Dynamic Moeling an Analysis of Large-scale Power Systems in the DQ0 Reference Frame Juri Belikov Tallinn University of Technology juri.belikov@ttu.ee December 12, 2017 Juri Belikov (TUT) Moeling an Ientification December 12, 2017 1 / 26
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, 2017 2 / 26
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, 2017 3 / 26
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 2016 44983 MW of win energy by 2016 Juri Belikov (TUT) Moeling an Ientification December 12, 2017 4 / 26
Renewable energy goals [EU]: present/future Main irectives: 2009/72/EC http://eur-lex.europa.eu/legal-content/en/txt/pdf/?uri=celex: 32009L0072&from=EN COM(2016) 767/F2 https://ec.europa.eu/transparency/regoc/rep/1/2016/en/ COM-2016-767-F2-EN-MAIN-PART-1.PDF EU Goals on renewable energy source (from COM(2016) 767/F2): 10.4% by 2007 17% 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 (http://unfccc.int/paris_agreement/items/9485.php). National plans by countries: https://ec.europa.eu/energy/en/topics/renewable-energy/national-action-plans Estonia Goals on renewable energy: 5.1% by 2010 (real 9.7%) 25% by 2020: https://www.mkm.ee/et/eesmargi-tegevuse/arengukava 27% by 2030: https://elering.ee/taastuvenergia-0 Real time online system: https://ashboar.elering.ee/en/system/prouction-renewable Juri Belikov (TUT) Moeling an Ientification December 12, 2017 5 / 26
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, 2017 6 / 26
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, 2017 7 / 26
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, 2017 8 / 26
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) 3 1 1 1 2 2 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, 2017 9 / 26
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, 2017 10 / 26
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, 2017 11 / 26
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, 2017 12 / 26
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, 2017 13 / 26
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, 2017 14 / 26
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 + 1 + 1 + 1 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. 12 19, Feb. 2011. Juri Belikov (TUT) Moeling an Ientification December 12, 2017 15 / 26
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, 2017 16 / 26
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, 2017 17 / 26
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, 2017 18 / 26
Synchronous machine: Physical vs. Simplifie Juri Belikov (TUT) Moeling an Ientification December 12, 2017 19 / 26
Examples: State-space representations (matrix form) 9-bus system x R 45, u, y R 27 200-bus x R 1119, u, y R 600 2383-bus x R 15675, u, y R 7149 Aξ Bξ Aξ Bξ Aξ Bξ Cξ Dξ Cξ Dξ Cξ Dξ 100 Sparsity (%) 95 90 4 5 6 9 14 24 30 39 abc q0 US state of Illinois 57 118 2383 2736 Polish system: winter 1999-2000 peak Nonzero elements 10 5 10 4 10 3 10 2 10 1 10 0 4 5 abc 6 9 q0 14 24 30 Number of buses 39 57 118 2383 2736 Sparsity (%) elements 100 95 90 10 5 4 5 6 9 14 24 30 39 57 abc q0 118 2383 2736 10 4 abc q0 Juri Belikov (TUT) Moeling an Ientification 10 3 December 12, 2017 20 / 26
Examples: 118-bus network (single-line iagram) Juri Belikov (TUT) Moeling an Ientification December 12, 2017 21 / 26
Examples: 118-bus network (matrices) Juri Belikov (TUT) Moeling an Ientification December 12, 2017 22 / 26
Examples: 118-bus network (Scenario 1) Imag 400 200 0-200 -400-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 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] 0.15 0.1 0.05 0-0.05 0 5 10 15 20 25 30 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%) q0 1 0.1629 0.0848 qs 1 0.1628 0.0848 q0 2 0.2390 0.1908 qs 2 0.2389 0.1907 q0 3, 4 0.2941 ± 314.1393j 0.2940 ± 314.1394j qs 3 0.6200 0.4121 Juri Belikov (TUT) Moeling an Ientification December 12, 2017 23 / 26
Examples: 118-bus network (Scenario 1) 200 100% 200 25% 200 50% 0.1 0.1 Imag 100 0-100 100 0-100 100 0-100 i,27 [A/MW] 0.05 0 0.05 8.2 8.3 8.4 8.5-200 -30-20 -10 0 Real -200-30 -20-10 0 Real -200-30 -20-10 0 Real -0.05 0 1 2 3 4 5 6 7 8 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 q0 0.1629 0.2390 29.356 ± 185.681j 100% qs 0.1628 0.2389 q0 0.1630 0.2390 13.916 ± 185.437j 25% qs 0.1629 0.2389 q0 0.1631 0.2390 2.668 ± 183.11j 50% qs 0.1630 0.2389 Juri Belikov (TUT) Moeling an Ientification December 12, 2017 24 / 26
Software package Toolbox for Moeling an Analysis of Power Networks in the DQ0 Reference Frame MATLAB Central File Exchange https://www.mathworks.com/matlabcentral/fileexchange/58702 Currently, the package contains: Manual & Tutorial https://a-lab.ee/projects/q0-ynamics 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, 2017 25 / 26
Thank you very much for your attention! Any questions? Juri Belikov (TUT) Moeling an Ientification December 12, 2017 26 / 26