Synthesis and optimization of utility system using parameter adaptive differential evolution algorithm☆

Zeqiu Li,Wenli Du,Liang Zhao*,Feng Qian*

Key Laboratory of Advanced Control and Optimization for Chemical Process,Ministry of Education,East China University of Science and Technology,Shanghai 200237,China

Keywords:Utility system synthesis MINLP Operation optimization

ABSTRACT Synthesis and optimization of utility system usually involve grassroots design,retro fitting and operation optimization,which should be considered in modeling process.This paper presents a general method for synthesis and optimization of a utility system.In this method,superstructure based mathematical model is established,in which different modeling methods are chosen based on the application.A binary code based parameter adaptive differential evolution algorithm is used to obtain the optimal configuration and operation conditions of the system.The evolution algorithm and models are interactively used in the calculation,which ensures the feasibility of configuration and improves computational efficiency.The capability and effectiveness of the proposed approach are demonstrated by three typical case studies.

1.Introduction

Utility system is an essential part of processing industries and satisfies the utility energy demand of production process.The utility energy cost of a large processing industry is usually large.A lot of utility energy cost can be saved by synthesis and optimization of the utility system.Mathematical optimization models based synthesis and operation optimization of utility system have been studied extensively over the last decades.Most of these models rely on specified performance models of units,which are developed by researchers and are usually different under different considerations.This prevents their further applications in industry.

Mathematical optimization models were based on linear programming(LP)formulation[1]at first.Papoulias and Grossmann[2]presented a mixed integer linear programming(MILP)approach,which allows the optimization procedure to select the optimal configuration of the system by using linear units model and objective function.The LP/MILP method ensures the global optimum of the solution.However,a lot of process characteristics are not linear.Bruno et al.[3]proposed a rigorous mixed integer nonlinear programming model(MINLP),in which nonlinear equations are extensively used for cost of equipment and plant performance.This method gives a more accurate and general method for optimal synthesis and operation of utility plant.Although the authors have indicated that this method can be used to optimize existing systems,there will be a large deviation due to a lot of simplification on unit models.Some researchers have developed more practical models[4,5],which extract the information from the field data.These methods are more suitable for optimizing existing systems.

Due to the advantage of global searching,differential evolutionary(DE)algorithm has received great attention in solving MINLP problems.Lin et al.[6]developed a co-evolution DE algorithm,which consists ofan integer variable evolution and a continuous variable evolution.DE algorithm was extended to optimize mixed integer engineering problems by handling the integer variables the same as continuous variables[7–10],and some integer approximation mechanisms were performed for the function evaluation step.Srinivas and Rangaiah[11]proposed a DETL method,which integrates a tabu list into DE to improve its computational efficiency.Recently,Datta and Figueira[12]developed a real integer discrete coded DE,which can handle the integer and discrete variables of a problem without any conversion.Its successful application to mechanical design motivates the application in chemical synthesis and optimization.

In present paper a general method for performing synthesis and optimization of a utility system is proposed.In this method,superstructure based mathematical model is established,in which different modeling methods are chosen based on the application.A binary code based parameter adaptive differential evolution algorithm is performed to obtain the optimal configuration and operation conditions of the system.The evolution algorithm and models are interactively applied in the calculation,which ensures feasible configuration and improves the computational efficiency.The capability and effectiveness of the proposed approach are demonstrated by two design and one operation optimization problems.

2.Utility System Modeling

Synthesis and optimization of utility system are to obtain the optimal equipment configuration and their corresponding operating conditions satisfying the demands of electricity,mechanical power and steam at various pressure levels.The objective is usually to minimize the total annual cost,energy lost or pollution parameter.The total annual cost is more important since it can balance the capital and operation costs.The problem is usually solved by optimizing the structural topology and parameter of a super-structural model.

The structural optimization is to determine the optimal equipment location from a set of candidates,including the selection of optimal size,type and level of corresponding unit.The operation optimization is to determine the optimal operation conditions for a given system configuration.In the view of mathematics,structural changes are usually discrete decisions while operation changes are usually continuous in applications.However,these two types of variables are often interrelated and interactive.For the synthesis and operation optimization of a utility system,the super-structural design will introduce some integer variables(binary variables)in the optimization model.Hence,the over all utility system synthesis and optimization are a mix-integer programming problem with a sequence of constraints representing system characters and feasible operating ranges.

2.1.Structural representation

The structural representation of utility system used in this paper is similar to that in Bruno et al.[3]and Li et al.[5].The main conventional utility plant equipment includes boiler,steam header,gas turbine,steam turbine,and letdown valve.The steam is the main carrier of energy transmission and distribution and used in a cycle.In the previous system[3],the electricity demands are supplied by an electric generator driven by a gas turbine or a steam turbine.Different from this self sufficient system,an external electric line is added in present work and the electricity demands can be satisfied by external or internal electric line,which improves the system optimization.The structural representation is shown in Fig.1.In this system steam can be generated in three different boilers:(1)heat recovery steam generator,recovering the heat in gas turbine or furnace exhaust gases;(2)fuel boiler,generating different level pressure steam by burning fuel;(3)waste heat boiler,recovering heat from process units such as chemical reactors.Then,the steam is collected by steam header and distributed to steam consumers,such as steam turbines,letdown valves and production process.The external and internal power demands can be satisfied by gas turbines and steam turbines.Gas turbines are usually used to constitute a cogeneration system,which is an energy efficient system.Steam turbines are usually used to supply mechanical power demands or generate electricity for electric motor.

2.2.Application-oriented unit models

To establish a model presenting the superstructure diagram,basic units should be modeled first.Application for synthesis and optimization of a utility system usually includes grassroots design,existing system retro fit or operation optimization.For grassroots design,the unit model should be established based on theory or experimental characteristics.For retro fit and optimization of existing system,a more actual unit model consistent with operation characteristic is preferred.Hence,each unit model is divided into two types.One is mechanism or experimental model for candidate units,another is actual model for existing unit in industrial field.Proper type is selected and used in synthesis and optimization process according to the application.The unit models are described as follows.

2.2.1.Steam turbines

In a large chemical plant,various steam turbines are inter-connected with each other through steam supply pipe lines as a network.For generality,the model in this paper is for multi-extraction steam turbines,and the simple back pressure and condensing steam turbines are considered as the specific conditions of multi-extraction steams when the number of extraction is zero.The structure diagram of multi extraction steam turbine is shown in Fig.2.

Fig.1.The super-structural diagram of utility system.

Fig.2.The schematic diagram of multi-extraction steam turbine.

2.2.1.1.Model of candidate units.With the law of conservation of energy and mass,the thermodynamic model of a steam turbine can be developed,

where ηisand ηmdenote is entropic efficiency and mechanical efficiency,respectively.The product of efficiencies and ΔHisreflects the real enthalpy change of steam through the turbine.With specifying the efficiencies in Eq.(1),the power generated by steam turbine can be obtained by Eqs.(1)to(3).The mechanical efficiency ηmis usually assumed to be a constant between 0.9 and 1.Although it changes case by case depending on the mechanical behavior of a steam turbine,its influence on the performance of steam turbines is ignored for the candidate units.ηisis set to a design value for grassroots design.Eqs.(1)to(3)constitute a complete candidate steam turbine model for grassroots design.

2.2.1.2.Model of actual units.In practice,there is no isentropic process and the isentropic efficiency changes with operating conditions.The hybrid model of steam turbine proposed by Li et al.[5]is used for retro fit and optimization of existing system.The mathematical formulations of the hybrid model are described as

Eq.(8)is the neural network model established with industrial data.The detailed descriptions of the modeling process can be found in[5].Eqs.(4)to(8)constitute a complete actual unit model of steam turbines for retro fit and optimization of existing system.The hybrid model introduces industrial information into the model.Hence,it can compensate the deviation of theoretical model.

2.2.2.Gas turbine

2.2.2.1.Model of candidate units.For gas turbine screening and system grassroots design,it is assumed that the gas turbine works at full or design load operation.Thus the energy balance and mass balance equations can be used to describe the turbine model.The simplified linearized equations from Shang[13]are adopted here:

By defining fuel-to-air ratio f=and steam-to-air ratio s=,the gas turbine model in Eq.(9)can be used to predict the power generation with specified fuel flow rate and gas turbine size.The parameter values in Eq.(9)are listed in Table 1.

2.2.2.2.Model of actual units.For the optimization of existing sites,ambient temperatures should be considered in the model.Eq.(10)[14]is used here to describe the relationship between rated powers and ambient temperatures.The related coefficients for temperature correction are given in Table 1.

Table 1Parameter values in gas turbine models[13,14]

Eqs.(9)and(10)constitute a complete model of gas turbine for the design and optimization.

2.2.3.Boiler

The steam boiler model presented by Shang[15]is applied here:

where Δhstmdenotes the heat load of steam from feed water temperature to its superheated temperature,a and b are regression parameters,which is related to the type of boiler.The boiler model in Eq.(11)relates the fuel consumption with boiler size Fbs,max,steam load Fbsand its operating conditions.The candidate boiler is assumed to be full load.Parameters a and b are set to 0.0126 and 0.2156,respectively.For existing boiler,parameters a and b are obtained by regression using the process data.

For simplicity,deaerator and condenser are not considered in this paper due to their less influence on the optimization of whole system,and the model of other equipment(including steam header and letdown valve)in the system is assumed to be linear without energy conversion.Hence,simple mass balance equations can describe the processes in these units,which will be described in the calculation part.

2.3.Synthesis and optimization model

By defining a set of binary variables ynfor describing equipment n,with yn=1 for existing equipment and yn=0 for non-existing one,the synthesis and optimization model of the utility system can be formulated as a MINLP problem for representing the superstructure model.

2.3.1.Objective function

There are several synthesis and optimization models for the utility system according to the objective.In this paper,the total annual cost is used as the objective,which includes the capital and operation cost of the utility system.The objective function of Bruno et al.[3]is modified as

where the operating costs include the price of utility consumption:fuel,demineralized water,cooling water,and electricity.The capital cost depends on the equipment selection,procurement and maintenance.Some recommended data and function of capital cost can be found in literature[3].

2.3.2.Mechanical power demand constraints

The mechanical power generated by turbines must be equal to the mechanical power demand in the process.

2.3.3.Heat demand constraints

For each steam header,the steam flow entering the header is equal to the steam leaving the header.For heat demand of each steam level,the total enthalpy supply of its corresponding steam is greater than or equal to the heat demand of the process.

2.3.4.Electricity demand constraints

The total electricity generated by gas turbines and imported from external electric line must be equal to the electricity demand of the process.

2.3.5.Boundary constraints of operation variables

All the operation variables must be in their operating ranges.

2.3.6.Logical constraints

For the mechanical power demand,only one steam turbine can be selected for each demand.

The MINLP model for the synthesis and optimization problem consists of minimizing the objective function(12)subject to constraints(1)to(11)and(13)to(18).

3.Method for Synthesis and Optimization

Differential evolution(DE)is one of the evolutionary algorithms,developed by Storn and Price[16]for solving unconstrained,continuous optimization problems.DE becomes popular in recent years with its successful application in chemical process optimization.The classic DE consists of three main steps:mutation,evaluation,and selection.There exist various DE versions,depending on the selection method for the three individuals used in mutation process and the combination mode of mutated and initial solutions that take place in the crossover step[17].

3.1.Handling integer variables

One of the simplest and most effective methods in dealing with MINLP problems is enumeration method,which converts the optimization of MINLP problems to solving a series of NLP sub-problems.Once the algorithm gets the optimum solution of NLP sub-problems,the enumeration method can obtain the global optimum solution of original problems.However,this method is impractical and less efficient,since the number of integer may be large.Another obvious and straightforward method for DE algorithm in dealing with MINLP problems is to tread discrete variables as continuous variables,while only rounding discrete variables at evaluating the fitness function step.However,this method is not so effective and it is difficult to find global optimum,because the algorithm cannot search adequately in a continuous space since discrete variables evolve at the same time[19].The discrete variable in the optimization models here is binary variables,denoting the selection of set or not.Hence,an independent binary variables mutation method proposed by Datta and Figueira[12]is adopted.The offspring individuals are generated based on a user-defined mutation probability Pm.Then,the mutantagainst the binary variable yjis as follows.

where i,r1,r2 and r3(i≠r1≠r2≠r3)denote the index of target vector,base vector,and two random vectors,respectively,F is the random constant in the range of(0,1),and the value of Pmis set to 15%,as recommended by Datta and Figueira[12].With this binary-coded method,the JADE can be extended for solving the mixed binary variables programming problems,named bcJADE.

3.2.Handling constraints

In utility system optimal synthesis and operation problems,there are logical constraints on the system configuration and equipment selections,and the equality or inequality constraints on the operations optimization.The feasible structure is the precondition of operation optimization.As bcJADE is a general optimizer for mixed binary variables programming problems but not customized for utility system synthesis,the individuals in the generation step could result in violation of logical constraints.Thus the examination of structure feasibility is imposed on each individual generated in the generation step.For the structure infeasible individuals,regeneration is carried out to obtain a structure feasible individual.Then the model and objective function are used to evaluate the solutions.

General constraints are handled by a popular penalty method.For the solution(xj,yj),the constraint violation is defined as

where h and g denote the equality and inequality constraints,respectively,and δ is a positive tolerance value for equality constraints.The constraint violation is multiplied by a penalty weight and added to the objective function.For safety production and operation feasibility,a very high value 1.0×107is used for the penalty weight to make sure to select feasible solutions in prior the selection step.

The overall methodology for the optimal synthesis and optimization of utility system is shown in Fig.3.It consists of two main parts:superstructure based modeling and bcJADE based optimization.The evolution algorithm and model evolution are interactively performed in the calculation,which ensures the feasibility of configuration and improves the computational efficiency.

Fig.3.Flowchart for optimal utility system synthesis and optimization.

4.Case Study

To verify the effectiveness and reliability of the proposed approach,three optimal synthesis and operation optimization problems are studied here.Cases 1 and 2 are taken from Bruno et al.[3],which are two examples for optimal grassroots design.They are used to illustrate the ability of the proposed model to choose the optimal configuration and operation conditions.Case 3 is an application example for the operation optimization of an ethylene plant[5],which is used to show the effectiveness and reliability of the proposed method for operation optimization.

In order to compare the results for these examples with that for original cases,the operating conditions and system demands are the same,as shown in Table 2 for Cases 1 and 2.Although the internal unit models are not described in this paper,the cost of internal units is included in the objective function.The values of parameters in the algorithm for all the cases are shown in Table 3.10 runs are executed for each case and the mean value of the solutions is adopted.A summary of the computational results is shown in Table 4.

Table 2 Utility demands for Cases 1 and Case 2

Table 3 Parameter values in the algorithm for solving three cases

Table 4 A summary of results for Case 1 and Case 2

4.1.Case 1

The aim of this case is to design a utility plant to supply three mechanical power demands,a part of electricity,steam at medium level,and lower pressure.According to the heat and power demands,a three-level steam system is selected.To supply the required mechanical power,any type of steam turbines or electric motors can be selected.To generate electricity,a gas turbine and any type of steam turbines or the external electric line can be selected.It should be noticed that the heat recovery boiler is automatically included if a gas turbine is chosen.The optimization results are shown in Fig.4.The same optimal configuration of the system is obtained.The high pressure steam(HS)is generated by a steam boiler,and the electricity demand is supplied by the HS turbines driving an electric generator,because the heating demands in this case are more than the other demand.Thus this steam turbine for electricity power is conducive to the steam distribution in different steam levels.The mechanical power no.1 and no.2 is also satisfied by steam turbines.The mechanical power no.3 is supplied by electric motor.A letdown valve is also needed for providing the required low pressure heating demand.This option is found more attractive than the single back pressure steam turbine.The annual cost for this optimal utility system is 1.1526×107USD·a?1.Using the proposed model with the optimal configuration and operation conditions in literature[3],a very close total cost,1.1526001× 107USD·a?1,is obtained.

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Fig.4.Optimal design of utility system for Case 1.

4.2.Case 2

This case is used to study the influence of heating demands on the system configuration.The heating demands are lower than those in Case 1 and other power demands are the same,so Case 2 is a retro fitting problem of Case 1.The optimal configuration and operation condition for Case 2 is shown in Fig.5.Since the steam demand is lower,less steam is available to generate mechanical power.Thus a condensing steam turbine is chosen to supply the mechanical power no.1.Notice that the type of steam turbine for mechanical power no.2 is changed.Also,a letdown valve is chosen to supply the lower pressure steam demand.Power demand no.3 is satisfied by external electric line.Then the steam consumption is reduced from84.7 t?h?1[3]to 79.9 t?h?1.The total annual cost is 1.0356×107USD·a?1.Compared to 1.0547×107USD·a?1of the configuration in literature[3],1.91 × 105USD·a?1is saved.This case study shows the capability of the proposed method to choose the optimal devices for power demand and find the optimal operation conditions.The optimization results benefits from the external electric line in superstructure model.

4.3.Case 3

Fig.5.Optimal design of utility system for Case 2.

This case is a real operation optimization problem of a steam system in an ethylene plant,which has been studied by Li et al.[5]using GAMS,a commercial software for mathematical programming and optimization.The optimal solutions are found by CONOPT solver in GAMS.All the main units in this case exist in the plant and the field data can be used in modeling.The same models are used in present work and optimized using bcJADE algorithm.The summary and comparisons of the optimal system are shown in Table 5.The solutions obtained by bcJADE are very close to the optimal solution.As suggested by Lampinen[20],an execution of the evolution algorithm is called global success if the solution[X,Y]is feasible and satisfying f(X,Y)?f(X*,Y*)≤0.0001,where[X*,Y*]is the optimum solution.The success rate(SR),best,mean,worst,and standard deviation(Std)are summarized in Table 6.The best solution obtained by the proposed algorithm reaches the global optimal point.And the best,mean,and Std of the solutions by bcJADE are all better than those by CONOPT,because CONOPT is highly dependent on the initial value.Intelligent optimization algorithms can effectively avoid this problem.Hence,bcJADE has the capability of reaching the area near the optimal solution and presents good performance of stability and success rates.It can be used as an alternative method for solving the synthesis and operation optimization problems of utility systems.

Table 5 A summary of results for Case 3

Table 6 Performance of the proposed algorithm

5.Conclusions

A general synthesis and optimization method using DE optimizer is proposed.Its generality of application is tested by three case studies for grassroots design,system retro fitting and operation optimization.The bcJADE algorithm is used to obtain the optimal configuration and operation conditions of the system.The interaction between the algorithm and model structure in the calculation can effectively ensure the feasibility of configuration and improves the computational efficiency.This method can be used as an alternative method for synthesis and optimization of utility systems.However,to find the optimal solution,bcJADE still needs a large size of population and a larger number of function evolutions.It is the future work to improve the efficiency of evolution algorithm for solving chemical engineering optimization problems.

Nomenclature

E set of extract stream

ed electric power demand,kW

Fbffuel flow rate of boiler,t?h?1

Fbs,maxmaximum steam load of boiler,t?h?1

Festeam flow rate at bleeding point,t?h?1

Fisteam flow rate at inlet,t?h?1

Fosteam flow rate at outlet,t?h?1

gt gas turbine

Hreereal enthalpy of extract steam,kJ?t?1

Hiesideal enthalpy of extraction steam,kJ?t?1

Hiosreal enthalpy of exhaust,kJ?t?1

ΔHisideal enthalpy change,kJ· t?1

ΔHrereal enthalpy change,kJ?t?1

Δhstmsteam heat load of boiler,kJ?t?1

I set of input stream

LHV low heating value,kJ?t?1

md mechanical power demand

Peextraction pressure,MPa

Poexhaust pressure,MPa

Qfheat flow from fuel combustion in a gas turbine,MW

qbfspecific heat load of fuel,kJ?t?1

Sinentropy of input flow rate,kJ?t?1?°C?1

t turbine

Zcaptotal capital cost,CNY