Fuzzy Predictive Control of a BoilerâTurbine System Based on a

Article pubs.acs.org/IECR

Fuzzy Predictive Control of a Boiler−Turbine System Based on a Hybrid Model System Morteza Sarailoo,* Zahra Rahmani,† and Behrooz Rezaie† Intelligent System Research Group, Electrical and Computer Engineering Department, Babol University of Technology, Babol 47148-71167, Iran S Supporting Information *

ABSTRACT: This paper proposes a fuzzy predictive control scheme for controlling power output of a boiler−turbine system in the presence of disturbances and uncertainties. A new model of the boiler−turbine system is introduced based on the modeling approaches of hybrid systems, namely, the mixed logical dynamical modeling approach. Nonlinear parts of the system are linearized using the piecewise affine approach. To overcome the deficiency of the model predictive control in presence of disturbance and uncertainty, a fuzzy predictive control scheme is proposed in which a fuzzy supervisor is utilized to adjust the main predictive controller. The proposed fuzzy predictive control scheme has advantages such as simplicity and efficiency in nominal conditions and strong robustness in the presence of disturbances and uncertainties. Simulation results demonstrate the effectiveness and superiority of the method.

1. INTRODUCTION Nowadays developing advanced techniques to extract enhanced models of industrial processes has attracted researchers as an accurate model of a system is compulsory in order to analyze behavior of the system and to design a proper controller for it. However obtaining accurate models of many systems is difficult because such systems actually consist of many different modes which are combined with each other with logic, conditions, or rules. Such systems are called hybrid systems; a hybrid system consists of different dynamical equations representing the system behavior at a certain condition(s).1 Some systems do not possess such behavior genuinely; however, due to acquiring a linearized model of those systems, such conditions could be defined, so those systems also could be considered as hybrid systems. During the past decade many work has been done to prepare systematic and accurate methods to model hybrid systems.2 The two most renowned methods are the piecewise affine (PWA) method3 and mixed logical dynamical (MLD) method.4 The PWA method is the easiest method for representing a hybrid system, and it is also widely used for linearization of nonlinear terms. The dynamics of system are represented in the PWA framework with the following structure:3 x(k + 1) = Ai x(k) + Bi u(k) + fi y(k) = Cix(k) + Diu(k) + gi

Figure 1. Bounded PWA model: in each section, linear dynamics describe system behavior.5

for designing a model-based controller based on the PWA model. To overcome this shortcoming of the PWA model, the MLD model of hybrid systems has been recommended.4 This model only uses two linear equations (eqs 2-1 and 2-2) and a linear inequality (inequality 2-3) which makes it suitable for designing a model-based controller. The general structure of MLD model is as follows:4

if Ωi = true (1)

where k is a discrete-time step. In addition, x, u, and y denote states, inputs, and outputs, respectively. Ωi indicates a set of conditions which defines ith section of space. Ai, Bi, f i, Ci, Di, and gi are proper time-invariant matrices related to section i. The concept of the PWA method with four distinctive sections is depicted in Figure 1. PWA model includes several different dynamical equations and the number of its dynamical equations varies case by case. This makes difficult to define a simple systematic procedure © 2014 American Chemical Society

x(k + 1) = Ax(k) + B1u(k) + B2 δ(k) + B3z(k)

(2-1)

y(k) = Cx(k) + D1u(k) + D2δ(k) + D3z(k)

(2-2)

E2δ(k) + E3z(k) ≤ E1u(k) + E4x(k) + E5

(2-3)

Received: Revised: Accepted: Published: 2362

August 13, 2013 January 7, 2014 January 14, 2014 January 14, 2014 dx.doi.org/10.1021/ie402649u | Ind. Eng. Chem. Res. 2014, 53, 2362−2381

Industrial & Engineering Chemistry Research

Article

where x is the system state and y and u are the output and the input signal, respectively. δ denotes logical auxiliary variables, and z denotes continuous auxiliary variables. A, B i (i = 1, 2, 3), C, Di (i = 1, 2, 3), and Ei (i = 1, ..., 5) are proper and time-invariant matrixes. In this paper, we consider a boiler−turbine system as a hybrid system and, based on the modeling methods, a hybrid model of the system is obtained. In the proposed model, the PWA method is only used for linearization purpose and based on this linearized model of system a MLD model is obtained for the boiler−turbine system. Accuracy of the obtained model is evaluated using the coefficient of determination. A boiler−turbine system is an energy conversion system that consists of a steam boiler and a turbine which uses chemical energy to generate electricity. The boiler−turbine system model first was represented at 1972 using nonlinear equations based on the boiler−turbine plant at Malmo, Sweden.6 The schematic diagram of the boiler−turbine unit is shown in Figure 2. After

the boiler−turbine system to analyze its dynamics. They showed that without a comprehensive understanding of nonlinearity, the operating range, and performance of a linear controller cannot be guaranteed. An advantage of avoiding nonlinearity is a single linear controller can be designed to work in this operating range. In 2009, Wu et al. designed a fuzzy H∞ state feedback tracking controller for the boiler−turbine system. They introduced a Takagi and Sugeno fuzzy model, and a controller was proposed based on the model. The important aspects of this method are that it uses a simple fuzzy controller and it is robust toward white noise disturbance.12 Finally in 2012, obstacles in designing a predictive controller for the nonlinear system of the boiler−turbine were surveyed.13 First, nonlinear predictive control was implemented by genetic algorithm. Then, to guarantee fast output stabilization, an H∞ fuzzy state-feedback control was applied with a designed switching principle. The structure of their proposed controller was based on the sequence of optimal inputs obtained from nonlinear predictive control. The procedures typically followed in the most of the previous analyses were only applicable to simple scenarios. Moreover, some of the previous analyses are complex, need huge computational efforts, and are not optimal. In addition, in most of the previous research, the disturbances or uncertainties were not considered. Therefore, it is a significant open challenge to study simple and systematic methods with optimal results while increasing the convergence of output and robustness of controllers in the presence of disturbance and uncertainty. In this paper due to using model predictive control (MPC), obtaining a relatively accurate model of the boiler−turbine system is mandatory. Therefore by using the PWA approach, a linearized model of the nonlinear dynamics of the boiler−turbine system is obtained. The PWA method divides the operating range into several sections in order to linearize the system in each section. According to the PWA model of the boiler−turbine system, the boiler−turbine system could be considered as a hybrid system16 with its dynamics changes based on the values of its variables. On the basis of the linearized dynamics of the system and in order to achieve a model for the boiler−turbine system over whole operating range, the MLD modeling approach is used. By using the MLD model of the boiler−turbine system, a model predictive controller is designed for controlling output power of the system based on the demands of consumers. The accuracy of the design controller based on the MLD model is investigated using a comparison between obtained values from the designed controller and their experimental values. The disturbance and uncertainty may degrade the performance of the MPC which should be avoided.17,18 A method to solve this problem is to modify the model used by the MPC which increases the complexity and the cost of implementation.19 In order to overcome this defect of MPC without increasing the complexity and the cost of implementation, and to reduce the system bias, a fuzzy supervisor is proposed to adjust the model predictive controller based on the measured states of the boiler− turbine system. It is shown that the proposed fuzzy predictive control scheme provides advantages such as simplicity and efficiency as well as strong robustness in the presence of disturbance and uncertainty. The rest of the paper has been organized as follows. In section 2, the modeling of the system based on the MLD modeling approach is presented. MPC and fuzzy predictive control are designed in section 3, and in this section, comparisons between MPC and fuzzy predictive control are provided in normal

Figure 2. Schematic diagram of the boiler−turbine unit.5

that, the boiler−turbine system was studied widely in many articles7−13 and its model6 has gone through a number of amends to provide a better and complete description of the boiler− turbine system.14,15 As stated before, the boiler−turbine system was studied widely in many articles and many different control approaches have been propounded for this system in recent years.7−13 Robert Dimeo and Kwang Y. Lee7 discussed the applicability of genetic algorithm in the control process of the boiler−turbine system. They used genetic algorithm in order to expand the proprotional−integral (PI) and state feedback controller for nonlinear and linearized model of the boiler−turbine system around its operating points. In 1996, Abdennour and Lee introduced an autonomous control system for the boiler−turbine system. In their work, a fuzzy approach was proposed for fault detection in valves and a fuzzy supervisor was used in order to adjust local LQG/LTR (linear quadratic Gaussian/loop transfer recovery) controllers in the fault conditions at feedwater valve. The result of simulations showed the improvement of system behavior in fault condition.8 Moon and Lee9 introduced three online selforganizing fuzzy logic controller loops for the system. In this method, the membership functions and database are produced automatically without access to the system model. A gainscheduled method for control of the boiler−turbine system with consideration of regulation of output power, dram pressure, and deviation of water level was proposed in 2004,10 and a family of linear parameter varying of linearized systems was used for gainscheduled from nonlinear dynamics of the boiler−turbine system. Then, a controller was designed by using set-valued method based on l1-optimization. Tan et al.,11 introduced a distance measure via the gap metric and applied this concept to 2363

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Figure 3. Values of f1 in the operating space of the boiler−turbine system.

Nonlinear equations of the boiler−turbine,15 which are represented in eqs 3 are used to obtain a linear model of the boiler−turbine system in the MLD framework by using the PWA approach for linearization.

conditions, in the presence of disturbance and uncertainty. Simulation results in this section reveal the effectiveness of the proposed method. Finally, concluding remarks are drawn in section 4.

dp = −0.0018u 2p9/8 + 0.9u1 − 0.15u3 dt

2. MODELING OF THE BOILER−TURBINE SYSTEM AS A HYBRID SYSTEM The boiler−turbine system can be modeled as a multi-input multi-output (MIMO) nonlinear system. This system is strongly coupled and is subject to various constraints on both inputs and outputs. In the literature that used the linearized form of the boiler−turbine system, they obtained a linearized model using the truncated Taylor series expansion of nonlinear equations around operating points.7,10−12 Linearization around operating points provides a relatively accurate description about behavior of a system only around these points; however for using this method, the nominal values of states and inputs at operating points should be known which are not readily available.

dp0

(3-1)

dt

= (0.073u 2 − 0.016)p9/8 − 0.1p0

(3-2)

dpf

=

(141u3 − (1.1u 2 − 0.19)p) 85

(3-3)

dt

where p, p0, and pf are drum pressure (kg/cm2), power output (MW), and fluid density (kg/cm3), respectively. The inputs to the system are u1, u2, and u3 which respectively indicate the fuel flow valve position, steam control valve position, and feedwater flow valve position and have a value in the interval [0 1]. 2364



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Figure 4. Values of f 2 in the operating space of the boiler−turbine system.

The outputs of the system are p, p0, and Xw which are respectively drum pressure, output power, and drum water level (in meters). Two first output are states of the system and readily available; whereas, water level is found through the following relationships:

Changes in positions of these valves have the following limitations: du1 ≤ 0.007/s dt du −2/s≤ 2 ≤ 0.02/s dt

du 3 ≤ 0.05/s dt

(4-1)

q ⎛ ⎞ X w = 0.05⎜0.13073pf + 100αcs + e − 67.975⎟ ⎝ ⎠ 9

(4-2)

αcs = (4-3)

(5-1)

(1 − 0.001538pf )(0.8p − 25.6) pf (1.0394 − 0.0012304p)

(5-2)

qe = (0.854u 2 − 0.147)p + 45.59u1 − 2.514u3 − 2.096

The limits on operation speed of a valve (eqs 4) are one of its main characteristics, and they mostly depend on the actuator of the valve and the physical structure of the valve. However, based on the features of the system, designers may set new limits for a valve within its actual limits range. In this paper defined limits in eqs 4 have been used.

(5-3)

where qe is the evaporation rate (kg/s) and αcs is the steam quality. For obtaining the MLD model4 of the boiler−turbine system by using Hybrid Systems Description Language (HYSDEL)20 2365



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Figure 5. Values of f 3 in the operating space of the boiler−turbine system.

and based on eqs 3−5, first the nonlinear terms in those equations must be linearized. The nonlinear terms are presented in eqs 6.

following equation is used to represent the linearized form of f1, f 2, and f 3:

f1 (u 2 , p) = u 2p9/8 ,

z = aix + biy + ci

f2 (u 2 , p) = u 2p , f3 (pf , p) =

(1 − 0.001538pf )(0.8p − 25.6) pf (1.0394 − 0.0012304p)

i = 1, ..., 16

(7-1)

In order to describe f4, linear lines with following equation are used: , and

f4 (p) = p9/8

y = aî x + bî

i = 1, ..., 4

(7-2)

where ai, bi, ci, âi, and bî are coefficients related to the ith section. In Figures 3−6, comparisons between nonlinear and linear form of the equations f1, f 2, f 3, and f4 have been provided over the operating space of the boiler−turbine system. On the basis of Figures 3−6, linearized forms of the nonlinear terms obtained by using PWA linearization technique have errors in some regions. However despite the dependence of MPCs on the model of system, these errors do not have substantial negative

(6)

Hence, it is considered that p, pf, and u2 respectively have values in the intervals [70 145], [200 600], and [0 1] which describe the whole operating space of the boiler−turbine system. Each variable is divided into four equal separate sections. Consequently, the nonlinear terms f1, f 2, and f 3 will consist of 16 sections and f4 will consist of 4 sections. Now the nonlinear terms in each section are described with a linear equation.16 The 2366



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Figure 6. Values of f4 in the operating space of the boiler−turbine system.

where Ts is sampling time. pui and pXw denote the new state variables. dui and dXw are pace of ith input and deviation of water level in the drum, respectively. Now based on the PWA model of the boiler−turbine system and using HYSDEL and hybrid toolbox,22 an MLD model of the boiler− turbine system is obtained. The MLD model of the system has the following properties (for codes refer to the Supporting Information): • Sampling time (Ts) is 1 s. • Seven states (seven continuous pf(k), p0(k), p(k), pXw(k), pui(k) i = 1, 2, 3 −0 binary), three inputs (three continuous ui(k) i = 1, 2, 3 −0 binary), one output (one continuous dXw(k) −0 binary). • 64 continuous auxiliary variables, 45 binary auxiliary variables, and 394 mixed-integer linear inequalities. Consequently the matrixes A, B1, B2, B3, C, D1, D2, D3, E1, E2, E3, E4, and E5 of the MLD model of the boiler−turbine system (eqs 2) have dimensions of 7 × 7, 7 × 3, 7 × 45, 7 × 64, 1 × 7,

effects on the performances of the control process because MPCs are robust against small model errors.21 Now according to the linearized form of the nonlinear terms, the dynamics of the boiler−turbine system can be rewritten in the PWA formation (eqs 2) with 64 sections. Before obtaining the MLD model of the boiler−turbine system based on the PWA model, in order to impose the constraints of inputs (eqs 4) and since one of our goals is to limit the variation of the water level in drum, four new states and four new equations are introduced as follows: pui(k + Ts) = ui(k),

i = 1, 2, 3

pXw(k + Ts) = Xw(k) dui(k) = (ui(k) − pui(k))/Ts ,

dXw(k) = Xw(k) − pXw(k)

(8-1) (8-2)

i = 1, 2, 3

(8-3) (8-4) 2367



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Table 1. Coefficient of Determination of the MLD Model variable

R2 value

p p0 pf Wx

0.978 0.999 0.954 0.989

Figure 9. Inner structure of the fuzzy system.

where ŷi and yi are acquired data from the system and estimated data from the MLD model, respectively. y ̅ is the mean of the acquired data from the system. The results of computation of the coefficient of determination for MLD model based on the PWA linearization technique are provided in Table 1. According to Table 1, it is obvious that using hybrid system modeling approach has relatively good accuracy in comparison with the nonlinear model (eqs 3). It is obvious that increase in the number of sections of PWA method causes increase in accuracy of the MLD model; however, it also increases the number of auxiliary variables and mixed-integer linear inequalities which leads to more computational complexity. Consequently the number of sections cannot be increased carelessly.21 In next section, a model predictive control and a fuzzy predictive control are designed based on the derived MLD model. Their behaviors in present of disturbance and uncertainty are also studied.

Figure 7. Block diagram of a conventional MPC method.

Table 2. Comparison between Maximum Computation Time of Different Prediction Horizons prediction horizon (Hp)

maximum computation time (s)

1 2 3 4

infeasible 0.017 0.891 101.80

3. MODEL PREDICTIVE CONTROL AND FUZZY PREDICTIVE CONTROL FOR THE BOILER−TURBINE SYSTEM In the boiler−turbine system, our goals are regulation of state variables at desirable values and avoiding deviation of the water level in the drum by considering the input constraints (eqs 4). In this section, MPC and fuzzy predictive control are designed for the boiler− turbine system and applied on the system in various conditions. MPC is one of the mature and commonplace control approaches which is applied in many practical and academic systems, and it strictly depends on the model of the system.24−31 In the previous section, a relatively accurate hybrid model of the boiler−turbine system was obtained. Hybrid model predictive control (HMPC) is very similar to conventional MPC, except for its cost function in which the logical and continuous auxiliary variables are also considered.17,32 The open loop optimization problem (OOP) in hybrid systems is to minimize the cost function J(·) by meeting the constraints imposed on the controller within a constant prediction horizon (Hp) and subject to the system model which here is the MLD model. The following equation represents the OOP in hybrid systems subject to the system model.18

Figure 8. Block diagram of MPC method with fuzzy supervisor. The fuzzy supervisor adds a new term (G) into the cost function by using the reference signal (xr). Therefore, xrf is the new reference signal.

1 × 3, 1 × 45, 1 × 64, 394 × 3, 394 × 45, 394 × 64, 394 × 7, and 394 × 1, respectively. To evaluate the accuracy of the obtained MLD model using PWA approximation, a coefficient namely “coefficient of determination” is used to show the fitness of the MLD model. The coefficient of determination varies from zero to one, where “zero” means the model has no relation with the system, and “one” means that the model describes the system perfectly; on the other hand, the larger value of coefficient of determination indicates a better fitness of the model. The most general form of the coefficient of determination is shown in eq 9.23 R2 ≡ 1 −

SSerr SSerr − SSreg

(9)

2

where R is the coefficient of determination. SSerr and SSreg are the residual sum of squares and the explained sum of squares, respectively. They can be written as below: SSerr =

∑ (yi − yi ̂ )2

SSreg =

≅

(10-1)

i

∑ (yi ̂ − y ̅ )2

J(u(k), y(k), z(k), x(k))

min

u(k), δ(k), z(k) Hp

∑

Hp − 1

Wx(x(k + i|k) − xr(k))

i=1

p

+

∑

Wu(u(k + i|k) − ur(k))

+ Wz(z(k + i|k) − zr(k))

p

+ Wy(y(k + i|k) − yr (k)) ) p

(10-2)

i

p

i=0

(11-1)

Table 3. Nominal Values of State Variables of the Boiler−Turbine System15 operation point

1

2

3

N (normal)

4

5

6

power output drum pressure fluid density

15.27 75.60 299.6

36.65 86.40 342.4

50.52 97.2 385.2

66.65 108 428

85.06 118.8 470.8

105.8 129.6 513.6

128.9 140.4 556.4

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Figure 10. Membership functions of fuzzy system.

H1uk ≤ b1 , G1xk + H2uk ≤ b2

follows the reference signal and the inputs and the other state variables do not violate the constraints. The state variables of the boiler−turbine system are considered as x1 = p, x2 = p0, x3 = pf, x4 = Xw, x5 = du1, x6 = du2, and x7 = du3. So the cost function is defined as follows:

(11-2)

where J(·)is the cost function. Wx, Wu, Wz, and Wy are the proper constant weighting matrices, and the index p indicates the norm which determines the type of the OOP (if p = 2, then the problem is a mixed integer quadratic program and, if p = 1, ∞, then it is a mixed integer linear program). In inequalities 11-2, b1, b2, G1, H1, and H2 are the proper matrices which define the constraints and guarantee the stability of the system. A traditional MPC scheme is shown in Figure 7. Now, based on the MLD model of the boiler−turbine system, a model predictive controller is designed based on the defined cost function (eqs 12) to ensure that the output power (p0)

J(x1(k), x 2(k), x3(k)) Hp

=

∑ i=1

⎡ ⎤ ⎡ 742 0 0 ⎤⎢ x1(k + i|k) − xr1(k) ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 441 0 ⎥⎢ x 2(k + i|k) − xr 2(k) ⎥ ⎣ 0 ⎦ 0 1 ⎣⎢ x 3(k + i|k) − x (k)⎦⎥ r3

∞

(12-1) 2369



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⎡ 60 ⎤ ⎡ x1 ⎤ ⎡ 150 ⎤ ⎥ ⎢ ⎥ ⎢x ⎥ ⎢ ⎢ 5 ⎥ ⎢ 2 ⎥ ⎢ 140 ⎥ ⎢ 210 ⎥ ⎢ x3 ⎥ ⎢ 590 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ≤uk≤1, ⎢ −0.01 ⎥ ≤ ⎢ x4 ⎥ ≤ ⎢ 0.01 ⎥ ⎢−0.007 ⎥ ⎢ x5 ⎥ ⎢ 0.007 ⎥ ⎥ ⎢ ⎥ ⎢x ⎥ ⎢ ⎢ −2 ⎥ ⎢ 6 ⎥ ⎢ 0.02 ⎥ ⎣ −0.05 ⎦ ⎢⎣ x 7 ⎥⎦ ⎣ 0.05 ⎦

Table 4. Definition of CN, MN, N, LN, Z, LP, P, MP, and CP symbol

(12-2)

Table 5. Fuzzy Rule Base

in which, the values of the weighting matrix have been determined based on the importance of the variables and the relationships between the variables. In eq 12-1 the prediction horizon (Hp) must be chosen wisely in order to keep the computation time of the OOP in all conditions (nominal, disturbance, and uncertainty) less than the sampling time. To select the proper prediction horizon, a comparison is provided between maximum computation times of different prediction horizons in Table 2. According to the Table 2 all prediction horizons more than one are feasible, but due to the shortage of time for solving the optimization problem, the prediction horizon is set to three. In order to find the proper inputs, the OOP should be solved. Due to the cost function (eq 12-1) being an infinite-norm, the OOP is a mixed integer linear program. In this paper the OOP is solved by using hybrid toolbox which uses the GNU Linear Programming Kit (GLPK) to solve the OOP.20,33 The cost function (eqs 12) has been considered in its simplest form to show the efficiency of proposed fuzzy method even without well-developed model predictive controllers. According to eq 12-1, three reference signals are needed to be defined for MPC. Due to the demand of consumers being our priority, the reference signal of the output power (p0) is set based on demand of consumers. According to Table 3, two other reference signals are defined based on the relation between the output power and other variables. The MPC method is very sensitive to the model of the system, and any difference between the system and its model results in undesirable outcome which appears as tracking error. In order to solve this problem without increase in computation effort and complexity of controller, a fuzzy predictive control scheme is proposed. The fuzzy supervisor uses the reference signal to add a new term into the MPC based on the measured variables. This term is used to approximate actual values of variables. The proposed scheme is shown in Figure 8. In the proposed scheme, the MPC part is not modified, but the reference signal is defined as follow:

e

MN

N

LN

Z

LP

P

MP

dP0 MP P Z N MN

Z CP Z CP Z

Z CP MP CP Z

LP MP P MP LP

Z Z Z Z Z

LN MN N MN LN

Z CN MN CN Z

Z CN Z CN Z

Figure 11. Single membership function.

in which x(k + i|k) + G(k) can be rewritten as x̂(k + i|k), so we have 3

J(x(k)) =

=

(14)

This equation can be rearranged as follows: 3

J(x(k)) =

∑ i=1

Wx((x(k + i|k) + G(k)) − xr(k))

∞

(16)

⎡ ⎤ ⎡ 742 0 0 ⎤⎢ x1(k + i|k) − xr1(k) ⎥ ⎢ ⎥⎢ x (k + i|k) − x (k)⎥ rf 2 ⎢ 0 441 0 ⎥⎢ 2 ⎥ ⎣ 0 0 1 ⎦⎢ ⎣ x 3(k + i|k) − xr 3(k) ⎥⎦

∞

(17)

where xrf2 is the fuzzy reference signal of power demand. Two other reference signals (xr1, xr3) are defined based on Table 3 and using the fuzzy reference signal of power output. Now, the inputs of the fuzzy system should be defined based on our goals which are decreasing output error and avoiding from oscillation of variables in presence of disturbance and uncertainty. So by analyzing the behavior of the system and its variable in different conditions, two variables e and dp0 are proposed as inputs of the fuzzy system (F), where

3 ∞

∑ i=1

where xr, xrf, and F are the actual reference signals, fuzzy reference signals, and fuzzy function (F(k) = 0, k ≤ 0), respectively. The logical explanation behind the proposed method is that the fuzzy supervisor adds a new term (G) to the cost function through the reference signal. So by replacing xrf(k) in place of xr(k) in a simple form of eq 11-1, it can be rewritten as

i=1

∞

J(x1(k), x 2(k), x3(k))

(13)

∑ ∥Wx(x(k + i|k) − xr(k) + G(k))

Wx(x(̂ k + i|k) − xr(k))

where x̂ is an approximation of the actual value of x at sampling time k + i. So, the fuzzy supervisor actually compensates errors caused by disturbance or uncertainty in computing of future values of variables. Consequently based on the aforementioned explanation, the cost function (eq 12-1) should be rewritten as follows:

3

G(k) = F(k) + F(k − 1)

∑ i=1

xrf (k) = xr(k) − G(k)

J(x(k)) =

meaning

CN the variable is completely less than the desired value MN the variable is high less than the desired value N the variable is less than the desired value LN the variable is a little less than the desired value Z the variable has desired value For CP, MP, P and LP, “less” shall be replaced with “more”.

(15) 2370

e(k) = xr2(k) − x 2(k)

(18-1)

dp0 (k) = x 2(k) − x 2(k − 1)

(18-2)



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Figure 12. continued

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Figure 12. Comparison between the results of the MPC (a-1, ..., e-1) and the fuzzy predictive control (a-2, ..., e-2) in nominal conditions.

error and avoiding from excessive output oscillation. The triangular membership functions in Figure 10 have a mathematic equation as follows:

Table 6. Comparison between Final Values of Input Signals from MPC and Their Nominal Values final value from MPC

nominal value operation point

3

N

4

3

N

4

u1 u2 u3

0.271 0.621 0.34

0.34 0.69 0.435

0.418 0.759 0.543

0.270 0.621 0.339

0.339 0.689 0.435

0.418 0.758 0.543

⎧ ⎪ (ci ⎪ ⎪ μj , i (x) = ⎨ ⎪ (c ⎪ i ⎪ ⎩ 0,

in which e(k) and dp0(k) represent tracking error (MW) and pace of output change (MW/s), respectively. In Figure 9, the fuzzy system including a Mamdani-type fuzzy inference system is depicted. This system uses minimum inference engine. Triangular membership functions are used to create fuzzy sets from input signals due to their simplicity and ease of use. A center average defuzzifier is also used to generate output signal. Membership functions of fuzzifier and defuzzifier are shown in Figure 10. In Table 4 the definitions of CN, MN, N, LN, Z, LP, P, MP, and CP have been provided. In Figure 10, the triangular membership functions for inputs and output have been defined specifically to reduce the output

1 (x − ai), ai ≤ x ≤ ci − ai) 1 (x − bi), ci < x ≤ bi − bi) otherwise

⎧j = e and i = MN, ..., MP ⎪ ⎪ for ⎨ j = dp0 and i = MN, N, Z, P, MP ⎪ ⎪ j = F and i = CN, ..., CP ⎩

(19)

where μj,i is a triangular membership function and ai, bi, and ci are proper constant parameters. Consequently, the weight of the ith (i = CN, ..., CP) membership function for output (Figure 10c) equals the minimum 2373



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Figure 13. Comparison between the results of the MPC (a-1, ..., e-1) and the fuzzy predictive control (a-2, ..., e-2) in the presence of a disturbance affecting the system at t = 1000 s.

weight of the all membership functions μj,i (j = e, dp0, i = MN, ..., MP) for inputs (Figure 10a and b) which are related to the ith output membership function according to Table 5. To convert the fuzzy sets to the output signal, the center average defuzzifier is used with following equation. F=

∑i μFi yi ̅ ∑i μFi

,

Now to provide a comparison between designed MPC and fuzzy predictive control, both controllers are applied to the boiler−turbine system in nominal conditions, in the presence of disturbance and in the presence of uncertainty. It has been considered that the boiler−turbine system is at its normal operating point and then the designed controllers are applied to the system. The demands of consumers are changed according to the following sequence N → 4 → N → 3 → N (refering to Table 3). Results of applying the designed MPC and fuzzy predictive control are shown in Figure 12. As it has been shown in Figure 12, both controllers perform very well in the nominal conditions. All constraints over inputs are satisfied, and state variables follow their reference signals. It is clear that the fuzzy predictive controller converges the output power toward the desirable value; however, it increases the overshoot (undershoot) of the variables. The fuzzy predictive control also degrades other state variables very slightly, but still all variables stay in the operating ranges. A comparison between the final values of input signals achieved from the MPC and their nominal values15 are shown in Table 6.

i = CN, ..., CP (20)

in which, yi̅ is the center (ci in Figure 11) of the ith (i = CN, ..., CP) triangular membership function (Figure 10c), and μFi is the weight of the ith (i = CN, ..., CP) membership function for the defuzzifier. The fuzzy rules are presented in Table 5. The fuzzy rules are chosen based on analyzing the system behavior with model predictive controller in the presence of disturbance and uncertainty. The general form of rules is as the following statement: If e is ... and dh is ... then F is ...

where “...” should be filled according to Table 5. 2376



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Figure 14. Comparison between the results of the MPC (a-1, ..., e-1) and the fuzzy predictive control (a-2, ..., e-2) in the presence of uncertainty (Δf = −0.05).

According to Table 6, it is obvious that the results of the designed MPC based on the MLD model of the boiler−turbine system are very close to the results of the nominal value from the experiment. In the scenario of disturbance it is assumed that one of sieves in the fuel line has been broken and its particles have hung through the fuel line, the error happened suddenly. Disturbance is imposed on the system at 1000 s. The results of simulations are shown in Figure 13. By looking at Figure 13, it can be seen that the disturbance causes a persistent error in power output and oscillation in the deviation of water level in the drum and input signals under the MPC; however, oscillations of the input signals stay within the defined ranges. According to the results of the fuzzy predictive control, in the fault condition the proposed fuzzy controller has removed the persistent error in the power output and the oscillations in the deviation of water level in the drum and input signals. In addition it keeps two other state variables closer to their reference signals and in the operating range. The boiler−turbine system is a complex system, so acquiring an accurate differential model of this system is difficult and

complex. This complexity causes the acquired model to have some differences from the actual system which are considered here as an uncertainty in the boiler−turbine system. Uncertainty in the system is considered as follows: dp0 dt

= (0.073u 2 − 0.016)p9/8 − (0.1 + Δf )p0

|Δf | ≤ 0.05

(21)

The results of simulations in the worst case (Δf = −0.05) are shown in Figure 14. As shown in Figure 14, in the presence of uncertainty the power output of the boiler−turbine system under the MPC has a great persistent error. Conversely, the proposed fuzzy predictive controller in the presence of uncertainty has good results and put the power output at the desirable value; also, it satisfies all constraints in the system. In Table 7, a comparison is provided to discuss effects of the traditional MPC and the proposed fuzzy predictive control on 2379



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Table 7. Comparison between the Results of the Traditional MPC and the Proposed Fuzzy Predictive Control in Different Conditionsa traditional MPC maximum computation time (s) persistent error of P0 (%) digression from nominal value (%) P Pf a

fuzzy predictive control

normal

disturbance

uncertainty

normal

disturbance

uncertainty

0.89 0.13

0.95 0.48

0.93 9.45

0.91 0.00

0.96 0.00

0.97 0.00

0.00 0.01

0.42 23.36

0.05 6.66

0.07 0.00

0.12 6.54

4.03 1.87

All values were measured while the boiler−turbine system was working at its normal capacity.

Also a hybrid modeling approach, namely the MLD model, was used in order to obtain a relatively accurate linear model of the boiler−turbine system over its operating range. For linearization of nonlinear dynamics of system, the PWA approach was used. The results of simulations showed that the proposed fuzzy predictive control scheme has advantages such as simplicity and efficiency in the nominal condition and strong robustness in the presence of disturbance and uncertainty in comparison with the conventional MPC method. In addition it gives the ability of control management. Moreover, the proposed method only uses the measured states which are necessary for the conventional MPC and due to the existence of a cost function, the control signals, and outputs are optimal.

the variables of the boiler−turbine system and maximum computation time in different conditions. According to Table 7 and Figures 12−14, the proposed approach in comparison with the traditional MPC has improved the results significantly without dramatic increase in the computation time. One of the most important features of the proposed fuzzy predictive control regardless of its significant robustness is its ability in prevention of unwanted and ruinous oscillations of input signals and variables in the boiler−turbine system. According to the results of the simulations, the proposed fuzzy method is clearly competitive with conventional MPC. In addition, the proposed method reduces effects of uncertainty and disturbance significantly. However in this paper, fuzzy predictive control was designed for certain situations, it could be easily developed based on more possible situations in order to make a more comprehensive fuzzy predictive controller. In the following, practical considerations for application of the proposed method are discussed. From a practical view, setting sampling time at 1 s is feasible but challenging. The main keys in this matter are the computation power of the processor and solving algorithm. On the basis of Table 7, the OOP can definitely be solved within the sampling time for this system using the hybrid toolbox,20 but the matter is the cost of required hardware. In the other words based on the available hardware and solving algorithm, solving the OOP problem may take more than 100 s or less than 100 ms. So, in order to decrease the cost of hardware two possible solutions are suggested. The first solution is developing an optimal and quick algorithm for solving the OOP. The other way is to increasing the sampling time. However the sampling time cannot be increased carelessly because MPC only measures variables of the system at the beginning of each sampling time, so between each two consecutive sampling times the controller does not measures variables and any change in those variables cannot be seen by controller until the next sampling time. The other challenge is the delay time of the actuators and measuring devices. In this paper the delay time has not been considered. Regarding measuring devices, this time is very small and can be neglected. But the delay time of the actuators is generally around 500 ms in practice. That means it takes about 500 ms for a valve to start to operate from a stationary position. However this delay will not significantly affect the control process because the proposed approach is based on traditional MPC and model predictive controllers generate input signals based on the current measured variables of the system. Therefore this delay only will cost a few sampling times and prolong the needed time to reach the desire values. After a valve starts to operate, it will respond to the input signals with a negligible delay.

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ASSOCIATED CONTENT

S Supporting Information *

Matlab and HYSDEL codes for obtaining the MLD model of the boiler−turbine system.This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: (+98)-911-157-1140. Author Contributions †

Z.R. and B.R.: These authors contributed equally.

Notes

The authors declare no competing financial interest.

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ABBREVIATIONS PWA = piecewise affine MLD = mixed logical dynamical PI = proportional−integral LQG = linear quadratic Gaussian LTR = loop transfer recovery MPC = model predictive control MIMO = multi-input multi-output HYSDEL = hybrid systems description language HMPC = hybrid model predictive control OOP = open loop optimization problem GLPK = GNU Linear Programming Kit REFERENCES

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4. CONCLUSION In this paper, a novel fuzzy predictive control scheme has been proposed in which a fuzzy supervisor is used to regulate the reference signal of the MPC according to the states of the system. 2380



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