Direct Computation of Generator Internal Dynamic States from Terminal Measurements

Transcription

1 Direct Computation of Generator nternal Dynamic States from Terminal Measurements aithianathan enkatasubramanian Rajesh G. Kavasseri School of Electrical En. an Computer Science Dept. of Electrical an Comp. En. Washinton State University, Pullman, WA North Dakota State University Faro, ND Abstract-- Estimatin the ynamic state variables of a synchronous enerator has been a lon stanin research problem in power system control esins. n this paper, we present a techniue to compute approximately the state variables of a synchronous enerator, purely from terminal measurements. The approximation follows from an analysis of the funamental euations escribin synchronous machine moels. The metho assumes the availablity of the measurements: real an reactive power, terminal voltae, an the fiel current. By exploitin the structure of the flux linkae euations, approximate formulas for the machine rotor anle (with respect to the phase of the terminal voltae) as well as for the internal flux linkaes are erive, an they are shown to be excellent approximations in simulations of etaile machine moels. nex Terms Synchronous machine moels, power system ynamics, power system moels, power system controls.. NTRODUCTON n this paper, we provie a systematic framework for approximate computation of the ynamic state variables of a synchronous machine connecte to an arbitrary external network. The techniue uses measurable output uantities of the enerator alon with its electrical parameters to compute the estimate. Once a reasonable estimate of the state is available, state feeback type control as oppose to output feeback can be employe to esin sophisticate controllers base on avance linear as well as nonlinear control esin techniues. We use the wor ``computation" as oppose to estimation to avoi potential confusion with the conventional state estimation problem in power systems, that computes the least suare estimate of the euilibrium state of a power system from a lare set of network wie measurements. Here, we eal with the problem of computin the ynamic internal state of a synchronous enerator in real- from measurements available at the terminal of the enerator. The computational metho is erive analytically by carefully processin the internal flux linkae euations of a enerator ynamic moel. Once the problem is properly pose, the uestion reuces to solvin a set of nonlinear euations analytically for certain unknown uantities. We analyze the limitations of how much information can be extracte on the internal variables of a enerator from external measurements. The results reporte in this paper form parts of the octoral issertation [1] at Washinton State University, Pullman, WA. The rest of this paper is oranize as follows.. Section provies a brief introuction to the problem an escribes the techniue evelope in this paper. Later, simulations are presente to illustrate an assess the performance of the state computation techniue alon with some iscussions.. FORMULATON Let us consier a synchronous enerator that is connecte to an external network. The available terminal measurements are assume to be 1) P, real power output, ) Q, reactive power output, 3), terminal bus voltae manitue, an 4) f, the fiel current. We may also assume that the fiel voltae E f is available which will be iscusse in a later paper. Note that the availability of P an Q is euivalent to the availability of the terminal current an the power factor anle φ. We will also assume that all the enerator ynamic parameters are known..1. Sinle Axis flux ecay moel There exists a rich history on the moelin of synchronous enerators [] an the ynamic moels can vary a reat eal in terms of their complexity []. For ease of presentation, we will start with a sinle axis flux ecay moel [3]. n this section, we show that the ynamic state of a sinle axis flux ecay moel can be compute exactly from terminal measurements. However, the problem is more challenin for other etaile moels as we will see in later sections. Usin the sinle axis moel, the ynamic euations of a synchronous enerator can be state [,3] as δ = ( ω 1) H ω = P T E 0 P = sin( δ θ ), = M ω s P KD( ω 1) ( x x ) = E +, Q = + E f = cos( δ θ ) () (3) (5) (1) (4) Here, the three state variables are: δ, the rotor anle, ω, the rotor freuency in per unit, an E the internal flux linkae. The external bus voltae has manitue an phase θ. The variables an are the components of the bus voltae after they are transforme into the internal Park coorinate frame that is rotatin synchronously with the rotor. The /04 $17.00 (C) 004 EEE 1

2 transformation epens on the ifference in phase δ-θ, between the terminal voltae an the internal rotor anle. Similarly, an are the Park transforme components of the terminal current. The state computation problem therefore reuires us to compute the values of δ, ω an E from available external measurements. Since the rotor freuency is essentially the erivative of the rotor anle δ, the uestion is that of finin δ an E. For computin the rotor anle δ, we can assume that the phase of the terminal voltae is available with respect to some anle reference frame. We will procee to compute the phase ifference δ-θ, which is then euivalent to finin the Park variables an by (4). Let us look at the basic euations relatin the measurements an the unknowns = = = P = Q (6) (7) (8) (9) n Euations (6)-(9), the riht han uantities,, P an Q are known from measurements, while the Park variables,, an are unknown. As note earlier, once the Park variables an are known, that will ive us the phase ifference δ-θ, an hence, the internal rotor anle. Historically, estimation of the internal rotor anle has been one of the most ifficult analytical uestions. The reason is very clear from Euations (6)-(9) since they only represent three inepenent euations an the euations cannot be solve for all four variables,, an. We must always satisfy = P +Q, an therefore, there are effectively only three euations in (6)-(9). An, we also nee to compute the other state variable E which is the internal flux linkae. We nee aitional measurements in orer to solve the problem, an this is exactly what we ain from the fiel current measurement f. Goin throuh the erivation of the sinle axis flux ecay moel as in [], we observe that = x = x + x a f (10) (11) Euations (10)-(11) toether with (6)-(9) can be solve for the Park variables an, an hence the rotor anle θ. Also, once,, an are known, the internal flux E can be solve from x E = + x x a a f ff (1) where x a = x x l an x ff = x f + x a []. Let us consier how to solve the euations (6)-(11) for the unknown variables. First, we substitute (10) an (11) into (9), an use (7) to et the uaratic euation for, 1 1 x x f X + X a Q + X = 0 (13) Normally, x > x. Therefore, the coefficients of the an the terms in (13) are positive. Also, uner typical operatin conitions, the constant term (Q + /X ) woul be neative, an hence, it is uarantee that there is a uniue positive real solution for from (13) which can be easily solve. Also, since by efinition, = cos(δ-θ), the Park variable woul be normally positive, which is easily solve from (13) for the sinle axis flux ecay moel. Once is known, the other Park variable can be easily solve for, by usin (6) by assumin that is positive. For enerator operation, normally, internal rotor anle δ woul be leain the terminal bus anle θ, an hence, it is reasonable to assume that = sin(δ -θ), is positive. Next, the Park currents an can be uniuely calculate from (8)-(9) since the Park voltaes an have been solve for. Therefore, the internal flux variable E can be compute by usin (1). The phase anle ifference between the rotor anle δ an the terminal bus anle θ, can also be solve for, from = sin(δ -θ), an = cos(δ-θ). Therefore, we have completely solve the ynamic internal state variables of the enerator moel, purely from terminal measurements, when it is represente by the sinle axis flux ecay euation. We want to emphasize that the state computation metho propose in this section is completely inepenent of the types of internal controls incluin exciters, overnors, or power system stabilizers. The metho is also inepenent of whatever external network the enerator is connecte to. The metho irectly uses the Park couplin euations to erive the formulas for internal flux linkae an the rotor anle uner the assumption of a sinle axis flux ecay moel.. Simulations Next, we test the state computation techniue usin numerical simulations, an we use the Kunur two area system liste in Example 1.6 of []. We will inore the machine saturation ata at first, an we also set the subtransient parameters T 0, T 0, x an x to be zero. The enerations are assume to be 665, 600, 760 an 00 MW for enerators G1, G, G3 an G4 respectively. The loas are 960 MW an 100 MW at buses 7 an 9 respectively. The tie-line power transferre from Area 1 to Area is approximately 80 MW. Exciters are represente in all four enerators usin the selfexcite DC exciter with ata liste on pae 814 in []. Power system stabilizers are present at all the units. Governors are also represente in all the enerators. Now, we use EPR transient stability simulation proram ETMSP [] for simulatin some continencies for this system, an use the simulate values of enerator terminal uantities P, Q,, an f. These terminal measurements are then use /04 $17.00 (C) 004 EEE

3 to compute the internal enerator state variables usin the formulas erive earlier in Section.1. Since the enerator internal flux variables are not available in ETMSP simulation outputs, we will compare the actual rotor anle from ETMSP with the rotor anle compute by the metho in Section.1 for all the four enerators. n the first simulation, the loa is represente as a mixe static loa consistin of 5% constant power, 5% constant current, an 50% constant impeance types. The continency is as follows: At t=1 sec., the transmission line connectin buses 7 an 8 is opene. At t=3 sec., the line connectin buses 8 an 9, is trippe. Then, at t=5 sec., the line between buses 7 an 8 is reclose. At t=7 sec., the other line between buses 8 an 9 is reclose. Each line trippin or reclosin chanes the topoloy of the system. However, since our metho is a ecouple computation proceure for each enerator, an is purely base on the terminal measurements of each enerator, the state computation metho of Section.1 provies uite reasonable results throuhout the continency. A comparison of the relative anle between the rotor an terminal δ -θ is plotte in Fiure 1 for the four enerators. n all the plots that follow, the soli plots represent the actual simulate values from ETMSP, an the otte lines represent the values erive from the state computation alorithm of section.1 uner the sinle axis flux ecay assumption. theta1 elta1 theta elta theta3 elta theta4 elta Fi. 1. Comparison of actual an compute enerator anles n Fi. 1, we note that the compute rotor anle closely matches the actual rotor anle at all four enerators. The ifferences between the otte an soli lines in the plots are uite small numerically. Let us next simulate a less severe continency wherein the line connectin buses 7 an 8 is opene at t=1 sec., an is reclose at t=5 sec. Aain, the compute results match very well with actual values in Fi.. theta1 elta1 theta elta theta3 elta3 theta4 elta Fi.. Comparison of actual an compute enerator anles n the next simulation, the overnor reference of the enerator G is increase by 300 MW at t= sec. This results in an increase in system freuency, an causes the overnors at all the other enerators to ramp own their power outputs. Owin to the overnor actions, all the phase anles start riftin, an the case is also chosen to be small-sinal unstable so that the phase variations become raually larer. Therefore, this simulation is an excellent test for the state computation alorithm uner rowin phase variations. Aain, the results from the alorithm in Section.1 match very well with actual values in Fi Effects of subtransient flux ecay ynamics Next, we will stuy the effects of the subtransient parameters on the accuracy of the alorithm. The alorithm as presente in this paper completely inores the subtransient flux /04 $17.00 (C) 004 EEE 3

4 theta1 elta1 theta elta theta3 elta3 theta4 elta4 theta1 elta1 theta elta theta3 elta3 theta4 elta Fi.3. Comparison of actual an compute enerator anles Fi.4. Comparison of actual an compute enerator anles theta1 elta1 theta elta theta3 elta3 theta4 elta Fi.5. Comparison of actual an compute enerator anles ynamics. Hence, the approximation suffers somewhat, as compare to the earlier results in this section. The alorithm can be improve with the availability of the fiel voltae E f an these results will be presente elsewhere. Even when the subtranients are inore, the anles compute by the metho of the paper still compare well with the actual values in Fiures 4 an 5 which are re-simulations of Fiures 1 an 3 after incluin the subtransient parameters in the ETMSP simulations..4. Effects of machine saturation Next, we will consier the effects of machine core saturation, an the saturation effects can be reaily absorbe into the state computation alorithm. A iscussion on the moelin of saturation in machine moels can be seen in Section 3.8. of []. The saturation effects can be effectively hanle by multiplyin the machine parameters x, x, x a,an x a by a saturation parameters K s that epens on the internal flux linkaes. Since the state computation metho oes compute approximate values for the internal flux linkaes, these can be irectly use for calculatin K s. The irect alorithm presente in Section.1 nees to be moifie to be an implicit iterative alorithm for solvin K s as shown below. Suppose for a iven set of terminal measurements, we start with K s 0 =1 an the iteration counter k=0. We can compute the Park variables for the voltae an current as in Section /04 $17.00 (C) 004 EEE 4

5 Next, evaluate the flux linkaes as follows. a at = = + x a l, + a a = + x l (13) (14) f ψ at is less than the saturation parameter ψ T1, then, K s remains at 1. Otherwise, we can compute the flux ψ as = A sat ( B ( )) (15) exp 1 sat at where A sat an B sat are also machine saturation parameters. Now, K s can be compute as K s at = + at T (16) an we will enote this new value of K s as the next iterate value K s k+1. We can upate the values of the machine reactances x, x, x a,an x a by multiplyin with the new value K s k+1, an increment the iteration counter from k to k+1. We can now o back to the alorithm of Section.1 to compute the values for Park voltaes an currents from K s. Typically, in our simulations, the saturation parameter K s converes to final value in 3 to 4 iterations on k, an hence, the computation is very fast. theta1 theta theta3 theta Fi.6. Comparison of actual an compute enerator anles We will once aain simulate the two continencies shown in Fiures 1 an 3 after incluin the saturation parameters in ETMSP simulations. These simulations are therefore, very realistic moels for synchronous machine simulations use in the inustry. We will also assume that the saturation parameters are available in the state computation alorithm. Fiures 6 an 7 are the re-simulations of Fiures 1 an 3 with both subtransient ynamics an machine saturation inclue in the simulations. The alorithm presente here inores the subtransient ynamics while the saturation is incorporate into the state computation proceure. theta1 theta theta3 theta Fi.7. Comparison of actual an compute enerator anles Simulations in Fiures 6 an 7 emonstrate that the compute values for the states are very oo approximations for the actual values even for realistic etaile machine moels.. CONCLUSONS The paper proposes a new methooloy for irect computation of the internal ynamic state variables of a synchronous enerator, purely from terminal measurements. The computational proceure is completely ecouple from one enerator to another, an is also inepenent of the internal controls in the enerator as well the external network. t is erive from an analysis of the machine flux ecay euations. Simulations show that the results are reasonable even for etaile machine moels. The results thus far, have not use the availability of the fiel voltae E f which provies aitional insiht into the internal machine flux linkaes, an /04 $17.00 (C) 004 EEE 5

6 the analysis will be extene to cover the availability of E f in a future paper. All the results in the paper are base on irect computation an we have not utilize any smoothenin of the compute values usin Kalman filter type ynamic state estimation approaches. The results open up a whole array of excitin new control esins for the synchronous enerator wherein we can assume the internal ynamic state variables of the enerator incluin the rotor anle to be known for the enerators in the system. The traitional control esins base on output feeback esins can be moifie to be state feeback esins, an some of their avantaes are shown in the octoral thesis [1]. With the evolution of wie-area measurement systems, an with the knowlee of internal ynamic state variables at each enerator, we can start to formulate truly ambitious wie-area control alorithms in the future.. REFERENCES [1] Rajesh G. Kavasseri, A new methooloy for the esin of power system stabilizers, Ph.D. Dissertation, School of Electrical En. an Computer Science, Washinton State University, Pullman, WA , Auust 00. [] P. Kunur, Power System Stability an Control, McGraw Hill, [3] M. lic an J. Zaborszky, Dynamics an Control of Lare Electric Power Systems, John Wiley an Sons nc., /04 $17.00 (C) 004 EEE 6