A Simplified Lagrangian Method for the Solution of Non-linear Programming Problem

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1 Chapter 7 A Simplified Lagrangian Method for the Solution of Non-linear Programming Problem 7.1 Introduction The mathematical modelling for various real world problems are formulated as nonlinear programming problem. Transportation problem, manufacturing problem, generation scheduling problem, etc. are modelled as non-linear optimization problem. A simplified solution method for non-linear programming problem is developed in this Chapter. The practical application of proposed approach is illustrated with fixed head hydrothermal power scheduling problem. Optimum scheduling of power plant generation is of great importance for electric utility systems. The hydrothermal scheduling plays an important role in the operation planning of an interconnected power plants. The short term hydrothermal scheduling problem is one of the most important daily activities for a utility company. This problem is concerned with optimization over a time horizon of one day. The operational cost of hydroelectric units is insignificant because the source of hydro power is the natural water resources. The total volume of water to be utilized for the scheduling period (1 day) and power demand in each interval (1 hour) are known with complete certainty. Due to the insignificant operating cost of A part of this chapter has been published in International Journal of Mathematical Modelling & Computations, Article in press. 128

2 hydro plants, the scheduling problem essentially reduces to minimizing the fuel cost of thermal plants constrained by generation limits, available water and the power balance condition for a given period of time. Short term hydrothermal scheduling involves the hour-by-hour scheduling of all generating units on a system to achieve minimum generation cost. The objective of fixed head hydrothermal scheduling is to minimize the fuel cost of thermal generating units by determining the optimal hydro and thermal generation schedule subject to satisfying the constraints. The head of the reservoir is assumed to be constant for the hydro plants having reservoirs with large capacity. The hydraulic constraint imposed in the problem formulation is that the total volume of water discharged in the scheduling horizon must be exactly equal to the defined volume. In fixed head hydrothermal scheduling problem, The fuel cost curve is modelled as a quadratic function of the thermal power generation and water discharge rate is modeled as quadratic function of hydro power generation. The power output of each hydro unit varies with the rate of water discharged through the turbines. Different mathematical methods for hydrothermal scheduling problem have been reported in the literature. Zaghlool and Trutt (1998) have developed an iterative technique based on computing LU factors of the Jacobian with partial pivoting for solving hydrothermal scheduling problem. Rashid and Nor (1991) have proposed a Lagrangian multiplier method for optimal scheduling of fixed head hydro and thermal plants. This method linearizes the coordination equation and solves for the water availability constraint separately from unit generations. Lagrangian relaxation techniques (Yan et al., 1993) have proposed in the literature to solve hydrothermal coordination problem. A Lagrangian relaxation technique (Guan et al., 1995) solves the dual problem of the original hydrothermal coordination problem. However the perturbation procedures are required to obtain primal feasible solution. These perturbation procedures may deteriorate the optimality of the solution obtained. Classical methods such as Newton s method (Zaghlool and Trutt, 1988) and mixed integer programming (Nilsson and Sjelvgren, 1995) are applied to solve the hydrothermal scheduling problems. 129

3 Recently, as an alternative to the conventional mathematical approaches, the heuristic optimization techniques such as simulated annealing (Basu, 2005), genetic algorithm (Sasikala and Ramaswamy, 2010), artificial immune algorithm (Basu, 2011c) have used to solve fixed head hydrothermal scheduling problem. Heuristic methods use stochastic techniques and include randomness in moving from one solution to the next solution. Due to the random search nature of the algorithm, these methods provide feasible optimal solution for the optimization problems. Neural network based approach has been developed for scheduling thermal plants in coordination with fixed head hydro units (Basu, 2003). This method provides near-optimal solution for hydrothermal scheduling problem. Narang et al. (2012) have developed an integrated technique of predator-prey optimization and Powell s method for optimal operation of hydrothermal system. In this hybrid method, predator-prey optimization is used as a base level search in the global search space and Powell s method as a local search technique. In this Chapter, we have formulated a model to study for the solution of nonlinear optimization problem. A new deterministic method based on Lagrangian multiplier is developed applied for the solution of short term fixed head hydrothermal scheduling problem. The scheduling period is divided into a number of subintervals each having a constant load demand. The proposed approach is validated by applying it to three test problems. 7.2 Problem Formulation The mathematical model of the non-linear programming problem is given by Minimize Z = T M t k F i (X Sik ) (7.1) where, T is the number of intervals, M is the set of decision variables, X Sik is the i th decision variable in interval k, t k is the time in interval k and F i (X Sik ) is the cost function in interval k, which is expressed as F i (X Sik ) = a i XSik 2 +b i X Sik +c i (7.2) 130

4 here a i,b i and c i are cost coefficients of i th decision variable subject to M X Sik + N X Hik = X Dk, for k = 1,2,...,T (7.3) where, M +N is the total number of decision variables, X Dk is the given constant during the k th interval. T t k q ik = V i s i, for i = 1,2,...,N (7.4) where, V i s i is constant during the scheduling period and q ik is defined as q ik = d i X 2 Hik +e i X Hik +f i, for i = 1,2,...,N; k = 1,2,...,T (7.5) here d i,e i and f i are constant coefficients. X min Si X Sik X max Si, for i = 1,2,...,M; k = 1,2,...,T (7.6) X min Hi X Hik X max Hi, for i = 1,2,...,M; k = 1,2,...,T (7.7) where, XSi min, XHi min, XSi max variables. and X max Hi X Sik 0, for i = 1,2,...,M; k = 1,2,...,T X Hik 0, for i = 1,2,...,N; k = 1,2,...,T arelowerandupperlimitsofthe i th decision (7.8) 7.3 Method of Solution The Lagrangian function is given by L(x,λ,γ) = [ T M T M t k F i (X Sik ) λ k X Sik + N X Hik X Dk ]+ [ N T ] t k q ik V i +s i i.e., L(x,λ,γ) = T M T M N t k (a i XSik+b 2 i X Sik +c i ) λ k ( X Sik + X Hik X Dk )+ N T ( (d i XHik 2 +e i X Hik +f i ) (7.9) 131

5 The minimum value is obtained by partially differentiating the equation (7.9) with respect to X Sik,X Hik,λ k, and equating to zero. L X Sik = 0 t k (2a i X Sik +b i ) λ k = 0 for i = 1,2,...,M; k = 1,2,...,T (7.10) L X Hik = 0 t k (2d i X Hik +e i ) λ k = 0 for i = 1,2,...,N; k = 1,2,...,T (7.11) M X Sik + L = 0 λ k N X Hik X Dk = 0 for k = 1,2,...,T (7.12) L = 0 T t k q ik V i +s i = 0 for i = 1,2,...,N (7.13) From the equation (7.10), we get therefore, where, A i = 1 2a i and B i = b i 2a i X Sik = λ k 2a i b i 2a i X Sik = λ k A i B i for i = 1,2,...,M; k = 1,2,...,T (7.14) From the equation (7.11), we get therefore, where, C i = 1 2d i and D i = e i 2d i X Hik = λ k 2d i e i 2d i X Hik = λ k C i D i for i = 1,2,...,N; k = 1,2,...,T (7.15) Substituting equations (7.15) and (7.16) in equation (7.12), we get M (λ k A i B i )+ N ( λ kc i D i ) = X Dk 132

6 therefore, λ k = X Dk + M B i + N D i M A i + for k = 1,2,...,T (7.16) N C i Substituting the equation (7.16) in equation (7.15) gives X Hik = C i [ X Dk + M B i + N M A i + N D i C i ] D i Substituting the equation (7.17) in equation (7.13), we get i.e., T e i [ C i t k { d ic 2 i γ 2 i ( ei C i 2C id i d i therefore, d i C 2 i γ 2 i for i = 1,2,..,N; k = 1,2,...,T (7.17) [ ( T C i X Dk + M t k { d B i + ) ] N D 2 i i γ M i A i + D N C i + i γ ( i X Dk + M B i + ] N D i M ) D A i + N i +f i } = V i s i T t k [X Dk + [ M A i + [ ei C i 2C id i d i N [ X Dk + M B i + N M A i + N C i C i D i C i ) [ X Dk + M B i + N M A i + N M B i + ] 2 N D i + ] [ M A i + N C i ] 2 + D i C i ] +(d i D 2 i e i D i +f i ) } = V i s i ] t t k [X Dk + M B i + N D i ]+ ] 2[T(di D 2 i e i D i +f i ) V i +s i ] = 0 for i = 1,2,...,N (7.18) The equation (7.18) is solved by the Newton s method described in section of Chapter 2. The solution of equation (7.18) gives the optimal values of for i = 1,2,...,N. 133

7 7.3.1 Computational procedure The computational procedure for implementing the proposed method for the solution of non-linear programing problem described in Section 7.3 is given by the following steps: Step 1: Calculate the initial decision variables and initial λ k from the following equations: X Sik = X Dk, for i = 1,2,...,M; k = 1,2,...,T M +N X Hik = X Dk, for i = 1,2,...,N; k = 1,2,...,T M +N λ k = 2a i X Sik +b i, for k = 1,2,...,T Step 2: Determine the initial value of using equation (7.11), = λ k 2d i X Hik +e i, for i = 1,2,...,N Step 3: Substitute the initial values of in equation (7.18). Step 4: Calculate the incremental values of using Newton s method. Step 5: Calculate the new value of is equal to old value of plus incremental value of, for i = 1,2,...,N Step 6: Substitute the new values in equation (7.18) and then go to Step 4. This iterative procedure is continued till the difference between two consecutive iterations is less than the specified tolerance. Step 7: Calculate λ k values from the equation (7.16) by substituting the optimal values of. Step 8: Determine the optimal values of decision variables using equations (7.14) and (7.15). Step 9: Calculate the objective function using optimal decision variables. 134

8 7.4 Computational Studies A mathematical approach developed in this Chapter is implemented on three fixed head hydrothermal scheduling problems. The scheduling period is divided into 24 intervals. The program is developed using MATLAB 6.5. The system data (Rashid and Nor, 1991) and results obtained through the proposed approach are given below: Problem-1 This problem consists of one thermal and one hydro plant. The cost equation of thermal plant in dollars per hour ($/hr) is given by F 1 (P T1 ) = PT1k P T1k $/hr where, P T1k is the thermal power generation during k th interval. The rate of discharge of hydro plant is given by q 1k (P H1k ) = PH1k P H1k Mcubicft.perhour where, P H1k is the hydro power generation during k th interval. Total volume of water available for discharge (24 hours) V = M cubic ft. Total spillage s= M cubic ft/24 hours. This short term hydrothermal scheduling problem requires that a given amount of water be used in such a way as to minimize the cost of running thermal plant in dollars ($). The objective is to 24 Minimize F 1 (P T1 ) = ( PT1k P T1k ) $ subject to satisfying power demand in each interval and total water discharge during the entire scheduling horizon should be equal to the given water availability. Problem-2 This problem consists of one thermal and two hydro plants. The cost equation of thermal plant is given by F 1 (P T1 ) = 0.01PT1k P T1k +15 $/hr 135

9 Water discharge equations are q 1k (P H1k ) = PH1k P H1k +0.2 Mcubicft.perhour q 2k (P H2k ) = PH2k P H1k +0.4 Mcubicft.perhour Total volume of water available for Hydro plant 1(24 hours) V 1 = 25 M cubic ft. Total volume of water available for Hydro plant 2(24 hours) V 2 = 35 M cubic ft. Total spillage s 1 = 0.25 M cubic ft/24 hours, s 2 = 0.35 M cubic ft/24 hours. The objective is to 24 Minimize F 1 (P T1 ) = (0.01PT1k P T1k +15) $ subject to satisfying power demand in each interval and total water discharge during the entire scheduling horizon should be equal to the given water availability. Problem-3 This problem has two thermal and two hydro plants. The cost equations of thermal plants in dollars per hour ($/hr) are given by F 1 (P T1 ) = PT1k P T1k +25 $/hr F 2 (P T2 ) = PT1k P T1k +30 $/hr Water discharge equations are q 1k (P H1k ) = PH1k P H1k Mcubicft.perhour q 2k (P H2k ) = PH2k P H1k Mcubicft.perhour Total volume of water available for Hydro plant 1 (24 hours) V 1 = 2600 M cubic ft. Total volume of water available for Hydro plant 2 (24 hours) V 2 = 2200 M cubic ft. Total spillage s 1 = 26 M cubic ft/24 hr, s 2 = 22 M cubic ft/24 hr. The objective is to Minimize F 1 (P T1 )+F 2 (P T2 ) $ where, 24 F 1 (P T1 ) = (0.0025PT1k P T1k +25) $ 136

10 24 F 2 (P T2 ) = (0.0008PT1k P T1k +30) $ subject to satisfying power demand in each interval and total water discharge during the entire scheduling horizon should be equal to the given water availability constraints. The power demands for these problems are given in Tables Results and Discussion The optimal generation schedule of Problem-1 obtained through the proposed method is given in Table 7.1. The solution converged in 4 th iteration. The optimal value of γ = and total amount of water utilized for hydro power generation exactly satisfied water availability constraint. The total fuel cost of thermal power generation is $. The optimal generation schedule of Problem-2 obtained through the proposed method is given in Table 7.2. The solution converged at 4 th iteration. The optimal values of γ 1 = and γ 2 = The total fuel cost of thermal power generation is $. The optimal generation schedule of problem-3 is given in Table 7.3. The solution converged at 4 th iteration. The optimal values of γ 1 = and γ 2 = The total fuel cost of thermal power generation is $. The optimal load distribution of thermal and hydro generating units for problems 1-3 obtained through the proposed method are shown in Figures and the convergence characteristics of the proposed method are shown in Figures , it is seen that the solution converges in 4 th iteration for all three problems. 137

11 Table 7.1 Optimal hydro and thermal generation schedule for Problem-1 Interval Demand Water discharge Hydro generation Thermal generation k P Dk q 1k P H1k P T1k

12 Table 7.2 Optimal hydro and thermal generation schedule for Problem-2 Interval k Demand Water discharge Hydro generation Thermal generation q 1k q 2k P H1k P H2k P T1k

13 Table 7.3 Optimal hydro and thermal generation schedule for Problem-3 Interval k Demand Water discharge Hydro generation Thermal generation q 1k q 2k P H1k P H2k P T1k P T2k

14 Power generation Number of Intervals Figure 7.1 Optimal generations over the scheduling period for Problem-1 Thermal Hydro Power generation Number of Intervals Figure 7.2 Optimal generations over the scheduling period for Problem-2 Thermal Hydro-2 Hydro-1 Power generation Number of Intervals Figure 7.3 Optimal generations over the scheduling period for Problem-3 Thermal-2 Thermal-1 Hydro-2 Hydro-1 141

15 Gamma Number of iterations Figure 7.4 Convergence characteristic of the proposed method for Problem Gamma Number of iterations Gamma-1 Gamma-2 Figure 7.5 Convergence characteristic of the proposed method for Problem Gamma Number of iterations Gamma-1 Gamma-2 Figure 7.6 Convergence characteristic of the proposed method for Problem-3 142

16 7.6 Conclusion A simplified mathematical approach for the solution of non-linear optimization problem is presented in this Chapter. The proposed method is implemented with standard test systems available in the literature. The practical constraints of fixed head hydrothermal scheduling such as water availability constraints, water spillage and load demand in each interval are taken into account in the problem formulation. Numerical results show that the proposed method has the ability to determine the optimal solution which satisfies the equality and inequality constraints and the method requires lesser number of iterations for the convergence. 143

Integrated PSO-SQP technique for Short-term Hydrothermal Scheduling

Integrated PSO-SQP technique for Short-term Hydrothermal Scheduling Shashank Gupta 1, Nitin Narang 2 Abstract his paper presents short-term fixed and variable head hydrotherml scheduling. An integrated