Hybrid modelling and control of a power plant three-stage reheater

Hybrid modelling and control of a power plant three-stage reheater J. Hlava Faculty of Mechatronics, Technical University of Liberec 461 17 Liberec, Studentská 2, Czech Republic, (e-mail: [email protected]) Abstract: The problem of modeling and control of a three-stage steam reheating process in a steam power plant is considered in this paper. Mathematical model of this process is marked by considerable nonlinearity. Moreover, switching of dynamic behavior occurs because individual reheat stages are bypassed depending on the value of manipulated variable. For both of these reasons, there is a little chance that replacing the traditional control system based on cascade PID/PI loops with advanced control methods such as MPC based on single linearized model may bring any performance improvement. However, both of these features make the steam reheating process a suitable object of modeling with a hybrid model in the form of a switched piecewise affine (PWA) system. Hybrid MPC based on this model then becomes a promising candidate approach to control. This paper describes the results of PWA modeling of the steam reheating process. A good agreement of responses between PWA model and nonlinear first principles model is shown. Following the development of this model, the design of a hybrid MPC controller is outlined. Keywords: Hybrid systems, model predictive control, piecewise affine systems, steam power plant control 1. INTRODUCTION Many fossil fueled power plants are now being rebuilt and retrofitted. They must meet the current environmental requirements. Moreover, they will operate in the conditions of a deregulated market where the requirement on fast following of changing load demands is of key importance. Load following operation implies the necessity of tighter control over a wide range of loads. In the wide range operation, the inherent nonlinearity of the power plant components becomes much more manifest than in the operation with roughly constant power output. Consequently, classical control systems based on PI/PID control loops and feedforward compensators are no more sufficient to meet the control performance specifications. Replacement with advanced control strategies is needed. Advanced control theory has much to offer and there have been many attempts to apply advanced control concepts to the control of various subsystems of the steam power plants. Large collection of research papers (Flynn, 2003) can be used as a reference. However, there have not been many attempts yet to take advantage of the potential offered by the hybrid systems theory. Although numerous applications of hybrid systems in many fields were reported (see e.g. the overview in Morari & Baric, 2006), the applications to the field of power plant control are rare. Moreover, these applications such as the hybrid control of cogeneration power plant described in (Ferrari-Trecate et al., 2004) and similar are usually focused on high level scheduling and economic optimization not on direct control of power plant subsystems. It is the objective of the present paper to show that hybrid systems theory is an appropriate tool for modeling and

control tasks in the power plant engineering field. In particular, this paper will be focused on modeling and control of one important subsystem of a steam power plant: three stage reheater process. This process features considerable nonlinearity and individual stages of this process are switched on or off depending on the current value of manipulated variable. It will be shown that this process can well be modeled as a piecewise affine (PWA) system and its control can be based hybrid model predictive control. 2. STEAM REHEATER Reheater is used in big steam power plant units (above 100 MW) to increase Rankine cycle efficiency. Process schematic of the reheat Rankine cycle is shown in Fig. 1. The temperature of the steam leaving high pressure stage of the turbine is increased again by the reheater before entering the low pressure (or intermediate pressure) stage of the turbine. The reheater is usually a large heat exchanger where steam is heated by hot flue gas and temperature control is accomplished by a flue gas flow proportioning system.

Fig. 1 Rankine cycle with reheat

Fig. 2 Schematic of the three stage reheater process The dynamic response of this process is slow due to the big heat capacity of the reheater metal wall. To speed up the response, this method of control is often combined with water spray attemperator at reheater output. In this paper, the even more complex structure of the reheater process shown in Fig. 2 is considered. It is used in some Czech steam power plants that are now being refurbished. The whole process is divided into three stages. Low pressure steam is first reheated by hot high pressure steam in counter current heat exchanger HE1. This exchanger is followed by additional two reheat stages where the steam is reheated by hot flue gas. Control of the temperature of the reheated steam ΤRH is done by three control valves. Diverting three way valves V1 and V2 change the relative portions of the steam that flow through exchangers HE1 and HE2 and through the bypass lines. If the power plant operates in a regime where the heat transferred to the steam by exchanger HE3 alone is high enough to achieve the reference value of ΤRH, exchangers HE1 and HE2 are fully bypassed. Otherwise portions of steam flow must go through HE1 and HE2. Valve V3 controls the cold feedwater flow to attemperator spray. It is used during transients to decrease the temperature of reheated steam quickly. It should be closed in the steady state operation because cold feedwater decreases cycle efficiency. Control valves operate in a split-range configuration. The control action is just negative (cooling). If the manipulated variable is 0 (i.e. zero cooling), the steam is reheated to the maximum possible extent: V3 is closed and both bypass lines are fully closed. If the manipulated variable is in the range 034%, valve V1 proportionally increases the steam flow through the first bypass line and when manipulated value reaches 34%, the first by pass line is fully open a no steam goes through HE1. In the same way, the second bypass line opens if manipulated variable is between 31 and 68%. Cold water spray control valve V3 opens if manipulated variable is between 65 and 100%. Thus, the steam flow from the inlet to the outlet of the reheater may go through three, two or one heat exchanger depending on the value of the manipulated variable. Moreover, steam flow through heat exchangers HE1 and HE2 varies from zero to maximum because of bypassing. As a result of it, switches of dynamic behavior occur and the dynamics of the heat exchangers become highly nonlinear due to the large flow variations. It is therefore very natural to model this process as a hybrid system. Hybrid modeling can account for dynamics switches.

It is also a well known fact, which was stated already in (Sonntag, 1981), that more general nonlinear systems can be globally approximated arbitrarily closely by PWA systems. Thus, the nonlinear dynamics of heat exchangers with varying flow can be modeled by PWA systems. As a first step to the PWA model it is necessary to discuss briefly the nonlinear first principles model of the reheater process. 3. MATHEMATICAL MODEL OF THE REHEATER First, a simplified control oriented nonlinear model with lumped parameters will be outlined. This model is based mainly on the hints given in (Hubka, 2009), (Colonna & van Putten, 2007) and (Lu & Hogg, 2000). The basis of the model is mass, momentum and energy conservation equations. The dynamics associated with energy storage (thermal process) are much slower than mass and momentum conservation dynamics (hydrodynamic process). Thus, the hydrodynamic process can be assumed to be in a quasi-stationary state and the equations for pressures and mass flows are reduced to algebraic form. Thus, input and output mass flowrates can be considered equal even during transients. Further, heat transfer correlations and thermodynamic fluid properties must be added. In this paper the IAPWS-IF97 thermodynamic properties of water and steam were used as implemented in FluidProp 2.3 software (Colonna & van der Stelt, 2004). Convective heat flux between metal wall and fluid (steam or flue gas) flow is expressed by (1) q = UA∆T where A is the heat transferring area, U is the convective heat transfer coefficient and ∆T is the temperature difference between fluid temperature and wall temperature. For controloriented modeling of convective heat transfer the following expression is usually adopted (Colonna & van Putten, 2007)

U = αw n (2) where w is fluid mass flow rate, α is constant. If heat transfer between metal wall and steam is considered n=0.8, in the case of heat transfer between flue gas and metal wall n=0.6. Similarly as in (Colonna & van Putten, 2007) and (Lu & Hogg, 2000) the modeling is based on averaged variables. It is assumed that the pressure drops take place in the inlet, the steam compressibility within the heat exchanger volume is negligible and outer tube is perfectly insulated. Exchanger HE1 is a tubular countercurrent steam to steam exchanger. Hot steam flows in the inner tube (volume V1h), cold steam

flows in the annulus between inner and outer tube (volume V1c). Under these assumptions, energy balance equations for the hot high pressure and cold low pressure steam and for the metal wall yield the following model of HE1

dT1co 1 = w1ci (h1ci − h1co ) + A1outα1out w10ci.8 (T1w − T1c ) (3) dt V1c ρ1c c p1c

(

)

dT1ho 1 = w1hi (h1hi − h1ho ) + A1inα 1int w10hi.8 (T1w − T1h ) (4) dt V1h ρ1h c p1h

(

)

(

)

dT1w A α w0.8 (T − T ) − A1outα1out w10ci.8 (T1w − T1c ) = 1in 1in 1hi 1h 1w ρ wV1w cw dt

(5)

T1c = 0.5(T1ci + T1co ); T1h = 0.5(T1hi + T1ho ) (6) Subscripts indicate the points where the respective variables are considered. Their meaning follows from Fig. 2. Steam density ρ1c, ρ1h and specific heat cp1c, cp1h are nonlinear functions of temperature and pressure. Subscript w denotes the constants and variables pertinent to the metal wall between hot and cold steam flows. A1out is the outer surface area of the inner tube in contact with cold steam, A1in is the inner surface area in contact with hot steam, α1out and α1in are respective constants from heat transfer equation (2). Pressure drop across reheater heat exchangers is low. Output pressure of the cold steam in HE1 can be expressed as

p1co = p1ci − ξ1c w12ci ρ1c

(7)

where ξ is the pressure loss coefficient. Similar expressions can be written for hot steam and for other heat exchangers. Model of HE1 must be complemented with the diverting and mixing. As the mass and energy storage associated with these processes is negligible in comparison with the heat exchanger itself, steady state models are sufficient.

w1cb = a1 w1cit ; w1ci = (1 − a1 )w1cit

(8)

w1co h1co + w1cb h1cb = (1 − a1 )h1co + a1h1cb (9) w1co + w1cb Coefficient a1 is used to express the position of valve V1. It varies from 0 (no bypass) to 1 (exchanger is fully bypassed). Bypassing and mixing after HE2 is modelled in the same way. The respective coefficient is denoted as a2. Temperatures T1m and T2m are calculated from the respective enthalpies using thermodynamic properties of steam. h1mix =

Flue gas to steam exchangers HE2 and HE3 are modelled in a similar way with some modifications. The dynamics of the gas stream can be neglected as the time constant involved is very small. The metal/steam heat transfer coefficient is very high. Consequently, the steam and metal wall temperatures are close to each other and can be considered equal. Under these assumptions it is possible to add both sides of equations (3) and (5) and to obtain the model of HE2 in the form

(V

2c

ρ 2 c c p 2 c + V2 w ρ w cw )

(

+ A2U g Tg − T2co

)

dT2co = w1cit (1 − a2 )(h2ci − h2co ) + dt (10)

where Tg is the average flue gas temperature and Ug is flue gas to metal heat transfer coefficient. Similar equation can be written for HE3. On the other hand, exchangers HE2 and

HE3 have much larger physical dimensions than HE1. To achieve better accuracy (in particular in response to input temperature variations) it is beneficial to divide them into several compartments of the same length and write equation (10) for each compartment separately. The remaining part of the process is the spray attemperator. It has a very small volume with negligible mass and energy storage. Its model is based on steady state balances.

w3ci = w2 m + w fw ; h3ci = ( w2 m h2 m + w fw h fw ) w3ci

(11)

The flow of the cold feedwater to the attemperator can be represented by a normalized coefficient a3

w fw = a3 w fw max ; a3 ∈ 0,1

(12)

From the control viewpoint, a1, a2, a3 represent the manipulated variable and their values are changed using the split range control technique as it was described in section 2. 4. APPROXIMATE PWA MODEL OF THE REHEATER The general form of a discrete-time PWA system is given by

x(k + 1) = M m x(k ) + N m u(k ) + f m

(13) y ( k ) = C m x( k ) + D m u( k ) + g m where each dynamics m=1,2..NĐ is active in a polyhedral partition Đ that is defined by guard lines described by

G mx x(k ) + G um u(k ) ≤ G cm (14) That means, the dynamics m represented by matrices and vectors [Mm, Nm, fm, Cm, Dm, gm] is active in the region of state-input space which satisfies constraints (14). There are several methods for experimental identification of PWA models. However, there is no general procedure to find a PWA approximation of a nonlinear system described by analytical state equations. The route to the PWA approximation is always closely associated with a particular system to be approximated. For example, (Hlava, 2009) elaborates a PWA model of a complex laboratory scale plant with hybrid dynamics, the structure of which was given in (Hlava & Šulc, 2008). The selection of representative operating points and respective local linearized models could be based in part on the natural partitioning of the state-input space given by the combinations of discrete valued control inputs. Further finer partitioning of the continuous valued dynamics was done in such a way as to keep the relative change in dynamic behaviour within each partition the same. PWA approximation of the reheater process must be done in a different way. This process is a subsystem of a steam power plant. Reheater inputs such as steam flows, temperatures and pressures are given by other power plant subsystems. The values of these inputs vary but their changes are not independent and their nominal values depend on the plant power output. To get some idea about the relation between plant power output and reheater input, the data for the power plant considered in this paper are summarized in Table 1. The plant power output is given in percent of the maximum. Pressures and mass flows grow nearly linearly with the power output, temperatures exhibit a maximum followed by a slow decrease.

Table 1 Reheater input variables as a function of plant power output

a31 =

A1outα1out w10ci.8 A α w 0.8 ; a32 = 1in 1in 1hi ; w1ci = (1 − a1 ) w1cit 2 ρ w cwVw 2 ρ w cwVw

(18)

Pout

T1ci

p1ci

w1cit

T1hi

p1hi

w1hi

[%]

[°C]

[MPa]

[kg/s]

[°C]

[MPa]

[kg/s]

A α w 0.8 + A1inα1in w10hi.8 a33 = − 1out 1out 1ci ρ w cwVw

30

300

1.275

47.5

400

6.2

54.5

Steam properties c p1c , c p1h , ρ1c , ρ1h are functions of respective

50

366

2.075

80

471.2

10

88.4

70

360

2.8

110.6

463.4

13.7

122.4

100

352

3.95

156.6

451.8

19.3

173.4

4.1. PWA Models of individual components

Model (3)-(5) together with (8) has 3 states and 7 inputs but just a1 is a manipulated variable. Other input variables change with power plant power output as given in Table 1. The model can generally be written as T x& (t ) = f (x(t ), υ(t ) ); x = [T1co T1ho T1w ] (15) T θ = [T1ci p1ci w1cit T1hi p1hi w1hi ], υ = [θ a1 ] Denoting the steady state values by an overbar, the steady state values of the state variables can be computed numerically from a set of nonlinear algebraic equations

f (x, υ ) = 0 (16) Approximation of nonlinearities in state and input of (15) by PWA relations results in a PWA system where the matrices in each region of state-input space are non-constant. They depend on the vector of scheduling variables θ. Using the terminology introduced in (Besselmann & Morari, 2008) this can be called a hybrid parameter-varying model. To make the problem more tractable, this model will be replaced with a PWA model with constant system matrices where the scheduling variables are fixed to values given by nominal operating points. Nominal models are defined at specified power output levels. Their selection depends on the required accuracy. The switching between models can be based on any single variable from Table 1 that is a monotonous function of plant power output and that is measured. A good choice is e.g. p1ci. At each nominal power output level further finer partitioning is introduced by dividing the range of the colder steam flow into several subranges (nominal models at e.g. a1=0.05, 0.15 etc.). At any steady state operating point x m , υm , m = 1,2,...N Ð , HE1 is described by

x& (t ) = A m (x(t ) − x m ) + b m (a1 (t ) − a1m ) = A m x(t ) + b m a1 (t ) + o m  ∂f   ∂f  Am =   ; Bm =   ; o m = − A m x m − b m a1m  ∂x  xm ,υm  ∂u  xm ,υm (17) Elements of matrix Am computed at operating point T1co T1ho T1w T1ci p1ci w1cit a1 T1hi p1hi w1hi m are given by

[

a11 = −

]

0. 8 1ci

0.5 A1outα1out w

a21 = 0; a22 = −

+ w1ci c p1co

V1c c p1c ρ1c

; a12 = 0; a13 =

w1hi c p1ho + 0.5 A1inα1in w10hi.8 V1h c p1h ρ1h

; a23 =

0.8 1ci

A1outα 1out w V1c c p1c ρ1c A1inα1in w10hi.8 V1h c p1h ρ1h

steady state values of output pressures and temperatures. Equation (7) was used to compute the output pressures that are necessary to calculate the values these properties. Elements of the vectors bm can be computed similarly as Am. They are not given here to save the space. The nonlinearity of the dynamics of heat exchanger HE 1 is considerable. If the dominant time constant of the models at various operating points (30%, 40% ..,100% power output levels with further finer partitioning done at several different values of a1) is computed, it varies from 10 to about 60 s for the particular heat exchanger considered. Output from the first stage of the reheater is temperature T1mix. It is computed from (9) using the thermodynamic properties of the steam T = Θ(h, p) to make the conversion from enthalpy to temperature. Pressure drops across both HE1 and its bypass are small and roughly comparable. This allows taking the pressure of the mixed steam p1mix as approximately equal to p1co. Temperature T1mix is then

T1mix = Θ((1 − a1 )h1co + a1h1cb , p1co ) (19) Linearized equation on which control is based includes the influence of state and manipulated variable.

T1mix = T1mix +

∂Θ (1 − a1 )c p1co (T1co − T1co ) + ∂h

(20) ∂Θ + (h1cb − h1co )(a1 − a1 ) ∂h The partial derivative is taken in respective operating point. Comparison with (13) gives the following values of elements of cm, dm, gm

∂Θ ∂Θ (1 − a1 )c p1co ; d = (h1cb − h1co ); c12 = c13 = 0 ∂h ∂h (21) ∂Θ ∂Θ g11 = T1mix − (1 − a1 )c p1coT1co − (h1cb − h1co )a1 ∂h ∂h The linearization of HE2 and HE3 at selected operating points is then done in a similar way but with some important differences. In the case of HE2, there are two input variables a2 and T2ci=T1mix. The output T2m is obtained in the same way as T1mix in (20). Spray attemperator is modeled as a purely static element. It does not have its own state variable. Its presence merely modifies the output equation of the model of HE2. Heat exchanger HE3 has no bypass and mixing. In its modeling just the response to the changes of T3ci is important.

c11 =

4.2. PWA Model of the whole Reheater

The order of the model of the whole reheater process depends on the number of compartments used to model HE2 and HE3. In the simplest case when just one compartment is used, a 5th T order PWA model with state vector xRH = [T1co T1ho T1w T2co TRH ]

is obtained. The whole process has one manipulated variable u ∈ 0,1 and this variable is projected to the values of input variables a1, a2 and a3 according to the split range control rule as it was specified in section 2 of this paper.

u if u ≤ 0.34, a1 = 1 if u > 0.34; a2 = 0 if u ≤ 0.31 0.34 (u − 0.31) a2 = if 0.31 < u ≤ 0.68, a2 = 1 if u > 0.68 (22) 0.37 (u − 0.65) a3 = 0 if u ≤ 0.65, a3 = if u > 0.65 0.35 Rule (22) considerably limits the number of possible combinations of values of a1, a2 and a3. The number of partitions is therefore much smaller than it might seem from the models of the individual components. If the nominal models are defined at 30%, 40%,..., 100% power output levels and the whole range of u is represented by 20 nominal points (unequally spaced to cover the overlaps in (22)), total number of 160 partitions is obtained. a1 =

Fig. 3), the error is slightly less than 2°C. This is quite acceptable. If necessary, this error can be decreased by using finer partitioning of the range nominal power outputs. Fig. 4 shows the response of the whole process. The responses are considerably slower because the dimensions and thermal capacity of HE2 and HE3 are higher than those of HE1. The responses are to power output changes (from 50 to 80% at 3000 s and from 80 to 95 at 6000 s and to change of u from 0.1 to 0.25 at 9000). Again the agreement is relatively good. It should also be noted that the step changes are used to compare the behaviour of the full non-linear model and its PWA approximation because they are standard input signals. However, both plant power output and valve position physically cannot be changed in steps.

Reheater process has one controlled variable TRH. Basically, it is a SISO system. However, temperatures T2co and T3ci are measured and constraints on their values exist. These constraints can be included in MPC setup. It can be seen from (18) and (21) that if a1=1 (full bypass), the dynamics of T1co completely disappear. The respective row in matrix A is zero as well as coefficient c11. This change of system order is undesirable because available methods for design of MPC for PWA systems assume that all partial models are of the same order. However, no partial model is defined at a1=1 but the selection of nominal values of u is done in such a way as to have a1 at some value close to 1. Thus the contribution of the dynamics of the respective exchanger to the overall dynamic behavior of the whole reheater is almost negligible but the order of all partial models remains the same.

Fig. 3 Responses of HE 1

Continuous time PWA model of the reheater process outlined here must finally be converted to discrete time. The discretization is done by discretizing each partial model separately assuming zero order hold at the inputs. Fig. 4 Temperature of the reheated steam TRH 4.3 Comparison with the Nonlinear Model

The responses of the PWA model are compared with those of the nonlinear system in Figs. 3 and 4. Fig. 3 shows the response of exchanger HE1. The response is to the plant power output changes (50 to 80% at 500 s, 80 to 95 at 1000 s, 95 to 70% at 1500 s) and manipulated variable change (u change from 0.1 to 0.25 at 2000 s, i.e. change of a1 from 0.3 to 0.74). The responses of PWA system were simulated using MPT Toolbox (Kvasnica et al., 2004). Sampling period is 5 s. Responses of the nonlinear model are plotted with dotted line; PWA model is plotted with solid line. It can be seen that agreement of responses to manipulated variable changes between PWA approximation and nonlinear model is very good. The quality of response to plant power output changes depend on how far is the output from nominal operating point. In the worst case when the power output is just between two nominal operating points (95% power output in

5. CONTROL DESIGN AND EXPERIMENTS Model predictive control (MPC) of PWA systems has recently attracted a considerable research attention (see e.g. Christophersen, 2007; Borrelli, 2003) and it has already reached a considerable level of maturity. Controller design is now made much easier by the existence of specialized Matlab Toolboxes. In this paper, MPT Toolbox (Kvasnica et al., 2004) was used. The main task of the reheater control system is to keep the temperature of the reheated steam constant in spite of the power plant power output changes. In the case of the particular reheater considered here the reference value of TRH is set to 580°C. This task is relatively complicated because power output changes considerably modify reheater dynamics. Limited space of this paper does not allow treating

this topic in a more depth. It can just be stated that the current results indicate that model predictive control based on PWA model clearly has the potential to outperform the traditional control. One comparison is shown in Fig. 5. It is the response to step change of power plant power output from 100% to 60%. However, the experiments are now ongoing and many questions regarding optimum selection of MPC parameters, sensitivity to unmodeled dynamics of valves and sensors and plant-model mismatch are yet not completely solved.

Fig. 5 Response of TRH to the change of power output 6. CONCLUSION At present, most applications of hybrid modelling and control can be found in such fields like automotive control, mechatronics, power electronics and similar. The purpose of this paper was to show that hybrid systems can also be beneficial in other less traditional applications such as modelling and control of steam power plant subsystems. The particular subsystem considered was three-stage reheater process. It was shown that this process, which is nonlinear and also exhibits certain hybrid features, can adequately be modeled as a PWA system. Consequently, the task of designing advanced control system for this process can be formulated as a design of a model predictive controller for a PWA system. This can be considered an advantage because MPC for PWA systems is now a more mature and easier to apply approach than full nonlinear MPC. The paper describes results of an ongoing project. Due to this fact and also because of the limited space of the paper, the presented model was limited to the most important dynamic features of the reheater process. It has been shown that they can well be approximated by a PWA system. However, a full model that could be expected to serve as a basis for practically applicable control system will have to include also the dynamics and nonlinearity of control valves and sensors. The design of hybrid MPC for reheater process was just briefly outlined in order to show that this approach is an appropriate choice for the reheater process. Detailed treatment of this topic will have to be a subject of a separate paper. It should also be noted that the reheater is a part of a supercritical power plant with a once-through boiler. Oncethrough boilers are known to pose many challenging control problems. Thus, the next research step will be to evaluate whether and to which extent the approach based on PWA

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