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NFN can be devised with the network incorporating all the local generalized predictive controllers (GPC) designed using the respective local linear models. Following this approach, the nonlinear generalized predictive controllers based on the NFN, or simply, the neuro-fuzzy generalized predictive controllers (NFG-PCs)are derived here. The proposed controller is then applied to control the superheated steam temperature of the 200MW power unit. Experimental data obtained from the plant are used to train the NFN model, and from which local GPC that form part of the NFGPC is then designed. The proposed controller is tested first on the simulation of the process, before applying it to control the power plant.
2. Neuro-fuzzy network modelling
Consider the following general single-input single-output nonlinear dynamic system:
'?1), y(t)?f[y(t?1),...,y(t?n'y),u(t?d),...,u(t?d?nu' e(t?1),...,e(t?ne)]?e(t)/? (1)
where f[.]is a smooth nonlinear function such that a Taylor series expansion exists, e(t)is a zero mean white noise andΔis the differencing operator,ny,nu,neand d are respectively the known orders and time delay of the system. Let the local linear model of the nonlinear system (1) at the operating pointo(t)be given by the following Controlled Auto-Regressive Integrated Moving Average (CARIMA) model:
?1?d?1?1 A(z)y(t)?zB(z)?u(t)?C(z)e(t) (2)
'''WhereA(z?1)??A(z?1),B(z?1)andC(z?1)are polynomials inz?1, the backward shift operator. Note that the coefficients of these polynomials are a function of the operating pointo(t).The nonlinear system (1) is partitioned into several operating regions, such that each region can be approximated by a local linear model. Since NFN is a class of associative memory networks with knowledge stored locally, they can be applied to model this class of nonlinear systems. A schematic diagram of the NFN is shown in Fig.2.B-spline functions are used as the membership functions in the
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NFN for the following reasons. First, B-spline functions can be readily specified by the order of the basis function and the number of inner knots. Second, they are defined on a bounded support, and the output of the basis function is always positive, i.e.,?kj(x)?0,x?[?j?k,?j]and?kj(x)?0,x?[?j?k,?j].Third, the basis functions form a partition of unity, i.e.,
??jjk(x)?1,x?[xmin,xmam]. (3)
And fourth, the output of the basis functions can be obtained by a recurrence equation.
Fig. 2 neuro-fuzzy network
The membership functions of the fuzzy variables derived from the fuzzy rules can be obtained by the tensor product of the univariate basis functions. As an example, consider the NFN shown in Fig.2, which consists of the following fuzzy rules: IF operating condition i (x1is positive small, ... , andxnis negative large),
THEN the output is given by the local CARIMA model i:
?i(t)?ai1y?i(t?1)?...?ainay?i(t?na)?bi0?ui(t?d)?... y ?binb?ui(t?d?nb)?ei(t)?...?cincei(t?nc) (4)
?i(t)?z?d?Bi(z?1)ui(t)?Ci(z?1)ei(t) (5) or Ai(z?1)yWhereAi(z?1),Bi(z?1)andCi(z?1)are polynomials in the backward shift operatorz?1, and d is the dead time of the plant,ui(t)is the control, and ei(t)is a zero mean independent random variable with a variance of ?2. The multivariate basis functionai(xk)is obtained by the tensor products of the univariate basis functions,
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ai???Aki(xk),i?1,2,...,p (6)
k?1nwhere n is the dimension of the input vector x, and p, the total number of weights in the NFN, is given by,
p??(Ri?ki) (7)
k?1nWhere kiand Ri are the order of the basis function and the number of inner knots respectively. The properties of the univariate B-spline basis functions described previously also apply to the multivariate basis function, which is defined on the hyper-rectangles. The output of the NFN is,
?a?yi?1ppiip?? y?ai?1?iai (8) ??yi?1i3. Neuro-fuzzy modelling and predictive control of superheated
steam temperature
Let?be the superheated steam temperature, and??, the flow of spray water to the high temperature superheater. The response of?can be approximated by a second order model:
G(s)????(s)?Kp(T1s?1)(T2s?1)e??s (9) The linear models, however, only a local model for the selected operating point. Since load is the unique antecedent variable, it is used to select the division between the local regions in the NFN. Based on this approach, the load is divided into five regions as shown in Fig.3,using also the experience of the operators, who regard a load of 200MW as high,180MW as medium high,160MW as medium,140MW as medium low and 120MW as low. For a sampling interval of 30s, the estimated linear local models A(z?1)used in the NFN are shown in Table 1.
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Fig. 3 Membership function for local models Table 1 Local CARIMA models in neuro-fuzzy model
Cascade control scheme is widely used to control the superheated steam temperature. Feed forward control, with the steam flow and the gas temperature as inputs, can be applied to provide a faster response to large variations in these two variables. In practice, the feed forward paths are activated only when there are significant changes in these variables. The control scheme also prevents the faster dynamics of the plant, i.e., the spray water valve and the water/steam mixing, from affecting the slower dynamics of the plant, i.e., the high temperature superheater. With the global nonlinear NFN model in Table 1, the proposed NFGPC scheme is shown in Fig.4.
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Fig. 4 NFGPC control of superheated steam temperature with feed-for-ward control.
As a further illustration, the power plant is simulated using the NFN model given in Table 1,and is controlled respectively by the NFGPC, the conventional linear GPC controller, and the cascaded PI controller while the load changes from 160MW to 200MW.The conventional linear GPC controller is the local controller designed for the“medium”operating region. The results are shown in Fig.5,showing that, as expected, the best performance is obtained from the NFGPC as it is designed based on a more accurate process model. This is followed by the conventional linear GPC controller. The performance of the conventional cascade PI controller is the worst, indicating that it is unable to control satisfactory the superheated steam temperature under large load changes. This may be the reason for controlling the power plant manually when there are large load changes.
Fig.5 comparison of the NFGPC, conventional linear GPC, and cascade PI controller.
4. Conclusions
The modeling and control of a 200 MW power plant using the neuro-fuzzy approach is presented in this paper. The NFN consists of five local CARIMA models.
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