The synchronous machine (detailed model)

ELEC0029 - Electric Power System Analysis The synchronous machine (detailed model) Thierry Van Cutsem t.vancutsem@ulg.ac.be www.montefiore.ulg.ac.be/~vct February 2018 1 / 6

Objectives The synchronous machine Extend the model of the synchronous machine considered in course ELEC0014: more detailed appropriate for dynamic studies includes the effect of damper windings applicable to machines with salient-pole rotors (hydro power plants) Relies on the Park transformation, also used for other power system components. 2 / 6

The two types of synchronous machines The two types of synchronous machines Round-rotor generators (or turbo-alternators) Driven by steam or gas turbines, which rotate at high speed one pair of poles (conventional thermal units) or two (nuclear units) cylindrical rotor made up of solid steel forging diameter << length (centrifugal force!) field winding made up of conductors distributed on the rotor, in milled slots even if the generator efficiency is around 99 %, the heat produced by Joule losses has to be evacuated! Large generators are cooled by hydrogen (heat evacuation 7 times better than air) or water (12 times better) flowing in the hollow stator conductors. / 6

The two types of synchronous machines 4 / 6

Salient-pole generators The synchronous machine The two types of synchronous machines Driven by hydraulic turbines (or diesel engines), which rotate at low speed many pairs of poles (at least 4) it is more convenient to have field windings concentrated and placed on the poles air gap is not constant: min. in front of a pole, max. in between two poles poles are shaped to also minimize space harmonics diameter >> length (to have space for the many poles) rotor is laminated (poles easier to construct) generators usually cooled by the flow of air around the rotor. 5 / 6

The two types of synchronous machines 6 / 6

The two types of synchronous machines Damper windings and eddy currents in rotor Damper windings (or amortisseur) round-rotor machines: copper/brass bars placed in the same slots at the field winding, and interconnected to form a damper cage (similar to the squirrel cage of an induction motor) salient-pole machines: copper/brass rods embedded in the poles and connected at their ends to rings or segments. Why? in perfect steady state: the magnetic fields produced by both the stator and the rotor are fixed relative to the rotor no current induced in dampers after a disturbance: the rotor moves with respect to stator magnetic field currents are induced in the dampers...... which, according to Lenz s law, create a damping torque helping the rotor to align on the stator magnetic field. Eddy currents in the rotor Round-rotor generators: the solid rotor offers a path for eddy currents, which produce an effect similar to those of amortisseurs. 7 / 6

Modelling of machine with magnetically coupled circuits Modelling of machine with magnetically coupled circuits Number of rotor windings = degree of sophistication of model. But: more detailed model more data are needed while measurement devices can be connected only to the field winding. Most widely used model: or 4 rotor windings: f : field winding d 1, q 1 : amortisseurs q 2 : accounts for eddy currents in rotor; not used in (laminated) salient-pole generators. 8 / 6

Modelling of machine with magnetically coupled circuits Remarks In the sequel, we consider: a machine with a single pair of poles, for simplicity. This does not affect the electrical behaviour of the generator (it affects the moment of inertia and the kinetic energy of rotating masses) the general case of a salient-pole machine. For a round-rotor machine: set some parameters to the same value in the d and q axes (to account for the equal air gap width) the configuration with four rotor windings (f, d 1, q 1, q 2 ). For a salient-pole generator : remove the q 2 winding. 9 / 6

Modelling of machine with magnetically coupled circuits Relations between voltages, currents and magnetic fluxes Stator windings: generator convention: v a (t) = R a i a (t) dψ a dt In matrix form: v b (t) = R a i b (t) dψ b dt v c (t) = R a i c (t) dψ c dt vt = RT i T d dt ψ T with RT = diag(r a R a R a ) 10 / 6

Modelling of machine with magnetically coupled circuits Rotor windings: motor convention: In matrix form: v f (t) = R f i f (t) + dψ f dt 0 = R d1 i d1 (t) + dψ d1 dt 0 = R q1 i q1 (t) + dψ q1 dt 0 = R q2 i q2 (t) + dψ q2 dt vr = Rr i r + d dt ψ r with Rr = diag(r f R d1 R q1 R q2 ) 11 / 6

Modelling of machine with magnetically coupled circuits Inductances Saturation being neglected, the fluxes vary linearly with the currents according to: [ ] [ ] [ ] ψt LTT (θ = r ) LTr (θ r ) it ψ r L T Tr (θ r ) Lrr ir LTT and LTr vary with the position θ r of the rotor but Lrr does not the components of LTT and LTr are periodic functions of θ r obviously the space harmonics in θ r are assumed negligible = sinusoidal machine assumption. 12 / 6

Modelling of machine with magnetically coupled circuits LTT (θ r ) = L 0 + L 1 cos 2θ r L m L 1 cos 2(θ r + π 6 ) L m L 1 cos 2(θ r π 6 ) L m L 1 cos 2(θ r + π 6 ) L 0 + L 1 cos 2(θ r 2π ) L m L 1 cos 2(θ r + π 2 ) L m L 1 cos 2(θ r π 6 ) L m L 1 cos 2(θ r + π 2 ) L 0 + L 1 cos 2(θ r + 2π ) L o, L 1, L m > 0 1 / 6

Modelling of machine with magnetically coupled circuits LTr (θ r ) = L af cos θ r L ad1 cos θ r L aq1 sin θ r L aq2 sin θ r L af cos(θ r 2π ) L ad1 cos(θ r 2π ) L aq1 sin(θ r 2π ) L aq2 sin(θ r 2π ) L af cos(θ r + 2π ) L ad1 cos(θ r + 2π ) L aq1 sin(θ r + 2π ) L aq2 sin(θ r + 2π ) L af, L ad1, L aq1, L aq2 > 0 14 / 6

Modelling of machine with magnetically coupled circuits L rr = L ff L fd1 0 0 L fd1 L d1d1 0 0 0 0 L q1q1 L q1q2 0 0 L q1q2 L q2q2 15 / 6

It is easily shown that: P P T = I P 1 = P T 16 / 6 The synchronous machine Park transformation and equations Park transformation Park transformation and equations is applied to stator variables (denoted. T ) to obtain the corresponding Park variables (denoted. P ): vp = P vt ψ P = P ψ T ip = P it 2 where P = cos θ r cos(θ r 2π ) cos(θ r + 2π ) sin θ r sin(θ r 2π ) sin(θ r + 2π ) 1 2 1 2 1 2 vp = [ v d v q v o ] T ψ P = [ ψ d ψ q ψ o ] T ip = [ i d i q i o ] T

Park transformation and equations Interpretation Total magnetic field created by the currents i a, i b et i c : projected on d axis: k (cos θ r i a + cos(θ r 2π ) i b + cos(θ r 4π ) i c) = k 2 i d projected on q axis: k (sin θ r i a + sin(θ r 2π ) i b + sin(θ r 4π ) i c) = k 2 i q The Park transformation consists of replacing the (a, b, c) stator windings by three equivalent windings (d, q, o): the d winding is attached to the d axis the q winding is attached to the q axis the currents i d and i q produce together the same magnetic field, to the multiplicative constant 2. 17 / 6

Park equations of the synchronous machine Park transformation and equations vt = RT it d dt ψ T P 1 vp = R a I P 1 ip d dt (P 1 ψ P ) vp = R a PP 1 ip P ( d dt P 1 )ψ P PP 1 d dt ψ P = RP ip θ r Pψ P d dt ψ P with: RP = RT P = 1 0 0 0 1 0 0 0 0 By decomposing: v d = R a i d θ r ψ q dψ d dt θ r ψ d, θ r ψ q : speed voltages v q = R a i q + θ r ψ d dψ q dt v o = R a i o dψ o dt dψ d /dt, dψ q /dt : transformer voltages 18 / 6

Park transformation and equations Park inductance matrix [ ψt ψ r [ P 1 ψ P [ ψ r [ ψp LPP LrP ] ] ψ r ] LPr Lrr = = = [ [ ] = LTT L T Tr LTT L T Tr LTr Lrr LTr Lrr ] [ it ir ] ] [ P 1 i P [ PL TT P 1 PLTr L T Tr P 1 Lrr ir ] [ ] ip ir ] [ = LPP LrP LPr Lrr ] [ L dd L df L dd1 L qq L qq1 L qq2 L oo L df L ff L fd1 L dd1 L fd1 L d1d1 L qq1 L q1q1 L q1q2 L qq2 L q1q2 L q2q2 ip ir ] (zero entries have been left empty for legibility) 19 / 6

with: The synchronous machine Park transformation and equations L dd = L 0 + L m + 2 L 1 L qq = L 0 + L m 2 L 1 L df = 2 L af L dd1 = 2 L ad1 L qq1 = 2 L aq1 L qq2 = 2 L aq2 L oo = L 0 2L m All components are independent of the rotor position θ r. That was expected! There is no magnetic coupling between d and q axes (this was already assumed in LTr and Lrr : zero mutual inductances between coils with orthogonal axes). 20 / 6

Park transformation and equations Leaving aside the o component and grouping (d, f, d 1 ), on one hand, and (q, q 1, q 2 ), on the other hand: v d v f 0 v q 0 0 = = R a R a R f R q1 R d1 R q2 i d i f i d1 i q i q1 i q2 + θ r ψ q 0 0 θ r ψ d 0 0 d dt d dt ψ d ψ f ψ d1 ψ q ψ q1 ψ q2 with the following flux-current relations: ψ d L dd L df L dd1 ψ f = L df L ff L fd1 ψ d1 L dd1 L fd1 L d1d1 ψ q ψ q1 ψ q2 = L qq L qq1 L qq2 L qq1 L q1q1 L q1q2 L qq2 L q1q2 L q2q2 i d i f i d1 i q i q1 i q2 21 / 6

Energy, power and torque Energy, power and torque Balance of power at stator: p T + p Js + dw ms dt = p r s p T : three-phase instantaneous power leaving the stator p Js : Joule losses in stator windings W ms : magnetic energy stored in the stator windings p r s : power transfer from rotor to stator (mechanical? electrical?) Three-phase instantaneous power leaving the stator : p T (t) = v a i a + v b i b + v c i c = vt T i T = vp T PP T ip = vp T i P = v d i d + v q i q + v o i o = (R a id 2 + R a iq 2 + R a io 2 dψ d dψ q dψ o ) (i d + i q + i o }{{} dt dt dt ) + θ r (ψ d i q ψ q i d ) }{{} p Js dw ms /dt p r s = θ r (ψ d i q ψ q i d ) 22 / 6

Energy, power and torque Balance of power at rotor: P m + p f = p Jr + dw mr dt + p r s + dw c dt P m : mechanical power provided by the turbine p f : electrical power provided to the field winding (by the excitation system) p Jr : Joule losses in the rotor windings W mr : magnetic energy stored in the rotor windings W c : kinetic energy of all rotating masses. Instantaneous power provided to field winding: p f = v f i f = v f i f + v d1 i d1 + v q1 i q1 + v q2 i q2 = (R f if 2 + R d1 id1 2 + R q1 iq1 2 + R q2 iq2) 2 dψ f dψ d1 dψ q1 + i f + i d1 + i q1 }{{} dt dt dt p Jr + i q2 dψ q2 dt } {{ } dw mr /dt P m dw c dt = θ r (ψ d i q ψ q i d ) 2 / 6

Energy, power and torque Equation of rotor motion: I d 2 θ r dt 2 = T m T e I : moment of inertia of all the rotating masses T m : mechanical torque applied to the rotor by the turbine T e : electromagnetic torque applied to the rotor by the generator. Multiplying the above equation by θ r : I θ r θr = θ r T m θ r T e dw c = P m θ r T e dt P m : mechanical power provided by the turbine. Hence, the (compact and elegant!) expression of the electromagnetic torque is: T e = ψ d i q ψ q i d Note. The power transfer p r s from rotor to stator is of mechanical nature only. 24 / 6

Energy, power and torque The various components of the torque T e T e = L dd i d i q + L df i f i q + L dd1 i d1 i q L qq i q i d L qq1 i q1 i d L qq2 i q2 i d (L dd L qq ) i d i q : synchronous torque due to rotor saliency exists in salient-pole machines only even without excitation (i f = 0), the rotor tends to align its direct axis with the axis of the rotating magnetic field created by the stator currents, offering to the latter a longer path in iron a significant fraction of the total torque in a salient-pole generator. L dd1 i d1 i q L qq1 i q1 i d L qq2 i q2 i d : damping torque due to currents induced in the amortisseurs zero in steady-state operation. L df i f i q : only component involving the field current i f the main part of the total torque in steady-state operation in steady state, it is the synchronous torque due to excitation during transients, the field winding also contributes to the damping torque. 25 / 6

The synchronous machine in steady state The synchronous machine in steady state Balanced three-phase currents of angular frequency ω N flow in the stator windings a direct current flows in the field winding subjected to a constant excitation voltage: i f = V f R f the rotor rotates at the synchronous speed: θ r = θ o r + ω N t no current is induced in the other rotor circuits: i d1 = i q1 = i q2 = 0 26 / 6

The synchronous machine in steady state Operation with stator opened Park equations: i a = i b = i c = 0 i d = i q = i o = 0 ψ d = L df i f and ψ q = 0 v d = 0 v q = ω N ψ d = ω N L df i f Getting back to the stator voltages, e.g. in phase a : 2 v a (t) = ω NL df i f sin(θr o + ω N t) = 2E q sin(θr o + ω N t) E q = ω NL df i f = e.m.f. proportional to excitation current = RMS voltage at the terminal of the opened machine. 27 / 6

Operation under load The synchronous machine The synchronous machine in steady state v a(t) = 2V cos(ω N t+θ) v b (t) = 2V cos(ω N t+θ 2π ) vc(t) = 2V cos(ω N t+θ+ 2π ) i a(t) = 2I cos(ω N t + ψ) i b (t) = 2I cos(ω N t + ψ 2π ) ic(t) = 2I cos(ω N t + ψ + 2π ) i d = 2 2I [cos(θ o r + ω N t) cos(ω N t + ψ) + cos(θr o + ω Nt 2π ) cos(ω Nt + ψ 2π ) + cos(θ o r + ω Nt + 2π ) cos(ω Nt + ψ + 2π )] = I [cos(θ o r + 2ω Nt + ψ) + cos(θ o r + 2ω Nt + ψ 4π ) + cos(θo r + 2ω Nt + ψ + 4π ) + cos(θ o r ψ)] = I cos(θ o r ψ) Similarly: i q = I sin(θ o r ψ) i o = 0 v d = V cos(θ o r θ) v q = V sin(θ o r θ) v o = 0 In steady-state, i d and i q are constant. This was expected! 28 / 6

The synchronous machine in steady state Magnetic flux in the d and q windings: The electromagnetic torque: ψ d = L dd i d + L df i f ψ q = L qq i q T e = ψ d i q ψ q i d is constant. This is important from mechanical viewpoint (no vibration!). Park equations: v d = R a i d ω N L qq i q = R a i d X q i q v q = R a i q + ω N L dd i d + ω N L df i f = R a i q + X d i d + E q v o = 0 X d = ω N L dd : direct-axis synchronous reactance X q = ω N L qq : quadrature-axis synchronous reactance 29 / 6

Phasor diagram The synchronous machine The synchronous machine in steady state The Park equations become: V cos(θr o θ) = R a I cos(θr o ψ) X q I sin(θr o ψ) V sin(θr o θ) = R a I sin(θr o ψ) + X d I cos(θr o ψ) + E q which are the projections on the d and q axes of the complex equation: Ē q = V + R a Ī + jx d Ī d + jx q Ī q 0 / 6

The synchronous machine in steady state Particular case : round-rotor machine: X d = X q = X Ē q = V + R a Ī + jx (Īd + Īq) = V + R a Ī + jx Ī Such an equivalent circuit cannot be derived for a salient-pole generator. 1 / 6

Powers The synchronous machine The synchronous machine in steady state Ē q = E q e j(θo r π 2 ) Ī d = I cos(θ o r ψ)e jθo r = i d e jθo r Ī q = I sin(θ o r ψ)e j(θo r π 2 ) = j i q e jθo r Ī = Ī d + Ī q = ( i d j i q )e jθo r V d = V cos(θ o r θ)e jθo r = v d e jθo r V q = V sin(θ o r θ)e j(θo r π 2 ) = j v q e jθo r V = V d + V q = ( v d j v q )e jθo r 2 / 6

The synchronous machine in steady state Three-phase complex power produced by the machine: S = V Ī = ( v d j v q )( i d + j i q ) = (v d j v q )(i d + j i q ) P = v d i d + v q i q Q = v d i q v q i d P and Q as functions of V, E q and the internal angle δ, assuming R a 0? v d = X q i q i q = v d X q v q = X d i d + E q i d = v q E q X d v d = V cos(θ o r θ) = V sin δ v q = V sin(θ o r θ) = V cos δ P = E qv X d sin δ+ V 2 2 ( 1 X q 1 X d ) sin 2δ Q = E qv X d Case of a round-rotor machine: P = E qv X sin δ Q = E qv X cos δ V 2 ( sin2 δ + cos2 δ ) X q X d cos δ V 2 X / 6

Nominal values, per unit system and orders of magnitudes Nominal values, per unit system and orders of magnitudes Stator nominal voltage U N : voltage for which the machine has been designed (in particular its insulation). The real voltage may deviate from this value by a few % nominal current I N : current for which machine has been designed (in particular the cross-section of its conductors). Maximum current that can be accepted without limit in time nominal apparent power: S N = U N I N. Conversion of parameters in per unit values: base power: S B = S N base voltage: V B = U N / base current: I B = S N /V B base impedance: Z B = VB 2/S B. 4 / 6

Nominal values, per unit system and orders of magnitudes Orders of magnitude (more typical of machines with a nominal power above 100 MVA) (pu values on the machine base) round-rotor salient-pole machines machines resistance R a 0.005 pu direct-axis reactance X d 1.5-2.5 pu 0.9-1.5 pu quadrature-axis reactance X q 1.5-2.5 pu 0.5-1.1 pu 5 / 6

Park (equivalent) windings The synchronous machine Nominal values, per unit system and orders of magnitudes base power: base voltage: base current: S N VB S N = I B (single-phase formula!) VB With this choice: Similarly: i dpu = i d I = cos(θr o ψ) = I pu cos(θr o ψ) IB I B i qpu = I pu sin(θ o r ψ) v dpu = V pu cos(θ o r θ) v qpu = V pu sin(θ o r θ) Ī = Īd + Īq = (i d j i q )e jθo r V = Vd + V q = (v d j v q )e jθo r All coefficients have disappeared hence, the Park currents (resp. voltages) are exactly the projections on the machine d and q axes of the phasor Ī (resp. V ) 6 / 6