Modeling and Optimization of the Condensing Steam Turbine Network

Dec 13, 2005 - An online optimization system was developed and applied to the condensing steam turbine network of a chemical plant. First, we develope...
0 downloads 0 Views 472KB Size
Download PDF
670

Ind. Eng. Chem. Res. 2006, 45, 670-680

Modeling and Optimization of the Condensing Steam Turbine Network of a Chemical Plant In-Su Han Department of Chemical Engineering, Pohang UniVersity of Science and Technology (POSTECH), Pohang, Gyeongbuk 790-784, Republic of Korea

Young-Hak Lee Automation and Systems Research Institute and School of Chemical and Biological Engineering, Seoul National UniVersity, Seoul 151-742, Republic of Korea

Chonghun Han* School of Chemical and Biological Engineering and Institute of Chemical Processes, Seoul National UniVersity, Seoul 151-742, Republic of Korea

An online optimization system was developed and applied to the condensing steam turbine network of a chemical plant. First, we developed a hybrid model of the condensing steam turbines with multiple steam injectors, by combining thermodynamic models into support vector machines. The developed hybrid model is capable of predicting the electric power generated by the steam turbines, with prediction errors of 1%2%, and computing several performance indicators, such as the overall efficiency and the power recovery rate. An optimization problem then was formulated by utilizing the developed model to maximize the total electric power recovery from the steam turbine network. Finally, an online optimization system was developed that consists of the optimization engine (to solve the optimization problem), the model manager (to update the models), and the optimization client (to inform the turbine operators of the optimization results). The energy cost has been considerably reduced, because the optimization system was applied to the steam turbine network. The proposed hybrid modeling method can be used to predict the performance and power generation rate of various types of steam turbines in the chemical process industry. 1. Introduction Steam turbines are used to drive electric generators or other rotating machinery (such as compressors, pumps, and fans). The effective operation of the steam turbines is very important, because they are strongly related to the energy usage in a chemical process. Typically, chemical processes are equipped with steam turbines of various types such as back-pressure turbines and condensing turbines. The primary function of a back-pressure steam turbine is to supply lower-pressure steam to other services while extracting mechanical work. On the other hand, that of a condensing turbine is to drive rotating machinery by exhausting steam to vacuum conditions. In a large chemical plant, these steam turbines constitute a network where the turbines are interconnected with one another through steamsupply lines, which make their operations a challenging task. Each steam turbine has a distinct design, indicating different performances in the utility network; therefore, the overall performance of the network relies on the distribution condition of steam to each steam turbine. This situation means that the total power generation of the entire steam turbine network can be maximized by optimally allocating steam to the steam turbines. Many optimization studies1-5 have been performed on the modeling and optimization of utility systems including boilers, turbines, steam headers, compressors, and expanders, which contribute greatly to the total energy usage in a chemical plant. The models in these studies only take into account the * To whom correspondence should be addressed. Tel.: +82-2-8801887. Fax: +82-2-888-7295. E-mail: [email protected].

macroscopic mass and energy balances of utility systems. Rodriguez-Toral et al.6 presented an equation-oriented mathematical model for water, steam, and air streams in utility systems and performed an optimization of a utility system using successive quadratic programming algorithms. Papalexandri et al.7 proposed a multiperiod optimization framework to determine optimal energy management schemes and operating conditions of utility systems under uncertainties of energy demands and efficiencies. They provided the mathematical models derived on the basis of macroscopic conservation laws for a steam turbine and production network. However, they were not able to evaluate the detailed performance such as a change in the efficiency of utility systems according to various operating conditions. Lee et al.8 reported a hierarchical optimization method consisting of top- and bottom-level optimizations for utility plants: the total utility plant is optimized in the top level and the optimal load allocation for boilers and turbines are performed in the bottom level. Recently, Yi and Han9 proposed the integration of complete re-planning and plan repairing, which is constructed by a rule-based system to handle the prediction errors for energy demands during a multiperiod operation. However, most of the previous works have focused on the formulation methods of optimization frameworks, rather than on the modeling of utility systems. In the optimization of utility systems, reliable and accurate models for utility systems are essential to guarantee reliable optimization results. However, the previous works have not taken a serious view of the models of utility systems in their optimization frameworks. In industry, compressors and turbines are usually optimized on the basis of macroscopic mass and energy balances, where the performances

10.1021/ie049425a CCC: $33.50 © 2006 American Chemical Society Published on Web 12/13/2005

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 671

(efficiencies) of compressors and turbines are assumed to be constant or expressed as relatively simple correlation equations, such as a second-order polynomial function of a single variable.10,11 The comparison of a fundamental model and an empirical efficiency curve model for the online optimization of the boiler network has been investigated. The fundamental model requires significant engineering effort in model development and online computation. The use of the empirical efficiency curve model may not lead to the true optimal conditions, because the performances of industrial boiler systems are affected by diverse operating conditions such as fuel heating value, ambient temperature and humidity, burner plugging, and so forth.12 Consequently, reliable and useful models for the utility systems, such as turbines, compressors, and boilers, are relatively few. Recently, Han and Han13 proposed a systematic modeling method for air-compression systems, which are frequently used in chemical plants. For industrial compression systems, modeling for the purpose of measuring and predicting performance is not straightforward, because it always varies, as a result of the changing operation and ambient conditions (such as ambient temperature, relative humidity, molecular weight of the gases into the expanders, intercooling and reheating temperatures, and aging of the components, including impellers, bearings, and casings). Hence, they used thermodynamic models to compute the ideal power generation/consumption rate and artificial neural networks and partial least-squares methods to predict the overall efficiency of a compression system. The modeling method can be applied to find better operating conditions and to use as a core model for the optimization of compression systems. They showed that modeling results were successful when the modeling method was applied to several industrial compression systems. In this study, we present an optimization system based on a hybrid-modeling framework for a steam turbine network and its application to an industrial chemical plant. First, we develop a reliable modeling framework for the condensing steam turbines with multiple steam injectors by combining a support vector machine (SVM)14 as a black-box modeling method into a thermodynamic model that represents steam expansion in a turbine. The SVM method is adopted to predict the overall efficiency and the exhaust pressure of a steam turbine under an operating condition. The thermodynamic model includes the steam property equations that calculate the enthalpy, entropy, and the quality of the steam entering and exiting the steam turbine. To monitor the performance of condensing steam turbines, the major performance indicators, including the overall efficiency and the power recovery rates, are defined in the modeling framework. The optimization problem is then formulated on the basis of the developed modeling framework to maximize the electric power generation of the whole steam turbines of a steam turbine network. A steam turbine optimization system that uses the proposed modeling framework and optimization problem is developed and applied to the steam turbine network of an industrial chemical plant. 2. Condensing Steam Turbines and Their Network Figure 1 illustrates a schematic of a condensing steam turbine with multiple steam injectors. The steam turbine consists of three major components: the turbine itself, the steam condenser, and the electric generator. Several types of steams, such as MLS (medium-low steam), LS (low steam), and LLS (low-low steam), are available for driving the turbines in the steam turbine network shown in Figure 2. The turbine mounts two or three steam injectors, which are designed to supply a different type

Figure 1. Schematic of a condensing steam turbine.

of steam to the turbine blades through which kinetic energy of steam is converted to work. Steams are partially condensed through the turbine blades and then are totally condensed to water in the condenser. A sufficient amount of cooling water should be supplied to condense the steam passing through the turbine casing and thereby maintain a vacuum at the turbine exhaust. The electric generator transforms rotational energy into electricity. Figure 2 shows the steam turbine network of the Samsung Petrochemicals Corp. in Ulsan, Korea. As shown in the figure, the condensing steam turbines are interconnected to each other via steam-supply lines. Various types of steams are generated by extracting heat from the hot-gas streams produced in the reaction sections of the plant. The steams are then stored in the headers called MMS (medium-medium steam), MLS, LS, and LLS headers. Steams are first supplied to satisfy the demands on the steam users such as dryers, distillation columns, and heat exchangers. The excess steams then are supplied to the steam turbines to generate electricity. In the steam turbine network, the total generation rate of steams is almost constant as long as the production load in the reaction sections is fixed. The flow rates of the steam entering the steam turbines are freely adjustable, within their constraints. 3. Modeling of the Condensing Steam Turbine The modeling objective is to accurately predict the electric power generated by the electric generator of a condensing steam turbine under its varying operating conditions. In the modeling, the actual electric power is predicted by a hybrid model that combines a thermodynamic model for the ideal power into empirical models for an unknown state variable and the overall efficiency. The ideal power is the total enthalpy change of the steam through the turbine under an isentropic condition. It is essentially equal to the actual electric power if all the following conditions are satisfied: (i) the steam entering the steam turbine expands under adiabatic and reversible expansion conditions (isentropic conditions); (ii) there are no energy losses due to frictions or leakages in the turbine, bearings, gears, and other auxiliary components; and (iii) the electric generator perfectly converts mechanical work to electricity. Figure 3 shows a simplified structure of the hybrid model of the condensing steam turbine. Given the current values of turbine operating variables x, Empirical Model I predicts the unknown state variable (the exhaust pressure (Pd) of a steam turbine in this study) and Empirical Model II, in turn, fits the overall efficiency (ηT) of the condensing steam turbine. The thermodynamic model calculates the ideal power based on the steam property equations

672

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006

Figure 2. Condensing steam turbine network in a chemical plant.

variables is represented using an SVM, to be presented later in this article. The steam quality (qV) is the mass fraction of the steam in the condensate exiting the turbine exhaust and can be calculated by

qV(i) )

Figure 3. Model structure for the condensing steam turbines.

and the unknown state variable. Finally, the actual electric power (WG) can be predicted by multiplying the ideal power ∆H by the predicted overall efficiency. The thermodynamic model part of the hybrid model framework captures the effect of the changes in the operating variables to which the empirical models may give some biased errors in predicting the exhaust pressure and the overall efficiency. Thus, we can expect to get more accurate and extrapolative results in predicting the exhaust pressure and the overall efficiency with this systematic modeling method for turbine systems. In the following sections, we will present the thermodynamic modeling method and the empirical modeling method in detail. 3.1. Thermodynamic Modeling. The actual electric power generated by an electric generator is calculated by multiplying the ideal power by the overall efficiency of a condensing steam turbine with NI steam injectors, as described by NI

WG ) ηT∆H ) ηT

∑ i)1

{FV(i)[HsV(TV(i),PV(i))

- qV(i)HV(Pd) -

(1 - qV(i))Hl(Pd)]} (1) In the equation above, the overall efficiency is the performance indicator integrating the thermal and mechanical efficiencies of a steam turbine and should be correctly correlated as a function of various operating variables such as steam flow rate (FV), exhaust pressure (Pd), and steam temperature (TV). The functional relation between the overall efficiency and operating

SsV(TV(i),PV(i)) - Sl(Pd) SV(Pd) - Sl(Pd)

(for i ) 1, 2, ..., NI)

(2)

The steam and water properties, such as the vapor enthalpy of superheated steam HsV, the vapor enthalpy of saturated steam HV, the liquid enthalpy Hl, the vapor entropy of superheated steam SsV, the vapor enthalpy of saturated steam SV, and the liquid entropy Sl, can be calculated from the steam property equations used from the industrial standard IAPWS-IF97.15 The exhaust pressure at the condenser is an unknown state variable that varies with various operating variables such as the flow rate and temperature of cooling water and the steam flowrates. Hence, we must first find the functional relationship between the exhaust pressure and these operating variables to predict the overall efficiency and, finally, the actual electric power. The exhaust pressure is also predicted using the SVM to appear in the following section of this article. The condensing steam turbine has multiple steam injectors, as shown in Figure 1; therefore, the prediction of the power recovery from each type of the steam entering the steam injectors is a matter of concern. The power recovery rate from a type of steam is defined as the electric power extracted from one ton/h of the steam passing through the turbine and can be estimated using the following equation:

δWG(i) ) ηT{HsV(TV(i),PV(i)) - qV(i)HV(Pd) (1 - qV(i))Hl(Pd)}(for i ) 1, 2,‚‚‚NI) (3) Note that the aforementioned equation is derived from the assumption that the equal overall efficiency can be applied to all of the steam injectors of a steam turbine, which means that the performance difference between the steam injectors is ignored. Based on eq 3, one can estimate the performance of each steam injector of a steam turbine, and it is possible to easily grasp a significant performance change by continuously monitoring the power recovery rates.

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 673

3.2. Empirical Modeling. SVM is used as an empirical modeling tool to predict the overall efficiency and unknown state variable of the condensing steam turbine. The SVM has been known as a very powerful modeling algorithm to solve classification problems since Vapnik and co-workers16 proposed this modeling strategy. Because the formulation of SVMs is based on structural risk minimization rather than on the empirical risk minimization that has been used by other conventional black-box modeling algorithms such as artificial neural networks (ANNs)17 and partial least-squares (PLS),18 the SVMs typically performs better than the conventional algorithms.19 An SVM uses a hypothesis space of linear functions in a high-dimensional feature space, trained by solving a convex quadratic optimization problem; thus, a global solution is always guaranteed in locating its model parameters. SVMs have a few tunable parameters, such as the capacity constant to control the complexity of functions and the type of a kernel function used for transforming the original input space into a high-dimensional feature space. In addition, SVMs have an outstanding advantage over the conventional algorithms: coping with the overfitting problem. Hence, guaranteeing the robustness of model, the frequency of model update in an optimization system is less, relative to other nonlinear regression algorithms. Proof of the fact that it protects the overfitting was presented by Vapnik.20 For these reasons, SVMs have been become increasingly popular alternatives to ANNs. Recently, the concept of SVMs has been extended to the domain of regression problems. Solving a regression problem with SVMs are called support vector regressions (SVRs), and applications of them to many regression problems have yielded excellent performances.21-23 Also, Han et al. applied SVRs to melt index modeling for industrial polymerization processes.24 In this study, a brief sketch of the SVR algorithm is provided, and a more-detailed description can be found in the Appendix. In a regression problem using a SVM, given a training data set of n samples, X ˆ ) [xˆ 1, xˆ 2, ‚‚‚, xˆ n]T (a matrix of the measured operating variables) and ηˆ T ) [ηˆ T1, ηˆ T2, ‚‚‚, ηˆ Tn]T (a vector of the measured overall efficiency) or Pˆ d ) [Pˆ d1, Pˆ d2, ‚‚‚, Pˆ dn]T (a vector of the measured exhaust pressure), the unknown functionality is approximated using a finite number of parameters as follows: n

〈ηT|Pd〉 ) f(x,R,R*) )

(Ri - R/i )K(x,xi) + r ∑ i)1

1

n

∑∑(Ri - R/i )(Rj - R/j )K(xˆ i,xˆ j) + 

(Ri + ∑ i)1

-

〈ηˆ Ti|Pˆ di〉(Ri ∑ i)1

R/i )

(5)

(6) (7)

n

R/i ) ∑Ri ∑ i)1 i)1

(8)

where the parameters C (which represents the capacity constant) and  (which represents the size of the epsilon insensitive zone of an -insensitive loss function14) are two free parameters that are used to control the generalization ability of the approximated function (eq 4). They should be adequately determined to obtain good prediction results. In this study, those parameters are determined on the basis of such a method that the parameter values are found when the resulting model best fits testing data (not used for the modeling), which is also called a crossvalidation method.18 The solutions of the optimization problem, as described by eqs 5-8, are the vectors r* and r. Most elements of these solution vectors vanish and the data samples corresponding to nonzero values of R/i and Ri are called support vectors. In eq 4, the bias term r can be calculated from m training samples pertaining to the support vectors obtained as the optimization solutions, and it is a unique constant that minimizes the error over the training data set: m

min L(r) )

∑ j)1

n

(〈ηˆ Tj|Pˆ dj〉 -  -

[Ri - R/i )K(xˆ,xˆi) - r]2 ∑ i)1

(9)

To solve the QP problem described in eqs 5-8 efficiently, several types of optimization algorithms have been proposed for SVMs. In this study, the sequential minimal optimization (SMO) algorithm with a decomposition method is used that decomposes a large QP problem into a series of smaller QP sub-problems.25 To construct the models for the overall efficiency and the exhaust pressure, various operating variables concerned with a steam turbine should be measured first. The measured values of the exhaust pressure Pˆ d can be used directly for modeling. However, the measured values of the overall efficiency ηˆ T for a steam turbine must be calculated using the following relation:

W ˆG ∆H ˆ

NI

)W ˆ G/

{Fˆ V(i)[HV(Tˆ V(i),Pˆ V(i)) - qˆ V(i)HV(Pˆ d) ∑ i)1 (1 - qˆ V(i))Hl(Pˆ d)]} (10)

In the above equation, the operating variables, such as the temperature and pressure of the steam, the exhaust pressure, and the power generation rate, must be measured for past operation periods of the steam turbines. 4. Formulation of the Optimization Problem The objective of the formulation of the optimization problem is to locate the optimal operating condition that maximizes the sum of the electric powers generated by all the steam turbines in the steam turbine network. Therefore, the objective function can be written as

∑

n

R/i )

(for i ) 1, 2, ..., n)

n

NT

2 i)1 j)1 n

0 e R/i e C

ηˆ T )

n

(for i ) 1, 2, ..., n)

0 eRi eC

(4)

Equation 4 is used for the prediction of the overall efficiency or the unknown state variable (the exhaust pressure). The kernel function K(x,z) maps the input space X implicitly to a feature space and can be chosen alternatively from various types of kernel functions such as a linear kernel (K(x,z) ) x‚z), a polynomial kernel (K(x,z) ) (xTz)d) or a radial bias function (RBF) kernel (K(x,z) ) exp(- |x - z|2/(2σ2)). In this study, the RBF kernel is adopted to model both the overall efficiency and the unknown state variable, because it is frequently used for its high-resolution power.22,23 The Lagrange multipliers R and R* are obtained by solving the following constrained quadratic programming (QP) problem:

min J(R,R*) )

subject to

NT

∑

N(k) I

∑

(k) (k) (k) (k) (k) {η(k) {FV(i) max J ) W(k) G (x ,ω ) ) T (x ,ω ) xD k)1 k)1 i)1 (k) (k) (k) (k) (k) [HsV(TV(i) ,PV(i) ) - qV(i) HV(P(k) d ) - (1 - qV(i))Hl(Pd )]}} (11)

674

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006

The equation above is subject to the following equality and inequality constraints: NS

(l) a(j) Fs(j) - Fu(l) - Ft(l) ) 0 ∑ j)1

(for l ) 1, 2, ..., NH)

(12)

NS

(l) a(j) Fs(j)HsV(Th(c(j)),Ph(c(j))) - Fu(l)HV(Th(l),Ph(l)) ∑ j)1

Ft(l)HV(Th(l),Ph(l)) ) 0

(for l ) 1, 2, ..., NH) (13)

(k) ) Ft(l) FV(i) (k) (for l ) b(i) ; i ) 1, 2, ..., N(k) I ; k ) 1, 2, ..., NT) (14) (k) ) Th(l) TV(i) (k) (for l ) b(i) ; i ) 1, 2, ..., N(k) I ; k ) 1, 2, ..., NT) (15) (k) ) Ph(l) PV(i) (k) (for l ) b(i) ; i ) 1, 2, ..., N(k) I ; k ) 1, 2, ..., NT) (16) (k) (k) W(k) G,L e WG e WG,U

(for k ) 1, 2, ..., NT)

(17)

(k) (k) (k) e FV(i) e FV(i),U FV(i),L

(for i ) 1, 2, ..., N(k) I ; k ) 1, 2, ..., NT) (18) N(k) I

(k) Fc,L

e

(k) (k) FV(i) e Fc,U ∑ i)1

Fs(i),L e Fs(i) e Fs(i),U

(for k ) 1, 2, ..., NT)

(19)

(for i ) 1, 2, ..., NS)

(20)

In the optimization problem above, the decision variables xD are the flow rates FV of the steam entering the steam injectors and the flow rates Fs of the steam between the steam interconnection lines. The steam header pressures are the major manipulated variables. However, the steam header pressures are not suitable for the optimization variables, because their set points are determined according to the reactor operating conditions. Equations 12 and 13 represent the mass and energy balances, respectively, which must be satisfied for each steam header in the steam turbine network shown in Figure 2. The energy balances are set up to describe the temperature changes between the steam headers, whereas the pressure of the steam is reduced through a letdown valve. Note that the energy balances are derived under the assumption that there is no energy loss from each steam header while letting down the steam from one steam header to another. The steam from each steam header is decreased by utilizing the pressure of the steam as an energy source of turbines or users. The energy loss from each steam header while decreasing the amount of steam is negligible. This is because it is much smaller than the steam energy entering the turbines or the users (that is, the other terms in energy balances). Equations 14-16 are subsidiary equations that are used to make the operating variables of the steam turbines correspond to those of the steam headers. Equations 17-19 represent the upper and lower bounds on the major operating variables of a condensing steam turbine, and they are set up by taking into account hardware constraints, operating conditions, and safety. The minimum and maximum loads on the electric power generation rates must be specified for the electric generators (eq 17). The flow rates of the steam into the steam

Figure 4. Architecture of the optimization system of the condensing steam turbine network.

injectors must be confined between their minimum and maximum values (eq 18), and the steam and condensate mixture exiting a steam turbine cannot exceed its maximum value (eq 19). The flow rate of the steam cannot exceed a prescribed maximum flow through an interconnection line connecting one steam header to another. Thus, the upper and lower bounds on the interconnection flows should be specified as described in eq 20. The matrix elements a and b in eqs 12, 13, 15, and 16 represent the linkage of the steam injectors to the steam headers or the interconnection lines, and their values for the steam turbine network shown in Figure 2 are given in the Nomenclature section near the end of this paper. 5. Optimization System Figure 4 shows the architecture of the optimization system developed by implementing the optimization problem mentioned in the preceding section. Basically, the optimization system was designed through the use of a server and client architecture, enabling us to access the optimal and current operating conditions of the steam turbine network anywhere. Samsung Petrochemicals Corp. uses the real-time database system for stacking most of the process operational data collected by distributed control systems (DCSs) or programmable logic controllers (PLCs) for all of the chemical equipment. The operational data concerning the steam turbines are rectified using the filters or the mapping functions provided by the real-time database system, and then they are used by the optimization system that implements the optimization problem. The optimization system was developed as an application program of the real-time database system and consists of several component modules that are designed for modularizing each step required to solve the optimization problem. The optimization engine is responsible for calculating the performance indicators and for locating the optimal operating conditions of the steam turbines, using a successive quadratic programming (SQP) algorithm26 known as an effective optimization algorithm for such a constrained nonlinear programming problem as that described by eqs 11-20. Data reconciliation is also performed using the optimization engine to conform to the mass balance on the steam turbine network. The optimization problem is solved automatically and cyclically, using a newly obtained

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 675 Table 1. Specification of the Condensing Steam Turbines turbine name

number of steam injectors

type of steam injected

maximum condensate flow [ton/h]

maximum generator power [kW]

Turbine 1 Turbine 2 Turbine 3

2 3 2

MLS, LS MLS, LS, LLS LS, LLS

52 86 108

5500 7200 8200

operating condition in the optimization engine. The primary role of the system manager is to convey the operational data to the optimization engine and to transfer the optimization results to the steam turbine network, by communicating with the realtime database system. In addition, it is used to adjust the control parameters of the optimization system, such as the optimization cycle and the model updating method. The model manager implements the thermodynamic and empirical modeling structure shown in Figure 3. Two models are newly built or rebuilt for each steam turbine on an offline basis. The performance of the optimization models is evaluated by checking the root-meansquare error (RMSE) of each model at each optimization cycle. If the RMSE is beyond the predefined threshold, the model manager is used to rebuild the models, to remove the processmodel mismatch. The optimization client serves to show the current and optimal operating values and the performance indicators on the process flow diagrams wherever the intranet is available. In this optimization system, the optimal operating conditions are continuously stored in the real-time database system from which the set points of the steam turbine operating conditions are transferred to the DCSs and PLCs.

Figure 5. Comparison between the measurements (dotted lines) and the predictions (solid lines) for turbine 1: (a) exhaust pressure, (b) overall efficiency, and (c) electric power.

6. Results and Discussion First, the proposed hybrid-modeling framework is verified by applying itself to the three steam turbines shown in Figure 2. The online optimization is then performed to maximize the total electric power generation of the steam turbine network of Samsung Petrochemicals Corp. Both modeling and optimization results are discussed in this section. 6.1. Model Validation. The models are verified by comparing the predicted values with the measured values of the three steam turbines. Table 1 summarizes the specifications of the three condensing steam turbines to be modeled. For each steam turbine, two empirical models are built to predict the exhaust pressure and the overall efficiency, respectively. It is known that the exhaust pressure is strongly dependent on the steam flow rates and the temperature and flow rate of the cooling water entering the condenser, and that the overall efficiency is affected by the steam flow rates and the exhaust pressure. According to this information, the independent variables used for both empirical models are selected as summarized in Table 2. The model manager builds the prediction models of the electric power and exhaust pressure from the operational data (training data) obtained during a period of 5-6 months (1 h sampling time) for the three steam turbines. The sampling period used for modeling is sufficiently large to cover most of the possible operating regions of the steam turbines, because the steam flowing into the turbines is frequently changed during the normal operation.

Figure 6. Comparison between the measurements (dotted lines) and the predictions (solid lines) for turbine 2: (a) exhaust pressure, (b) overall efficiency, and (c) electric power.

Figures 5-7 compare the predicted outputs with the measured outputs, using the steam turbine models built for Turbines 1, 2, and 3, respectively, during a period of 200 h (200 testing data points). As shown in the figures, the predicted outputs match the measured outputs for the entire observations quite well. Table 3 summarizes the prediction performance of these models. The prediction error for the exhaust pressure of Turbine 1 is somewhat large but acceptable. For all the turbines, the

Table 2. Variables Used To Build Empirical Models I and II Empirical Model I turbine name Turbine 1 Turbine 2 Turbine 3

independent variable,

x(k) I

(1) (1) (1) FV(1) ,FV(2) ,F(1) c ,Tc (2) (2) (2) (2) FV(1),FV(2),FV(3),F(2) c ,Tc (3) (3) (3) (3) FV(1),FV(2),Fc ,Tc

Empirical Model II

dependent variable

independent variable, x(k) II

dependent variable

P(1) d P(2) d P(3) d

(1) (1) FV(1) ,FV(2) ,P(1) d (2) (2) (2) FV(1),FV(2),FV(3) ,P(2) d (3) (3) (3) FV(1),FV(2),Pd

η(1) T η(2) T η(3) T

676

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006

Figure 8. Power gains of the steam injectors, showing the deviation of the electric power by increasing the flow rate of the steam into an injector. Table 4. Major Optimization Variables and Their Constraints Bounds [ton/h] Figure 7. Comparison between the measurements (dotted lines) and the predictions (solid lines) for turbine 3: (a) exhaust pressure, (b) overall efficiency, and (c) electric power. Table 3. Prediction Performance of the Steam Turbine Models root-mean-squared error, RMSEa turbine name

Pd [mm Hg]

ηT [%]

WG [kW]

Turbine 1 Turbine 2 Turbine 3

3.02 1.08 0.59

0.75 0.63 0.47

42.1 68.8 51.5

a

Defined by RMSE )

x

n ∑i)1 (yˆ i-yi)/n.

prediction errors for the overall efficiency of the three turbines are 25 ton/h, as shown in Figure 8. As can be expected from the fact that the LS performance of Turbine 1 is better than that of Turbine 2, the flow rate of the LS into Turbine 2 reached its upper bound and, conversely, the flow rates of the LS into Turbines 1 and 3 decreased. Figures 11 and 12 show the changes in the overall efficiency and the power recovery rate, respectively, during the optimization. As shown in Figure 11, both Turbines 2 and 3 show that the overall efficiencies changed little. However, Turbine 1 shows ∼7% higher overall efficiency than that before the optimization was started. The performance of Turbine 1 is sensitive to the changes in the MLS flow rate, compared to the other turbines as shown in Figure 8; therefore, the increase in the MLS into the turbine sharply raised its overall efficiency. As can be seen near the 50th observation in Figure 12, all the power recovery rates steadily increase just after the optimization was started. Near the 100th observation, even though the production load

had changed significantly, the optimization was consistently well-performed. It shows that the proposed online optimization system is well-operated for tracking the changing optimum, in response to nonstationary disturbances. Note that a comparison of the power recovery rates between two turbines just indicates the performance difference of the turbines excluding their condensers, because the power recovery rate is evaluated at an exhaust pressure, which is changed somewhat by the condenser. 7. Conclusions The hybrid-modeling framework for a condensing steam turbine with multiple steam injectors was developed. It then was used for online optimization of a steam turbine network. The modeling framework consists of a thermodynamic modeling component, to calculate the ideal power, and an empirical modeling component, to calculate the overall efficiency of a condensing steam turbine. To model the overall efficiency and the exhaust pressure, we used the support vector machine (SVM), which is known as a state-of-the-art black-box modeling technique. The models are able to predict the electric power generated by the steam turbines with very high accuracies. The optimization results have been successful because they have brought us a savings of ∼4.5% of the total cost for purchasing electricity, because the optimization system was applied to an industrial steam turbine network. The modeling and optimization

678

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006

method can be used for a similar utility system that might consist of steam headers, turbines, and steam generation systems. Acknowledgment This work was supported in part by the Brain Korea 21 Project. The authors gratefully acknowledge the support provided by Woochang Lee, Mooho Lee, and Chundong Chang (all with Samsung Petrochemicals Corp., Ulsan, Korea). The authors also gratefully acknowledge the partial financial support of the Korea Science and Engineering Foundation through the Advanced Environmental Biotechnology Research Center (No. R11-2003-006-02003-0) and the Basic Research Program (No. R01-2004-000-10345-0). This work is also partially funded by Korea Institute of Science and Technology (Development of Dried Liquid Fuel Cells), by Hyundai Motor Company & Korea Energy Management Corporation (Development of Dynamic Model and Optimized Operation Technology of Polymer Electrolyte Membrane Fuel Cell for bus), and by 2005 Research Fund of Seoul National University. Appendix A: Support Vector Regression-Basic Foundations14 Suppose we are given a set of training data {(xi,yi)|i ) 1, 2, ..., n} of input vectors xi and associated targets yi. The goal is to fit a function f(x) that approximates the relation inherited between the data set points, and it can be used later to infer the output y for a new input data point x. Any practical regression algorithm has a loss function L(y,f(x)), which describes how the estimated function deviated from the true function. Many forms for the loss function can be found in the literature (linear, quadratic, exponential loss function, etc.). In this work, Vapnik’s loss function is used, which is known as an -insensitive loss function and is defined as

{

(if |y - f(x)| e) 0 L(y,f(x)) ) |y - f(x)| -  (otherwise)

(A1)

where  is a predefined constant ( > 0) that controls the noise tolerance. With the -insensitive loss function, the goal is to find the function f(x) that has, at most, a deviation of  from the actually obtained targets yi for all training data, and, at the same time, is as flat as possible. In other works, the regression algorithm does not care about errors as long as they are less than , but it will not accept any deviation larger than this. For pedagogical reasons, we begin by describing the case of linear functions f, taking the form

f(x) ) ω‚x + r

(A2)

where ω ∈ ℵ, ℵ is the input space, r ∈ R, and ω‚x is the dot product of the vectors ω and x. Flatness in the case of eq A2 means that one seeks a small value of ω. One way to ensure this is to minimize the norm, i.e., |ω|2. Thus, the regression problem can be written as a convex optimization problem:

1 minimize |ω|2 2

(A3)

subject to

Figure A1. In support vector regression (SVR), a tube with radius  is fitted to the data. The tradeoff between model complexity and points lying outside of the tube (with positive slack variables ξ) is determined by minimizing eq A6.

precision, or, in other words, that the convex optimization problem is feasible. Sometimes, however, this may not be the case, or we also may want to allow for some errors. Analogously to the “soft margin” loss function that was adapted to SVM by Vapnik, slack variables (ξi and ξ/i ) can be introduced to cope with infeasible constraints of the optimization problem in eq A3. Hence, we arrive at the following formulation: n 1 minimize |ω|2 + C (ξi + ξ/i ) 2 i)1

∑

(A6)

subject to

y - (ω‚x + r) e  + ξi (ω‚x + r) - y e  + ξ/i ξi, ξ/i g 0 The constant C > 0 determines the tradeoff between the flatness of function f and the amount up to which deviations larger than  are tolerated. This corresponds to working with the -insensitive loss function, which has been described previously. As shown in Figure A1, only the points outside the shaded region contribute to the cost, insofar as the deviations are penalized in a linear fashion. It turns out that, in most cases, the optimization problem (eq A6) can be solved more easily in its dual formulation. Moreover, the dual formulation provides the key for extending SVM to nonlinear functions. Hence, we use a standard dualization method utilizing Lagrange multipliers. The key idea is to construct a Lagrange function from the objective function (it will be called the primal objective function) and the corresponding constraints, by introducing a dual set of variables. It can be shown that the Lagrange function has a saddle point, with respect to the primal and dual variables at the solution. The primal objective function with its constraints is transformed to the Lagrange function as follows: n n 1 L ) |ω|2 + C (ξi + ξ/i ) - (λiξi + λ/i ξ/i ) 2 i)1 i)1

∑

∑

n

Ri( + ξi - yi + (ω‚x + r)) ∑ i)1 n

y - (ω‚x + r) e 

(A4)

(ω‚x + r) - y e 

(A5)

The implied assumption in eqs A4 and A5 is that such a function f actually exists that approximates all data points with 

R/i ( + ξ/i + yi - (ω‚x + r)) ∑ i)1

(A7)

Here, L is the Lagrangian and Ri, R/i , λi, and λ/i are Lagrange multipliers. Hence, the dual variables in eq A7 must satisfy positivity constraints, i.e.,

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 679 (/) R(/) i , λi g 0

(A8)

/ Note that R(/) i refers to Ri and Ri . It follows from the saddle point condition that the partial derivatives of L, with respect to the primal variables (ω,r,ξi,ξ/i ), must vanish for optimality.

n

∂rL )

∑ i)1

(R/i - Ri) ) 0

(A9)

Several useful conclusions can be drawn from these conditions. First, only samples (xi, yi) with a corresponding R(/) i )C lie outside the -insensitive tube. Second, RiR/i ) 0, i.e., there can never be a set of dual variables Ri,R/i that are both simultaneously nonzero. This allows us to conclude that

 - yi + ω‚xi + r g 0 and ξi ) 0  - yi + ω‚xi + r e 0

(if Ri e C) (if Ri > C)

(A17) (A18)

n

∂ ωL ) ω -

(Ri - R/i )xi ) 0 ∑ i)1

(A10)

(/) ∂ξ(/)L ) C - R(/) i - λi ) 0

(A11)

i

Substituting eqs A9, A10, and A11 into eq A7 yields the dual optimization problem:

minimize

1

n

n

(Ri - R/i )(Rj - R/j )(xi‚xj) + ∑ ∑ 2 i)1 j)1 n



∑ i)1

n

(Ri + R/i ) -

xi(Ri - R/i ) ∑ i)1

(A12)

subject to n

(Ri - R/i ) ) 0 ∑ i)1

(Ri, R/i ∈ [0,C])

(A13)

In deriving eq A12, the dual variables, ξi and ξ/i , are eliminated through the condition described by eq A11, which (/) can be reformulated as λ(/) i ) C - Ri . Equation A10 can be rewritten as follows: n

ω)

(Ri - R/i )xi ∑ i)1

thus, n

f(x) )

A final note must be made regarding the sparsity of the SV expansion. From eq A15, it follows that, only for |f(xi) - yi| g, the Lagrange multipliers may be nonzero, or, in other words, for all samples inside the shaded region in Figure A1, the Ri,R/i vanish: for |f(xi) - yi| < , the second factor in eq A15 is nonzero; hence Ri,R/i must be zero, such that the KKT conditions are satisfied. Therefore, there is a sparse expansion of ω in terms of xi (i.e., not all xi are needed to describe ω). The training samples that come with nonvanishing coefficients are called support Vectors.

(Ri - R/i )(xi‚x) + r ∑ i)1

(A14)

This is the so-called support vector (SV) expansion, i.e., ω can be completely described as a linear combination of the training patterns xi. In a sense, the complexity of a function’s representation by SVs is independent of the dimensionality of the input space ℵ and is dependent only on the number of SVs. Moreover, the complete algorithm can be described in terms of dot products between the data. Even when evaluating f(x), the value of ω does not need to be computed explicitly. These observations will come in handy for the formulation of a nonlinear extension. The Krash-Kuhn-Tucker (KKT) conditions are the basics for the Lagrangian solution. These conditions state that, at the solution point, the product between dual variables and constraints must vanish, i.e.,

Ri( + ξi - yi + ω‚xi + r) ) 0

(A15a)

R/i ( + ξi + yi - ω‚xi - r) ) 0

(A15b)

(C - Ri)ξi ) 0

(A16a)

(C - R/i )ξ/i ) 0

(A16b)

Nomenclature a ) element of the matrix used for representing the mass and (2) (3) (4) (5) (5) (5) (6) energy balances (1 for a(1) (5), a(8), a(3), a(9), a(2), a(6), a(11), a(4), (6) (6) (9) (10) (1) (2) (2) (3) (4) (4) a(7), a(13), a(10), a(12), -1 for a(1), a(2), a(3), a(4), a(5), a(6), a(5) (7), (8) (8) (9) (9) (10) a(8) , a , a , a , a , a , and 0 for others) (8) (9) (10) (11) (12) (13) b ) element of the matrix used for representing the mass and (1) (2) (2) (2) energy balances (b(1) (1) ) 2, b(2) ) 3, b(1) ) 5, b(2) ) 6, b(3) ) (3) (3) 7, b(1) ) 10, b(2) ) 11) C ) capacitance constant c ) index used for connecting the steam headers to the interconnection lines (c(1) ) 1, c(2) ) 2, c(3) ) 2, c(4) ) 3, c(5) ) 4, c(6) ) 4, c(7) ) 5, c(8) ) 8, c(9) ) 8, c(10) ) 8, c(11) ) 9, c(12) ) 9, c(13) ) 10) d ) natural number (d ) 1, 2, 3, ...) Fc ) flow rate of the coolant entering a steam turbine condenser [ton/h] Fs ) flow rate of the steam through an interconnection line [ton/h] Ft ) flow rate of the steam from a steam header to a steam turbine injector [ton/h] Fu ) flow rate of the steam from a steam header to all its steam users [ton/h] FV ) flow rate of the steam entering a steam injector [ton/h] Hl ) enthalpy of liquid water [kJ/kg] HV ) enthalpy of saturated steam [kJ/kg] HsV ) enthalpy of superheated steam [kJ/kg] L ) Lagrangian function L ) loss function NI ) number of steam injectors of a steam turbine NH ) number of the steam header in a steam turbine network (NH ) 11) NT ) number of the steam turbines in a steam turbine network (NT ) 3) NS ) number of the steam interconnection lines in a steam turbine network (NS ) 13) n ) number of measurements Pd ) exhaust pressure of a steam turbine [kg/(cm2 g)] Ph ) steam header pressure [kg/(cm2 g)] PV ) pressure of the steam entering a steam injector [kg/(cm2 g)] qV ) steam quality at the exhaust of a steam turbine r ) bias term of the support vector function

680

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006

Sl ) liquid entropy [kJ/(kg K)] SV ) entropy of saturated steam [kJ/(kg K)] SsV ) entropy of superheated steam [kJ/kg] Tc ) temperature of the coolant entering a turbine condenser [K] Th ) steam header temperature [K] TV ) temperature of the steam entering a steam injector [K] WG ) electric power generated by a steam turbine [kW] X ) matrix of measured operating variables x ) vector of measurements for an operating variable xD ) vector of decision (optimization) variables y ) output variable z ) vector of measurements for an operating variable Greek Symbols R, R* ) Lagrange multipliers r, r* ) vector of Lagrange multipliers ∆H ) enthalpy change through a steam turbine [kJ/h] δWG ) power recovery rate of a steam injector [kWh/ton]  ) size of the epsilon insensitive zone of a -insensitive loss function ηT ) overall efficiency of a steam turbine λ, λ* ) Lagrange multipliers σ ) standard deviation in representing the RBF kernel function ω ) vector of model parameters ξi, ξ/i ) slack variables to cope with infeasible constraints of the optimization problem Superscripts k ) index of a steam turbine ∧ ) measured value Subscripts i ) index of a steam injector or a measured sample j ) index of a support vector or a steam interconnection line L ) lower bound l ) index of a steam header m ) number of support vectors U ) upper bound Operators 〈A|B〉 ) select A or B A‚B ) dot product of the vectors A and B Literature Cited (1) Yi, H.-S.; Han, C.; Yeo, Y.-G. Optimal multiperiod planning of utility systems considering discrete/continuous variance in internal energy demand based on decomposition method. J. Chem. Eng. Jpn. 2000, 33, 456. (2) Hui, C. W. Determine marginal values of intermediate material and utility using a site model. Comput. Chem. Eng. 2000, 24, 1023. (3) Kim, J.; Lee, B.; Lee, E.-S. Optimal production sequence and processing schedules of multiproduct batch processes for heat integration and minimum equipment costs. Korean J. Chem. Eng. 2001, 18, 599. (4) Yi, H.-S.; Kim, J. H.; Han, C.; Jung, J. H.; Lee, M. Y.; Lee, J. T. Periodical replanning with hierarchical repairing for the optimal operation of a utility plant. Control Eng. Pract. 2003, 11, 881.

(5) Shang, Z.; Kokossis, A. A transshipment model for the optimization of steam levels of total site utility system for multiperiod operation. Comput. Chem. Eng. 2004, 28, 1673. (6) Rodriguez-Toral, M. A.; Morton, W.; Mitchell, D. R. The use of new SQP methods for the optimization of utility systems. Comput. Chem. Eng. 2001, 25, 287. (7) Papalexandri, K. P.; Pistikopoulous, E. N.; Kalitvenzeff, B. Modelling and optimization aspects in energy management and plant operation with variable energy demandssapplication to industrial problems. Comput. Chem. Eng. 1998, 22, 1319. (8) Lee, S. J.; Lee, M. H.; Chang, K. S.; Han, C. Optimization of a utility plant operation based on hierarchical method. Hwahak Konghak 1998, 36, 415. (9) Yi, H. S.; Han, C. The integration of complete replanning and rule based repairing for optimal operation of utility systems. Korean J. Chem. Eng. 2001, 18, 442. (10) Saxena, M. N. Optimize gas turbine-driven centrifugal compressors. Hydrocarbon Process. 2000, (November), 61-64. (11) Bolch, H. P. A Practical Guide to Compressor Technology; McGraw-Hill: New York, 1996. (12) Yip, W. S.; Marlin, T. E. The effect of model fidelity on real-time optimization performance. Comput. Chem. Eng. 2004, 28, 267-280. (13) Han, I.-S.; Han, C. Modeling of Air-Compression Systems in a Chemical Plant. Ind. Eng. Chem. Res. 2003, 42, 2209. (14) Cristianini, N.; Shawe-Taylor J. An Introduction to Support Vector Machines and Other Kernel-based Learning Methods; Cambridge University Press: Cambridge, U.K., 2000. (15) Wagner, W.; Cooper, J. R.; Dittmann, A.; Kijima, J.; Kretzschmar, H.-J.; Kruse, A.; Mares, R.; Oguchi, K.; Sato, H.; Sto¨cker, I.; Sifner, O.; Takaishi, Y.; Tanishita, I.; Tru¨benbach, J.; Willkommen, T. The IAPWS Industrial Formulation 1997 for the thermodynamic properties of water and steam. ASME J. Eng. Gas Turbines Power 2000, 122, 150. (16) Bosen, B. E.; Guyon, I. M.; Vapnik, V. N. A training algorithm for optimal margin classifiers. In Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory, Pittsburgh, PA, 1992. (17) Hagan, M. T.; Demuth, H. B.; Beale, M. Neural Network Design; PWS Publishing: Boston, 1996. (18) Geladi, P.; Kowalski, B. R. Partial least-squares regression: A tutorial. Anal. Chim. Acta 1986, 185, 1. (19) Gunn, S. R. Support Vector Machines for Classification and Regression; University of Southampton: Southampton, U.K., 1998. (20) Vapnik, V. The Nature of Statistical Learning Theory; Springer: New York, 1995. (21) Chuang, C.-C.; Su, S.-F.; Jeng, J.-T.; Hsiao, C.-C. Robust support vector regression networks for function approximation with outliers. IEEE Trans. Neural Networks 2002, 13, 1322. (22) Smola, A. J.; Scholkopf, B. A Tutorial on Support Vector Regression; Royal Holloway University of London: London, 1998. (23) Collobert, R.; Bengio, S. Support vector machines for large-scale regression problems. J. Mach. Learn. Res. 2001, 1, 143. (24) Han, I.-S.; Han, C.; Chung, C.-B. Melt index modeling with support vector machines, partial least squares, and artificial neural networks. J. Appl. Polymer Sci. 2005, 96, 967-974. (25) Flake, G. W.; Lawrence, S. Efficient SVM Regression Training with SMO. Mach. Learn. 2002, 46, 271. (26) Rao, S. S. Engineering OptimizationsTheory and Practice; Wiley: New York, 1996.

ReceiVed for reView July 1, 2004 ReVised manuscript receiVed October 14, 2005 Accepted November 3, 2005 IE049425A