NLPP.pdf

NLPP USING KUNH TUCKER METHOD Presenter Tempa Zangmo (05210355)  1. Introduction 2. Importance OUTLINE 3. Flowchart 2 4. Numerical problem using KKT 5. MATLAB Code 6. References ...

NLPP USING KUNH TUCKER METHOD Presenter Tempa Zangmo (05210355)  1. Introduction 2. Importance OUTLINE 3. Flowchart 2 4. Numerical problem using KKT 5. MATLAB Code 6. References  o Optimization is a technique to find an alternative with the INTRODUCTION cost effectiveness under the given constraints, by maximizing and minimizing the desired and undesired factors 3 o Maximization is the process to find out maximum results without regard to cost o Optimization problems have maximization and minimization of a function by choosing input values within the allowed sets and calculate the value of function  Nonlinear programming (NLP) involves optimizing (maximizing or minimizing) an objective function subject to equality and/or inequality constraints The Karush-Kuhn-Tucker (KKT) conditions are a set of necessary 4 conditions for a solution in nonlinear programming to be optimal, especially when inequality constraints are involved. These conditions generalize the method of Lagrange multipliers to handle inequality constraints.  The KKT conditions are crucial in solving nonlinear programming IMPORTANCE problems, particularly when inequality constraints are present. They provide a way to determine whether a candidate solution is 5 optimal by checking if it satisfies these conditions. In many cases, the KKT conditions are both necessary and sufficient for optimality.  START Define the Lagrangian function Find Partial Derivatives ALGORITHM 6 Consider different cases Solve for the variables Interpret the results END  Consider a power system with two generators, each with a quadratic cost function for power generation. The objective is to minimize the total cost of generating a specific amount of PROBLEM power while ensuring that the total power produced meets the demand. The power output of each generator is denoted by P1 7 and P2. Generator 1 Cost Function: 𝑪𝟏 𝑷𝟏 = 𝟎. 𝟏𝑷𝟐𝟏 + 𝟖𝑷𝟏 + 𝟓𝟎𝟎 Generator 2 Cost Function: 𝑪𝟐 𝑷𝟐 = 𝟎. 𝟐𝑷𝟐𝟐 + 𝟏𝟎𝑷𝟐 + 𝟑𝟎𝟎  THE GOAL IS TO MINIMIZE THE TOTAL GENERATION COST 𝐶𝑡𝑜𝑡𝑎𝑙 = 𝐶1 𝑃1 + 𝐶2 𝑃2 Constraints: 𝑷 𝟏 + 𝑷 𝟐 = 𝟏𝟓𝟎 𝑴𝑾 Inequality constraint: 𝑷𝟏 ≤ 𝟏𝟎𝟎 𝑴𝑾 Step 1: formulation of the Lagrange function 𝑳 𝑷𝟏 , 𝑷𝟐, 𝝀, µ = 𝑪𝟏 𝑷𝟏 + 𝑪𝟐 𝑷𝟐 + 𝝀(𝑷𝑫𝒆𝒎𝒂𝒏𝒅 −𝑷𝟏 −𝑷𝟐 ) 10 = 0.1𝑃12 + 8𝑃1 + 500 + 0.2𝑃22 + 10𝑃2 + 300 + 𝜆. 𝑃1 +𝑃2 −150 + µ(100 − 𝑃1 ) 𝐿 𝑃1 , 𝑃2, 𝜆, µ = 800 + 8 𝑃1 + 0.1𝑃12 + 10𝑃2 + 0.2𝑃22 + 𝜆. 150 −𝑃1 −𝑃2 + µ(100 − 𝑃1 )  1. Partial derivative with respect to 𝑷𝟏 : Step 2: KKT Conditions 𝝏𝑳 = 𝟖 + 𝟎. 𝟐 × 𝑷𝟏 − 𝝀 − µ = 𝟎 𝝏𝑷𝟏 2. Partial derivative with respect to 𝑷𝟐 : stationary 𝝏𝑳 = 𝟏𝟎 + 𝟎. 𝟒 × 𝑷𝟐 − 𝝀 = 𝟎 𝝏𝑷𝟐 11 3. Partial derivative with respect to 𝜆: 𝝏𝑳 = 𝟏𝟓𝟎 − 𝑷𝟏 − 𝑷𝟐 = 𝟎 𝝏𝝀 4. Partial derivative with respect to µ 𝝏𝑳 = 𝟏𝟎𝟎 − 𝑷𝟏 ≥ 𝟎 𝝏µ 0 (𝐼𝑛𝑎𝑐𝑡𝑖𝑣𝑒 𝐼𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦) Now we solve the three resulting equations Step 3: Case 1: 𝜇 = simultaneously, 𝝏𝑳 1. From = 𝟖 + 𝟎. 𝟐 × 𝑷𝟏 − 𝝀 − µ = 𝟎 𝝏𝑷𝟏 𝝀 = 𝟖 + 𝟎. 𝟐𝑷𝟏 𝝏𝑳 2. From = 𝟏𝟎 + 𝟎. 𝟒 × 𝑷𝟐 − 𝝀 = 𝟎 𝝏𝑷𝟐 12 𝝀 = 𝟏𝟎 + 𝟎. 𝟒𝑷𝟐 3. Equating 𝜆 gives: 𝟎. 𝟐𝑷𝟏 = 𝟎. 𝟒 𝑷𝟐 + 𝟐 4. Simplifying the above equation 𝑷𝟏 = 𝟐𝑷𝟐 + 𝟏𝟎  5. Substitute into the power balance constraint 𝑷𝟏 + 𝑷𝟐 = 𝟏𝟓𝟎 2𝑷𝟐 + 𝟏𝟎 + 𝑷𝟐 = 150 𝑷𝟐 = 𝟒𝟔. 𝟔𝟕𝑴𝑾 𝑷𝟏 = 𝟏𝟎𝟑. 𝟑𝟑𝑴𝑾 (Exceeds Pmax) 13  𝑰𝒇 𝑷𝟏 = 𝑷𝒎𝒂𝒙 = 𝟏𝟎𝟎𝐌𝐖: STEP 4: Case 2: 𝜇 > 0 𝝏𝑳 From : 𝝏𝑷𝟏 (active inequality) 𝝀 + µ = 𝟐𝟖 𝝏𝑳 From 𝝏𝑷𝟐 𝝀 = 𝟏𝟎 + 𝟎. 𝟒 𝑷𝟐 14 Substitute 𝝀: 𝟐𝟖 − µ = 𝟏𝟎 + 𝟎. 𝟒 𝑷𝟐 , 𝑷𝟐 = 𝟒𝟓 + 𝟐. 𝟓µ Using Power balance 𝟏𝟎𝟎 + 𝟒𝟓 + 𝟐. 𝟓µ = 𝟏𝟓𝟎 µ = 𝟐, 𝑷𝟐 = 𝟓𝟎𝑴𝑾 𝝀 = 𝟐𝟖 − µ =26  Optimal Power Output 𝑷𝟏 : 100 MW Optimal Power Output 𝑷𝟐 : 50 MW Costs: 𝑪𝟏 (100) = Nu. 2300 FINAL RESULT 𝑪𝟐 (50) = Nu. 1300 15 Total Cost: Nu. 3600 The optimal solution minimizes the total generation cost to Nu. 3600 while satisfying the demand and the constraints. KKT conditions effectively handle the non- linear optimization problem.  The Karush-Kuhn-Tucker (KKT) conditions are a fundamental set of necessary conditions for solving constrained optimization problems, particularly when dealing with nonlinear programming. The KKT method extends the Lagrangian approach by considering both CONCLUSION equality and inequality constraints. In this problem, the KKT conditions help identify the optimal solution by ensuring that: 16 Stationarity: The gradient of the Lagrangian function with respect to the decision variables is zero. Primal Feasibility:The solution satisfies the original constraints of the problem. Dual Feasibility: The Lagrange multipliers associated with the inequality constraints (\(\mu\)) are non-negative. -.  Complementary Slackness: If an inequality constraint is active, its corresponding Lagrange multiplier is positive (𝜇 > 0). If the constraint is inactive, the multiplier is zero (𝜇= 0). 17 By applying the KKT conditions, we systematically evaluate different scenarios where constraints are either binding or non-binding to determine the optimal solution. This method provides both the optimal decision variables and insights into how constraints impact the objective function, ensuring that the solution is both feasible and efficient within the system's operational limits  V. K. Singh and S. K. Singal, "Optimal operation of run of river small hydro power plant," *Bio Physical Economics and Resource Quality*, vol. 3, pp. 1-11, 2018. S. A. H. Soliman and A. A. H. Mantawy, *Modern Optimization Techniques with Applications in Electric Power Systems*. Berlin, Germany: Springer, 2011. REFERENCES D. P. Bertsekas, "Nonlinear programming," *J. Oper. Res. Soc.*, vol. 48, no. 3, pp. 334-334, 1997. 18 D. P. Kothari, "Power system optimization," in *Proc. 2012 2nd National Conf. on Computational Intelligence and Signal Processing (CISP)*, Mar. 2012, pp. 18-21. [Computerphile], "Kuhn Tucker's Method with Numerical Example | Optimization Techniques," YouTube,Aug. 10, 2023. [Online].Available: https://youtu.be/mHEJ0rQlk_c. THANK YOU