Nonlinear Modelling of Electric Power Plants

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NONLINEAR MODELLING OF ELECTRIC POWER PLANTS P. Bourdon*, H. Dang Van Mien* and D. Normand-Cyrot** :;'/:'In/ririlr: rll'

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ABSTRACT: A non 1 inear identif ication method by simple state-space models is proposed in the discrete-time case . Th1S approach . 1S suggested by theoretical results similar to the approx1mat1on property known for a long time for Volterra ser1es . The proposed techn ique uses extens ions with polynom1al 1nput.s of b 111near systems called state-aff ine systems. Th1S modell1ng requues a small computing time and enables us to obta1n non-Ilnear models which can be easily implemented on microprocessors and remaln valld over a wide range of operating conditions. Several physical examples concerning electric power plants are also reported. KEYWORDS: I.

I dent if icat ion; nonl inear systems; power plants. In the discrete time case. state-affine systems which are polynomial control extens ions of b il inear systems ver lfy an approximation property (Pliess and one of the authors. 1980) . Let us note the link between these results and the approximation by Volterra series which was at the origin of a great deal of work in nonlinear identification. Various physical examples. dealing with electrical. hydraulic and even nuclear power plants are given in the single input single output case (cf. the authors. 1980. 1983). in the single input several outputs case (cf . Pliess and the authors. 1982). This sucessfull application confirms the usefulness to consider. in identification problems, bilinear models or their extensions . This has already been very effective in other contexts (see Beghelli and Guidorzi. 1976. Espana and Landau, 1975, 1978) .

Introduction

The nonlinear identification problem is far from being satisfactor ily solved in spite of all the techniques that have been developed over the last two decades (cl.surveys of Billings. 1980). Amongst all these various techniques. none can be reconunended. the choice being strongly dependent on the physical process to be identified. For example. the one uSlng physical equations is often not easy since the the number of parameters to be adjusted by means of exper iments rapidly grows with the complexity of the process . Another conunon approach. known for a long time. is based on the volterra serie s representation. This procedure which necessitates the difficult measurement of the volterra kernels is also quite involved (cf . Schetzen. 1980 ) . The aim of this paper is to propo s e a new nonl inear ident i f icat lon techn ique by slmple state- s pace models and to glve practical applications dealing wlth electric power plants. Because of some obv ious advantages in computation and s imulation. a di s crete tlme approach is chosen. The proposed procedure applies to processes that have a linear representation around ea c h operating point. When a physi c al proces s can be represented by linear model. thlS one remains valid only for small variations of the controls or around an operating pOint . The main lnterest of a nonlinear modelling is therefore to obtain an overall model which reproduces the dynamic and static evolution of the proce~s over a wide range of operating conditions . The technique here developed achieves this obJective. at least for the studied physical examples. This approach which uses state- s pace models is theoretically suggested by an approximation result obtained by Fliess (1976) and Sussmann (1976). "Any nonlinear system c an be arbltrar ily approximated in the continuous time c ase by f ln i te d lmens ional bi 1 inear systems . "

This work has been supported by the Research Center of Electricity of France. 11- Identification procedure

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Single input single output case

Given a physical process. we assume that there exists a set of stable equilibr ium states ( a set of operating points) which can be character ized by the value of parameters. These parameters must be directly (physical measurement or lnd1rectly (computation case) determined. For any value of these parameters, denoted'
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111- Industrial applications 10) Thermal power plant (Fig.l The nonlinear model is compared to the real process in parallel to a linear variation of the power between 60 MW and 120MW. The input signal is pseudorandom, the steam flow rate stays between -2. 5m 3 /s . and +2. 5m 3 /s. The output is the superheat temperature ( Fig.2 ) . 2 0 ) Hydraulic power plant (Fig. 3

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Single Input s everal outputs case

The direct extension of this method to several outputs systems lead to models of great dimension. In order to obtain reduced models (of mInimum state space dimension) a reduction algorithm, theorically developed by Fliess (1976), has to be used . This technique which is explained in Bourdon (1982) , is based on an exten s ion to nonlinear systems of Hankel matr1x notion. Remarks: (i) The choice of the products of inputs Pi ( {, e) in (L) is suggested by the dependance between phys ical character istics of the process (static gain, time constant, damping coef f ic ients) and the oper at ing conditions which are represented by the parameters ( i . (ii)When a state space representation (L) is obtained, it is usefull to improve the results by mod ify ing the output matr ix A in order to minimize a quadratic error cr iter ium between the real data and the simulation results.

In order to apply this identification technique to an oscilling system, we have studied the wave propagation in an hydraulic system. In that case, the input e is the water flow rate turbined and the output, the amplitude of the wave. The simulations are performed for different values of the main water flow rate of the river. Fig.4 is the compar ison between the nonlinear model and the real process according to a linear variation of the main flow and pseudo-random signal of the water flow turbined between +100m 3 / s. and -100m 3 /s . 3 0 ) Steam eXChanger of a nuclear power plant(Fig.5 ) We have studied the a c tion of the sodium flow rate on the steam pressure, the steam flow rate, the thermal power given to the turbine, the temperature of the secondary sodium exchanger . The obtained nonlinear model is of the form (2) with four products of inputs. The matrices are 17 dimensional square matrices. The reduction algorithm leads to 12 dimensional square matrices. The output matr ix is adjusted by a least square method. In Fig.6, the load of the plant is fixed at 15%, 40%, 70% and 90% of the nominal electric power plant. The efficiency of the adjustment of the output matrix is illustrated especially in Fig . 6c and 6d. The physical model and the reduced overall nonlinear model are compared for steps responses (negative and positive) to the sodium flow rate . A lot of simulations which cannot be included, have been carried out at the E.D . F. computer center. These results show the validity of the overall model submitted to an increase in the load between 10% and 90% of the nominal electr ic power, to negative and positive sodium flow steps untill 20% of the normal flow as well as to represent a nonlinearity due to the input sign. In the same way, have been tested other controls such as the water flow rate (F ig. 7 ) IV-Conclusion This identification technique leads to substitute many linear models with an

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of Elec tri c Powe r Plan t s

overall nonlinear model given by recursive equations and thus greatly simplifies simulation work. With this method, two kinds of nonlinearities can be represented, with respect to the operating mode and with respect the control , for example in the case of a dissymetry linked to the input sign. This approah is also very effective as far as its computing time and its implementation on microprocessors are concerned. At this time. the identified model dealing with parts of the nuclear plant is implemented on microprocessor Texas 9900 and used to test adaptive control algor ithms. Many developments suh as modelling of the whole nuclear plant which is a multi-input multioutput system, are used for a simulator in real time. In this multivariable case, minimal realizations are required. This is achieved by means of a reductIon algorithm of nonlinear models which is developed in connection with a generalized Hankel matrix notion to nonlinear systems.

V-References

[lJ Beghelli S., and R. Guidorzi (1976). Billnear systems identification from input-output sequences. IV IFAC Symposium on IdentifIcation and System Parameter Estimation, Tbilisi. 17591"164.

[2J Billings S.A. (1980). IdentifIcation of nonlinear systems-a survey, IEEE Proc., 127, D. 272-285. [3J Bourd o n P. (1982). Techniques non lineaire s en temps discret d'identification et de realisation minimale par modeles a et at aff ine. These de docteur-ingenieur . Universite Paris XI. Orsay. [4J Bourdon P., H. Dang Van Mien. M. F1iess and D . Normand-Cyrot (1982), ApplIcations to nuclear power plants of nonlinear identification and r eallZat ion techn iques. VI I FAC Symp. on Identification and System Parameter Estimation . Washington, 2. 1514-1519. [SJ Dang Van Mien H. (1970), Regulation numerique de la temperature de surchauffe d'un generateur de vapeur. Centrale de Pont-sur-Sambre. Revue Automatisme, 12, 617-626.

[6J Dang Van Mien H. (1973). Utilisation pratique de l' algor ithme de B.L.HO. Application a la reduction des systemes multidimensionnels. Amelioration par la methode des moindres carres . Bull . E.D.F. Etudes et Recherches, serie C, Math. Info .• 2. 23-46. [7J Dang Van Mien H. and D. NormandCyrot (1980). Nonl inear state-af fine identification methods. applications to electric power plants, Proc. IFAC Symp. Aut. Cont. Power Gener at ion D istr ibut ion and Protection. Pretoria. (J.F. Herbst ed.) Pergamon Press. Oxford. 449-462. [8J Dang Van Mien H. and D. NormandCyrot (1983). Nonlinear state-affine identification methods, applications to electr ic power plants, Automatica to appear. [9J Espana M. and I .D. Landau (1975). Bilinear approximation of the distillation processes, Ric. Autom . , 6. nOl o [lOJ Espana M. and I . D. Landau (1978). Reduced order bilinear model for distillation columns. Automatica, 14, 345355. [11) Fliess M. (1976), Un outil algebrique:les series formelles non commutatives, in "Mathematical Systems Theory" (G.Marchesini and S.K. Mitter. eds.) Lect. Notes Econom. Math. Syst. 131. p . 122-148, Springer-Verlag. Berlin. [12) Fliess M. et D. Normand-Cyrot (1980) • Vers une approche algebr ique des systemes non lineaires en temps discret. IVeme Conf. Int. Analyse et Optimisation des systemes. INRIA. Versailles. [13) Kalman R.E .• Falb P.L. and Arbib M.A. (1969). Topics in Mathematical System Theory. McGraw-Hill. New York. [14) Normand-Cyrot D. (1978), Utilisation de certaines familles algebriques de systemes non linealres a quelques problemes de filtrage et d'identification. These de 3eme cycle. Universite Paris VII. Paris . [15) Schetzen M. (1980). The Volterra and Wiener theories of nonlinear systems. Wiley, New York. [16) Sussmann H.J. (1976). Semigroup representations. Bilinear approximation of input-output maps and generalized inputs. in "Mathemat ical Systems Theory" (G. Marchesini and S.K. Mitter. eds.). Lect. Notes Econom. Math. Syst .. 131. p.172-191. Springer-Verlag. Berlin.

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