Thermal Unit Commitment Problem
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1 Thermal Unit Commitment Problem Moshe Potsane, Luyanda Ndlovu, Simphiwe Simelane Christiana Obagbuwa, Jesal Kika, Nadine Padayachi, Luke O. Joel Lady Kokela, Michael Olusanya, Martins Arasomwa, Sunday Ajibola 07 January 2012 (Optimization Group) TUC Problem 07 January / 35
2 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
3 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
4 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
5 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
6 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
7 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
8 Contents Outline 1 Introduction 2 Problem description 3 Constraints 4 Heuristic Solution 5 Deterministic Solution 6 Results 7 Questions (Optimization Group) TUC Problem 07 January / 35
9 Introduction Introduction The industrial problem arises because electricity utilities need a method to determine the most efficient way to meet electricity demand and this is done by solving the unit commitment problem. The TUC Problem involves only thermal units. The TUC Problem is defined as determining the combination of generators and their output levels, in so doing to minimise the cost of generation while meeting the constraints. The load over a certain time horizon is given and we chose some effective methods of meeting this load with the required spinning reserve. (Optimization Group) TUC Problem 07 January / 35
10 Problem Description Problem Description min x F (P i,t, U i,t ) = i I C i,t (P i,t, U i,t ) i T s.t. P i,t = L t, t T, i I U i,t P i L t + R t t T, i I P i,t Π(i, t), t T, t T, Ui, t (0, 1), P i,t R (Optimization Group) TUC Problem 07 January / 35
11 Constraints Constraints 1 Maximum and Minimum Power generation. 2 Minimum Up Time. 3 Minimum Down Time. 4 Shut Down Cost. Maximum and Minimum power generation P min i P i (t) P max (t), U i (t) > 0, P i (t) = 0, U i (t) < 0 t = 1,..., T, T is the time horizon committed. i = 1,..., I, I is the number of units. (Optimization Group) TUC Problem 07 January / 35
12 Constraints The minimum and maximum power generation reduces to: U i,t Pi min (t) P i (t) U i,t Pi max (t), U i (t) > 0, where U i,r < 0; U i,t [0, 1] Example: When the units are off then, U i,t = 0. Resulting in 0 P i,t 0 Which simply means P i,t = 0. When the units are on, U i,t = 1 P min i (t) P i (t) Pi max (t) (Optimization Group) TUC Problem 07 January / 35
13 Minimum Up time Constraints This constraint signifies the minimum time for which a committed unit should be turned on. Note : once the unit is running, it cannot be turned off immediately. U i,t U i,r U i,r 1, where r = t τ + + 1,..., t 1, t T, i I. t = 1,..., T, T is the time horizon committed. i = 1,..., I, I is the number of units. (Optimization Group) TUC Problem 07 January / 35
14 Constraints Minimum Up time continued Example: Assuming that we have t = 5, minimum up time(mu)=3, then, r = 3.4. First units binary variable: { U1,5 U U 1,t = 1,3 U 1,2 U 1,5 U 1,4 U 1,3. Second unit variable: U 2,t = { U2,5 U 2,3 U 2,2 U 2,5 U 2,4 U 2,3. Thus such observation shows the general behaviour to be: { Ui,5 U U i,t = i,3 U i,2 U i,5 U i,4 U i,3. (Optimization Group) TUC Problem 07 January / 35
15 Constraints Minimum up time continued Figure below goes further to show the period that the possible combinations of minimum up time are Figure: Period of possible combinations (Optimization Group) TUC Problem 07 January / 35
16 Constraints Minimum Down Time This constraint signifies the minimum time for which a de-committed unit should be turned off. Note : Once the unit is de-committed, there is a minimum time before it can be recommitted. 1 U i,t (1 U 1,r ) (1 U 1,r 1 ) U i,t U i,r 1 + U i,r U i,r 0; U i,t [0, 1] where r = t τ + + 1,..., t 1, t T, i I. t = 1,..., T, T is the time horizon committed. i = 1,..., I, I is the number of units. (Optimization Group) TUC Problem 07 January / 35
17 Minimum Down Time Continued Assuming that we have, t = 4, minimum down time(md) = 2. From the above we get that, r = 3. Observing the first unit binary variable: Second variable: U 1,t = {U 1,4 1 U 1,2 U 1,3 U 2,t = {U 2,4 1 U 2,2 U 2,3 (Optimization Group) TUC Problem 07 January / 35
18 Minimum Down Time Continued Minimum Down Time Continued Thus such observation shows the general behaviour to be: U 2,t = {U i,4 1 U i,2 U i,3 Figure below goes further to show the period that the possible combinations of minimum down time are Figure: Period of possible combinations (Optimization Group) TUC Problem 07 January / 35
19 Minimum Down Time Continued Shut Down Cost Objective function was, F (P i,t, U i,t ) = C i,t (P i,t, U i,t ) i I i T F (P i,t, U i,t ) = C i,t (P i,t, U i,t ) + SD i,t i I i T { 0, if not shut down SD i,t = SD cost, if shut down. t = 1,..., T, T is the time horizon committed. i = 1,..., I, I is the number of units. (Optimization Group) TUC Problem 07 January / 35
20 Heuristic Solution Heuristic Approach (Optimization Group) TUC Problem 07 January / 35
21 Available Methods Heuristic Solution Dynamic programming Benders decomposition mixed integer programming Lagrangian relaxation Simulated annealing Tabu search The high dimensionality and combinatorial nature of the unit commitment problem curtail attempts to develop any rigorous mathematical optimization method capable of solving the whole problem for any real-size system. (Optimization Group) TUC Problem 07 January / 35
22 Heuristic Solution Lagrangian Relaxation Algorithm Why choose the LR algorithm 1 Specific for the UCP. 2 Flexible in dealingg with different types of constraints. 3 Flexible to incorporating additional coupling constraints that have not been considered so far. 4 Flexible because no priority ordering is imposed 5 Computationally much more attractive for large system since the amount of computation varies with the number of units (Optimization Group) TUC Problem 07 January / 35
23 Heuristic Solution How the algorithm works: The problem has three components; 1 Cost function 2 Set of constraints involving a single unit 3 Set of coupling constraints, one for each hour in the study period involving all unit. (Optimization Group) TUC Problem 07 January / 35
24 Heuristic Solution An approximation solution can be obtained by adjoining the coupling constraints onto the cost using lagrangian multipliers. The cost function (primal objective function) of the UCP is relaxed to the power balance and the generating constraints via two sets of lagrangian dual function. The dual problem is the decoupled into small subproblems which are solved separately with the remaining constraints. Meanwhile the dual function is maximized with respect to the lagrangian multipliers usually by a series of iterations. (Optimization Group) TUC Problem 07 January / 35
25 Now Heuristic Solution 1 loading contraints N Pload t Pi t Ui t = 0 i=1 2 unit limits Ui t Pi min Pi t Ui t Pi max 3 unit minimum up- and down-time constraints (Optimization Group) TUC Problem 07 January / 35
26 Objective function Heuristic Solution min x F (P i,t, U i,t ) = i I C i,t (P i,t, U i,t ) i T s.t. P i,t = L t, t T, i I U i,t P i L t + R t t T, i I P i,t Π(i, t), t T, t T, Ui, t (0, 1), P i,t R The procedure attempts to reach the constrained optimum by maximizing the lagrange multipliers, while minimizing with respect to the other variables in the problem. That is (Optimization Group) TUC Problem 07 January / 35
27 Heuristic Solution This is achieved in two basic steps q (λ) = max q(λ)] λ t whereq(λ) = min L(P, U, λ). (1) Pi t,ut i 1 Find a value for each λ t which moves q(λ) towards a larger value 2 Assuming that the λ t found are now fixed, find the minimum of L by adjusting the values of P t and U t. (Optimization Group) TUC Problem 07 January / 35
28 Heuristic Solution We rewrite the objective function by taking the coupling constraints and adding them into the objective function to come up with the lagrangian function L = i,t C i,t (P i,t, U i,t ) + t λ t (L t i P i,t ) + t u t (L t + R t i U i,t P i ) Now drop the constant terms, thus the equation above simplifies to L = i,t C i,t (P i,t, U i,t ) t λ t i P i,t t u t i U i,t P i ). = i ( t C i,t (P i,t, U i,t ) t λ t i P i,t t u t i U i,t P i ) λ(t) demand lagrange multiplier U(t) - spinning reserve langrange multiplier (Optimization Group) TUC Problem 07 January / 35
29 Inner System Heuristic Solution Low-level: i = 1, 2,..., I min L P i,t,u i,t (2) with, L = i ( t C i,t (P i,t, U i,t ) t λ t i P i,t t u t i U i,t P i ) (3) subject to, P i,t (i, t) U u,t [0, 1]. thus L i (λ, u) is the optimal lagrangian for low level with given and u. (Optimization Group) TUC Problem 07 January / 35
30 Deterministic Solution Deterministic Method Branch and Bound Method (Optimization Group) TUC Problem 07 January / 35
31 Deterministic Solution What is a deterministic solution? One which guarantees the optimal solution The current state of the solution determines the next state It is a more reliable method (Optimization Group) TUC Problem 07 January / 35
32 Deterministic Solution Some general solution methods considered for solving MIQPs Benders Decomposition Outer Approximation Lagrangian Decomposition Branch and Bound Method (Optimization Group) TUC Problem 07 January / 35
33 Deterministic Solution Why Branch and bound? BB algorithm searches the complete space for optimal solution For a convex problem, the convergence to a global optimum can be proved Can be used for general discrete and continuous problems Most popular in Optimization Literature (Optimization Group) TUC Problem 07 January / 35
34 Deterministic Solution Figure: BB Tree (Optimization Group) TUC Problem 07 January / 35
35 Deterministic Solution General Procedure Choosing the branching variable: Randomly Choosing a value U from the continuous relaxation closest to an integer Bounding: Lower Bound- Continuous relaxation of the objective function Upper Bound-Using a heuristic (Optimization Group) TUC Problem 07 January / 35
36 Deterministic Solution General Procedure 3 rules for fathoming the nodes: If the problem is infeasible If the lower bound of node A is greater than or equal to the upper bound of node B The solution is an integer Stopping Condition: ub lb < ɛ When all the nodes have been fathomed (Optimization Group) TUC Problem 07 January / 35
37 Constraints Deterministic Solution Maximum and minimum power generated The power generated while the machine is switched on must satisfy the load The maximum power while the machine is switched on must exceed the sum of the load and the reserve at each time period (Optimization Group) TUC Problem 07 January / 35
38 Deterministic Solution Figure: Time against Units (Optimization Group) TUC Problem 07 January / 35
39 Deterministic Solution Figure: Power against Period (Optimization Group) TUC Problem 07 January / 35
40 Remarks Deterministic Solution In this instance, the bigger generators were utilized first In reality, a combination of both big and small generators will ensure efficiency (Optimization Group) TUC Problem 07 January / 35
41 Q and A Thank You!!! Any Questions? (Optimization Group) TUC Problem 07 January / 35
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