International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September ISSN

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1 International Journal of Scientific & Engineering Research, Volue 4, Issue 9, Septeber-3 44 ISSN he unscented Kalan Filter for the Estiation the States of he Boiler-urbin Model Halieh Noorohaadi, Masoud Suresrafil Abstract In any cases interesting dynaic are not linear by nature, so the traditional Kalan filter cannot be applied in estiating the state of such a syste. In these inds of systes, both the dynaics and the easureent processes can be nonlinear, or only one of the. In this paper, an extension to the traditional Kalan filter will be described, which can be applied for estiating nonlinear dynaic systes, that is called Unscented Kalan filter (UKF) based on the unscented transforation of the joint distribution. hen this ethod is used for the estiation of the states of the Boiler- urbin Model. Siulation results show the effectiveness of this ethod. Index ers Kalan Filter (KF), Unscented Kalan Filter (UKF), Extended Kalan Filter (EKF), Boiler- urbin Model INRODUCION stiating the state of a dynaic syste is a fundaental Unscented Kalan Filter (UKF) [], the enseble Kalan filter (EnKF) [8] and high order EKFs. he EnKF is especially Eproble in the process industries. State estiation oftenplay an iportant role in accoplishing this goal in process control and perforance onitoring applications. De- designed for large scale systes, for instance, oceanographic odels and reservoir odels [3]. he UKF sees to be a pending on the type of process and the operating region of the proising alternative for process control applications [4-6]. process, soe processes can be approxiated with alinear odel and the KF (Kalan filter) can be used for state estiation. heori call y the alan fi l ter i s an estiator for what i s it is accurate up to second order in estiating ean and co- he UKF propagates the pdf in a siple and effective way and called the linear quadratic proble, which is the proble of estiating the state of a linear dynaic syste, so for nonlinear dy- nonlinear state estiation in process systes and the perforvariance [8]. he present paper focuses on using the UKF for naic, the ost successful techniques for state estiation are ance is evaluated in coparison with the EKF.he paperproposes a siple ethod to incorporate state constraints in Bayesian filters such as particle filters or extended and unscented Kalan filters []. Bayes filters recursively estiate the UKF. posterior probability distributions over the state of a syste. In a boiler-turbine unit, stea fro the boiler enters the highpressure cylinder of a condensing turbine and, after passing he ey coponents of a Bayes filter are the prediction and observation odels, which probabilistically describe the teporal evolution of the processand the easureents returned through it, returns to the boiler, entering through an interediate superheater. he secondary superheated stea is fed by the sensors, respect tively. ypically, these odels are paraetric descriptions of the involved processes. he ost into the ediu-pressure cylinder of the turbine and then into the low-pressure cylinder and the condenser. he water is coon way of applying the KF to a nonlinear syste is in reoved fro the condenser by a pup. It then passes the for ofthe extended K alan filter (EKF). In the EKF, the through the low-pressure and high-pressure feed-water heaters and a deaerator and enters the boiler. Usually, a boiler pdf is propagated through a linear approxiation of the syste around the operating point at each tie instant. In doing so, the EKF needs the Jacobian atrices which ay be difficult cannot operate at loads below a certain value for a nuber of to obtain for higher order systes, especially in the case of reasons (for exaple, the conditions of cooling of the tubes of tie-critical applications. Further, the linear approxiation of heating surfaces); therefore, soe-ties ore stea is generated than is required for the turbine (for exaple, during the the syste at a given tie instant ay introduce errors in the state which ay lead the state to diverge over tie. In other start-up of a unit) [7]. In such cases the excess stea is words, the linear approxiation ay not be appropriate for duped into the condenser through a reduction device. hen soe systes. In order to overcoe the drawbac EKF, other Bell and Astro produce a 3rd order non-linear MIMO odel nonlinear state estiators have been developed such as the with fuel flow, control valve position, and feedwater flow as control inputs, and dru pressure, power output, and dru water level deviation as outputs. Boiler- turbin unit is a ultivariable, tie varying and nonlinear syste with strong cou- Author:Halieh Noorohaadi.Departaent of Electrical Engineering, Anar Branch, Islaic Azad University, Anar, Iran, E-ail: pling between the paraeters. h.noorohaadi@gail.co Co-Author:Masoud Suresrafil, Departaent of Electrical Engineering, In this paper, the states of boiler- turbin are estiated with Anar Branch, Islaic Azad University, Anar, Iran, E-ail: unscented alan filter. In section, he Unscented alan asood.suresrafil5@gail.co filter is introduced. hen in section 3, the Boiler- turbin unit is described. Section 4 discusses the siulation results followed by 3

2 International Journal of Scientific & Engineering Research, Volue 4, Issue 9, Septeber-3 45 ISSN conclusions in section 5. UNSCENED KALMAN FILER. Unscented transfor he unscented transfor (U) (Julier et al., 995; Julier and Uhlann, 4b;Wan and van der Merwe, ) can be used for foring a Gaussian approxiation to the joint distribution of rando variables x and y. In U we deterinistically choose a fixed nuber of siga points, which capture the desired oents (at least ean and covariance) of the original distribution of x exactly. After that we propagate the siga points through the nonlinear function g and estiate the oents of the transfored variable fro the [8]. he advantage of U over the aylor series based approxiation is that U is better at capturing the higher order oents caused by the non-linear transfor, as discussed in (Julier and Uhlann, 4b). Also the Jacobian and Hessian atrices are not needed, so the estiation procedure is in general easier and less error-prone. he unscented transfor can be used to provide a Gaussian approxiation for the joint distribution of variables x and y of the for n x P Cu ~ N, y μ u C Cu Wc ( x )( y u) i u s u (). Copute the set of n + siga points fro the A P where coluns of the atrix ( n+ ) P P AA ( ) x x + [ n+ P ], i,,, n ( n P) x [ + ] i, i n+,, n and the associated weights: () W n + + ( α + β) n + () W c i () (3) ethod.. Propagate each of the siga points through nonlinearity as 3. Calculate the ean and covariance estiates for y as (6) ( i) y g x i n 4. Estiate the crosscovariance between x and y as i i i (5) he square root of positive definite atrix P is defined as o calculate the atrix A we can use, for exaple, lower triangular atrix of the Cholesy factorialization.. Unscented Kalan Filter he unscented Kalan filter (UKF), aes use of the unscented transfor described above to give a Gaussian approxiation to the filtering solutions of non-linear optial filtering probles of for x f x, + q,,,., n u w y i i n ( i) ( i) ( i) u c ( u)( u) S W y y (, ) y h x + r (8) (7) Paraeter is a scaling paraeter, which is defined as α + n n he positive constants α,β and are used as paraeters of the (4) n Where x R is the state, is the easureent, q N(, Q ) is the Gaussain Noise process, r N(, R ) y R is the Gaussian easureent noise. he prediction and update steps are in the following way: 3

3 International Journal of Scientific & Engineering Research, Volue 4, Issue 9, Septeber-3 46 ISSN Prediction: Copute the predicted state ean predicted covariance P as ( X ) Xˆ f, P xw ˆ [ ] X + c P P xˆ w[x ˆ ] + Q and the (9) So, the prediction and updating steps are in the for of: Prediction: [ ] x + c P P P ] P Q R () Update: Copute the predicted ean µ and covariance of the easureent S, and the cross-covariance of the state and easureent C: X,, c P P + Y ( ) h x, μ Y w () In this way, the noise ust be in additive for. S Y w[y ] + R C X W[Y Update: ] r Y hx (, x, ) hen copute the filter gain K and the updated state ean and covariance P: Y w K S [ ] () Y wy CS P P KSK + K [ y ].3 Augented Unscented Kalan filter It is possible to odify the UKF procedure described above by foring an augented state variable, which concatenates the state and noise coponents together, so that the effect of process and easureent noises can be used to better capture the odd-order oent inforation. his requires that the siga points generated during the predict step are also used in the update step, so that the effect of noise ters are truly propagated through the nonlinearity (Wu et al., 5). If, however, we generate new siga points in the update step the augented approach give the sae results as the non-augented, if we had assued that the noises were additive. If the noises are not in the additive for, the augented version should produce ore accurate estiates than the non-augented version, even if new siga points are created during the update step. So: 3 x q ˆ X, x f X, ˆ xw P ˆ w [ ˆ ] x x C xw ˆ [Y ] (3) (4) Where we have denoted the coponent of siga points corresponding to easureent noise with atric x. Lie the r state transition function f also the easureent function h is now augented to incorporate the effect of easureent noise, which is passed as a second paraeter to the function. hen copute the filter gain K and the updated state ean and covariance P: K CS + K [ y ] P P KSK (5) Note that non-augented for UKF is coputationally less deanding than augented for UKF, because it creates a

4 International Journal of Scientific & Engineering Research, Volue 4, Issue 9, Septeber-3 47 ISSN saller nuber of siga points during the filtering procedure. hus, the usage of the non-augented version should be preferred over the non-augented version, if the propagation of noise ters doesn t iprove the accuracy of the estiates. 3. he nonlinear odel of Boiler-urbin unit he odel is based on the boiler-turbine plant P6/G at the Sydvensb Kraft AB plant in Malo, Sweden. he boiler is oilfired and the rated power is 6 MW. Data acquired during a series of experients in 969 for the basis for the syste identification. Both physics and epirical ethods were used to produce this boiler-turbine dynaic odel [9]. Since 969, the odel has undergone a nuber of alterations. Subsequent iproveents in the plant odel have resulted in better odels that yield iproved predictive abilities for the plant. his resulted in a nd order non-linear syste of differential equations with fuel flow and control valve setting as the Fig. Scheatic diagra of the Boiler-urbin unit control variables and dru pressure and power Output as the Output variables. hen Bell and Astro produce a 3rd order non-linear MIMO odel with fuel flow, control valve position, and feedwater flow as control inputs, and dru pressure, power output, and dru water level deviation as outputs. Boiler- turbin unit is a ultivariable, tie varying and nonlinear syste with strong coupling between the paraeters. he dynaic of the syste is in the for of: 9/8 x.8ux +.9u.5u x u x x x3 (4 u3 (.u.9) x) / 85 y x y x 3 9/8 (.73.6). y3.5(.373x3+ acs ) (6) a cs (.538x3)(.8x5.6) x3( x) q.854u.47 x u.54u.96 e 3 (7) Where X, x, x3 are dru pressure (g/c ), power output(mw), fluid density (g/3) respectively. he noralized inputs to the syste u l fuel flow valve position, u stea control valve position, and u3 feedwater flow valve position, and all valve position variables are constrained to lie in the interval [, ]. he outputs to the syste are y (dru pressure), y (output level), y3 is the dru water level in eter. And the variable acs, qe are stea quality and the evaporation rate (g/s): Fig. he siulation of the Boiler- urbin unit Figure, shows the scheatic diagra of a boiler turbin unit, and also in figure, the siulation of boiler turbin is brought that is siulated in Matlab siulation. 4. Siulation Results 3 For the Boiler- urbin, as discussed in the previous section, the states x, x, x3 are estiated with Augentd Kalan Filter. In

5 International Journal of Scientific & Engineering Research, Volue 4, Issue 9, Septeber-3 48 ISSN this way, noise ust be added to the syste. In this ethod, linearization is not needed. So this ethod is better than the traditional alan filter. In this paper, Unscented Kalan filter is discussed, and this ethod is considered for boiler- turbin syste. In copare with traditional alan filter, linearization is not needed. When linearization is used, only the equibliriu point is exained, but Unscented alan filter estiates the states around all the points without any linearization and is ore accurate than the alan filter and the tie for siulation in less than the other ethod. REFERENCES [] S. hrun, W. Burgard, and D. Fox. Probabilistic Robotics. MI Press, Cabridge, MA, Septeber 5. ISBN [] S. Julier, J.K. Uhlann, Unscented filtering and nonlinear estiation, Proceedings of the IEEE 9 (4) 4 4. Fig.3 he estiation of the state x [3] G. Evensen, Data Assiilation, he Enseble Kalan Filter, Springer, Berlin, 7. [4] A. Roaneno, J.A.A.M. Castro, he unscented filter as an alternative to the EKF for nonlinear state estiation: a siulation case study, Coputer and Cheical Engineering 8 (4) [5] B. Ain, U. Orguner, Aydin Ersa, State estiation of induction otor using unscented Kalan filter, IEEE ransactions on Control Applications (3) [6] W. Li, H. Leung, Siultaneous registration and fusion of ultiple dissiilar sensors for cooperative driving, IEEE ransactions on Intelligent ransportation Systes 5 (4) Fig.4 he estiation of the state x [7] Robert Dieo, Kwang Y. Lee, Boiler-urbine Control Syste Design Using a Genetic Algorith, IEEE ransactions on Energy Conversion, Vol., No. 4, Deceber 995 [8] Rababu Kandepu, Bjarne Foss, Lars Island, Applying the unscented Kalan filter for nonlinear state estiation, Journal of Process Control xxx (8) xxx xxx [9] K.J.Astro, K.Elund, A siplified non- linear odel of a dru boiler turbin unit, Int.J.control, 97, Vol.6, No. Fig.5 he estiation of the state x 3 In the ost papers the equibliriu for the Boiler_ urbin syste [x,x,x3,u,u,u3][8,66.65,48,.34,.69,.433] is considered. he above figures show that the estiation of the states x, x, x3 are accurate. 5. Conclusion 3