MLD Model of Boiler-Turbine System Based on PWA Linearization Approach

In this paper we consider boiler-turb ine’s nonlinear dynamics and linearize the nonlinear parts based on the piecewise affine method in order to obtain a mixed logical dynamical model of the system. By using piecewise affine approach for linearization of the system’s nonlinear equations, the obtained linearized model switches between different modes based on its parameters, so acquired piecewise affine model can be categorized in the switching hybrid system class. We model the linearized boiler-turbine system in a mixed logical dynamical model of the hybrid systems using hybrid system’s description language and hybrid toolbox. Mixed logical dynamical model describes system by two linear equ ations and one linear inequality with a reasonable accuracy and considering the constraints in the system. We provide a comparison between the acquired mixed logical dynamical model using piecewise affine linearizat ion method and the actual boiler-turbine system through simulat ion and show the efficiency of the mixed logical dynamical model to describe the Boiler-Turbine system.


Introduction
In the study of power systems having a desirable model of system is important. A boiler-turb ine system is an energy conversion system that consists of a steam boiler and a turbine which uses chemical energy to generate electricity. Boiler-turbine system model first was represented at 1972 using nonlinear equations based on the boiler-turb ine plant P16/ G16 at the Sydsvenska Kraft AB Plant in Malmo, Sweden [1]. A schemat ic d iagram of the boiler-turbine unit is shown in Figure.1. After that the boiler-turbine's nonlinear model was studied widely in many art icles and was undergone a number of amends to provide a better description of system [2,3]. In o rder to design a controller some researchers used nonlinear form of system's equations [5,6,10 ] and in other works the linear fo rm o f the system was used [4,[7][8][9].
The boiler-turbine system can be modeled as a multi-input mu lt i-output (M IM O) non linear system. Th is system is strongly coupled and is subject to various constraints on both inputs and outputs. In literatures that used the linearized form of boiler-turbine system, they obtained a linearized model using a truncated Ty lo r series expans io n of n on lin ear equations around operating points [4,[7][8][9]. Linearization around operating points provides a relatively accurate description about system behaviour around these points, however for using of this method we should have nominal states' and inputs' values at operating points which are not readily availab le. Because of improvement in co mputational methods and hardware, today the control's technology is moving toward intelligent and computer based controller methods namely model predictive control. For using such control methods, we need to provide a model of system which describes the system's behaviour properly. A good way for providing a proper model of different classes of systems is using hybrid system's modelling approach specially mixed logical dynamical (M LD) method which describes system using two linear equations and one linear inequality. In o rder to obtain a MLD model of a system first the system's equations need to be linearized, so we use piecewise affine (PWA) approach for linearization in wh ich the space is divided into different pieces and in each section a linear equation is used for description of system behaviour in that section [11].
Here we use the boiler-turbine's nonlinear equations which represented in [3], to obtain a linear model of boiler-turbine system in the mixed logical dynamical framework using piecewise affine approach for linearizat ion. The rest of the paper is as follows. In Section 2, we introduce the mathematical model of the boiler-turbine system and use piecewise affine method for linearization. Next in section 3, we acquire a M LD model and provide simulations and comparisons of the achieved MLD model and the actual system in identical condition. Finally, concluding remarks are drawn in Sect ion 4.

Mathematical and Linear Model of Boiler-Turbine System
In this section, we consider the boiler-turbine system's continuous-time dynamical equations as follo w[3]: The output to the system are p, p 0, and Xw wh ich are respectively drum pressure, output power, and drum water level (in meters). Two first output are system's states and readily available, whereas water level is found through the following relat ionships: where q e is the evaporation rate ( sec / kg ) and cs  is the steam quality.
For obtaining the MLD model of the boiler-turb ine system according equations (1), (3) and by using Hybrid Systems Description Language (HYSDEL), first the nonlinear terms in these equations must be omitted. These nonlinear terms are presented at equation (4).
As we mentioned earlier, for linearization we use the PWA approach [11] Figure.2. Now according to linearized form of nonlinear terms, the boiler-turbine system's dynamics can be written in the PWA configuration which Equation (6) shows general formu lation of that [11].
where, k is the discrete-time step. x, u, and y denote states, inputs, and outputs, respectively. i  indicates a set of conditions which defines i-th section of space (for boiler-turbine we have 64 section). A i , B i , h i , C i , D i , and g i are proper time-invariant matrix related to section i. The concept of PWA model is depicted in Figure.3.
In next section according to the linear discrete-time equations of boiler-turbine system, which were obtained in this section, and using HYSDEL we will acquire the M LD model of system.

Mixed Logical Dynamical Model and Simulation
There are many different methods for modelling of hybrid system, namely Piecewise affine, M ixed logical dynamical, Linear co mplementarity, and Max-min-p lus-scaling [12].
Among these models, MLD method is more co mmon and has less complexity than other methods which models hybrid system using two linear equations (7-1) and (7-2) and one linear inequality . M LD approach has the following structure [13]: are the output and the input signal, respectively. Before obtaining the MLD model o f boiler-turbine system based on the PWA model (equation (6)), in order to impose the inputs' constraints that were defined at equations (2); we introduce 3 new states (8-1) and 3 new equations (8-2) as follow: (8-1) where T s is samp ling t ime. and denote the new state variable and input's pace, respectively.
Now the M LD model (7) of boiler-turbine system is obtained by using HYSDEL and hybrid toolbo x [14,15]. The MLD model of the system by using PWA method for linearization has the follo wing properties: 1. Sampling time (T s ) is 1s. 3. 63 continuous au xiliary variab les, 45 binary au xiliary variables, 392mixed-integer linear inequalities.
In following, we provide a co mparison between approximated MLD model and actual system under identical conditions and inputs signals. The results of simulation are shown in Figure.4.
By looking at Fig.4, it is obvious that using piecewise affine approach for linearizat ion describes actual system behavior at a reasonable accuracy level. In order to increase the accuracy of the MLD model of boiler-turbine system we should increase the number of part itions of the piecewise affine approach. However according to the MLD models' properties, we should consider that increase in partitions' number causes increase in the nu mber of au xiliary variables and mixed-integer linear inequalit ies related to the MLD model. Consequently, we need mo re co mputational effort to use this MLD model in order to design a controller for the actual boiler-turbine system

Conclusions
As we showed in this paper hybrid systems modelling approaches can be used not only for systems with inherently hybrid behaviours but also nonlinear systems, by using local linearization method namely PWA. We showed using MLD model method for modelling boiler-turb ine system as a hybrid system makes us able to impose constraint directly into the system model and provides a very simp le, systematic, and relatively accurate model o f system. This accuracy comes fro m the PWA method which we used for linearization of nonlinear terms but increase in accuracy also causes increase in co mputational effort for designing a proper controller based on the MLD model. So we should make a balance between accuracy and computational cost according to our purposes and available equip ments.