Systems Optimization. Exercise V

Transcription

1 Fachgebiet Simulation und Optimale Prozesse Fakultät für Informatik und Automatisierung Institut für Automatisierungsund Systemtechnik Systems Optimization Summer Semester 206 Exercise V Dr. Abebe Geletu Mr. Xujiang Huang Technical University of Ilmenau Department of Simulation and Optimal Processes (SOP). Energy optimal building design (adapted from [?]) An architect considers to design a building with a cost optimal heating and cooling. The design specifications are: the building should be rectangular and be partially buried under the ground the total floor space needed is at least 20, 000 m 2 the floor dimensions should not be longer than 50 m the ratio of length to the width of the floor should be.68 each story must be 3.5 m high cost of heating and cooling the exposed areas of the building is 00 $/m 2 the annual heating and cooling energy costs of the exposed areas of the building should not exceed $225, 000 Objective: to determine the dimension of the building with minimum excavation costs. (Excavation costs are assumed to be proportional to the volume of the building below the ground.)

2 n d h l w Table : Design parameters Number of stories Depth of building below ground Hight of building above ground Length of building in plan Width of building in plan 2 Mathematical Model: min {f(n, d, h, l, w) = dlw} (n,d,h,l,w) subject to: d + h = 3.5 n l =.68w 00 (2hl + 2hw + lw) 225, 000 nlw l 50 w 50 n, d 0, h 0, l 0, w 0. (a) Re-write the above optimization problem as an optimization problem involving only the d, h, w as decision variables. (b) Solve the resulting optimization problem using Matlab s fmincon.m, GAMS and compare your results, and determine all design parameters. 2. Optimal Design of a pressure vessel. (Use Matlab and GAMS) A pressure chamber is a closed vessel for transportation and storage of gases and liquids. The pressure in a such chamber can alter during operation is usually much greater than the ambient pressure. Areas of application are of pressure chambers in steam boilers, as hot water tankers, in distillation columns and for the storage of chemicals in chemical plants, in nuclear power plants, for the transport and storage of liquefied gases and petroleum products, as an engine cylinder, etc., The design, manufacture and use of pressure vessels requires high precision and precautions to avoid potential future risks, and for the reduction of manufacturing costs. A not precise designs of pressure vessels can lead to life-threatening and harmful explosions. Therefore, the engineer must make the best design decision. For example, the thickness of the vessel wall and the material costs are considered as decision variables. Problem Formulation (see e.g. [?]) Following the pressure vessel which is capped at both ends by semi-spherical heads. Objective : To minimize the entire material and production costs.

3 3 Figure : Some applications of pressure vessels (Image source: [?]) Figure 2: Design of a pressure vessel (figure taken from [?]) Variablen: x - Outer radius of the hemispherical shells x 2 - Outer radius the cylindrical shell x 3 - Interior radius of the hemispherical and cylindrical shells x 4 - length of the cylinder Objective function: f(x) = x x 2 x x 2 x x 2 x x 2 x 3 Constraints: g (x) = x x 3 0 g 2 (x) = x x 3 0 g 3 (x) = πx 2 3x πx2 3 +, 296, g 4 (x) = x Bounds on the design variables: x x x x Exercises

4 2.. Solve the problem using Matlab 2.a Solve the optimization problem using the function fmincon.m from Matlab s 4 Optimization-Toolbox (see, e.g. [?]). 2.b Display the iteration steps of Matlab (see e.g. [?]) 2.c How many iterations are need to solve the problem? 2.d How big is the CPU time to solve the problem? 2.e How does the number of iteration change when chaining the stopping tolerance? 2.f Which step-length rules are implemented under fmincon.m? 2.2. Solve the same optimization problem using GAMS (see e.g. [?]) and compare your result with one obtained from Matlab. 3. (Adpated from Venkatraman [?]) A transmission system for gas pipeline is given in the figure below. The pressure at pumping stations P and P 2 are given by p and p 2, respectively; r = P P 2 compression ratio, D is the pipe diameter, L is the pipe length. The annual operational costs are given by C(D, p, L, r) = 7.8D 2 p +450, 00+36, 900D ( + r 0.28 ) (Euro/yr). L L Figure 3: Pumping stations The volume flow rate Q (cm 3 /hr)is given by the following equation [ (p 2 Q(D, p, p 2, L, D) = 3.39 p 2 ] )D5, fl with a friction factor of f = 0.008D /3. Find the design parameters in order to guarantee a daily flow rate of 000cm 3 /day at a minimum annual operational costs. 4. (Minimization of the Weight of a Tension-Compression String, see [?]) Consider a tension-compression spring given below. The objective is to minimize the spring weigh under constraints on deflection, shear stress, surge frequency, bounds on outer diameters and design variables. The optimization problem is Figure 4: Pumping stations

5 { min f(x) = (x3 + 2) x 2 x 2 } x subject to: g (x) = x 2x x 4 0, 4x 2 2 g 2 (x) = x x ( x 2 x 3 ) + x4 g 3 (x) = 40.45x x 2 2 x 3 0, 508x 2 0, g 4 (x) = x 2 + x 0, x 2.0, 0.25 x 2.3, 2.0 x (i) Solve this optimization problem using fmincon.m by telling Matalb to compute all gradients for you. (Hint: set the parameters GradObj, off and GradConstr, off ) (ii) Solve the same problem using GAMS and compare the results. 5. (Hanging chain). A chain is suspended from to two hooks at (x 0, y 0 ) and (x f, y f ). The chain consists of a set of links of stiff steel. The total length of the chain is given by L. Figure 5: The hanging chain (image sources [?], [?]) Objective: To find the shape of the chain that guarantees the stable equilibrium of the chain despite external forces and the chain s own weight. Assuming that the chain is represented by a curve, suppose the shape of the chain is given by a function y(x). Hence, we need to find the function y(x) that guarantees the minimum potential energy.

6 Optimization problem: { min J[y] = y xf x 0 } y + [y (x)] 2 dx :Potential energy subject to: xf + [y (x)] 2 dx = L; :total length of hanging chain x 0 y(x 0 ) = y 0, y(x f ) = y f. :boundary conditions 6 (a) Use a finite-difference discretization to transform the problem into a constrained nonlinear optimization problem. (b) Use L = 60 m, (x 0, y 0 ) = (0, 20), and (x f, y f ) = (40, 20) to and solve the discretize problem using fmincon.m. References [] M. A. Bhatti: Practical optimization methods. Springer-Verlag, New York, [2] A. D. Belegundu and T. R. Chandrupatla: Optimization concepts and applciations in engineering (2nd Ed.). Cambridge University Press, 20 [3] L. C. Cagnina, S. C. Exquivle. Solving eignineering optimization problems with simple constrained particle swarm optimizer. Informatica, V. 32, , [4] Coello Coello, C. A. Use of a self-adaptive penalty approach for engineering optimization problems. Computers in Industry, 4(2), [5] Optimization Toolbox T M User s Guide, R204a. The MathWorks, 204. [6] A. Geletu. GAMS - Modeling and Solving Optimization Problems. TU-Ilmenau, Faculty of Mathematics and Natural Sciences, Department of Operation Research & Stochastrics, [7] A. Geletu. Solving Optimization Problems using the Matlab Optimization Toolbox - a Tutorial. TU-Ilmenau, Faculty of Mathematics and Natural Sciences, Department of Operation Research & Stochastrics, [8] Indiamart, Oriental Engineering Co. URL: [9] Plowman Brothers Ltd. URL: [0] [] [2] Singiresu S. Rao: Engineering Optimization Theory and Practice Fourth Edition JOHN WILEY & SONS, INC. [3] P. Venkatraman: Applied optimization with matlab programming. Wiely & Sons, 2009.