Dynamic Modeling and Analysis of Large-scale Power Systems in the DQ0 Reference Frame

Transcription

1 Dynamic Moeling an Analysis of Large-scale Power Systems in the DQ0 Reference Frame Juri Belikov Tallinn University of Technology December 12, 2017 Juri Belikov (TUT) Moeling an Ientification December 12, / 26

2 Sources of energy Fossil fuel (non-renewable) energy sources: Oil, gas, coal, etc. Limite an can eventually run out Renewable energy sources: Sun, win, biomass, ties, waste, etc. Unlimite Juri Belikov (TUT) Moeling an Ientification December 12, / 26

3 Motivation Fossil fuel problems: Non-renewable Environmental hazars: Greenhouse gas emissions (carbon, nitrogen, an sulfur ioxie, etc.), air an water pollution Price fluctuations Overepenence Resources are running out: Fossil fuels are finite Possible solution: Shift energy prouction from fossil to renewable energy sources Juri Belikov (TUT) Moeling an Ientification December 12, / 26

4 Power systems: Current trens increasing interconnection more renewable sources more small an istribute power sources a shift from a centralize approach to a istribute approach 310 MW of win energy by MW of win energy by 2016 Juri Belikov (TUT) Moeling an Ientification December 12, / 26

5 Renewable energy goals [EU]: present/future Main irectives: 2009/72/EC L0072&from=EN COM(2016) 767/F2 COM F2-EN-MAIN-PART-1.PDF EU Goals on renewable energy source (from COM(2016) 767/F2): 10.4% by % by 2015 >27% by 2030 (Current estimation is 24.3%. EU countries has some work to o). Reaching this treshhol is in accorance with Paris agreement 2016 ( National plans by countries: Estonia Goals on renewable energy: 5.1% by 2010 (real 9.7%) 25% by 2020: 27% by 2030: Real time online system: Juri Belikov (TUT) Moeling an Ientification December 12, / 26

6 Distribute approach: Challenges How o we manage an control many inepenent energy sources an make them work together? Security Efficiency Reliability Dynamics & Stability Design Sensing Juri Belikov (TUT) Moeling an Ientification December 12, / 26

7 Moeling 1: Transient moels Network: linear moel Units: nonlinear moels x = Ax + BV t I = Cx + DV t ξ = f (ξ, I ) V = g(ξ, I ) Avantage: etaile an accurate Disavantage: too complex Juri Belikov (TUT) Moeling an Ientification December 12, / 26

8 Moeling 2: Quasi-static moels (time-varying phasors) Network: Power-flow equations N P n(t) = y n,k V n(t) V k (t) k=1 cos( y n,k + δ k (t) δ n(t)) N Q n(t) = y n,k V n(t) V k (t) k=1 sin( y n,k + δ k (t) δ n(t)) Units: nonlinear but time-invariant α 2 t 2 δ = Pm(t) 3P(t) K t δ V = g(ξ, I ) Avantage: simple moels an well-efine operating point small-signal stability analysis Disavantage: moels are only vali uner assumption of slowly varying signals Juri Belikov (TUT) Moeling an Ientification December 12, / 26

9 Mathematical Tools: DQ0 transformation Let x represent the quantity to be transforme (current, voltage, or flux), an use the compact notation x abc = [x a, x b, x c] T, x q0 = [x, x q, x 0 ] T. The q0 transformation with respect to the reference frame rotating with the angle ω st can be efine as x q0 = T ωs x abc, (1) with T ωs = 2 cos (ω st) cos ( ω st 2π ) cos ( ω 3 st + 2π ) 3 sin (ω st) sin ( ω st 2π ) sin ( ω 3 3 st + 2π ), (2) where ω s = 2πf s an f s {50, 60} Hz being the system nominal frequency. symmetric balance Juri Belikov (TUT) Moeling an Ientification December 12, / 26

10 DQ0 Transformation (cont.) Avantages: + Sinusoial (AC) signals are mappe into constant (DC) or slowly varying signals at steay-state + Inherits avantages of both quasi-static an abc moels + The analysis an controller esign are significantly simplifie Disavantage: Network is assume to be symmetric Table: Comparison of approaches for ynamic moeling Moel Operating Small- High Non-symmetric point signal frequencies networks time-varying phasors X X abc X X q0 X Juri Belikov (TUT) Moeling an Ientification December 12, / 26

11 Elementary passive components: Inuctor L Consier a network with a single three phase inuctor in the native abc reference frame. v a(t) i a(t) L unit v b (t) v c (t) i b (t) i c (t) L L A moel of the symmetric three-phase inuctor is given by L t I abc,12 = V abc,1 V abc,2. (3) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

12 Elementary passive components (cont.): Inuctor L The ifferentiation of I q0 = T ωs I abc results in which after simple algebraic manipulations yiels t I q0 = Tωs I abc + T ωs t t I abc, (4) t i,12 = ω si q,1 + 1 ( ) v,1 v,2, L t i q,12 = ω si,1 + 1 L (v q,1 v q,2 ), t i 0,12 = 1 L (v 0,1 v 0,2 ). This equation escribes a state-space moel of a symmetric three-phase inuctor. (5) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

13 Elementary passive components: Capacitor C an resistor R The moel of a symmetric three-phase capacitor C is given as ( ) ( ) 1 Vq0,1 V q0,2 = W Vq0,1 V q0,2 + t C I q0,12. (6) An for a symmetric three-phase resistor R the moel is given by the simple static relations where I 3 enotes the 3 3 ientity matrix. V q0 = I 3 RI q0, (7) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

14 Transmission network: Frequency omain moel In symmetric power networks, a ynamic moel base on q0 signals can be escribe as I (s) N 1 (s) jn 2 (s) 0 I q(s) = jn 2 (s) N 1 (s) 0 V (s) V q(s), I 0 (s) 0 0 Y bus (s) V 0 (s) where Y bus (s) is the frequency epenent noal amittance matrix an N 1 (s) := 1 2 N 2 (s) := 1 2 ( ) Y bus (s + jω s) + Y bus (s jω s), ( ) Y bus (s + jω s) Y bus (s jω s). Remark: If the general Y (s jωs) can be approximate by a constant matrix when s 0, then the ynamic moel is quasi-static, an the network may be moele by means of time-varying phasors. Juri Belikov (TUT) Moeling an Ientification December 12, / 26

15 Transmission network: More etails Network topology (by MatPower 1 ) ieal transformer L ik R ik shunt element y i shunt element y k bus i bus k τ ik : 1 Figure: Stanar branch connecting buses i an k. Noal amittance matrix: C i s 1 + C Y ik (s) = i R i s + 1 L i s R i L k F l1 l 2 s + R l1 l 2 τ 2 ( ) if i = k, i k T i l 1 l Ll1 2 l 2 s + R l1 l 2 1 ( ) if i k. τ l1 l Ll1 2 l 2 s + R l1 l 2 1 R. D. Zimmerman, C. E. Murillo-Sanchez, an R. J. Thomas, MATPOWER: Steay-state operations, planning, an analysis tools for power systems research an eucation, IEEE Trans. Power Syst., vol. 26, no. 1, pp , Feb Juri Belikov (TUT) Moeling an Ientification December 12, / 26

16 Synchronous machine: Simplifie moel The ynamic behavior of the angle δ such that δ = θ ω st + π/2 is escribe by 2 t 2 δ = poles ( P 3φ + 3P ref 1 ) 2Jω s D t δ, (8) which is the classic swing equation with the roop control mechanism. The term J is the rotor moment of inertia, poles is the number of machine poles (must be even), P ref is the single-phase reference power, an D represents the roop control sloop parameter. The three-phase power can be compute by Let δ = φ 1, then the state equations become P 3φ = 3 2 (v i + v qi q + 2v 0 i 0 ). (9) t δ = ω ωs, t ω = poles ( 32 2Jω Ve (cos(δ)i + sin(δ)i q) + 3P ref 1D ) (ω ωs), s (10) an the outputs are efine by v = V e cos (δ) v q = V e sin (δ) v 0 = 0. (11) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

17 Synchronous machine: Physical moel Recall a more sophisticate (physical) moel of a synchronous machine. The moel presente herein captures the interaction of the irect-axis magnetic fiel with the quarature-axis mmf, an the quarature-axis magnetic fiel with the irect-axis mmf, as well as the effects of resistances, transformer voltages, fiel wining ynamics, an salient poles. Table: Nomenclature: Synchronous machine λ, λ q, λ 0 λ f v, v q, v 0 i, i q, i 0 v f, i f L, L q, L 0 L af L ff R a, R f J T m flux linkages fiel wining flux linkage stator voltages stator currents fiel wining voltage an current synchronous inuctances mutual inuctance between the fiel wining an phase a self-inuctance of the fiel wining armature an fiel wining resistance rotor moment of inertia mechanical torque Juri Belikov (TUT) Moeling an Ientification December 12, / 26

18 Synchronous machine: Physical moel (cont.) The state equations of a synchronous machine in the q0 reference frame (with respect to ω st) are given by t φ 1 = 2RaL ff L 2 φ 1 + φ 2 φ 5 + 2RaL af β L 2 φ 4 + sin(φ 6 )v cos(φ 6 )v q, β t φ 2 = Ra L q φ 2 φ 1 φ 5 + cos(φ 6 )v + sin(φ 6 )v q, t φ 3 = Ra φ 3 + v 0, L 0 t φ 4 = 3R f L af L 2 φ 1 2R f L β L 2 φ 4 + v f, β ( t φ 5 = poles T m + 3L2 β 6L ) ff L q 2J 2L 2 φ 1 φ 2 + 3L af β Lq L 2 φ 2 φ 4, β t φ 6 = φ 5 ω s, where L 2 β = 2L L ff 3L 2 af. In this moel, the state variables are selecte as φ 1 = λ, φ 2 = λ q, φ 3 = λ 0, φ 4 = λ f, φ 5 = ω, δ = φ 6 an the inputs as v, v q, v 0, v f, T m. (12) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

19 Synchronous machine: Physical vs. Simplifie Juri Belikov (TUT) Moeling an Ientification December 12, / 26

20 Examples: State-space representations (matrix form) 9-bus system x R 45, u, y R bus x R 1119, u, y R bus x R 15675, u, y R 7149 Aξ Bξ Aξ Bξ Aξ Bξ Cξ Dξ Cξ Dξ Cξ Dξ 100 Sparsity (%) abc q0 US state of Illinois Polish system: winter peak Nonzero elements abc 6 9 q Number of buses Sparsity (%) elements abc q abc q0 Juri Belikov (TUT) Moeling an Ientification 10 3 December 12, / 26

21 Examples: 118-bus network (single-line iagram) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

22 Examples: 118-bus network (matrices) Juri Belikov (TUT) Moeling an Ientification December 12, / 26

23 Examples: 118-bus network (Scenario 1) Imag Real Figure: Eigenanalysis: root locus of largest eigenvalues when active power consumption is change. Diamons ( ) an crosses ( ) correspon to quasi-static an q0 moels, respectively. i,27 [A/MW] Time [s] Figure: Comparison of time omain responses. The lines correspon to quasi-static ( ), abc ( ), an q0 ( ) moels. Table: Largest Eigenvalues: Increase in Active Power Consumption Moel Eig. # Initial (4242 MW) Step (50%) q qs q qs q0 3, ± j ± j qs Juri Belikov (TUT) Moeling an Ientification December 12, / 26

24 Examples: 118-bus network (Scenario 1) % % % Imag i,27 [A/MW] Real Real Real Time [s] Figure: Eigenanalysis: root locus of largest eigenvalues when active power consumption is change. Diamons ( ) an crosses ( ) correspon to quasi-static an q0 moels, respectively. Figure: Comparison of time omain responses. The lines correspon to quasi-static ( ), abc ( ), an q0 ( ) moels. Table: Largest Eigenvalues: Changes in Damping Factor Moel K Eigenvalues q ± j 100% qs q ± j 25% qs q ± j 50% qs Juri Belikov (TUT) Moeling an Ientification December 12, / 26

25 Software package Toolbox for Moeling an Analysis of Power Networks in the DQ0 Reference Frame MATLAB Central File Exchange Currently, the package contains: Manual & Tutorial Construct the minimal state-space moel of a power network from given??(??) matrix Construct state-space moels of common units Derive feeback-connecte system Small-signal stability analysis Compute step response of very large systems 104 states Various examples of ifferent networks ranging from 2 to 2736 buses (mainly base on MatPower) Graphical user interface/tutorial Etc. Juri Belikov (TUT) Moeling an Ientification December 12, / 26

26 Thank you very much for your attention! Any questions? Juri Belikov (TUT) Moeling an Ientification December 12, / 26