Quasi optimal feedforward control of a very low frequency high-voltage test system

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1 Preprints of the 9th Worl Congress The International Feeration of Automatic Control Quasi optimal feeforwar control of a very low frequency high-voltage test system W. Kemmetmüller S. Eberharter A. Kugi Automation an Control Institute, Vienna University of Technology, Gußhausstrasse 7-9, 4 Vienna, Austria ( {eberharter, kemmetmueller, kugi}@acin.tuwien.ac.at. Abstract: This paper presents the esign of a quasi optimal feeforwar controller for a new type of high-voltage test systems. Compare to existing solutions, this test system has the avantage of a compact lightweight construction an is thus particularly suitable for on-site tests. For system analysis an controller esign, a tailore mathematical moel is erive, which escribes the (slow envelope ynamics of the occurring amplitue moulate signals. The propose (quasi optimal control concept ensures a simultaneous minimization of the mean power losses an of the istortion of the output voltage. Simulation results for a valiate mathematical moel show the feasibility of the approach. Keywors: high-voltage test systems, envelope moel, optimal control, feeforwar control.. INTRODUCTION In recent years, a force expansion of regenerative energy prouction by means of large scale win parks, photovoltaic or biomass power plants is reporte. These mostly ecentralize systems are often connecte to the electricity istribution network by high- an ultra-high-voltage cables. In orer to guarantee a fail-safe energy supply, the cables have to pass strict quality tests, such as factory acceptance tests an on-site tests of alreay installe cables. For the on-site cable tests, the test evices have to provie a compact an lightweight construction. Because of this, a new type of cable test metho with very low frequencies (VLF in the range of. Hz to. Hz was recently establishe, which reuces the amount of reactive power an, therefore, the size an weight of the test evices, see Putter et al. (; IEEE Power Engineering Society (4; Pietsch an Hausschil (5; Krüger et al. (99; Muhr et al. (; Coors an Schierig (8. The company Mohaupt High Voltage GmbH evelope a new type of VLF test system base on the so calle Differential Resonance Technology (DRT, which allows a mobile VLF testing of cables up to 5 kv rms an higher, see Mohaupt et al. (; Mohaupt an Bergmann (. The functional principle of this new test system is epicte in Figure. Therein, the resonant circuit is tune to its resonance by the two pulse with moulate input voltages u p an u p generate by the power moule. The choice of the two input frequencies, ω p = ω r ω an ω p = ω r ω, results in an amplitue moulate (AM high-voltage u r across the resonant capacitor. This high-voltage signal comprises a carrier frequency ω r an the esire low frequency ω of the test signal. By The research for this paper has been sponsore by the FFG- Austrian Research Promotion Agency in the BRIDGE-project no Further thanks go to Peter Mohaupt an the employees of Mohaupt High Voltage GmbH for supporting the measurement campaigns. a efine switching of the thyristors in the emoulator (SVU - switche valve unit, the esire high voltage test signal with the low frequency ω is generate. In orer u p 3 = = power moule u p = u r AM high-voltage low frequency envelope signal (ω ω r ω ω r ω exciter transformer u r SVU ul resonantcircuit Fig.. Functional principle of the DRT test system. loa (cable to fulfill the high quality stanars of cable test voltages, e.g. the istortion factor (see IEC 66- an IEC 66-3, an to allow an energy efficient operation, a suitable control of the DRT system is neee. This paper eals with the evelopment of a feeforwar control for the amplitue an shape of the test voltage of the DRT system. The control strategy is base on an envelope moel of the DRT test system, which is erive in Section. In Section 3, the moel will be use to analyze the behavior an ynamics of the DRT system, especially the behavior uring the emoulation of. Base on this formulation, an optimal control problem is solve by minimizing the mean power losses of the DRT system. The results of the optimal control problem serve as a basis for the esign of a realtime-capable quasi optimal feeforwar control, which is valiate by simulations. Conclusions are rawn in Section 4. Copyright 4 IFAC 63

2 . MATHEMATICAL MODELING The following section is concerne with the mathematical moeling of the DRT test system. In a first step a simplifie moel, consiering the essential physical behavior of the system, will be erive. Base on this moel an envelope moel is evelope. For more etaile information on the moeling an the analysis of the DRT system, see Eberharter et al. (3.. Simplifie moel An equivalent circuit iagram of the DRT system escribe in Section is shown in Figure. The moel is u p u p R p L p i p R p L p i p L s L s R s R s i r L r R r C r i u r R (t u Fig.. Circuit iagram of the DRT system with a continuously ajustable emoulator resistor R (t. base on the following assumptions an simplifications: (i The overall emoulator is escribe by a continuously variable resistor R (t. By means of a efine switching sequence of the thyristors, the value of R (t can be varie between R on an. (ii The exciter is supplie by the tunable voltages of the power moule u p an u p. The primary currents i p an i p of the exciter transformers an the resonator current i r are given by the following ifferential equations [ [ L ip R p i p u p i p = R p i p u p, ( t i r (R s R r i r u r with the primary an seconary resistance R p an R s of the exciter transformers, the resistance of the resonant circuit R r, the resonant voltage u r an the inuctance matrix L, which is given by [ Lp L ps L = L p L ps. ( L ps L ps L s L r Therein, L p an L s escribe the primary an seconary inuctance of the exciter transformer, L r is the resonant inuctance an L ps = k L p L s is the coupling inuctance with the coupling factor k <. The resonator voltage u r is escribe by u r = ( i r i, (3 t C r with the current i through the emoulator. Given a high-voltage cable with a constant capacitance C l an an ohmic resistance R l, the output voltage can be written as t = ( u l i. (4 C l R l As mentione before, a simplifie moel of the emoulator is summarize in this section, for more etails see R l C l Eberharter et al. (3. The behavior of the thyristors is escribe by switchable ieal ioes with a negligible on-resistance an vanishing threshol voltage. Thus, the positive an negative branches of the emoulator are given by ajustable resistors R an R an the current i through the emoulator reas as u R i = if u u (5 if u <, R with the emoulator voltage u = u r an R, R [R on,. Equation (5 can be rewritten as ( ( i = u R g R off R g, R off (6 with g an g efine as { g u if u = if u <, g = u g. (7 The primary voltages u p an u p of the exciter coils are generate by two full-briges in the power moule. They have a pulse-with moulate rectangular shape with a fixe amplitue u p an ajustable uty cycles χ an χ, which can be varie in the range of χ, χ. The esire amplitue moulation of the resonant voltage u r is achieve by the choice of the cycle times T p an T p of the power moule voltages accoring to T p = =, T p = = ω p ω r ω ω p. Envelope moel ω r ω. (8 For the system analysis an the evelopment of a control strategy, it is reasonable to use an envelope moel, which only escribes the time evolution of the envelopes, i.e. the time profile of the mean value an the amplitue of the high frequency signal prouce by the serial resonant circuit, see, e.g., Caliskan et al. (996; Saners an Verhulst (985; Egretzberger an Kugi (. Therefore, the system state variables of the mathematical moel are approximate by a slowly time varying mean component X (t, a cosine X c (t an a sine component X s (t in the form x(t = X (t X c (t cos(ω r t X s (t sin(ω r t, (9 where ω r is the high frequency of the carrier signal. With this efinition, the system state x(t can be written as x(t = X(tw(t, ( with [ X(t = [X (t X C (t X S (t, w(t = cos(ω r t sin(ω r t. ( The time erivative of ( results in ẋ(t = Ẋ(tw(t X(tẇ(t, ( where ẇ(t can be expresse as [ [ [ ẇ = ω r sin(ω r t = ω r cos(ω r t = Ωw. ω r cos(ω r t ω r sin(ω r t (3 64

3 As will be shown, a regular state transformation i Σ = i p i p, i = i p i p an i r = i r with the new system inputs u Σ = u p u p an u = u p u p is useful in view of a system analysis. Applying this state transformation to the mathematical moel of Section. an using (-(3, the envelope moel of the currents i Σ, i r, i results in [ [ [ [ [ Ir Ir Rs R L Σr = L t I Σr Ω r Ir Ur Σ I Σ R p I Σ U Σ (4a L p t I = L pi Ω R pi U, (4b wherein L Σr is the reuce inuctance matrix [ Ls L L Σr = r L ps. (5 L ps L p The vectors U Σ, U, U r, I Σ, I an I r combine the corresponing envelope coefficients as efine in (. The envelope moel of the voltages u r an can be written accoring to (3 an (4 in the form t U r = U r Ω ( I r I (6a C r t U l = U l Ω ( Rl U l I. (6b C l The envelope coefficients of the new system inputs U Σ an U with χ = χ = χ are calculate by an U Σ, = U Σ,c = 4 sin (χ cos (ω t u p U Σ,s = 4 ( cos (χ cos (ω t u p U, = U,c = 4 ( cos (χ sin (ω t u p U,s = 4 sin (χ sin (ω t u p. (7a (7b (7c (8a (8b (8c To complete the escription of the envelope moel, the current through the emoulator I, which is a function of the emoulator voltage U an the emoulator resistors R an R, nees to be calculate. Given (6, the current I of the envelope moel can be written as ( I = U R ( G R G, (9 with G = [ G, G c, G s, G = [ G, G c, G s an U = U r U l. The envelope coefficients of g an g are calculate by means of the perioic Fourier transformation (Papoulis (96 applie to (7. In orer to evaluate the resulting integrals, the conitions u an u < must be escribe in terms of the envelope coefficients U,, U,c an U,s. Here, it is assume that U, (t, U,c (t an U,s (t are constant for the integration interval,..., ω r. The nonlinear terms of G result in ( U, arccos = Û U, G Û U, (a ( U, arccos G c = Û U,c Û U, U, U,c (b Û ( U, arccos G Û s = U,s Û U, U, U,s, (c Û with Û = U,c U,s. The nonlinear terms of G can be calculate by G = U G. These results are vali in the interval Û U, Û. In the case U, > Û, the envelope coefficients of G an G are given by G = U,, G c = U,c, G s = U,s (a G =, G c =, G s = (b an for U, < Û they rea as G =, G c =, G s = (a G = U,, G c = U,c, G s = U,s. (b The envelope moel is valiate by a comparison with the voltage (kv current (A voltage (kv û r Û r time (s î rîr U l, Fig. 3. Comparison of the envelope moel with the simplifie moel of Section.. moel of Section., which has alreay been calibrate by measurements, cf. Eberharter et al. (3. Both moels were simulate with a resistive-capacitive loa of R l = 3 MΩ, C l = 5 nf an the power moule pulse withs χ =.8. For the generation of the output voltage, a simple emoulation strategy was use, where all thyristors of the positive branch are switche on for the positive half-wave of an all thyristors of the negative branch are switche on for the negative half-wave. To allow a better comparison of both moels, the results 65

4 of the envelope moel are compare with the envelopes of the signals of the simplifie moel in Figure 3. Due to the nonlinearities of the emoulator, a istortion of the sinusoial shapes of the currents an voltages in the resonant circuit occurs. Nevertheless, as can be seen in Figure 3, this eviation is negligible such that the envelope moel is a very goo approximation of the simplifie moel. 3. CONTROL STRATEGY To evelop a suitable control strategy, the behavior of the DRT system is analyze by means of the envelope moel of Section.. Subsequently, an optimization problem is formulate which is the basis for the evelopment of a quasi optimal feeforwar control. In this paper, it is presume that the voltage u r ieally tracks a esire voltage u r. Thus, u r can be use as a virtual control input an the series resonant circuit an the power moule can be neglecte. Consequently, the output voltage has to be controlle by changing the new (virtual control inputs u r, R an R. 3. System analysis For the system analysis the simple emoulation strategy of the last section is consiere again. Furthermore, the analysis is only carrie out for the positive half-wave of, since the negative half-wave can be treate in similar manner. The following questions will be iscusse in etail: (I How to choose the resonant voltage u r an accoringly the amplitue Ûr to minimize the error between the output voltage an a esire sinusoial trajectory u l? (II How large is the amplitue Ûl of the expecte output voltage ripple (istortion factor? In orer to aress these questions the following assumptions are mae: (i The average value of u r is zero, i.e. U r, = an thus U, = U l,. (ii The voltage ripple of the output voltage is neglecte, i.e. U l,c = U l,s = an thus Û = Ûr. (iii The esire output voltage is sinusoial u l (t = U l, = Û l, sin (ω t. Using these assumptions an R = R on, R = in (6b for U l, an (9 for I,, (6b can be solve with G = U, G for G, G (Û, U, G ( (Ûr, C Ul, l U l, = ( R R l R R U l,. (3 With the escription of G from (a, (3 can be solve numerically for Ûr as long as the right han sie of (3 is greater than zero. Figure 4 epicts a esire sinusoial output voltage u l (t = U l, an the require amplitue Ûr for a loa of C l = 5 nf an R l = 3 MΩ. As can be seen, the amplitue Ûr has to be slightly larger than the amplitue Ûl, of the esire output voltage in orer to compensate for the ohmic losses in the emoulator. Furthermore, Figure 4 shows that no solution for Ûr can be foun for t > 3 s. This means that a control of the output voltage by simply ajusting the voltage u r oes not work for the whole perio of the sinusoial output voltage. Thus, an active change of R an R is necessary for t > 3 s. voltage (kv 35 3 U l, Û r 5 U l,,min 5 no solution possible time (s Fig. 4. Require amplitue Ûr for a esire sinusoial output voltage Ul,. The secon part of the analysis is concerne with the output voltage ripple Ûl. For this purpose, a transformation to amplitue an phase angle in the form U l,c = Û l cos(ϕ l, U l,s = Ûl sin(ϕ l, U,c = Û cos(ϕ l ϕ an U,s = Û sin(ϕ l ϕ is applie to (6b an (9. Uner the assumptions mae at the beginning of this subsection, this results in [( Û l = Ĝ Û cos(ϕ t C l R on Û l R l C [( l t ϕ l = Û l C l Ĝ Û sin(ϕ R on (4a ω r, (4b with Ĝ = (G c (G s. In orer to get a first approximation of the output voltage ripple Ûl, a quasistationary solution of (4 is sufficient. This is justifie by the fact that the ynamics of the voltage ripple is much faster than the ynamics of U l,. This approximation leas to [( C l R on R Ĝ off Û Û l = (. (5 ωr R l C l Remark. Since in general Û epens on Ûl, (5 constitutes an implicit nonlinear equation for Ûl. However, in view of assumption (ii at the beginning of this subsection, Û can be fairly well approximate by Ûr. Figure 5 epicts the voltage ripple Ûl for a loa of C l = 5 nf, R l = 3 MΩ an the voltages Ul, an Ûr shown in Figure 4. It can be seen that the maximum voltage ripple is less than.5 of the amplitue of the esire output voltage Ul,. This is a goo result in view of the require high quality stanars of cable test voltages. However, it is important to note that Figure 5 only epicts a first approximation of the expecte voltage ripple an that the amplitue of this ripple may vary ue to other effects, e.g., parasitic effects cause by stray capacities in the housing of the DRT system. 66

5 Ûl (V time (s Fig. 5. Output voltage ripple Ûl for U l, an Ûr given in Figure Optimal control In the previous system analysis it was shown that a control of only by means of u r is not possible. Thus, a efine switching of the thyristors in the emoulator, i.e., changing the values of R an R is necessary. In this section, an optimal control problem is formulate to calculate the optimal control input u = [ Ûr, R,, R, such that U l, tracks a given trajectory Ul,, while Ûl an the mean power losses p in the emoulator are minimize. Since minimizing Ûl is equal to minimizing p, the resulting static optimization problem can be formulate as min p (u, (6 u with the mean power losses p given by p = ω r ωr p t = U l,i, U r,ci,c U r,si,s (7 an the nonlinear constraints written as ( C l U l, Ul, R l R Ul, R R G (u =, R on R R R on R. (8 R The static optimization problem (6-(8 is solve using the Matlab function fmincon with the Interior-Point algorithm. Figure 6 epicts the numerical results for a resistive-capacitive loa of R l = 3 MΩ, C l = 5 nf, U l, = kv rms, R on = 5 kω, = 9.3 MΩ an a sampling time T s =. s. In the upper part of Figure 6 it can be seen that Û r has to be slightly larger than Ul, uring the loaing phase of C l, which confirms the results of the system analysis in Section 3.. The ischarging phase of the loa capacity is characterize by Û r =. In the lower part of Figure 6, the optimal values of the emoulator resistors R an R are shown. During the loaing phase of C l, the resistors are assigne to their bounary values [R on,. To achieve a sinusoial output voltage, the corresponing emoulator resistor, i.e. R in the positive half-wave an R in the negative half-wave, has to change its value from to R on uring the ischarging of C l. The value of the opposite emoulator resistor jumps from R on to at the beginning of the ischarging phase. voltage (kv R (MΩ Û r U l, R, R, time (s Fig. 6. Numerical results of the optimal control problem for R l = 3 MΩ, C l = 5 nf. The results of this section thus give optimal trajectories for the control inputs. The solution of the optimal control problem, however, is time consuming an not suitable for a realtime implementation. Therefore, a simplifie, quasioptimal control strategy is evelope in the next section, which can be easily implemente in realtime. 3.3 Quasi optimal control The quasi optimal feeforwar control relies on the results of the previous section an the assumptions mae for the system analysis. For reasons of brevity, the control concept is illustrate for the positive half-wave of only, see Figure 7. It is separate into three phases A, B an C, where the superscript refers to he positive halfwave. The three phases can be escribe as follows: R on R R Û r t AB U l, A B C t BC Fig. 7. Phases of the quasi optimal feeforwar control. Phase A : C l is loae an U l, is controlle by means of Û r with Ûr U l,. The emoulator resistors are chosen as R = R on an R =. The amplitue Ûr is calculate using (3, where the numeric solution is simplifie by a polynomial approximation of G in (a. Phase B : C l is ischarge an U l, is controlle by means of the emoulator resistor R with Û r U l,. To minimize the power losses in the DRT system, Û r is chosen to zero an R is set to R =. The value of R is given by (see (3 R = Ul, C l U l,. (9 R l Ul, t t 67

6 voltage (kv error ul (V time (s 8 Û r 6 u 4 l 5 R (MΩ p (kw 5 R R time (s Fig. 8. Simulation results of the quasi optimal feeforwar control for R l = 3 MΩ, C l = 5 nf. Phase C : C l is ischarge an U l, is again controlle by means of Ûr, with Ûr U l,. The system input Ûr is calculate as escribe in phase A, using, however, R = an R = R on. The switching times t AB an t BC between the control phases are given by the conition G =. Thus, substituting U l, = Ûl, sin(ω t into (3 an solving for G yiels the switching time ( ( t AB = atan ω C l, (3 ω R l with R = R on an R =. By analogy, t BC is calculate with R = an R = R on to ( ( t BC = atan ω C l ω. (3 R on R l For a valiation of the quasi optimal feeforwar control, the control concept is applie to the mathematical moel of Section. an simulate in Matlab/Simulink. Figure 8 epicts the simulation result for a resistive-capacitive loa of R l = 3 MΩ, C l = 5 nf, U l, = kv rms an a sampling time T s = 3 ms. As can be seen, there is a goo agreement between the output voltage an the esire sinusoial voltage u l with an output error less than of amplitue of u l. Since the feeforwar control was evelope base on the envelope moel the remaining error can be primarily ascribe to the eviation between the simplifie moel an the envelope moel. 4. CONCLUSIONS AND FUTURE WORK In this work, a quasi optimal feeforwar control of a VLF test system for the testing of high-voltage cables was presente. The feeforwar control is base on an optimal control problem, which minimizes the power losses in the test system an the eviation of the test voltage of the ieal sinusoial shape. In future work, the control concept will be implemente an teste on a prototype esigne for cable tests up to kvrms an maximum loas of C l = 75 nf. The control concept has to be extene by the control of the resonant voltage u r by the power moule inputs χ an χ. Moreover, physical constraints, like the maximum voltage of the thyristors in the SVU, have to be incorporate into the esign. Finally, an estimator is uner evelopment, which ientifies unknown system parameters as, e.g., the capacitance C l of the test cable. REFERENCES Caliskan, V., Verghese, G., an Stankovic, A. (996. Multifrequency averaging of c/c converters. In Proc. IEEE workshop on computers in power electronics, 3 9. Portlan, USA. Coors, P. an Schierig, S. (8. Hv ac testing of super-long cables. In Proc. of the IEEE Int. Symposium on Electrical Insulation, Vancouver, Canaa. Eberharter, S., Kemmetmüller, W., an Kugi, A. (3. Mathematical Moeling an Analysis of a Very Low Frequency HV Test System. IEEE Transactions on Power Electronics-accepte. Egretzberger, M. an Kugi, A. (. A ynamical envelope moel for vibratory gyroscopes. Microsyst. Technol., 6, pp IEEE Power Engineering Society (4. IEEE Guie for Fiel Testing of Shiele Power Cable Systems Using Very Low Frequency (VLF. IEEE St. 4.. Krüger, M., Feurstein, R., an Filz, A. (99. New very low frequency methos for testing extrue cables. In Proc. of the 99 Int. Symposium on Electrical Insulation, Toronto, Canaa. Mohaupt, P. an Bergmann, A. (. A new concept for test equipment for testing large hv an uhv cables on-site. e&i Elektrotechnik un Informationstechnik, 7(, pp Mohaupt, P., Geyer, H., Bergman, B., Bergman, S., Bergman, A., Kemmetmüller, W., Eberharter, S., an Kugi, A. (. Extension an optimization of the loa range of rt test systems for testing extra-long hv an uhv cables. e&i Elektrotechnik un Informationstechnik, 9/7-8, Muhr, M., Sumereer, C., an Woschitz, R. (. The use of the, hz cable testing metho as substitution to 5 hz measurement an the application for p measuring an cable fault location. In Proc. of the th Int. Symposium on High Voltage Engineering. Bangalore, Inia. Papoulis, A. (96. The Fourier Integral an its Applications. McGraw-Hill, eition. Pietsch, R. an Hausschil, W. (5. Hv on-site testing with partial ischarge measurement. In Brochure of the CIGRE Working Group D Putter, H., Götz, D., Petzol, F., an Oetjen, H. (. The evolution of vlf testing technologies over the past two ecaes. In Proc. of the Transmission an Distribution Conference an Exposition, 4. Orlano, Floria. Saners, J.A. an Verhulst, F. (985. Averaging Methos in Nonlinear Dynamical Systems. Springer, New York. 68