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Multi-period steam turbine network optimisation. Part I:Simulation based regression models and an evolutionary

algorithm for finding D-optimal designs

Tor-Martin Tveit *, Carl-Johan Fogelholm

Energy Engineering and Environmental Protection, Helsinki University of Technology, P.O. Box 4400, FIN-02015 HUT, Finland

Received 18 May 2005; accepted 21 October 2005Available online 2 December 2005

Abstract

In this work, a methodology for building multi-period optimisation model of steam turbine network is presented. The optimi-sation model can estimate and evaluate the effect of changes to the thermal energy demand of processes. The subject is divided intotwo parts. In Part I, a method for finding regression models for steam turbine networks using a simulation model and an evolution-ary algorithm for finding D-optimal designs is presented. In Part II, the method presented in Part I is used to develop and solve amulti-period MINLP model of a steam turbine network in a utility system. There are two major problems that are addressed in PartI; Firstly, the evolutionary algorithm for finding D-optimal design is applied to try to determine which values should be simulated inorder to generate the data for the regression model. Secondly, a theoretical model of steam turbine performance is used to model thefeasible operation of the steam turbine. This is necessary to be able to make an efficient evolutionary algorithm for finding D-opti-mal designs. The use of regression models makes it possible to build optimisation models of utility systems that are compact andtransparent. There is a natural trade-off between the flexibility of a model and the accuracy. The major drawback of the method-ology, is that the models developed must be considered ad hoc-models, and are not as flexible compared to models where all theprocess units are modelled in full detail. An advantage of the methodology developed in this work is that it gives more possibilities offinding an acceptable simplification of the optimisation problem, as the methodology is not bound by a certain set of thermody-namic rules or specific mathematical form of the relations in the models.

Part II demonstrates how the methodology can be applied when building a multi-period optimisation model to estimate and eval-uate how changes to the processes will affect the utility system. 2005 Elsevier Ltd. All rights reserved.

Keywords: Multi-period optimisation; Steam turbine network; D-optimal design

1. Introduction

A common problem in the process industry is to cor-rectly estimate and evaluate how changes to the processwill affect the utility system. Utility systems generatesheat (usually as steam) and electricity, and are an impor-tant and essential part of many industrial processes. A

good utility system will potentially reduce both the neg-

ative environmental impact of the processes and reducethe costs of operation. The operation of steam turbinesis an important part of a utility system, and subse-quently the optimisation of steam turbine networks isan important part of improving the efficiency of the util-ity system. The methodology developed in this work canbe useful when evaluating how process changes willaffect the optimal operation and design of the utility sys-tems. In this work, a methodology for building multi-period optimisation model of steam turbine network is

1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.applthermaleng.2005.10.025

* Corresponding author. Tel.: +358 945 136 35; fax: +358 945 13418.

E-mail address: tor-martin.tveit@hut.fi (T.-M. Tveit).

www.elsevier.com/locate/apthermeng

Applied Thermal Engineering 26 (2006) 9931000

mailto:tor-martin.tveit@hut.fimailto:tor-martin.tveit@hut.fi

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presented. The methodology will be useful for consul-tants and process engineers involved in energy efficiencyin the process industry, as the methodology can be usedwhen evaluating how process changes will affect theoptimal operation and design of the utility systems.The optimisation model can estimate and evaluate theeffect of changes to the thermal energy demand of pro-cesses. The novel part of this work is the combinationof the regression models, mathematical programmingand evolutionary algorithm with physical insight of thesteam turbine behaviour.

The work is based on the earlier work [1,2]. The sub-ject is divided into two parts. In Part I, a method forfinding regression models for steam turbine networksusing a simulation model and an evolutionary algorithmfor finding D-optimal designs is presented.

In Part II, the method presented in Part I is used todevelop and solve a multi-period optimisation modelof a steam turbine network in a utility system. Theoptimisation model is used to model and evaluate howdifferent process changes will affect the performance ofthe utility system.

2. Background

There are traditionally two approaches to find theoptimal operation and to improve utility systems. Oneapproach is the thermodynamic approach, where thethermal efficiency is the focus. See for instance the workby Nishio et al. [3] or by Chou and Shih [4] for examplesof this approach. A problem with the thermodynamicapproach is that the methodologies have difficulties withhandling trade-offs, for example between the investmentcosts and thermal efficiency. The other approach is touse mathematical programming (optimisation), which

can handle trade-offs more efficiently. For an overviewof how optimisation have been applied in process sys-tems engineering see the paper by Biegler and Gross-mann [5]. However, for sufficiently large and complexsystems the mathematical programming approach runsinto problems. Mathematical formulations of processsynthesis problems are often nonlinear and contain inte-ger variables (so-called mixed integer nonlinear pro-gramming (MINLP) problems). MINLP problems canbe solved to a global optimum with algorithms that existtoday. However, for complex problems these algorithms

soon become prohibitively inefficient. Some MINLPproblems are NP-complete (nondeterministic polyno-mial) problems, which means that no efficient algorithmhas been found for these problems. When solving largeproblems of these kinds it is therefore necessary to sim-plify the problems, or to decompose them into smallersub-problems. As a result of the simplifications, themathematical programming problems might not beaccurate enough to be useful in the design of a realplant. A lot of work has been made to overcome theseproblems. Manninen and Zhu [6] and Hostrup et al.[7] reduced the model sizes by using thermodynamicinsights and analysis. Iyer and Grossmann [8] simplifiedthe optimisation problem by using only linear relations.Using only linear relations means that the nonlinearitiesmust be simplified or ignored, which can reduce theaccuracy of the models significantly. Another exampleis the work by Bruno et al. [9]. They developed a rigor-ous MINLP model for the synthesis of power plants.The focus was on the design and involved fixing thesteam pressure levels. One way of formulating optimisa-tion models of steam turbine networks is to use the con-cepts developed by Mavromatis and Kokossis [10,11].They used ideas from pinch and total-site analysis com-bined with the Willans line. The Willans line [12,13] is a

Nomenclature

l l k2cn2=ffiffiffiffiffiffiffiffiffiffi

2Dhsp

n matrix containing all candidate pointsa regression model coefficient

b regression model coefficientC(n,p) the number of combinations of size p from acollection of size n, i.e. Cn;p n!

p!np!c velocity vector (m/s)h specific enthalpy (kJ/kg)k flow-through factorm mass flow (kg/s)n polytropic exponent, number of candidate

pointsp pressure (bar), number of variables in a modelR2 the square of the Pearson correlationS set of all possible model matrices

v specific volume (m3/kg)W work/load (MW)X model matrix

x variabley response variable

Superscripts and subscripts

* optimal0 transpose of a matrixk number of factors in a 2k design0 design statea state before turbine stagex state after turbine stagen normal component of vectors isentropic

994 T.-M. Tveit, C.-J. Fogelholm / Applied Thermal Engineering 26 (2006) 9931000

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model matrix, X, that minimises the determinant of thedispersion matrix:

j X0X1 j minX2S

j X0X1 j 4

where S is the set of all C(n,p) possible model matrices.

Among several exchange algorithms for findingD-optimal designs, Fedorovs algorithm is most known[19]. An overview of some exchange algorithms can befound in the paper by Nguyen and Miller [20]. As thenumber of candidates increases, the number of possiblecombinations becomes prohibitively large to be solvedusing the exchange algorithms, and other strategies forsolving the problem must be applied. Stochastic searchmethods based on natural processes like genetic algo-rithms and simulated annealing have successfully beenapplied to this problem. For instance, Broudiscouet al. [21] used a genetic algorithm for selecting the D-

optimal design, while Duffull et al. [22] used simulatedannealing.In Section 5, an evolutionary algorithm for finding

D-optimal designs for developing regression modelsfor steam turbine behaviour based on simulations ispresented. An important part of the algorithm is themodelling of the feasible region for the steam turbineoperation. In the next section, an mathematical expres-sion of this feasible region is presented.

4. Feasible region for steam turbine operation

For a turbine stage there is a relationship between themass flow, m, the entrance state (e.g. pressure, pa, andspecific volume, va) and the back pressure, px. Eq. (5)shows an expression of this relationship for a fixed bladeconstruction [23].

m

m0

2 l

2pava0

l20pa0va

1 pxpa

n1n

1 px0pa0

n1n

0BB@

1CCA 5

where the subscript 0 refers to the design state, n is thepolytropic exponent and l

k2cn2= ffiffiffiffiffiffiffiffiffiffi2Dhs

p. cn2 is the

normal component of the velocity vector of the steamat the exit, Dhs is the isentropic enthalpy difference forthe stage and k2 is the flow-through factor. Under theassumptions that the polytropic exponent, n, is constantand equal to 1 and l is constant and that the steam isideal steam (i.e. p

ava

= pa0va0), Eq. (5) can for super-

heated steam be written as

m

m0

2 p

2a

p2a0

1 pxpa

21 px0

pa0

20B@

1CA 6

Solving Eq. (6) for pa

gives:

pa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m

m0

2p2a0 p2x0 p2x

s7

From Eq. (7) it is possible to calculate the extraction

pressure in a turbine as a function of the mass flowthrough the next stage. Many processes require thesteam to be above a certain pressure, so the mass flowthrough the turbine stages is bounded from below.The lower bound can be calculated using Eq. (8).

mlower;i maxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2a0;i p2x0;ip2x;i p2a;lower;i

qm0;i

p2a0;i p2x0;i;mi1

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