Energy Time Dichotomy in the Optimal Control of the Electrical Drives

Transcription

1 Proc of the 3r IAS/WSAS Int Conf on nergy, nvironment, cosystems an Sustainable Development, Agios Nikolaos, Greece, July 4-6, 7 54 nergy ime Dichotomy in the Optimal Control of the lectrical Drives CORNLIU BOAN, VASIL HORGA, FLORIN OSAFI Dept of Automatic Control an Inustrial Informatics "Gh Asachi" echnical University of Iasi B angeron 53A, 66 Iasi ROANIA cbotan@actuiasiro, horga@tuiasiro, fostafi@actuiasiro Abstract: A comparison beteen time optimal control an minimum energy control for electrical rives is performe It is consiere especially the rives ith DC an inuction motors Some simulation an experimental results are presente Key-Wors: electrical rives, optimal control, minimum time, minimum energy Introuction here are ifferent requirements for the transient perio an especially for the start of the electrical rives he main emans refer to the prouctivity an to the energy consumption If one of these aspects has a main importance for the technological process, the optimal control from the corresponing point of vie can be aapte If the interest is for a great prouctivity, the minimum time control of the electrical rive can be introuce, tacking also into account that this control can be easy achieve he optimal control from the energetic point of vie is important because in the transient perio the energy losses (especially the Joule losses) are very high Although the optimal control [], [] of the electrical rives represents an important ay for energy saving an there are numerous stuy eicate to optimal control of the electrical rive systems, for ifferent types of motors, criteria, or use methos (e mention, for instance [], [3], [4], [5], [6], but many other papers can be inicate), the number of applications is noaays very small We appreciate that a cause of the reluctance in this irection is the complexity of the algorithm Hoever, the metho propose by authors [7], [8] for DC or brushless motors allo to carry out an easy implementation an therefore, it is useful to apply the optimal control, since the iminution of the energy losses in the motor inings is significant in the electromechanical transient process, by comparison ith classical cascae control he mentione papers put in evience a kno fact [] that in many cases the emans of energy an time conitions are contraictory Unfortunately, the esign of electrical rives rarely tacke into account these aspects an oes not offer users the possibility of choice of an aequate operating moe, in concorance ith nees of the technological process an economical avantages he paper presents certain consierations referring to the optimal control of the electrical rives from the energetic an time response point of vie, respectively hese consierations are presente in a simple moe, because the currents (an not voltages) are aopte as control variables his proceure represents not only a computing simplification, but has a practical reason Inee, although the electrical rives are usually supplie from voltage sources, the motors are controlle by means of currents, eg the internal current loop in cascae control of the DC motors, or the frequent utilization of the current source inverters for inuction motors In this respect e mention that our previous results [8], [] have inicate a high similitue beteen the to cases (the control by mean of currents an voltages) acking into account that the use of current as control variables leas to an easier implementation (base on cascae control ith small transformations) this ay as aapte in the paper he optimal control using one control variable For the sake of a concrete presentation, the equations for a DC motor ill be consiere an some remarks referring to the inuction motor ill be inicate in the Section 3 he mechanical equilibrium equation is Ω ( ci ), () J

2 Proc of the 3r IAS/WSAS Int Conf on nergy, nvironment, cosystems an Sustainable Development, Agios Nikolaos, Greece, July 4-6, here Ω is the rotor spee, J is the inertia, I the rotor current, the loa torque an c a machine constant In the sequel e shall use the normalize (relative) values of the variables In this respect, e introuce the normalize variables i I/ I, m /, Ω/ Ω, t/, () N N N N here the inex N refers to the rate corresponing values, an JΩ / (3) N N N is a nominal time of the system In the steay state for rate values, ci N (4) N acking into account (), (3) an (4), the equation () becomes i m (5) he solution to this equation is () () i() θθ m() θθ (6) If the final esire value of the spee is impose, an if e consier (), equation (6) leas to ( ) i( θ) θ m( θ) θ (7) Remark : In many control problem, the aopte value for current epens on the loa torque mt ( ) he equation (7) inicates that in certain cases it is necessary to kno especially the mean value m m( θ) θ (8) the sequel, e shall consier m( ) const If this conition is not satisfie, the above remark, referring to the mean value have to be consiere an m ill be replace ith m From the energetic point of vie, the interest is to minimize the Joule losses, tacking into account that these ones are significantly greater than other losses in the transient perio For this reason, the criterion I i () (9) is aopte an e can formulate the optimal control problem P: Fin i( ) an ( ) hich minimizes the criterion (9) an satisfies the equation (5), an conitions (), ( ) From the prouctivity point of vie, the interest is to minimize the transfer time or the inex I () In all equations, the inex refers to a value obtaine for minimum time problem an the inex refers to the optimal energy consumption problem It is obviously kno that in minimum time problem, the restrictions have to be introuce, that is i() i, () here i is a maximum acceptable value of the current he minimum time problem is P: Fin the control variable it ( ) an the state variable ( t) so that the criterion () to be minimize, subject to (5) an () he solution to this last problem can be easy fin using the Pontriagyn minimum principle an it is i () i, [, ] () on the interval [, ] his fact implies to beforehan kno at least the shape of m( ), [, ] an to measure or estimate the magnitue of the loa torque at the beginning of the control process his aspect occurs in the optimal energy control problem, but not in the minimum time problem In many applications, the loa torque can be consiere constant on the short interval of the start, an in this case, of course m mfor simplicity, in he minimum time is i m an the normalize issipate energy is (3)

3 Proc of the 3r IAS/WSAS Int Conf on nergy, nvironment, cosystems an Sustainable Development, Agios Nikolaos, Greece, July 4-6, i m i (4) In (4), W / WN, here W is the Joule losses on the interval [, ] an WN rinn is the normalize energy losses he control variable on t epen on the loa torque (it is alays i ), but the transfer time epens on m he Hamiltonian in P problem is H i +λ( i m) an from the Hamilton conitions, it results λ () λconst an i() iconst (5) We obtain from the conition I / i : i m, (6) the minimum Joule losses m (6) 4 an the corresponing transfer time / m (7) Remark : We conclue from (6) that the optimal current must ensure a electromagnetic torque having a ouble value of the loa torque his is a general feature of the electrical machine [], vali for any motor type, using one or to control variable Remark 3: If the loa torque has a small value, the transfer time (7) increases very much In many applications this time must be limite to a maximum value an in this case i m+ an variation of the current is Δ i i i >, since m < Corresponing, the copper losses increase ith Δ i ie m Δi (9) he comparison beteen P an P problems can be performe referring the time transfer an the energy losses in the to cases Firstly, it is easy to remark that the solutions for both problems coincie if i he ifferences beteen the to cases epen on the ratio μ m i One obtain from (3) an (8) / i m i, () m μ ith μ <, since the motor cannot start if i< m his variation is presente in Fig / 3 /4 / 3/4 Fig Similarly, from (4) an (7), it results 4 mi ( m) 4 μ( μ ) () n i an this variation is inicate in Fig / / Fig he Fig an sho again the coincience of the solutions for P an P problems if i m From energetic point of vie, any other choice of i is unfavorable he choice / < μ < (or m< i < m) is not recommene from both point of vie If the prouctivity consierations have priority, one can aopt i > m ( μ < / ), but the energy losses increase in this case μ μ

4 Proc of the 3r IAS/WSAS Int Conf on nergy, nvironment, cosystems an Sustainable Development, Agios Nikolaos, Greece, July 4-6, Optimal control ith combine criterion acking into account the contraictions beteen time an energy emans, it is not lack of interest to consier an optimal control ith combine criterion: P3: Fin i( ) an ( t) hich minimize Ic i () α+, () subject to (5) an (), ( ) In () α is the eight coefficient for minimum time control problem: for α, one obtain the P problem an for α very great, the problem becomes the minimum time one (a restriction for i have to be introuce in this case) he Hamiltonian problem is H α+ i () + + λ[ i() m() ] Using a similar ay as in Section, it results an optimal constant current an from the conition I/ i, one obtain the optimal control current i m+ m +α (3) c an then +α (4) c / m an ( ) i m+ m +α (5) c c c hese energy losses are greater then in the P case (7): ( ) m m +α m +α c an c if α Also, from (8) an (4), it results / +α > c m m Of course, for P3 problem, the time transfer is improve, but the energy losses increase by comparison ith P case 4 he optimal control using to control variables 4 he DC motor case If the machine iron is nonsaturate the equation () becomes in this case (after normalization) () t kii m, (6) here i I / I N an i I / I N correspon to the stator an the rotor currents, respectively For minimum time problem, the Hamiltonian is H +λ( )( ii m) he canonical equation H / λ leas to λ ( ) λconst Since H linearly epens on i an i respectively, the minimum principle is satisfie if the currents have the maximum acceptable values i an i herefore, no ifference appears by comparison ith one control variable case For the minimum energy problem, the criterion is () (), (8) I ri + ri here r an r are the stator an rotor inings resistances he authors have prove [] that the conition (6) also hols in this case an it is achieve if ri ri, (9) that is the rotor an stator losses have to be equal Of course, such a conition can be carrie out only for small m In other cases, the stator currents reach its maximum value i an the behaviour is similar as in the Section 4 he inuction motor case In this case, the criterion for minimum energy problem is similar ith (8), ith the same significances of the variables We mention only the result prove in []: the optimal control is obtaine if (6) hols he components i an i q (in a -q frame) of the stator current are establishe [] Again, the obtaine values can be aopte only for small loa torque Otherise, the maximum value for i component (ie the maximum value of the flux) is achieve an the control is performe as in the Section For the minimum time problem, the conclusion is similar as in the DC motor case: the both components of the stator current must have the maximum acceptable values In conclusion, if the loa torque is small, it is not lake of interest to consier to control variables (the expression are inicate in []) in the minimum energy problem Otherise, an also in the minimum time problem, it is sufficient to aopt only one control current, as in Section

5 Proc of the 3r IAS/WSAS Int Conf on nergy, nvironment, cosystems an Sustainable Development, Agios Nikolaos, Greece, July 4-6, System implementation an experimental results he above establishe control is an ieal optimal control We say that the control is ieal because of the assumption referring to the currents: namely, e suppose these can have abrupt, non-inertial variations he fact that e kno the ieal optimal values of the currents suggests an easy implementation: the controller must ensure the esire value of the currents ith a great accuracy A better accuracy ensures a better proximity to ieal optimal control Of course, a task for loa torque estimation an for computing of the esire current values has to be introuce in the minimum energy problem his task is not necessary in the minimum time problem Our attention ill be focuse in sequel on the proper optimal controller, escribe above here are ifferent ays in this irection an e shall present only a structure base on a cascae control ith PI controllers: the output of the spee controller is transmitte to the input of the current controller via a saturation bloc he level of the saturation is variable an correspons to the esire value of the current his specific part (the internal loop of the cascae) is presente in Fig3, here S represents the saturation bloc, C is a PI controller, P is the controlle plant, an i represents the output of the external (spee) controller he level of saturation of the block S is i for minimum time problem or is establishe in epenence of the loa torque m, in the minimum energy problem i m i S + C P _ Fig 3 Structure for optimal current control he next figures present a selection of simulation an experimental tests, performe for the propose structure, for DC or inuction motors an for ifferent operation conitions We have suppose in all cases that the loa torque is knon For a rive system ith DC motor (having rate ata: 3kW, V, 75A, rpm) ith m5nm, 5 ra/s, J8Nms /ra, certain results obtaine by simulation tests are presente in Fig4 he first case correspons to the optimal energy control (I/c, an 75 N ), an the secon figure is for time optimal control ith limit value for current I 3 I N Of course, the rive system is faster in the last case, but the energy consumption increase ith 5% i Fig4 Behaviour of the rive system for minimum energy control an minimum time control Fig 5 shos to situations hen the loa torque has a variation at the moment t / (a step variation from 5 N to N ) A control variant is to aopt I/c on each interval (first figure) Other variant (optimal) is to aopt I me /c on all interval In this case, the energy losses ecrease ith about % [ra/s] [ra/s] [ra/s] [ra/s] Fig 5 Behaviour of the optimal system for the variable loa torque he Fig 6 presents the experimental results for an optimal energy control for an electrical rive system ith a inuction motor (ith rate ata 4kW, 38V, 8,64A, 43 rpm)

6 Proc of the 3r IAS/WSAS Int Conf on nergy, nvironment, cosystems an Sustainable Development, Agios Nikolaos, Greece, July 4-6, Fig 6 Behaviour of the rive system ith inuction motor for optimal energy control (experimental results): spee, active current an stator current One can remark a goo behaviour of the rive systems in all cases Of course, from energetic point of vie, the better situation is the optimal control Note that in certain situation, the cascae control can achieve similar performances, namely in the case hen current is limite to the value I I N an N he smallest energy losses are obtaine not because of controllers esign base on usual criteria, but because of limitation of the current at the same value as in the optimal control But, for other loa torque, the optimal control leas to a value up to 5% smaller as for the cascae control 6 Conclusions he minimum energy control an minimum time control of the electrical rives have contraictory emans Generally, the eneavor of the iminish of the transfer time leas to the increase of the energy losses In all cases, the minimum time control can be achieve using only one variable control For optimal energy control, the use of to variables present interest only for very small loa torque he structure of the optimal control is not complicate an is base on a usual cascae one, if e aopt the currents as control variable his assumption has a practical justification, base on the fact that motor currents are frequently controlle in the variable spee rives here is a great similitue in optimal control of ifferent motor types he performe simulation an experimental tests sho a very goo behaviour of the optimal control systems an inicate an opportunity for energy consumption ecrease in the transient perio of an electrical rive References [] BDO Anerson, JB oore, Optimal Control Linear Quaratic ethos, Prentice Hall, 99 [] Athans an PL Falb, Optimal Control, c Gra Hill, Ne York, 966 [3] C De Capua an C Lani, easurement Station Performance Optimization for esting on High fficiency Variable Spee Drives, I rans on Instrumentation an easurement, vol 48, no 6, 999, pp [4] I Wallace, DW Novotny, RD Lorenz an D Divan, Verification of nhance Dynamic orque per Ampere Capability in Saturate Inuction achines, I ransactions on Inustry Applications, vol 3, no 5, Sept/Oct 994, pp 93- [5] Ghribi, Y Dube an K Al-Haa, Linear Quaratic Control of a DC otor, Proceeings of the 3 r uropean Conference on Poer lectronics an Applications, P 89, Aachen 989, pp 6-65 [6] C Botan, F Ostafi an A Onea, Discrete Optimal Control ith Fixe n-point for an lectrical Drive System, Proceeings of th International Poer lectronics an otion Control Conference, P PC, Riga, sept 4, vol 4, pp [7] gami, H orita an suchiya, fficiency Optimize oel Reference Aaptive Control System for a DC otor, I ransactions on Inustrial lectronics, vol 37, no, Feb 99, pp 8-33 [8] C Botan, A Onea, F Ostafi, A Solution to the LQ-Problem ith Finite Final ime, WSS Int Conf on Automation an Information (AIA ), Skiatos, Greece, in Avance in Automation, ultimeia an Vieo Systems, an oern Computer Science, Kulev VV, D Attellis C, astorakis N,WSS Press,, pp6-66 [9] C Boţan, F Ostafi, A Onea, A Solution to the Optimal racking Problem for Linear Systems, WSAS ransactions on Systems, Vol, ISSN 9-777,, pp [] A Onea an C Botan, Real ime LQ-Optimal Control of a DC Drive System, Proceeings of the 4 th Annual Conference of the I Inustrial lectronics Society, ICON 98, Aachen, 998, pp47-4 [] C Botan, V Horga, F Ostafi, Albu, Ratoi, General Aspects of the lectrical Drive Systems Optimal Control, submitte to th uropean Conference on Poer lectronics an Applications - P 7, September 7, Aalborg, Denmark