1. Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery Lecture 11

2. Optimization of thermal processes2007/2008 Overview of the lecture Introduction to dynamic programming Multistage decision problems Objective function in multistage problem Transformation of nonserial problems into serial ones Types of multistage decision problems Suboptimization and principle of optimality

3. Optimization of thermal processes2007/2008 Dynamic programming - introduction Route map (find the shortest path from 1 to 6): 1 2 3 4 5 6 9 3 6 21 12 21 9 15 6 9 Greedy algorithm (local optimum solution) Distance: 24 Optimum path Distance: 21 Distance from 4 to 6 Apparently, the greedy algorithm is not the best solution. Let’s find a better way.

4. Optimization of thermal processes2007/2008 Dynamic programming - introduction 1 2 3 4 5 6 9 3 6 21 12 21 9 15 6 9 Combinatoric approach : let’s check all possible paths from 1 to 5 and choose the shortest. Path no. Nodes Distance 1-2-5-6 1 2 3 4 5 6 7 8 1-3-5-6 1-2-3-5-6 1-2-4-6 1-2-3-4-5-6 1-2-3-4-6 1-3-4-5-6 1-3-4-6 21 24 39 45 There can be quite a lot paths in general. Is there a better way?

5. Optimization of thermal processes2007/2008 Dynamic programming - introduction 1 2 3 4 5 6 9 3 6 21 12 21 9 15 6 It turns out that it’s better to start with the analysis from the destination point (contrary to the greedy algorithm discussed at the beginning). Thus, the idea is to start with node 6 and backtrack. Let’s denote: Minimal distance from node m to the destination Distance from node m to n The fundamental relation is: m n n+i... Continue optimal policy 9

6. Optimization of thermal processes2007/2008 Dynamic programming - introduction 1 2 3 4 5 6 9 3 6 21 12 21 9 15 6 Start at the destinationBacktrack to 5 Only one path from 5 to 6 Backtrack to 4 Two paths possible: from 4 to 5 from 4 to 6 Take the minimum In a similiar way, we bactrack to previous nodes. 9

7. Optimization of thermal processes2007/2008 Dynamic programming - introduction 1 2 3 4 5 6 9 3 6 21 12 21 9 15 6 1225 And now, we move forward again. 9

8. Optimization of thermal processes2007/2008 Dynamic programming - introduction 1 2 3 4 5 6 9 3 6 21 12 21 9 15 6 56 And the path is really optimal solution: 5612 No other choice In this approach we don’t have to consider all possible paths. We can rely on the principle of optimality stated by Bellman: An optimal policy has the property that whatever the initial state and initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. This is the foundation of dynamic programming. 9

9. Optimization of thermal processes2007/2008 Multistage decision problems Dynamic programming is a technique devised for optimization of multistage decision problems. It was developed by Richard Bellman in the early 1950s. Richard Bellman (1920-1984) Multistage decision problem – example (serial system) heater reactor Distillation tower temperatureReaction rate Number of trays inflow outflow Choose the optimal values to minimize the cost of the process

10. Optimization of thermal processes2007/2008 Multistage decision problem (initial value problem) n-1n12 i Stage nStage n-1Stage iStage 2Stage 1... Dynamical programming decomposes a multistage decision problem (N- -variable problem) as a sequence of single-stage decision problems (N single-variable problems) The decomposition has to be done carefully, though (see the introduction) Why the stages are labeled in decreasing order? inputoutput

11. Optimization of thermal processes2007/2008 Multistage decision problem (initial value problem) Stage transformation function InputOutput Decision Return Input state variablesOutput state variables Component of the objective function Single-stage decision problem In a sequence of these blocks: Design equations

12. Optimization of thermal processes2007/2008 n-1n12 i... output Objective function in multistage problem Whether a given multistage problem can be solved by dynamic programming depends on the nature of the objective function: Since the method works as a decomposition technique, it requires the separability and monotonicity of the objective function What does it mean?

13. Optimization of thermal processes2007/2008 Objective function in multistage problem Separability means that the objective function may be represented as the composition of the indivual stage returns. For instance: Additive objective function Multiplicative objective function The objective function is said to be monotonic if for all values of a and b that make: the following inequality is satisfied: Fortunetely, this conditions are satisfied is for many practical problems

14. Optimization of thermal processes2007/2008 Conversion to a serial system – steam power plant Pump Feed Water heater Boiler Super heater Steam turbine Electric generator Nonserial system Loop (some steam is taken to heat the feedwater) PumpBoiler and turbine system Electric generator Serial system The new serial system consists of only three components. This procedure can be extended to convert multistage system with more than one loop to equivalent serial system. What is the drawback of this procedure?

15. Optimization of thermal processes2007/2008 Types of multistage decision problems n-1n12i... Given input Initial value problem n-1n12 i... Given output Final value problem For both input and output specified, the problem is called boundary value problem

16. Optimization of thermal processes2007/2008 Statement of dynamic programming problem which optimizes Find and satisfies the design equations To solve this problem with dynamic programming we make use of the principle of optimality and suboptimization.

17. First, we suboptimize the last component, as it has no effect on the previous ones. According to principle of optimality we select x 1 such that R 1 is an optimum for the input s 2 irrespective of what happens to other stages: Optimization of thermal processes2007/2008 Suboptimization Suppose we have a system with three components: 12 3 and the objective is to minimize the function: Design equations Optimum

18. According to the principle of optimality, we can replace the term with the optimum value from the previous suboptimization: Optimization of thermal processes2007/2008 Suboptimization Next we suboptimize the two last components: 12 3 Design equations Optimum Optimization with respect to only one variable – x 2 or

19. Optimization of thermal processes2007/2008 Suboptimization And finally, we suboptimize the three last components: 12 3 Design equations Optimum Using again the principle of optimality: The value of s 4 is known, so this problem can be explicitly solved. Then we can find the remaining decision variables.

20. Optimization of thermal processes2007/2008 Thank you for your attention