Microsoft Word - ETASR_V12_N4_pp8850-8855 Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8850-8855 8850 www.etasr.com Dridi et al.: Application of the Levenberg-Marquardt Algorithm in Solving the Economic Emission … Application of the Levenberg-Marquardt Algorithm in Solving the Economic Emission Dispatch Problem Integrating Renewable Energy Tawba Dridi Faculty of Sciences of Tunis University of Tunis El Manar Tunis, Tunisia tawba_dridi@hotmail.fr Houda Jouini Faculty of Sciences of Tunis University of Tunis El Manar Tunis, Tunisia houda.jouini@gmail.com Abdelkader Mami Faculty of Sciences of Tunis University of Tunis El Manar Tunis, Tunisia abdelkader.mami@fst.utm.tn Abderrahman El Mhamedi IUT of Montreuil, University of Paris 8 Paris, France a.elmhamedi@iut.univ-paris8.fr El Mouloudi Dafaoui IUT of Montreuil, University of Paris 8 Paris, France e.dafaoui@iut.univ-paris8.fr Received: 21 April 2022 | Revised: 16 May 2022 | Accepted: 18 May 2022 Abstract-The Economic Emission Dispatch (EED) is a multi- objective optimization problem that seeks to find the optimal balance between the reduction of the generation costs and the pollutant emissions of power thermal plants while respecting power balance and several operational restrictions. This balance could be carried out by proper scheduling power generation of the committed units to fulfill the power demands considering emissions. This paper presents a novel application of the conventional Levenberg-Marquardt Algorithm (LMA) optimization approach to solve the EED problem with the integration of renewable energy. Wind and solar energy were chosen to be injected into the system’s power balance constraint. The combined EED objective function with Valve Point Effects (VPE) consideration was modeled using price penalty and weight factors. This study showed the effectiveness of the chosen optimization technique and the influence of injecting renewable energy along with traditional power resources on reducing total cost and pollutant emissions. The proposed method was applied to the IEEE 9-bus test system and tested in Matlab. Keywords-economic emission dispatch; Levenberg-Marquardt algorithm; wind; solar; valve point effects I. INTRODUCTION More than 80% of total consumed energy is derived from fossil fuels such as oil, coal, and gas [1]. Fossil fuels are expensive to extract, finite, and their use emits a variety of harmful pollutants [2]. Thermal power plants can respond to sharp and sudden power demands quickly. Economic Dispatch (ED) is a necessary task in fuel-based power production systems. It is a computational optimization problem where the overall required generation is distributed among the commuted thermal power units to minimize the total generation cost, no matter the harmful emissions produced. However, alternative strategies are required to respond to the increasing demands for environmental protection [3]. Economic Emission Dispatch (EED) is one of these strategies that has received considerable attention. EED aims to simultaneously reduce total generation costs and pollutant emissions, such as SO2, NOx, and CO2. Emissions could be considered within the economic dispatch problem by being incorporated as a constraint [4, 5], or as a weighted quantity within the objective function of the problem. Renewable energy is environment-friendly and sustainable with low production cost. Once built, renewable facilities cost very little to operate. However, it is difficult for renewable energy to generate power on the same large scale as fossil fuels. The most common renewable resources are solar and wind. Both these energy sources are intermittent, as they depend on climate parameters such as solar radiation, temperature, and wind speed and they are challenging to schedule [6]. Several tools have been explored considering renewable energy in the EED problem. In [7], EED with solar and wind power was solved using harmony search. The results were carried out for one level of power demand and the influence of integrating renewable energy was studied. Artificial neural networks were applied in [8] to predict and solve EED with the integration of renewable energy, reaching an optimum in real-time. The effectiveness of the PSO-based strategy was shown in [9] by solving the EED problem by considering VPE and including wind energy sources for a 10- unit system. The results were investigated for both cases, with and without wind power penetration. Among the various existing optimization techniques to solve power system problems, this study investigated the use of the Levenberg-Marquardt Algorithm (LMA) [10-12]. LMA is a hybrid numerical optimization approach that uses both the Corresponding author: Tawba Dridi Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8850-8855 8851 www.etasr.com Dridi et al.: Application of the Levenberg-Marquardt Algorithm in Solving the Economic Emission … Gauss-Newton algorithm and the steepest descent search to converge to the optimal solution. This technique is particularly effective in solving systems with non-linear equations [12]. This paper proposes a novel application to LMA for solving the static lossy non-convex multi-objective EED problem with solar and wind power penetration. The results showed the influence of integrating renewable energy to reduce total production costs and pollutant emissions. This method was tested and simulated on an IEEE 9-bus system via the Matlab environment. II. EED FORMULATION WITH WIND AND SOLAR POWER Two objective functions were considered: minimization of the total generation cost and of the NOx emissions of the committed power units. A. Objective Functions The EED problem is defined as a constrained multi- objective optimization problem that minimizes simultaneously total power cost and emissions of pollutant gases while satisfying power balance and operational constraints. 1) Cost Function with VPE The basic form of the thermal generation units’ cost function was approximated by a smooth quadratic function, as in [13]. However, in practice, multiple steam valves exist in large turbines, whose role is to maintain the power balance in thermal units by being opened and closed to reach a specific load. This operation contributes to the non-convexity of the fuel cost function, which is known as Valve-Point loading Effects (VPE) and is modeled as an additional sinusoidal component [14]. The fuel cost curve with VPE is illustrated in Figure 1. Fig. 1. Fuel cost curve with and without valve point effects. The total generation cost of thermal power units can be expressed as: Minimize��� � ∑ ���� �� � (1) where Fr is the total generation cost function, C(PGi) is the cost of the th i generation unit, PGi is the real power output of the th i generation unit, and NG is the number of dispatchable generation units. ���� � � � � �� � � �� � � �� ����� ��� � � � �� �� (2) where ai, bi, and ci are the cost coefficients of the th i generation unit, and di, ei are the valve-point effects coefficients of fuel cost for the th i generation unit. 2) Emission Function NOx is a serious global concern. In this study, it is one of the numerous dangerous gases generated by the production units. The emission function is a quadratic function, described as [15]: Minimize��� � ∑ �� !��� �� � (3) where ET is the total emission function measured in kg/h, and ENOx(PGi) is the NOx emission of the th i generation unit. �� !��� � " � # �� � $ �� � (4) where ai, βi, and γi are the NOx emission coefficients of the th i generation unit. The above bi-objective combined economic emission dispatch problem was converted into a single optimization problem by introducing a price penalty factor. The weighted objective function is represented as: �% � &����� � ℎ�1 � & ����� (5) where ω is the weight factor in the range of [0,1], used to balance cost and emissions. h is the price penalty factor which is the ratio between the maximum fuel cost and the maximum emissions of the corresponding generator [16]: ℎ � )�*�+ ,-. /01.�*�+ ,-. (6) B. Modeling Renewable Energy 1) Solar Power The maximum power a solar panel can deliver (Ps) is determined as [7-8]: �2 � ���341 � ���56 � 56789 : (7) where Ec is solar radiation, Tjref is the reference temperature of the panels in 25°C, and Tj is the cell junction temperature (°C). k1 represents the characteristic dispersion of the panels and is between 0.095 to 0.105, and k2=0.47%/°C is the drift in panels temperature. Including a third parameter P3 in the solar power equation improves the results: �2 � ���341 � ���56 � 56789 :��; � �3 (8) Having only 3 constant factors P1, P2, and P3 and a simple equation, this simplified model can predict the maximum power generated by a group of solar panels, given the panel temperature. A thermal solar power plant is made up of a solar system that generates heat and feeds it to turbines in a thermal cycle to generate electricity. 2) Wind power The power provided by a wind turbine Pw is expressed as [7-8]: �< � � � =�>?@ ;10B; (9) Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8850-8855 8852 www.etasr.com Dridi et al.: Application of the Levenberg-Marquardt Algorithm in Solving the Economic Emission … where A is the area traversed by the wind (m 2 ), ρ is the air density (1.225kg/m 3 ), V is the wind speed (m/s), and CP is the efficiency factor which depends on the wind speed and the architecture of the system. Wind turbines should be built considering a mechanical adjustment to develop nominal power from a nominal wind speed in a way to avoid mechanical overloads. C. Problem Constraints The objective function is minimized under the following operational constraints. 1) Power Balance Constraint The total generation power must meet demand while also accounting for transmission losses. Solar and wind power are injected into the system power balance constraint, where Pren is the sum of solar and wind power and Pd is the power demand. ∑ �� � �78� � �C � �D � 0�� � (10) The transmission lines’ network power loss PL could be calculated by solving the power flow problem [17]. The total real power losses can be calculated using the total net injected real power at all buses using [18]: �D = ∑ EF�G � + G6� − 2G G6 �I�(J − J6)��KF � (11) where Nl is the number of transmission lines, gk is the conductance of the k th line that connects bus i to bus j, Vi is the voltage magnitude at bus i, and δi is the voltage angle at bus i. 2) Real Power Operating Limits To ensure reliable operation, each unit's power generation should be provided concerning its minimal and maximal boundaries. �� � � ≤ �� ≤ �� �M! (12) Using the Lagrange function, the above-constrained optimization problem was converted into an unconstrained problem. The objective function henceforth is expressed as: N(�� ,P) = �% + P∑ (�� + �78� − �C − �D)�� � (13) III. LEVENBERG-MARQUARDT ALGORITHM LMA is a mathematical-based optimization approach based on a combination of the directions of the Gauss-Newton algorithm and the gradient descent search. V(x) is assumed to be the function to minimize considering the parameters vector x. Newton’s update for this vector is [19]: QFR� = QF − 4S�G(Q):B�SG(Q) (14) where S�G(Q) is the Hessian matrix and SG(Q) is the gradient. The function V(x) is assumed to be a sum of squares function: G(Q) = ∑ � �(Q)� � (15) where e(x) is the difference between the target and the network output. Then it can be shown that: ∇G(Q) = U�(Q)�(Q) (16) and S�G(Q) = U�U + V(Q) (17) The Jacobian matrix J(x) is: U(Q) = ⎣⎢ ⎢⎢ ⎢⎢ ⎢⎡ Z8[(!) Z![ Z8\(!) Z!\ . . . Z8^(!) Z!^ Z8\(!) Z![ Z8\(!) Z!\ . . . Z8^(!) Z!^. . . . . . . . . .. . Z8^(!) Z![ Z8^(!) Z!\ . . . Z8^(!) Z!^ ⎦ ⎥⎥ ⎥⎥ ⎥⎥ ⎤ (18) and V(Q) = ∑ � (Q)S�� � � (Q) (19) The Gauss-Newton method assumed that S(x)=0, so the equation becomes: QFR� = QF − 4U�(Q)U(Q):B�U�(Q)�(Q) (20) The Levenberg-Marquardt modification to the Gauss- Newton method is modeled as follows, where the characteristic μk is generally set to 0.01 as a starting point: QFR� = QF − 4U�(Q)U(Q) + bFc:B�U�(Q)�(Q) (21) QFR� = QF − 4d(Q) + bF:B�∇G(Q) (22) This algorithm sets μk at 0.01 as a starting point and then it’s multiplied by 10 whenever a step results in an increased V(x), otherwise, if V(x) decreases μk is divided by 10. To adapt LMA to this problem, Q = 4�� ,P: and G(Q) = N(�e ,P). A. Levenberg-Marquardt Algorithm • Step 1: Read the given data cost coefficients (ai, bi, ci), VPE coefficients (di, ei), NOx coefficients (αi, βi, γi), power demand (PD), and unit’s power limits (�� � �,�� �M!). • Step 2: Read the forecasted wind and solar power (Pren), where Pren<0.3PD. • Step 3: Run power flow analysis and calculate transmission lines' power losses (PL). • Step 4: Set the initial values of the Lagrangian multiplier λ 0 , the active generated power �� f , and the Levenberg- Marquardt characteristic μ0. • Step 5: Calculate the Lagrange function L(PGi, λ). • Step 6: Calculate the Jacobian matrix: U(�e ,P) = ⎣⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎡ gD[g*h[ gD[ g*h\ . . . gD[ g*h^ gD[ gi gD\ g*h[ gD\ g*h[ . . . gD\ g*h^ gD\ gi . . . . . . . . . .. . . . . gD^ g*h[ gD^ g*h[ . . . gD^ g*h^ gD^ gi ⎦⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎤ (23) • Step 7: Calculate the Hessian matrix: d(�e ) = U�(�e )U(�e ) (24) Engineering, Technology & Applied Science Research Vol. 12, No. 4, 2022, 8850-8855 8853 www.etasr.com Dridi et al.: Application of the Levenberg-Marquardt Algorithm in Solving the Economic Emission … • Step 8: Update the power generation following (23): j�e (kl[)P(FR�) m = j �e (k) P(F) m − n� o �e P p (25) • Step 9: Calculate the new value of Lk+1(PGi, λ) for each generating unit. • Step 10: Update the Levenberg-Marquardt characteristic μk: If Lk+1≥Lk set μ=μ×10 else if Lk+1