Computer Application in Power system: Chapter six - optimization and security
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Chapter six focuses on optimization and security in power systems, highlighting the conflict between minimizing operating costs and ensuring security. It discusses various optimization techniques, including long-term and short-term scheduling, and details the constraints involved in minimizing costs related to power generation. The chapter also emphasizes the importance of security assessments to monitor potential violations and manage system operations effectively.
![FORMULATION OF THE OPTIMIZATION
PROBLEM
With reference to power system operation the optimization
problem consists of minimizing a scalar objective function
(normally a cost criterion) through the optimal control of a
vector [u] of control parameters, i.e.
𝑚𝑖𝑛𝑓( 𝑥 , [𝑢])
Subject to
Equality constraints of the power flow equation
𝑔 𝑥 , 𝑢 = 0
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Computer Application in Power system: Chapter six - optimization and security
- 1. ASTU SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTING DEPT. OF POWER AND CONTROL ENGINEERING COMPUTER APPLICATION IN POWER SYSTEM (PCE5307) CHAPTER SIX OPTIMIZATION AND SECURITY BY: MESFIN M.
- 2. INTRODUCTION To provide a secure energy supply at a minimum operating cost is a very complex process that relies heavily upon on- line computer control. Optimization and security are often conflicting requirements and should be considered together. In an interconnected power system, the objective is to find the real and reactive power scheduling of each power plant in such a way as to minimize the operating cost. 1/13/2020
- 3. CONT.… The aim of optimal power system operation is to try and make the best use of resources subject to a number of requirements over any specified time period. Here are some examples of power system optimization studies, with time scales given in brackets. Long-term scheduling for plant maintenance and availability of resources (months/years) Short-term scheduling for unit commitment (days). Economic allocation of generation base points (minutes) 1/13/2020
- 4. CONT.… The economic criterion which appears to have universal acceptance is that of minimizing production costs of which only those of fuel and maintenance vary significantly with generation output. The security objective determines local plant loading limits. It also imposes limitations on network structures and loading patterns on a system scale which often conflicts with the economic objective. 1/13/2020
- 5. FORMULATION OF THE OPTIMIZATION PROBLEM With reference to power system operation the optimization problem consists of minimizing a scalar objective function (normally a cost criterion) through the optimal control of a vector [u] of control parameters, i.e. 𝑚𝑖𝑛𝑓( 𝑥 , [𝑢]) Subject to Equality constraints of the power flow equation 𝑔 𝑥 , 𝑢 = 0 1/13/2020
- 6. CONT.… Inequality constraints on the control parameters (parameter constraints). 𝑢𝑖,𝑚𝑖𝑛 ≤ 𝑢𝑖 ≤ 𝑢𝑖,𝑚𝑎𝑥 Dependent variables and dependent functions (functional constraints) 𝑥𝑖,𝑚𝑖𝑛 ≤ 𝑥𝑖 ≤ 𝑥𝑖,𝑚𝑎𝑥 ℎ𝑖 𝑥 , 𝑢 ≤ 0 1/13/2020
- 7. CONT.…. The optimal dispatch of real and reactive powers can be assessed simultaneously using the following control parameters: voltage magnitude at slack node voltage magnitudes at controllable P, V nodes taps at controllable transformers controllable power PGi phase shift at controllable phase-shifting transformers other control parameters. 1/13/2020
- 8. CONT.… Let us assume that only part (PGi) of the total net power (PNi) is controllable for the purpose of optimization. The objective function can then be defined as the sum of instantaneous operating costs over all controllable power generation: 𝑓 𝑥 , 𝑢 = 𝑖 𝑐𝑖 (𝑃𝐺𝑖) Where 𝑐𝑖 is the cost of producing 𝑃𝐺𝑖. 1/13/2020
- 9. UNCONSTRAINED PARAMETER OPTIMIZATION The mathematical tools that are used to solve unconstrained parameter optimization problems come directly from multivariable calculus. The necessary condition to minimize the cost function 𝑓 𝑥1, 𝑥2, … , 𝑥 𝑛 is obtained by setting the derivative of 𝑓 with respect to variables equal to zero: 𝜕𝑓 𝑥𝑖 = 0 𝑖 = 1, … , 𝑛 Or ∇𝑓 = 0 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 1/13/2020
- 10. EXAMPLE Find the minimum of 𝑓 𝑥1, 𝑥2, 𝑥3 = 𝑥1 2 + 2𝑥2 2 + 3𝑥3 2 + 𝑥1 𝑥2 + 𝑥2 𝑥3 − 8𝑥1 − 16𝑥2 − 32𝑥3 + 110 Soln. Find the gradient Check the values using the hessian matrix 1/13/2020
- 11. CONSTRAINED PARAMETER OPTIMIZATION; EQUALITY CONSTRAINTS This type of problem arises when there are functional dependencies among the parameters to be chosen. The problem is to minimize the cost function. 𝑓 𝑥1, 𝑥2, … , 𝑥 𝑛 Subjected to the equality constraints 𝑔𝑖 𝑥1, 𝑥2, … , 𝑥 𝑛 = 0 Such problems may be solved by the Lagrange multiplier method. This provides an augmented cost function by introducing k- vector l of undetermined quantities. 1/13/2020
- 12. CONT.…. The constrained cost function becomes. ℒ = 𝑓 + 𝑖=1 𝑘 𝜆𝑖 𝑔𝑖 The resulting necessary conditions for constrained local minima of ℒ are the following. 𝜕ℒ 𝜕𝑥𝑖 = 𝜕𝑓 𝜕𝑥𝑖 + 𝑖=1 𝑘 𝜆𝑖 𝜕𝑔𝑖 𝜕𝑥𝑖 = 0 𝜕ℒ 𝜕𝜆𝑖 = 𝑔𝑖 = 0 1/13/2020
- 13. OPERATING COST OF THERMAL PLANT The factor influencing power generation at minimum cost are operating efficiencies of generator, fuel cost and transmission losses. The most efficient generator the system doesn’t guarantee minimum cost as it may be located in an area where fuel cost is too high Also, if the plant is located far from load center, transmission losses may be considerably higher hence the plant may be overly uneconomical. 1/13/2020
- 14. CONT.…. Hence, the problem is to determine the generation of different plants such that the total operating cost is minimum. The input to the thermal plant is generally measured in Btu/hr, and the output is measured in MW. 1/13/2020
- 15. CONT.…. In all practical cases, the fuel cost of generator i can be represented as a quadratic function of real power generation. 𝐶𝑖 = 𝛼𝑖 + 𝛽𝑖 𝑃𝑖 + 𝛾𝑖 𝑃𝑖 2 An important characteristics is obtained by plotting the derivative of the fuel cost curve versus the real power. This is known as incremental fuel cost curve 𝑑𝐶𝑖 𝑑𝑃𝑖 = 2𝛾𝑖 𝑃𝑖 + 𝛽𝑖 The incremental fuel cost curve is a measure of how costly it will be to produce the next increment of power. 1/13/2020
- 16. ECONOMIC DISPATCH NEGLECTING LINE LOSS The simplest economic dispatch problem is the case when transmission loss is neglected. This problem model doesn’t consider system configuration and line impedance. The model assumes the system is only one bus with all generators and loads connected to it. 1/13/2020
- 17. CONT.…. since transmission loss is neglected, the total demand is the sum of the generation. A cost function Ci is assumed to be known for each plant. The problem is to find the real power generation for each plant such that the objective function (i.e., total production cost) as defined by the equation is min. subjected to the constraint. 𝐶𝑡 = 𝑖=1 𝑛 𝑔 𝐶𝑖 = 𝑖=1 𝑛 𝑔 𝛼𝑖 + 𝛽𝑖 𝑃𝑖 + 𝛾𝑖 𝑃𝑖 2 1/13/2020
- 18. CONT.…. Is to minimize subjected to the constraint. 𝑖=1 𝑛 𝑔 𝑃𝑖 = 𝑃 𝐷 A typical approach is to augment the constraints into objective function by using the Lagrange multipliers. ℒ = 𝐶𝑡 + 𝜆 𝑃 𝐷 − 𝑖=1 𝑛 𝑔 𝑃𝑖 1/13/2020
- 19. CONT.…. The minimum of the constrained function is found at the point where the partials of the functions to its variable are zero. 𝜕ℒ 𝜕𝑃𝑖 = 0 𝜕ℒ 𝜕𝜆 = 0 First condition, 𝜕 𝐶𝑡 𝜕𝑃𝑖 + 𝜆 0 − 1 = 0 Since 𝐶𝑡 = 𝐶1 + 𝐶2 + ⋯ + 𝐶 𝑛 1/13/2020
- 20. CONT.…. Then 𝜕𝐶𝑡 𝜕𝑃𝑖 = 𝑑𝐶𝑖 𝑑𝑃𝑖 = 𝜆 Therefore, the condition for optimum dispatch is 𝑑𝐶𝑖 𝑑𝑃𝑖 = 𝜆 Or 2𝛾𝑖 𝑃𝑖 + 𝛽𝑖 = 𝜆 𝑃𝑖 = 𝜆 − 𝛽𝑖 2𝛾𝑖 1/13/2020
- 21. CONT.…. Second condition 𝑖=1 𝑛 𝑔 𝑃𝑖 = 𝑃 𝐷 Solving for 𝜆 we have 𝑖=1 𝑛 𝑔 𝜆 − 𝛽𝑖 2𝛾𝑖 = 𝑃 𝐷 𝜆 = 𝑃 𝐷 + 𝑖=1 𝑛 𝑔 𝛽𝑖 2𝛾𝑖 𝑖=1 𝑛 𝑔 1 2𝛾𝑖 1/13/2020
- 22. CONT.…. Using iterative method Δ𝜆(𝑘) = Δ𝑃(𝑘) 1 2𝛾𝑖 Where Δ𝑃(𝑘) = 𝑃 𝐷 − 𝑃𝑖 And therefore 𝜆(𝑘+1) = 𝜆(𝑘) + Δ𝜆(𝑘) The process continued until Δ𝑃(𝑘) is less than specified accuracy 1/13/2020
- 23. EXAMPLE The fuel cost function for three thermal plants in $/hr are given by 𝐶1 = 500 + 5.3𝑃1 + 0.004𝑃1 2 𝐶2 = 400 + 5.5𝑃2 + 0.006𝑃2 2 𝐶3 = 200 + 5.8𝑃3 + 0.009𝑃3 2 Where 𝑃1, 𝑃2, and 𝑃3 are in MW. The total load 𝑃 𝐷 is 800MW. Neglecting line losses and generator limits, find the optimal dispatch and the total cost in $/hr, by A. analytical method. B. By iterative Method 1/13/2020
- 24. CONT.…. The economic dispatch with generator limit, the same procedure is followed except checking the limit for the generators. If it exceeds the upper limit fix the generator value at the maximum value and solve for the remaining generators power. Example: do the above example with Pd is at 975 MW and 200 ≤ 𝑃1 ≤ 450 150 ≤ 𝑃2 ≤ 350 100 ≤ 𝑃3 ≤225 1/13/2020
- 25. SECURITY ASSESSMENT The aim of security assessment is: the detection of operating limit violations through the continuous monitoring of power flows, voltages, etc. contingency analysis, a far more demanding task, which first considers all the possible outages in order of severity and uses that information to alter the pre-contingency operating state to try and reduce the effect of the disturbance. 1/13/2020
- 26. END OF CHAPTER SIX Thank You NEXT CHAPTER FIVE 1/13/2020