSO 

cost [$/h] 
630.06 
629.59 
616.9529 
Emission[ton/h] 
0.2079 
0.2079 
0.2004 


(a) 
(b) 
Figure 3. Convergence of cost and emission objective functions: (a) without wind form,(b) with wind form


(a) 
(b) 
Figure 4. Pareto front using MOPSO: (a) without wind farm, (b) with wind farm
· With wind farm
In this study, the wind farms comprising ten wind generators is connected in bus 24 of the IEEE 30 bus system . The results of simulation are given in Table 4.
Table 5 gives the best compromise solution. From the results, it can be seen that the fuel cost, is reduced by connect the wind farm.
Table 4. The best solution with wind farm of MOPSO

MOPSO with wind farm 

Best cost 
Best Emission 

cost [$/h] 
594.6563 
630.2102 
Emission [ton/h] 
0.2203 
0.1945 
Pg1 [pu] 
0.1009 
0.3951 
Pg2 [pu] 
0.2963 
0.4431 
Pg3 [pu] 
0.7140 
0.5914 
Pg4 [pu] 
0.9318 
0.3642 
Pg5 [pu] 
0.4335 
0.5226 
Pg6 [pu] 
0.3224 
0.4828 
Table 5.Best compromise solutions with wind farm of MOPSO
Cost [$/h] 
603.0989 
Emission[ton/h] 
0.2014 
Conclusions
A modified NewtonRaphson algorithm for load flow including SR_FSWG is developed. The efficiency of the proposed MOPSO algorithm to solve multiobjective EED problem is verified by comparison with NPGA and SPEA algorithms. IEEE30bus 6generators is considered in simulation results.
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