Power system modelling under the phasor approximation

Transcription

1 ELEC Power system dynamics, control and stability Thierry Van Cutsem t.vancutsem@ulg.ac.be October / 16

2 Electromagnetic transient vs. phasor-mode simulations Electromagnetic transient vs. phasor-mode simulations electromagnetic transient simul. dψ/dt terms included virtually any dynamics included computes the full-wave evolutions of voltages and currents time step smaller than s network modeled with differential eqs. of inductors and capacitors simul. under phasor approximation dψ/dt terms neglected (in network and connected equipment) sinusoidal evolution with varying magnitude and phase angle; ignore dynamics shorter than 1 cycle computes the evolutions of magnitudes and phase angles of voltages and current phasors time step larger than 1/4 cycle (except after a discontinuity) network modeled by admittance matrix and voltage/current phasors 2 / 16

3 Electromagnetic transient vs. phasor-mode simulations electromagnetic transient simul. simul. under phasor approximation three-phase representation all imbalanced conditions not suitable for large-scale studies simulated time up to 10 s single-phase representation by default, balanced conditions (+ corrections for inverse and zero sequences) suitable for large-scale studies (e.g buses for the continental European grid) simulated time from a few seconds to minutes used to design components used for system-wide stability studies + hardware-in-the-loop (angle, frequency, voltage) Examples: Examples: EMTP-RV, PSCAD, RTDS,... Eurostag, Power Factory, PSS/E,... RAMSES 3 / 16

4 Electromagnetic transient vs. phasor-mode simulations Dynamics considered in phasor-mode simulation 4 / 16

5 Network modelling under the phasor approximation Network modelling under the phasor approximation Network equations based on bus admittance matrix Ī = Y V (1) Ī : vector of complex currents injected into the network at the various nodes V : vector of complex voltages at the various buses Y : bus (or nodal) admittance matrix Which frequency consider in the admittances? In dynamic regime, each synchronous machine defines a local frequency in most cases those various frequencies remain close to f N for large deviations with respect to the nominal value f N, the admittances entering the Ȳ matrix can be updated with the average system frequency otherwise, the admittances are simply computed at frequency f N this is consistent with the approximation ω 1 pu made in the Park equations of the synchronous machines. 5 / 16

6 Network modelling under the phasor approximation The synchronous reference frame Under the phasor approximation, the voltage at the i-th bus 1 takes on the form: v i (t) = 2V i (t) cos(ω N t + φ i (t)) = 2 re [V ] i (t)e j φ i (t) e j ω N t = 2 re [ (v xi (t) + jv yi (t)) e j ω N t ] v xi (t) + j v yi (t) is the voltage phasor, in rectangular coordinates, expressed with respect to (x, y) axes rotating at the angular speed ω N. im y V ie jφi e jωn t v yi φ i v xi ω N t x re These (x, y) axes are said to make up a synchronous reference frame. 1 this is just an example. Similar expressions hold true for currents 6 / 16

7 Network modelling under the phasor approximation Main limitation of the synchronous reference frame : after a disturbance the system settles at a new frequency f f N all phasors rotate at the angular speed 2πf ω N phasor components such as v xi and v yi oscillate at frequency f f N, although the system is at equilibrium from a practical viewpoint. The synchronous reference is not suitable for long-term simulation, since tracking the above oscillations requires using a small enough time step size it is thus used in short-term simulation (where frequency has not yet returned to steady state). In fact, any reference frame (x, y) can be used it does not need to rotate at the angular speed ω N ; it can rotate at any known speed which is convenient the only constraint is that all voltage and current phasors must refer to the same axes. 7 / 16

8 The center of inertia reference Network modelling under the phasor approximation Consider axes (x, y) rotating at the angular speed : θ coi = d ( m i=1 M ) m iθ ri i=1 dt m i=1 M = M i θ ri m i i=1 M i where: m is the total number of synchronous machines θ ri is the angular speed of i-th synchronous machine (i = 1,..., m) (rad/s) M i is the inertia of the i-th machine expressed on a common base power S B M i = 2H i S Ni S B θ coi is the angular speed of the center of inertia (rad/s). When the system settles at a frequency f, all synchronous machines rotate at the speed 2πf, and so do the reference axes (ω coi = 2πf ) phasor components such as v xi and v yi tend to constant values simulation is less demanding and a larger time step size can be used this reference frame is suitable for long-term simulation. 8 / 16

9 Network modelling under the phasor approximation Network equations in an (x, y) reference frame With all voltage and current phasors referred to the (x, y) axes, the network equations take on the form: i x + j i y = Y (v x + j v y ) = (G + j B) (v x + j v y ) with: v x = v x1. v xn v y = v y1. v yn i x = i x1. i xn i y = i y1. i yn G: conductance matrix B: susceptance matrix Decomposing into real and imaginary parts and assembling into a single equation: [ ] [ ] [ ] ix G B vx = i y B G v y N buses 2N equations involving 4N variables 9 / 16

10 Incorporating synchronous machines Incorporating synchronous machines Passing from individual machine to network reference frame The (d, q) reference frame simplifies a lot the machine model but it is a local reference. It is required to get back to the common reference. The reference axes (x, y) rotate at the angular speed ω ref c is an arbitrary constant δ is often called rotor angle Ṽ = (v x + jv y )e j(ω ref t+c) = (v q + jv d )e j(δ+ω ref t+c) v x + jv y = (v q + jv d )e jδ = (v q + jv d ) (cos δ + j sin δ) Decomposing into real and imaginary parts: v d = sin δ v x + cos δ v y v q = cos δ v x + sin δ v y 10 / 16

11 Incorporating synchronous machines Similarly for the currents: i d = sin δ i x + cos δ i y i q = cos δ i x + sin δ i y The machine stator equations 2 : v d = R a i d ψ q = R a i d (L l i q + ψ aq ) v q = R a i q + ψ d = R a i q + (L l i d + ψ ad ) can be expressed in terms of v x, v y, i x, i y as follows: ( sin δ v x + cos δ v y ) + R a ( sin δ i x + cos δ i y ) + L l (cos δ i x + sin δ i y ) + ψ aq = 0 (cos δ v x + sin δ v y ) + R a (cos δ i x + sin δ i y ) L l ( sin δ i x + cos δ i y ) ψ ad = 0 2 See lecture Dynamics of the synchronous machine - Section Model simplifications 11 / 16

12 Incorporating synchronous machines Adjusting the motion equation to change from the variable θ r to the variable δ θ r = ω ref t + c + δ + π 2 d dt θ r = ω ref + d dt δ Hence, the previous equation : 1 d ω N dt θ r = ω is replaced by : 1 ω N d dt δ = ω ω ref ω N with the synchronous reference : with the COI reference : where ω, ω i and ω coi are in per unit. 1 d ω N dt δ = ω 1 1 d ω N dt δ = ω ω coi where ω coi = The motion equation 2H d dt ω = T m T e is unchanged. m i=1 M iω i m i=1 M i 12 / 16

13 Incorporating induction machines Incorporating induction machines The induction machine model relies on (d, q) axes rotating at ω s, the stator angular speed. In order the model to properly account for the effect of frequency, it is appropriate to use the center of inertia reference and approximate ω s by ω coi (the average angular frequency of the whole system) make the q axis coincide with the x axis make the d axis coincide with the y axis. The machine stator equations 3 : become : v ds = R s i ds + ω s ψ qs = R s i ds + ω s L ss i qs + ω s L sr i qr v qs = R s i qs ω s ψ ds = R s i qs ω s L ss i ds ω s L sr i dr v y R s i y ω coi L ss i x ω coi L sr i qr = 0 v x R s i x + ω coi L ss i y + ω coi L sr i dr = 0 3 See lecture Dynamics of the induction machine 13 / 16

14 Incorporating static loads Incorporating static loads Consider a load whose active and reactive powers vary with the terminal voltage V according to known algebraic relations P(V ) and Q(V ). voltage magnitude: V = v 2 x + v 2 y Note: in the formulation (1) the bus currents are oriented into the network P(V ) + j Q(V ) = V Ī = (v x + j v y )(i x j i y ) Decomposing into real and imaginary parts: ( ) P vx 2 + vy 2 + v x i x + v y i y = 0 ( ) Q vx 2 + vy 2 v x i y + v y i x = 0 14 / 16

15 Incorporating static loads Extension Consider a load whose active and reactive powers vary with the terminal voltage V and frequency f according to known algebraic relations P(V, f ) and Q(V, f ). How is the frequency evaluated in phasor-mode simulation? during transients, there is not a single frequency the frequency at a bus can be obtained numerically as the derivative of the voltage phase angle at that bus instead, the average system frequency ω coi can be used, as for the induction machine (thus, the same value is used for all loads). 15 / 16

16 Overall model of components connected to network Overall model of components connected to network V = [V x1 V y1... V xn V yn ] T network V xj V yj j th bus I xi I yi i th injector M injectors x i = [I xi I yi... ] T State vector of i-th injector : x i = [i xi i yi...] T (2) Differential-algebraic model of i-th injector: Γ i ẋ i = f i (x i, v xj, v yj ) (3) with dim x i = dim f i. Γ i : square matrix with zero elements, except: [Γ i ] kl = 1 if the k-th equation gives [ẋ i ] l 16 / 16