Simulation of Power Systems Dynamics using Dynamic Phasor Models. Power Systems Laboratory. ETH Zürich Switzerland

Transcription

1 X SEPOPE 2 a 25 de maio de 26 May 2 rt to 25 th 26 FLORIANÓPOLIS (SC) BRASIL X SIMPÓSIO DE ESPECIALISTAS EM PLANEJAMENTO DA OPERAÇÃO E EXPANSÃO ELÉTRICA X SYMPOSIUM OF SPECIALISTS IN ELECTRIC OPERATIONAL AND EXPANSION PLANNING Simulation of Power Sytem Dynamic uing Dynamic Phaor Model Turhan Demiray demirayt@eeh.ee.ethz.ch Göran Anderon anderon@eeh.ee.ethz.ch Power Sytem Laboratory ETH Zürich Switzerland SUMMARY Dynamic phaor modeling technique i motly applied on power electronic device. In thi paper application of the dynamic phaor modeling technique i extended to other major power ytem component. Starting from the detailed -domain model the dynamic phaor model for thee component are derived and implemented. Further an efficiency and accuracy comparion of the dynamic phaor modeling technique with other conventional modeling technique - like the tandard model in three-phae ABC reference frame (e.g. in EMTP) and model in DQ reference frame (e.g. in SIMPOW) - i given. A ytematic comparion i warranted by imulating thee differently modeled component in a Matlab baed common imulation framework. Furthermore reduced order dynamic phaor model of the component are derived and reult are compared with other production-grade tranient tability program. Simulation reult for two tet cae are hown. KEYWORDS Dynamic Phaor Tranient Stability Hybrid Sytem

2 . Introduction The dynamic behavior and tability of power ytem are mot often tudied with the o called tranient tability program. Traditionally in mot of thee program the phaor model approach i ued. Recent development particularly the introduction of more power electronic baed equipment e.g. HVDC and FACTS component how that there i a hortage in the analyi method with fundamental frequency phaor model. For uch component a full domain imulation might be needed. Due to the limitation of computer torage and computation a complete repreentation of a large power ytem in an electromagnetic tranient program i very difficult and won't give additional information. To overcome thi problem the concept of -varying Dynamic phaor i propoed [] to model power electronic baed equipment imilar approache have been reported in [2]. It ha everal advantage over phaor baed method. Firtly it ha a wider bandwih in the frequency domain than the quai-tationary phaor. Secondly dynamic phaor can be ued to compute fat electromagnetic tranient with larger tep o that it dratically reduce the imulation compared with conventional domain EMTP like imulation. Phaor Dynamic Approach provide a middle ground between the approximation inherent in a phaor baed inuoidal quai-teady-tate repreentation and the analytic complexity and computational burden aociated with repreenting network voltage and current in -domain. Section 2 give a brief overview of the different modeling technique which are commonly ued in production-grade program for modeling of power ytem component. It further contain an introduction to the concept of -varying Fourier coefficient which i ued a an analytical tool for deriving dynamic phaor model. Section 3 focue on the tructural overview of the ued imulation framework and implemented tool uch a Automatic Code Generator and Dynamic Phaor Model Generator which are ued in thi work to automate the ource code creation of component model. Section 4 comprie the implemented model and ued tet cae. And finally ection 5 give a comparion of imulation reult. 2. Different Modeling Technique 2.. Model in Three-Phae (ABC) Reference Frame True phyical model in three-phae reference frame are the tarting point for the modeling technique which will be treated in the following ection. All electrical quantitie of the electrical network uch a voltage current etc. and all model equation are given in the three-phae (ABC) reference frame. Such model are commonly ued in EMTP like detailed -domain imulation program. In ABC reference frame any kind of equipment can be modeled eaily e.g. power electronic baed equipment uch a FACTS and HVDC. But the efficiency of the imulation can uffer from the periodicity due to the preence of AC phae quantitie (5 6 Hz) even during teady tate condition Model in DQ Reference Frame The DQ reference frame i an at the ytem frequency rotating reference frame which i commonly ued in the derivation of ynchronou machine equation a decribed in [3]. One of the advantage in analyzing the ynchronou machine equation in DQ reference frame i that under balanced teady-tate operation tator quantitie have contant value and for other mode of operation thi quantitie vary lowly with (2-3 Hz). Thi advantage lead alo to fater imulation under balanced condition a the variation of variable in DQ reference frame are much lower than the original variable in three-phae ABC reference frame. Therefore in ome program (e.g. SIMPOW [4]) the DQ tranformation i applied to all component in the ytem. All variable and equation of the model in three-phae ABC reference frame are tranformed to DQ reference frame. The DQ tranformation i a ingle reference frame tranformation a the reference frame rotate with ytem frequency. Therefore the imulation in DQ reference frame will be efficient around ytem frequency. But if there are unbalanced condition in the ytem or other harmonic thi efficiency can

3 decreae dratically. Mathematically the DQ or Park tranformation i given by the following equation in (2.) where θ = ωt = 2π ftand f i the ytem frequency. 2π 2π co( θ) co( θ ) co( θ + ) () 3 3 id t ia () t (2.) iq ( t) π π in( θ) in( θ ) in( θ ) ib( t) = + idq( t) = Tdq i( t) i () t ic () t Here we ued the phae current a an example of a phae quantity. The invere tranformation i given by i () t = T idq() t with T = T. The derivative of the ABC phae quantitie can be written a dq (2.2) ω di di di = + + dq dq ω idq J idq Dynamic Phaor Model If power electronic baed equipment are ued in the power ytem the aumption of only having frequencie near to the fundamental frequency i uually not valid any more a hown in Figure. For example in the cae of TCSC - depending on the conduction angle - other harmonic have a ignificant participation in the tranient waveform of voltage and current. V C I L V C I L α τ α... Firing angle σ... Conduction angle σ V... Capacitor voltage C I... Inductor current L Harmonic Content Ratio ( V k / V ) k = 3 k = 5 k = 7 k = conduction angle [degree] Figure : TCSC: Tranient waveform and Harmonic content ratio a a function of the conduction angle The main idea behind dynamic phaor modeling i to approximate a real valued periodic waveform x( τ ) in the interval τ ( t T t] with a Fourier erie form [5] (2.3) x( τ ) = X ()co t ( kω τ) + X ()in t ( kω τ) k = kc k where ω = 2 π / T and X () t and X () t are the -varying Fourier coefficient. X () t and kc X k () t can be determined by the following averaging operation. k kc (2.4) t X t x k d x t () = ( τ) co( ωτ) τ = () kc kc T t T t X t x k d x t () = ( τ) in ( ωτ) τ = () k k T t T 2

4 A key factor in developing dynamic phaor model i the relation between the derivative of x( τ ) and the derivative of X kc () t and X k () t which i given in (2.5). Another important property i that the product of two -domain variable equal a dicrete convolution of the two dynamic phaor et of the variable. (2.5) 3. Simulation Framework dx dx dx k c = ω kc dx k = + ω k To be able to make a ytematic comparion between the efficiency and accuracy of the mentioned modeling technique we have implemented all the major component of a power ytem (Synchronou machine Tranformer Line Load TCSC etc.) in a common imulation framework where all equation of the power ytem are olved imultaneouly. In thi chapter a tructural overview of the implemented imulation framework i given. 3.. Hybrid Sytem Repreentation Power ytem are an important cla of hybrid ytem a they exhibit interaction between continuou dynamic and dicrete event [6][7]. Component of uch hybrid ytem can be modelled by a et of Differential Switched Algebraic and State-Reet (DSAR) equation (3.) = f( x y) (3.) = g x y () ( ) - ( i ) g x y ydi < + ( i ) g x y ydi > ( ) = i =... d ( ) + x = hx ( y) y = j... e e j { } X X c x f x x = z f = h j = hj λ λ where x are continuou dynamic tate z are dicrete tate y are algebraic tate y d and ye are o called event variable which trigger event if they change ign ( y d ) and/or pa through zero ( y e ) λ are parameter. The overall ytem i built by connecting the interface variable of the component together. Thee y = y c y = y y = ). connection can be formulated by imple algebraic equation ( ( ) y Dicrete + x = h( x y ) Dicrete + x = h2( x y ) t t2 Continuou Continuou Continuou = f( x y) = f( x y) = f( x y) = g xy () ( ) = g xy () ( ) = g xy (2) ( ) Figure 2: Simulation Flow - Differential Switched Algebraic and State-Reet (DSAR) equation 3

5 A hown in Figure 2 between event the ytem behavior i governed by the differential algebraic equation = f( x y) and () i g ( x y ) =. At t = t an event occur and the correponding potevent value of dicrete and continuou dynamic tate x + + are calculated according to x = hx ( y) where upercript + mean pot-event value and mean pre-event value of the correponding variable. After x + are calculated pot-event value of algebraic variable y + are calculated by olving the nonlinear equation () + + g ( x y ) = at t = t. After pot-event value x + and y + are determined they erve a initial condition for the new continuou ection. The ytem behavior in thi ection i governed by the equation = f( x y) and g () ( x y ) =. The overall ytem equation are given by (3.2) where in thi formulation x y contain all model variable and f g all model equation. The connection equation cy ( ) can be included in gxy= ( ). (3.2) = f( x y) = g( xy ) = cy ( ) xn+ =Ψ ( h f( xn+ yn+ ) f( xn yn)...) = gx ( n+ yn+ ) = cy ( n+ ) ( ) F xn+ yn+ = If we dicretize the differential equation with an implicit integration method x =Ψ ( h f( x y ) f( x y )...) we end up in a nonlinear et of implicit algebraic equation n+ n+ n+ n n χ xn+ yn+ F( xn+ yn+ ) = F( χ) = with = [ ]. Thi equation can be olved iteratively according to χ = i χ i Fχ ( χi) + F( χi) where F ( ) χ χ i the Jacobian of F( χ) with repect to χ (3.3). (3.3) Ψ( h f( x y ) f( x y )...) x F( χ) g( x y ) cy ( n+ ) n+ n+ n n n+ = n+ n+ ( Ψ f fx I) ( Ψ f fy) Fχ ( χ) = gx gy. c y From (3.3) we can ee that the imulation kernel need the evaluation of f g and the partial derivative fx fy gx g y from every model to olve the nonlinear algebraic equation et F( χ ) =. ΨΨ f are dependent on the applied numerical integration method and cc y are dependent on the connection. Becaue of it numerical tability the Trapezoidal Integration Method with variable tep control i ued a numerical integration method in thi framework Automatic Code Generator The imulation kernel i implemented in Matlab and the model are pecified in o called Model Definition File (MDF) where the uer imply define the equation and variable of the model a they are given in (3.). A propoed in [7] we have alo implemented an Automatic Code Generator (ACG) in Matlab. ACG take the MDF and create the Matlab code for the model uing the Symbolic Differentiation facility in Matlab for the analytical calculation of fx fy gx g y. All model of the major power ytem component are implemented in ABC and DQ reference frame in thi framework. To automate the creation of the dynamic phaor model from their ABC or DQ model we built in an interface called Dynamic Phaor Model Generator. It take the MDF of the model in ABC or DQ form. Uing (2.3) and (2.5) the continuou dynamic tate and algebraic tate are replaced with the approximated counterpart in the model equation (3.) which lead to f( x y) f( X X c co ( ωt) X in ( ωt)... Y Y c co ( ωt) Y in ( ωt)...) gxy ( ) gx ( X co ω t X in ω t... Y Y co ω t Y in ω t...) ( ) ( ) ( ) ( ) c c By uing (2.4) we can tranform the et of f g equation of the model into a new et of equation and get the definition of the phaor dynamic model in a new et of function (3.4) 4

6 (3.4) = f( x y) = g( xy ) X = F ( X X Y Y ) k ω X X = F ( X X Y Y ) + k ω X = Gk c ( Xk c Xk Ykc Yk ) = G ( X X Y Y ) kc kc kc k kc k k k k k c k kc k k c k kc k kc k where k i et of phaor which are ued for the approximation in (2.3) [e.g. k = { 2} ]. Figure 3 how the chematic view of the Automatic Code Generation. Model Definition File in ABC or DQO k = 2 Dynamic Phaor Model Generator Model Definition File Model Definition File of Dynamic Phaor Model Automatic Code Generator (Matlab Program uing Symbolic Toolbox) Automatic Code Generator (Matlab Program uing Symbolic Toolbox) Matlab Code of the model Simulation Kernel (Matlab Program) f g h fx fy gx gy Matlab Code of the dynamic phaor model Simulation Kernel (Matlab Program) f g h fx fy gx gy 4. Model and Tet ytem Figure 3 We implemented the ynchronou machine model in DQ reference frame a decribed in [3 p. 86]. Starting from thi model we derived then the dynamic phaor model of the ynchronou machine by chooing k={2} in (3.4). The reaon for thi i to include poitive (k=) zero (k=) negative (k=2) equence quantitie [8] to imulate alo unbalanced condition in our model. We ued the tandard Π model for tranmiion line and parallel RLC model for load. But other model can alo be implemented eaily. The implemented TCSC model i baed on the work decribed in []. The domain equation for TCSC i are given in (4.) where q = if one of the thyritor i conducting and q = if both thyritor are blocking. i l The main idea here i to approximate the capacitor v voltage by vt () Vkc co( ωt) + Vk in( ωt) with k k = 35 to include the participation of major harmonic a illutrated in Figure. A a econd tep a teady-tate approximation i applied on Vkc and V k for k = 35 with Vkc = f V c V ρk( σ ) and = ( ( )) where ρk ( ) Vk f V c V ρk σ conduction angle. (4.) dv dv c C = il i C = Il c Ic ωv di di c L = q v L = f c V c V ρ3( σ) ρ5( σ) ( ) σ i the teady-tate harmonic content ratio a a function of the ( ) and dv = + ω di = C Il I V c ( ρ ( σ) ρ ( σ) ) L f V c V 3 5 SMIB (Single Machine Infinite Bu) and Two Area ytem are ued a tet ytem with the data given in [3]. 5

7 BUS2 BUS3 BUS 7 BUS 8 BUS BUS 5 BUS 6 BUS 9 BUS BUS BUS 3 LI 7-8 A LI 8-9 A LI 6-7 LI 9- LI - BUS LI 5-6 LI 7-8 B LI 8-9 B LINE2 GEN GEN 3 TR TR 3 TR2 LOAD 7 LOAD 9 TR 2 TR 4 C 3 C 9 GEN LINE3 FEEDER BUS 2 BUS 4 GEN 2 GEN 4 5. Simulation Reult Figure 4: Single line diagram of SMIB and Two Area Sytem A mentioned in ection 4 all dynamic phaor model of the component are derived from DQ model with k = {2}. But a derivation from ABC model i alo poible. 5.. Simulation with Detailed Time Domain Model In both tet ytem a one-phae to ground fault i applied a a diturbance at. and removed at.7. In Figure 5 and Figure 6 the electrical torque of the ynchronou machine GEN (SMIB) and GEN (Two Area ytem) are hown. In both figure we have the overall imulation interval and a zoomed ection during the unbalanced condition. For verification of our reult SMIB ytem ha alo been implemented in PLECS [9] and imulation reult are compared with thoe in our framework. PLECS i a toolbox for high-peed imulation of electrical and power electronic circuit under MATLAB/Simulink. We ee a very good overall agreement between all reult epecially in the zoomed ection. In the table the computation for thee imulation are compared. The imulation in DQ reference frame are the fatet. In thee two tet cae the unbalanced condition only lat for.7 all implemented component are ymmetrical component and no power electronic device are ued. A tated in ection 2.2 under ABC DQ k= (2) SMIB 2 ec 2 ec 32 ec Two Area Sytem 453 ec 272 ec 35 ec Table balanced condition and with frequencie near to the fundamental (ytem) frequency the choice of DQ reference frame i the bet choice for efficient computation. But if the computation during the unbalanced condition are compared the imulation with dynamic phaor model with k={2} are more efficient. The reaon for thi i during unbalanced condition zero and negative equence quantitie appear a ytem inherent harmonic in DQ reference frame (zero equence 5Hz and negative equence Hz). The tep ize during imulation mut be kept mall enough to imulate thee ocillation. But if dynamic phaor model are ued thee mentioned ytem inherent harmonic are already included in the model equation with k= (zero equence) and k=2 (negative equence) and imulation i marginally influenced dqo k = 2 PLECS dqo k = 2 PLECS Te [pu] - GEN.5 Te [pu] - GEN Figure 5: SMIB One phae to ground fault at BUS2 at. and cleared at.7 6

8 The dynamic phaor approach i a projection of a ignal on a multiple-orthogonal reference frame (2.4) where ignal co( kω t) and in( kω t) are orthogonal bae ignal (vector) dqo k = dqo k = Te [pu] - GEN Te [pu] - GEN Figure 6: Two Area Sytem One phae to ground fault at BUS8 at. and cleared at Simulation with dynamic phaor TCSC model To tet the implemented TCSC model decribed in ection 4 we modified the Two Area Sytem in Figure 4. We have plit BUS8 into BUS8 and BUS2 and placed the TCSC between thee node. In our imulation the firing angle of the thyritor are changed from 68.3 to 59 at.2 and back again at.5. Achieved reult are given in Figure 7. The imulation with ABC model wa about 23 and with the phaor dynamic model about PLECS Phaor Dynamic PLECS Phaor Dynamic Te [pu] - GEN Te [pu] - GEN Figure 7: Two Area Sytem Changing of firing angle from 68.3 to 59 at.2 and back again at Simulation with Reduced Order Model A next tep we made an eigenanalyi and a -cale decompoition to reduce the order of the dynamic phaor model. We looked at the eigenvalue and their participation factor. Dynamic of continuou tate with very low participation in the low mode ha been neglected which mean for thoe tate the X kc and X k are et to zero in equation (3.4). Same approach ha alo been applied on the DQ reference frame model which lead to fundamental frequency model ued in tranient tability program. We ued ame tet ytem (SMIB) and the ame fault. The imulation reult with reduced order model are compared with thoe in NEPLAN and SIMPOW [4 ] in Figure 8. The left part of the figure how the poitive-equence electrical torque where the dotted curve labeled PLECS i calculated from the actual domain ignal by etting k= in equation (2.4). The right part of the figure how the negative equence electrical torque during unbalanced condition where again the dotted curve labeled PLECS are calculated from the actual domain ignal by etting k=2 in equation (2.4). Even though we are uing reduced order model the reult how an overall agreement. Uually in tranient tability program the negative and zero equence network are not imulated directly but their effect in the poitive-equence network are taken into account by appropriately combining the equivalent negative and zero equence impedance depending on the 7

9 fault-type a decribed in [3]. With the ue of phaor dynamic model with k = {2} it wa alo poible to imulate the zero and negative equence quantitie throughout in the ytem during unbalanced condition in a ytematic way. The imulation with detailed model wa 28 and with reduced order phaor dynamic model SIMPOW k = 2 PLECS.3.2. k = 2: 2c PLECS: 2c k = 2: 2 PLECS: 2 Te [pu] Te 2c [pu] Te 2 [pu] Concluion Figure 8: SMIB One phae to ground fault at BUS2 at. and cleared at.7 In thi paper we have applied the phaor dynamic approach to major power ytem component and compared the accuracy and efficiency with different modeling technique uch a model in ABC and DQ reference frame. Simulation with ABC reference frame model are accurate but not o efficient. On the other hand imulation with DQ reference frame model are accurate and alo efficient under balanced condition and with frequencie near to the fundamental (ytem) frequency. If we have unbalanced condition or other frequencie (harmonic) in the ytem than the fundamental frequency (e.g. uage of power electronic baed equipment) a ingle reference tranformation like DQ tranformation become inefficient for imulation. In uch cae phaor dynamic model how a better performance becaue of their multi reference frame characteritic but here the accuracy depend on the appropriate election of the et of k in equation (2.3). ACKNOWLEDMENT The author would like to thank BCP for their financial upport Dr. Luigi Buarello and Giatgen Cott for timulating dicuion and Ian A. Hiken & Marek Zima for making the ource code of their file available. Thee file have been very ueful information while implementing thi imulation framework. BIBLIOGRAPHY [] P. Mattavelli G. C. Verghee and A. M. Stankovic "Phaor dynamic of thyritor-controlled erie capacitor ytem" Power Sytem IEEE Tranaction on vol. 2 pp [2] W. Hammer and G. Anderon "Dynamic Modeling of Capacitor Commutated Converter" invited paper 4 SEPOPE IX Rio de Janeiro Brazil 24. [3] P. Kundur N. J. Balu and M. G. Lauby Power ytem tability and control. New York: McGraw-Hill 994. [4] STRI "Power Sytem Simulation & Analyi Software SIMPOW Uer Manual [5] S. R. Sander J. M. Noworolki X. Z. Liu and G. C. Verghee "Generalized averaging method for power converion circuit" Power Electronic IEEE Tranaction on vol. 6 pp

10 [6] I. A. Hiken and M. A. Pai "Hybrid ytem view of power ytem modelling" preented at Circuit and Sytem 2. Proceeding. ISCAS 2 Geneva. The 2 IEEE International Sympoium on 2. [7] I. A. Hiken and P. J. Sokolowki "Sytematic modeling and ymbolically aited imulation of power ytem" Power Sytem IEEE Tranaction on vol. 6 pp [8] A. M. Stankovic and T. Aydin "Analyi of aymmetrical fault in power ytem uing dynamic phaor" Power Sytem IEEE Tranaction on vol. 5 pp [9] Plexim "PLECS Uer Manual - High-peed imulation of electrical and power electronic circuit under MATLAB/Simulink." [] BCP "NEPLAN oftware manual 9