Chapter 2
Why a converter that measures the grid angle stops working when the grid gets weak.
A grid-following converter injects a commanded current at an angle it has measured. Model this converter as a current source behind a grid reactance Xg. The angle loop then obeys one equation: vq = XgId − Vg sin θp. Three facts follow from it.
This card summarises the results proved below. Its symbols are defined in §2.1 to §2.5.
Kpll = Vg cos θ0 is the structural twin of Chapter 1's synchronizing torque coefficient Ks = (E′Vinf/X) cos δ0 (R06). The whole book turns on that parallel.
From Chapter 1, three results. The per-unit system (R01). The synchronizing torque coefficient Ks = (E′Vinf/X) cos δ0, with Vinf the infinite-bus voltage of Chapter 1, and the condition |δ0| < 90° (R06). The small-signal single-machine result ωn = √(Ksω0/(2H)) with ζ = KD/(4Hωn) (R07). This chapter cites those results by identifier. It does not restate their proofs.
From the reader: linear time-invariant systems, second-order transfer functions, eigenvalues, phasors, and Kirchhoff's laws.
A power system that runs on synchronous machines has a frequency because rotors have mass. Chapter 1 showed the consequence. A rotor is a mechanical integrator. Its angle moves when mechanical power in and electrical power out differ. That one fact produced the swing equation, the inertia constant H, the synchronizing torque coefficient Ks, and a pair of complex eigenvalues that a planner can compute and a protection engineer can trust.
An inverter has no rotor. It has semiconductor switches, a filter inductor, and a capacitor on its direct-current side. It cannot integrate power into an angle by physics. It must be told what angle to use.
The standard answer for three decades is grid-following control. The converter measures the grid voltage. A phase-locked loop extracts the angle of that voltage. The converter then injects a current at a commanded magnitude and a commanded phase relative to that measured angle. The scheme needs no model of the grid, and the 10 Hz loop of §2.4 settles in about 0.09 s on a stiff grid. It has been the standard inverter control of the last three decades (book plan §1.1); the grid-forming plant of Chapter 3 is the alternative under test.
The scheme has one structural weakness. It needs a grid angle that already exists. A phase-locked loop cannot lock to a signal that is not there.
Here is the consequential question. As synchronous machines retire, the voltage that each converter measures is set more and more by other converters, which are themselves measuring it. The short-circuit power behind the point of connection falls. The grid reactance Xg seen by any one plant rises. What breaks first, and at what number?
The symbols in the next paragraph are defined in §2.4 and §2.5. The paragraph states the destination.
This chapter answers that with one scalar. Build the current-source model, write the angle-loop equation, and every weak-grid symptom that the static model contains falls out of the single quantity Kpll = Vg cos θ0: the loss of terminal voltage, the loss of export power, and the loss of loop bandwidth and damping. Grid strength enters the control loop of that model through this one number and nowhere else. The number is also the terminal voltage magnitude, and at unity power factor it is also the delivered power in per unit. One collapse therefore takes voltage, power, and control bandwidth together. One symptom is not among these: the oscillatory instability that a fuller model predicts. Section 2.6.3 says why, and what the static model cannot claim.
The route to that result runs through five pieces of machinery. Sections 2.1 to 2.4 build them: the rotating reference frame (§2.1), the averaged converter (§2.2), the inner current loop (§2.3), and the phase-locked loop (§2.4). Section 2.5 puts the converter behind a grid impedance and derives the bound. Section 2.6 reads off the damage, works two numerical cases, and states plainly what the model does not explain.
Every symbol below is used in this chapter and is not in the book-wide notation table of the plan. Each one is defined here and again in prose at first use. All other symbols in this chapter come from the book-wide table.
A three-phase converter deals in sinusoids. A controller deals in set points. The dq frame is the device that turns the first into the second. Introduce it before anything else, because every model after this point is written in it.
Start with three phase quantities va, vb, vc. Call them the natural frame, labelled a, b, c. Assume the set is balanced: the three add to zero at every instant.
The Clarke transform TC maps a balanced three-phase set onto two stationary axes α and β:
The Park transform TP(θ) rotates those two axes by an angle θ onto the direct axis d and the quadrature axis q:
Conventions fixed here and used for the whole book.
Apply the two transforms in sequence to a balanced set at the rated angular frequency ω0, and choose the rotation angle θ = ω0t. Put va = Vm cos(ω0t), with vb and vc lagging by 120° and 240°. Here Vm is the peak value of the set, in pu. The result is two constants:
Equation (2.3) is the whole point of the construction. Three time-varying signals become two numbers that do not move. A proportional-integral controller can then drive a sinusoid to zero steady-state error, which it cannot do on the sinusoid itself.
Check equation (2.3) numerically at three instants, at f0 = 50 Hz and Vm = 1 pu. At t = 0 ms the phases read +1.0000, −0.5000, −0.5000 pu, and the Clarke transform gives vα = +1.0000, vβ = 0.0000 pu. At t = 4 ms they read +0.3090, +0.6691, −0.9781 pu, and the Clarke transform gives vα = +0.3090, vβ = +0.9511 pu. At t = 8 ms they read −0.8090, +0.9135, −0.1045 pu, giving vα = −0.8090, vβ = +0.5878 pu. Rotating each pair by θ = ω0t returns vd = 1.0000 pu and vq = 0.0000 pu at all three instants, as equation (2.3) requires. Figure 2.1 draws the same statement.
One caution before proceeding. Equation (2.3) holds only because the rotation angle matched the true angle of the set. If the two differ by an error angle, vq is not zero. The size of vq per unit of angle error is the quantity this chapter tracks. Section 2.5 names it Kpll.
A two-level voltage-source converter has six switches. Each switch is either on or off. The bridge output voltage is therefore a train of rectangular pulses, not a sinusoid. Modelling every pulse is expensive and, for this chapter, unnecessary.
Replace the pulse train by its average over one switching period. Write that average as a modulation index times half the direct-current-link voltage. The modulation index components are md and mq. The converter alternating-current terminal voltage is then vd,conv = mdvdc/2 and vq,conv = mqvdc/2 in SI, or, in per unit, vd,conv = mdvdc/(2Vpk,base) with Vpk,base the peak phase-voltage base, √2 Vbase/√3 from R01.
Let Lf and Rf be the filter inductance and resistance between the converter bridge and the point of connection. Let vd, vq be the point-of-connection voltage and id, iq the current the converter injects. In a frame rotating at ω0, the filter obeys
The terms ±ω0Lfi are the cross-coupling terms. They exist because the frame rotates, not because the circuit couples the axes. A stationary inductor has no such term.
Assumptions. (1) The three phases are balanced. (2) The switching average is taken over one switching period, so the model is valid only for signals well below fsw. (3) The direct-current-link voltage vdc is held at vdc0 by a source behind the capacitor Cdc, and its dynamics are outside the loops studied here.
Omissions. Switching harmonics and their sidebands; dead time; converter losses other than Rf; any LCL filter resonance; unbalance and negative-sequence components; direct-current-link dynamics. Chapter 3 returns to Cdc and vdc, because matching control makes the capacitor a state.
Status: model, derived here from Kirchhoff's voltage law in a rotating frame. Structure follows [S06, Yazdani and Iravani 2010, Voltage-Sourced Converters in Power Systems, Ch. 8 — verify].When does the averaging assumption fail? The usual engineering rule keeps the closed-loop bandwidth of anything built on this model about one decade below the switching frequency. A converter switching at fsw = 2500 Hz with a current loop at 500 Hz has a ratio of 5.0, which is already tight; a ratio near 10 is the common design target [S06, Yazdani and Iravani 2010, Voltage-Sourced Converters in Power Systems, Ch. 8 — verify]. Below that ratio the average stops representing the pulse train, and a study must move to an electromagnetic-transient simulation. That move is out of scope for this book.
Equation (2.4) has one control input on each axis. The bridge voltage vd,conv is a command. The current id is the state. Design a controller that makes id follow a reference id,ref.
Two obstacles sit in the way. The cross-coupling terms tie the two axes together. The point-of-connection voltage vd acts as a disturbance. Cancel both by feeding them forward.
Apply the control law
and the mirror law on the q axis. The last two terms cancel the disturbance and the cross-coupling in equation (2.4). What remains on each axis is a first-order plant 1/(sLf + Rf) under proportional-integral control. Choose kp,c = αcLf and ki,c = αcRf. The controller zero then cancels the plant pole, and the closed loop becomes exactly first order:
αc is the closed-loop bandwidth of the inner current control, in rad/s. Its time constant is τc = 1/αc.
Assumptions. (1) The feedforward terms in equation (2.5) are exact. (2) Lf and Rf are known. (3) Modulation and measurement delays are neglected. (4) The command never hits the current limit.
Omissions. Pole-zero cancellation is never exact, so a real loop keeps a slow residual pole. Computation and pulse-width-modulation delay add about 1.5 sample periods of lag [S06, Yazdani and Iravani 2010, Voltage-Sourced Converters in Power Systems, Ch. 8, location unconfirmed — verify]. Both matter for Cited Result 2.9; neither changes the steady-state results of §2.5.
Status: model, derived here. Design rule follows [S06, Yazdani and Iravani 2010, Voltage-Sourced Converters in Power Systems, Ch. 8 — verify].Take a worked tuning. Set αc = 2π · 500 Hz = 3141.59 rad/s. The time constant is τc = 1/3141.59 = 0.318 ms. The phase-locked loop of §2.4 will be designed at 10 Hz. The ratio of the two bandwidths is 500/10 = 50. That ratio is the timescale separation this chapter relies on. It lets §2.5 treat the injected current as an algebraic constant while the angle loop moves.
Timescale separation is an assumption, not a law. Section 2.6 states where it breaks.
A semiconductor switch has a maximum junction temperature and a maximum instantaneous current. Exceeding either destroys the device within a fraction of one fundamental cycle; the device physics behind that time is out of scope (plan §1.4). Every converter therefore carries a hard current limit Imax, enforced by clamping the reference:
The quoted range is an industry figure, not a computed one [S18, IEEE Std 2800-2022, Standard for Interconnection and Interoperability of Inverter-Based Resources, current-injection clauses; S18 states requirements, not device ratings, so a human must confirm or replace this source — verify]. Compare it with a synchronous machine. A machine delivers 5 to 7 times rated current into a close fault, set by its subtransient reactance and limited only by heating over seconds [S01, Kundur 1994, Power System Stability and Control, Ch. 3; locator doubtful, verifier to confirm — verify]. Compare the two directly: 1.1 to 1.3 pu against 5 to 7 pu. The converter's fault current is between 0.157 (1.1/7) and 0.260 (1.3/5) of the machine's. Measured as headroom above rating, it is 0.1 to 0.3 pu against 4 to 6 pu. The converter also has no thermal buffer: the limit is a junction temperature reached within a fraction of a cycle, not a winding temperature reached over seconds.
Equation (2.7) is the single most consequential difference between the two device classes. It returns in Chapter 3 as a hard bound on how much inertia a grid-forming converter can supply (R33), and in Chapter 4 as a sizing constraint (R39).
Section 2.1 fixed the convention that the d axis lies on the measured voltage. Something must enforce that convention in real time. That something is the phase-locked loop.
The idea is one sentence. Rotate the frame at an estimated frequency. Measure vq. If vq is not zero, the frame is not aligned, so correct the estimated frequency.
The loop has two states: the estimated angle θpll and the integrator output that forms the estimated frequency ωpll. With proportional gain kp,pll and integral gain ki,pll, the loop obeys
The measured vq is the error signal. The loop drives it to zero. kp,pll has unit 1/s and ki,pll has unit 1/s2, because vq is in per unit.
Assumptions. (1) The voltage set is balanced, so no second-harmonic term appears in vq. (2) The measurement is noise-free. (3) The loop starts inside its capture range.
Omissions. Unbalance rejection, harmonic filtering, and frequency-limiting logic. All exist in a real product. None changes the small-signal result of Theorem 2.8.
Status: definition, stated here. Structure follows [S06, Yazdani and Iravani 2010, Voltage-Sourced Converters in Power Systems, Ch. 8 — verify].Take the stiff-grid case first. Let the grid behind the point of connection be stiff, so the point-of-connection voltage is a fixed phasor of magnitude Vg at angle θg in the stationary frame, whatever current the converter injects. Let the grid run at rated frequency: dθg/dt = ω0. Let θp = θpll − θg be the angle of the loop's frame measured from that phasor, so that dθp/dt = ωpll − ω0.
Apply the Park transform (2.2) with θ = θpll to the phasor vα = Vg cos θg, vβ = Vg sin θg. The second row gives vq = Vg(−sin θpll cos θg + cos θpll sin θg) = Vg sin(θg − θpll) = −Vg sin θp. For small θp, vq ≈ −Vgθp.
Subtract ω0 from both sides of equation (2.8). The left side becomes dθp/dt:
dθp/dt = kp,pllvq + ki,pll∫vq dt.
Substitute vq ≈ −Vgθp and differentiate once with respect to time: d2θp/dt2 + Vgkp,pll dθp/dt + Vgki,pllθp = 0. With Δθp the deviation of θp from zero, the closed loop has the characteristic polynomial
Equation (2.9) is a clean second-order system. With Vg = 1 pu it reads s2 + kp,plls + ki,pll = 0, so the designer picks a natural frequency and a damping ratio directly:
Design the loop for ωn,pll = 2π · 10 Hz = 62.8319 rad/s and ζpll = 1/√2 = 0.7071. Equation (2.10) gives ki,pll = 62.83192 = 3947.84 1/s2, which this chapter prints as 3948 1/s2, and kp,pll = 2 · 0.7071 · 62.8319 = 88.86 1/s. These two gains are fixed for the rest of the chapter. Nothing in §2.5 or §2.6 changes them. That is the point: the designer sets them once, and the grid then moves the loop underneath them.
Equation (2.9) also carries the warning. The coefficient in front of both gains is Vg, the voltage the loop actually sees. On a stiff grid that voltage is 1 pu and can be forgotten. On a weak grid it is not, and it cannot.
Everything so far assumed a stiff grid. Remove that assumption now. Replace the ideal voltage source at the point of connection by a Thevenin equivalent: a source of magnitude Vg behind an impedance. First, measure the strength of that equivalent.
Let Ssc be the three-phase short-circuit apparent power at the point of connection with the converter disconnected. Let Srated be the rated apparent power of the converter. The short-circuit ratio is
Take the grid Thevenin impedance to be purely inductive, so Rg = 0 and Zg = Xg. The short-circuit level is computed at the nominal voltage of the point of connection, which is 1 pu on the converter base, not at the actual source magnitude Vg. Express Xg in per unit on the converter base Srated. Then Ssc = 12/Xg = 1/Xg in per unit, and Srated = 1 pu, so
Equation (2.12) holds whatever the actual value of Vg, because the definition of Ssc uses the nominal voltage and not Vg. The theorems of this chapter therefore carry Vg as a symbol; the examples set Vg = 1 pu. The nominal-voltage definition follows [S19, IEEE Std 1204-1997, Guide for Planning DC Links Terminating at AC Locations Having Low Short-Circuit Capacities, definition of short-circuit ratio — verify].
Equation (2.11) is the planner's number and equation (2.12) is the control engineer's. A planner reports a short-circuit level in megavolt-amperes. A control engineer needs a per-unit reactance. Equation (2.12) converts one into the other.
Strength classes. One common convention groups connections as high short-circuit ratio above 3, low between 2 and 3, and very low below 2 [S19, IEEE Std 1204-1997, Guide for Planning DC Links Terminating at AC Locations Having Low Short-Circuit Capacities, strength-classification clause — verify]. The boundaries are conventions, not physics. This chapter computes what happens at each value instead of arguing about labels.
Status: definition, stated here. Class boundaries cited, not derived.Now place the converter behind that impedance. Section 2.3 showed that the inner current loop is about 50 times faster than the angle loop. Exploit that ratio. On the angle-loop timescale, treat the injected current as constant and equal to its reference.
The converter injects a constant current in its own phase-locked-loop frame: Id on the d axis and Iq = 0 on the q axis. Operation is therefore at unity power factor as the loop sees it. The grid is a source of magnitude Vg at angle θg in the stationary frame, behind a reactance Xg = 1/SCR. The loop's frame sits at angle θpll. Let θp = θpll − θg be the angle of the loop's frame measured from the grid source phasor.
Derivation of the point-of-connection voltage. Three lines give it.
Add lines 2 and 3. The point-of-connection voltage in the loop's frame is
Assumptions. (1) The inner current loop is infinitely fast, so id = Id at every instant (timescale separation, §2.3). (2) The grid impedance is purely inductive. (3) The converter provides no reactive current: Iq = 0. (4) The system is balanced and at fundamental frequency. (5) The grid source runs at the rated frequency, so dθg/dt = ω0 and dθp/dt = ωpll − ω0.
Omissions, in order of consequence.
Split equation (2.13) into real and imaginary parts. The real part gives the terminal voltage on the d axis; the imaginary part gives the loop's own error signal:
The second half of equation (2.14) is the loop equation. It is the whole of §2.5 and §2.6 in one line. Read it slowly. The loop drives vq to zero. The term Vg sin θp is the only term it can use to cancel XgId. That term is bounded above by Vg.
Under Model 2.6, with Vg > 0 and Id ≥ 0:
Part 1 and 2. Set vq = 0 in equation (2.14). This gives sin θp = XgId/Vg. The sine function has range [−1, 1]. A real solution therefore exists if and only if XgId/Vg ≤ 1, that is XgId ≤ Vg. Substitute Xg = 1/SCR from equation (2.12) to get Id ≤ SCR · Vg.
When the bound holds strictly, the equation has two solutions in [0, 180°): θ0 = arcsin(·) and 180° − arcsin(·). Part 4 shows that the second has Kpll < 0, so the loop's negative feedback becomes positive feedback there and the point is not operable. The first branch, with cos θ0 > 0, is the operating point. At equality the two branches merge at θ0 = 90°, where Kpll = 0. Above equality no steady state exists at all.
Part 3. Delivered active power in the loop frame is P = vdId + vqIq by Definition 2.1. At the locked point vq = 0 and Iq = 0, so P = vdId = Vg cos θ0 · Id. Substitute Id = Vg sin θ0/Xg from Part 1:
sin(2θ0) peaks at 1 when 2θ0 = 90°, that is θ0 = 45°. There P = Vg2/(2Xg) = SCR · Vg2/2 by equation (2.12). Beyond θ0 = 45°, sin(2θ0) decreases, so P decreases although Id keeps rising. At θ0 = 90°, sin(2θ0) = 0 and P = 0.
Part 4. Differentiate the loop equation of (2.14) with respect to θp at θp = θ0. XgId is constant on the angle-loop timescale by Model 2.6, so it contributes nothing:
The minus sign is the negative feedback the loop needs, and Kpll = Vg cos θ0 is its magnitude. Both Ks and Kpll are dimensionless per-unit numbers once the angle is in radians. Part 2 gave the terminal voltage as Vg cos θ0, the same expression. At θ0 = 90°, which is the current bound, cos θ0 = 0 and Kpll = 0. On the second branch, θ0 > 90°, cos θ0 < 0, so Kpll < 0 and the feedback sign flips. □
Three readings are worth making explicit.
First: the bound is on current, and the power limit arrives earlier. A plant operator thinks in megawatts. The theorem does not bound megawatts at SCR · Vg. It bounds amperes. Power peaks at θ0 = 45°, which by Part 2 is the current Id = Vg sin 45°/Xg = SCR · Vg/√2. At SCR = 1.2 and Vg = 1 pu that is Id = 1.2/1.4142 = 0.8485 pu, against the current bound of 1.2000 pu. Push current past 0.8485 pu and delivered power goes down. A control loop that commands more current to get more power therefore commands more current and receives less power.
Second: Kpll is the twin of Ks. Set the two side by side.
| Quantity | Synchronous machine (R06) | Grid-following converter (R18) |
|---|---|---|
| Angle variable | δ, a physical rotor position | θp, an estimate inside a controller |
| Restoring coefficient | Ks = (E′Vinf/X) cos δ0 | Kpll = Vg cos θ0 |
| Sign condition | Ks > 0 iff |δ0| < 90° | Kpll > 0 iff |θ0| < 90° |
| What it restores | Power balance on a shaft | Alignment of a measurement frame |
| What it carries | ωn = √(Ksω0/(2H)) (R07) | ωn,pll = √(Kpllki,pll) (Theorem 2.8) |
| Energy behind it | H · Sbase; for H = 3.5 s on 200 MVA that is 700 MJ (Chapter 1, R03) | None. The loop is a computation. |
The algebra matches line for line. The last row does not. A machine's angle is backed by kinetic energy. A phase-locked loop's angle is backed by nothing. That difference is what Corollary 2.10 makes precise and what Chapter 3 sets out to repair.
Third: this is a static result. Theorem 2.7 asks whether a steady state exists. It does not ask whether the loop reaches that state, or stays there. Section 2.6 takes the next step; Cited Result 2.9 states plainly that the static bound is not the operating limit a real plant meets.
Theorem 2.7 delivered one scalar, Kpll. Section 2.4 built a loop whose behaviour depends on the voltage it sees. Join the two.
Linearise the loop of Definition 2.4 about the operating point of Theorem 2.7, under Model 2.6 including assumption (5), with fixed gains kp,pll and ki,pll. The closed-loop characteristic polynomial in the angle deviation Δθp is
so that
Both fall as Kpll falls, and Kpll = Vg cos θ0 falls as the short-circuit ratio falls at fixed Id. Kpll is the only channel through which grid strength enters the loop. Eliminating ωn,pll between the two parts of equation (2.18) gives the scaling law
Write θp = θ0 + Δθp and vq = 0 + Δvq. Equation (2.16) gives Δvq = −KpllΔθp to first order. By Model 2.6 assumption (5), dθp/dt = ωpll − ω0. At the operating point vq = 0, ωpll = ω0, and the integrator output in equation (2.8) is zero. Subtract that operating point from equation (2.8). The loop reads, in deviation form,
d(Δθp)/dt = kp,pllΔvq + ki,pll∫Δvq dt.
Substitute Δvq = −KpllΔθp and differentiate once with respect to time:
d2(Δθp)/dt2 + Kpllkp,pll d(Δθp)/dt + Kpllki,pll Δθp = 0.
Its characteristic polynomial is equation (2.17). Compare with the standard form s2 + 2ζωns + ωn2 and read off equation (2.18). Then substitute ωn,pll = √(Kpllki,pll) into the expression for ζpll: the factor Kpll in the numerator divides by √Kpll in the denominator and leaves √Kpll, which is equation (2.19).
Note also that for any Kpll > 0 the polynomial (2.17) has both coefficients positive, so both roots lie strictly in the left half-plane. This linearised loop never becomes unstable; it only gets slower and less damped. Cited Result 2.9 addresses what is missing. □
Given. SCR = 1.2. Vg = 1.0 pu. Id = 1.0 pu, Iq = 0 (unity power factor in the loop frame). Phase-locked-loop gains from §2.4: ki,pll = 3948 1/s2, kp,pll = 88.86 1/s.
Step 1 — grid reactance. Equation (2.12): Xg = 1/SCR = 1/1.2 = 0.8333 pu.
Step 2 — operating angle. Theorem 2.7 Part 2: sin θ0 = XgId/Vg = 0.8333 · 1.0 / 1.0 = 0.8333. So θ0 = arcsin(0.8333) = 0.9851 rad = 56.44°.
Check the bound first: XgId = 0.8333 ≤ Vg = 1.0000, so a steady state exists. Equivalently Id = 1.0000 ≤ SCR · Vg = 1.2000 pu.
Step 3 — terminal voltage. |v| = vd = Vg cos θ0 = 1.0 · cos(56.44°) = 0.5528 pu. Cross-check without the angle: cos θ0 = √(1 − 0.83332) = √(1 − 0.6944) = √0.3056 = 0.5528. The two agree.
Step 4 — delivered power. P = vdId = 0.5528 · 1.0 = 0.5528 pu. Cross-check with equation (2.15): (Vg2/(2Xg)) sin(2 · 56.44°) = (1/1.6667) · sin(112.89°) = 0.6000 · 0.9213 = 0.5528 pu. The two agree.
Step 5 — the model's power ceiling. Theorem 2.7 Part 3: Pmax = SCR · Vg2/2 = 1.2 · 1.0/2 = 0.6000 pu, reached at θ0 = 45°, which is Id = 1.2/√2 = 0.8485 pu. The plant is already past the peak: it carries 1.0000 pu of current and gets 0.5528 pu of power, where 0.8485 pu of current would have given 0.6000 pu.
Step 6 — the loop gain. Kpll = Vg cos θ0 = 0.5528 pu. This is the same number as the terminal voltage of Step 3 and the power of Step 4. That is Theorem 2.7 Part 4, not a coincidence of rounding.
Step 7 — what the loop now does. Equation (2.18) with the unchanged gains:
ωn,pll = √(Kpllki,pll) = √(0.552771 · 3947.84) = √2182.25 = 46.71 rad/s
in hertz: 46.71 / (2π) = 7.435 Hz
ζpll = 0.7071 · √0.5528 = 0.7071 · 0.7435 = 0.5257 (scaling law (2.19))
Cross-check the damping ratio against the second half of equation (2.18), carrying full precision: Kpllkp,pll/(2ωn,pll) = 0.552771 · 88.8577 / (2 · 46.7146) = 49.1180 / 93.4292 = 0.5257. The two agree.
Result. The loop was designed for 10.00 Hz and ζpll = 0.7071. It now runs at 7.435 Hz and ζpll = 0.5257. Bandwidth fell by 25.7 %. Damping ratio fell by the same 25.7 %, because equation (2.19) makes both scale as √Kpll.
Conclusion in words. The designer changed nothing. The grid moved the loop.
Rounding note. Carrying the four-digit intermediates 0.5528 and 3948 into the square root returns √2182.45 = 46.72 rad/s and 7.44 Hz. Carrying the six-digit factors 0.552771 and 3947.84, as Step 7 does, returns 46.71 rad/s and 7.435 Hz. The difference is rounding, not physics. This chapter prints the full-precision values throughout.
Status: computed here from Theorem 2.7 and Theorem 2.8. No external number is used.Given. Vg = 1.0 pu, Id = 1.0 pu, Iq = 0, and the same fixed gains ki,pll = 3948 1/s2, kp,pll = 88.86 1/s. Every row repeats Steps 1 to 7 of Example 2.1 at a different SCR.
| SCR | Xg (pu) | θ0 (deg) | Kpll (pu) | ωn,pll (rad/s) | ζpll | P at Id=1 (pu) | Pmax (pu) |
|---|---|---|---|---|---|---|---|
| 10 | 0.1000 | 5.7392 | 0.9950 | 62.6742 | 0.7053 | 0.9950 | 5.0000 |
| 3 | 0.3333 | 19.4712 | 0.9428 | 61.0087 | 0.6866 | 0.9428 | 1.5000 |
| 2 | 0.5000 | 30.0000 | 0.8660 | 58.4716 | 0.6580 | 0.8660 | 1.0000 |
| 1.2 | 0.8333 | 56.4427 | 0.5528 | 46.7146 | 0.5257 | 0.5528 | 0.6000 |
| 1.05 | 0.9524 | 72.2472 | 0.3049 | 34.6949 | 0.3905 | 0.3049 | 0.5250 |
Reading 1 — the two power columns are different objects. Column P is what this plant delivers at its rated current. Column Pmax is the largest power the model can deliver at any current. At SCR = 10 the plant delivers 0.9950 pu against a ceiling of 5.0000 pu, so current is the binding constraint. At SCR = 1.2 it delivers 0.5528 pu against a ceiling of 0.6000 pu, so the ceiling binds. The crossover, where Pmax = 1.0000 pu, is exactly SCR = 2.
Reading 2 — P and Kpll are the same column. Both equal Vg cos θ0 at Id = 1 pu. This is not a coincidence in the table; it is Theorem 2.7 Parts 2, 3 and 4 in one place. The consequence: a plant that loses export power on a weak grid loses loop bandwidth and damping ratio at the same moment, by the square root of the same factor, equation (2.19). At SCR = 1.2 the terminal voltage and the power are down 44.7 % (1 − 0.5528) from the stiff-grid value, and the bandwidth and the damping ratio are down 25.7 % (1 − √0.5528 = 1 − 0.7435), as Example 2.1 prints.
Reading 3 — the decline is slow, then steep. From SCR = 10 to SCR = 2, ζpll falls from 0.7053 to 0.6580. That is (0.7053 − 0.6580)/0.7053 = 0.0671, or 6.7 %, over a factor of five in grid strength. From SCR = 2 to SCR = 1.05, ζpll falls from 0.6580 to 0.3905. That is (0.6580 − 0.3905)/0.6580 = 0.4066, or 40.7 %, over a factor of under two. The square-root law (2.19) explains the shape: √Kpll is flat near Kpll = 1 and steep near Kpll = 0.
Reading 4 — the warning to the planner. A margin study at SCR = 3 sees a loop 2.9 % off its design damping ratio and reports no problem. The same study at SCR = 1.2 sees 25.7 % off. Linear extrapolation from the strong-grid region underestimates the weak-grid result. Sample the low end of the range, not the middle.
Deviation note. The chapter plan prints Kpll = 0.3047 and ζpll = 0.3903 at SCR = 1.05, and ζpll = 0.6865 at SCR = 3. Full-precision recomputation returns 0.3049, 0.3905 and 0.6866. The table above prints the recomputed values, as the drafting rule requires.
Status: computed here. Every entry follows from Theorem 2.7 and Theorem 2.8.Theorem 2.8 ended with an observation that limits its reach. For every Kpll > 0 the polynomial (2.17) has both coefficients positive, so both roots stay in the left half-plane. Figure 2.4 draws that fact.
Restore the current-loop dynamics that Model 2.6 assumption (1) discarded, that is the pole at αc in equation (2.6), together with the direct-current-link voltage loop. The combined system then loses stability through a pair of complex poles crossing into the right half-plane, at a current and a short-circuit ratio below the static existence bound of Theorem 2.7. The mechanism is an interaction between the phase-locked loop and the current loop, and the phase-locked-loop bandwidth is a leading parameter: a faster loop destabilises earlier on a weak grid.
Status: cited, not proved in this book. [S08, Zhou, Ding, Fan, Zhang and Gole 2014, "Impact of Short-Circuit Ratio and PLL Parameters on the Small-Signal Behavior of a VSC-HVDC Converter," IEEE Trans. Power Delivery 29(5), 2287–2296, eigenvalue-study sections — verify] [S07, Zhang, Harnefors and Nee 2010, "Power-Synchronization Control of Grid-Connected Voltage-Source Converters," IEEE Trans. Power Systems 25(2), 809–820, weak-grid limit section — verify]
Why it sits here. Model 2.6 named the omission of current-loop dynamics as its second omission. This result is the repayment. Without it a reader could conclude from Figure 2.4 that a weak grid only makes a loop sluggish. That conclusion is wrong, and the model that produced it says so about itself.
What this chapter therefore does not claim. Theorem 2.7 does not explain any observed weak-grid oscillation. It gives a ceiling that no design may exceed. The usable region is strictly smaller, and its boundary needs the model this chapter did not build.
Under Model 2.6, and with the phase-locked loop fast compared with the swing mode of Chapter 1 — here 7.4 Hz to 10 Hz against 1.43 Hz in Example 1.1 (R08) — a population of grid-following converters contributes no synchronizing torque coefficient and no inertia constant to the system of Chapter 1, whatever its installed capacity.
□
Corollary 2.10 states the gap. It does not state the remedy. If an angle must be an internal state backed by energy, then some converter must stop following and start setting. That is the terminal definition of grid-forming control, and it opens Chapter 3 (R24).
State the boundary plainly, so that no reader carries a result past its assumptions.
Each exercise names its type. Each answer target states the number the solution must reach. Recompute every number; do not copy a target.
A converter rated 200 MVA connects where the short-circuit level is 300 MVA. State the short-circuit ratio. State whether the plant can export its full rating at unity power factor under Model 2.6.
SCR = 300/200 = 1.5000, so Xg = 1/1.5 = 0.6667 pu by equation (2.12). At Id = 1 pu: sin θ0 = 0.6667, so θ0 = arcsin(0.6667) = 41.81°. Terminal voltage and Kpll are both Vg cos θ0 = 0.7454 pu, and P = 0.7454 pu. This model's ceiling is Pmax = SCR · Vg2/2 = 0.7500 pu.
Answer: no. The plant cannot export 1.0 pu. The ceiling 0.7500 pu is below rating, and even that needs θ0 = 45°, not the 41.81° of this operating point. Exporting 1.0 pu requires reactive support that holds the terminal voltage up, which Model 2.6 omission 1 excludes. Cited Result 2.9 (R22) warns that the true stable limit is lower still.
Derive P = vdid + vqiq in per unit from the three-phase instantaneous power, using the amplitude-invariant Park transform of Definition 2.1. Show where the factor 3/2 goes.
Instantaneous three-phase power in SI is p = vaia + vbib + vcic. Invert equations (2.1) and (2.2) for a balanced set and substitute. The cross terms cancel, and the result is p = 1.5(vdid + vqiq), with v and i as SI peak-referred dq quantities.
The factor 3/2 comes from the 2/3 in the amplitude-invariant Clarke transform. It disappears in per unit. With a peak-referred voltage base and a peak-referred current base whose product is (2/3)Sbase, dividing p by Sbase gives P = vdid + vqiq exactly. The factor is absorbed by the bases, which is why the transform of Definition 2.1 and the bases of R01 must be chosen together: the peak phase-voltage base √2 Vbase/√3 and the peak current base √2 Ibase multiply to (2/3)Sbase. Had the power-invariant factor √(2/3) been chosen instead, the 3/2 would not have appeared, but a 1 pu phase peak would no longer give a 1 pu dq magnitude.
Show that Kpll equals the magnitude of ∂vq/∂(angle error) at the operating point. Explain why this makes grid strength enter the loop through exactly one scalar.
Linearise the loop equation of (2.14) about the locked point. Write θp = θ0 + Δθp, where Δθp is the angle error. The term XgId is constant on the loop timescale by Model 2.6 assumption (1), so its derivative is zero. Only −Vg sin θp survives differentiation, giving ∂vq/∂Δθp = −Vg cos θ0 = −Kpll. This is equation (2.16).
Grid strength enters through one scalar because the grid touches equation (2.8) only through vq, and vq is zero at the operating point. A quantity that is zero at the operating point contributes nothing but its slope to the first-order model. That slope is Kpll. Every other grid property — Xg, Vg, Id — reaches the loop only by setting θ0, and therefore only through Kpll.
Sweep SCR from 1.01 to 10 and find the short-circuit ratio at which ζpll first falls below 0.300, for the gains of Example 2.1 at Id = 1 pu and Vg = 1 pu. Report the value the sweep returns and state the resolution used.
Closed form first, as a check on the sweep. Equation (2.19) gives ζpll = 0.7071√Kpll, so ζpll = 0.300 at Kpll = (0.300/0.7071)2 = 0.1800. Then θ0 = arccos(0.1800) = 79.63°, and SCR = 1/sin θ0 = 1.0166.
Sweep result at a resolution of 0.001 in SCR, ascending from 1.010: the first grid point with ζpll ≥ 0.300 is SCR = 1.017, where Kpll = 0.1821 and ζpll = 0.3017. The grid point below it, SCR = 1.016, gives Kpll = 0.1768 and ζpll = 0.2973. The crossing therefore lies between 1.016 and 1.017, which brackets the closed-form 1.0166. Report the bracket and the step, not a single digit beyond the resolution.
Reading: under this static model the damping ratio holds above 0.300 almost to the existence bound of SCR = 1.000. That is exactly why Cited Result 2.9 (R22) is needed. Real plant meets trouble far above SCR = 1.017.
Retune kp,pll and ki,pll so that Example 2.1 recovers ζpll = 0.7071 and ωn,pll = 62.83 rad/s at SCR = 1.2. Then report what the new gains do at SCR = 10.
Equation (2.18) shows both gains enter only as products with Kpll. To restore the design pair at Kpll = 0.5528, divide both gains by 0.5528: kp,pll = 88.86/0.5528 = 160.7 1/s and ki,pll = 3948/0.5528 = 7142 1/s2. Check with equation (2.18): ωn,pll = √(0.5528 · 7141.82) = √3948.00 = 62.83 rad/s, and with equation (2.19) ζpll = 0.7071 · √(0.5528/0.5528) = 0.7071 · 1.0000 = 0.7071. Correct.
At SCR = 10, Kpll = 0.9950. Both equations (2.18) and (2.19) scale from the SCR = 1.2 point by the factor √(0.9950/0.5528) = √1.8000 = 1.3416. So ωn,pll = 62.83 · 1.3416 = 84.30 rad/s, which is 84.30/(2π) = 13.42 Hz, and ζpll = 0.7071 · 1.3416 = 0.9487.
The trade this chapter names. A loop tuned for the weak case runs at 13.42 Hz on a strong grid, against the 10.00 Hz it was designed for. The ratio is 13.42/10.00 = 1.3416. Cited Result 2.9 (R22) reports that a faster phase-locked loop destabilises earlier on a weak grid. So raising the gains to fix the weak-grid bandwidth works against the weak-grid stability limit. One fixed pair of gains cannot serve both ends of the range. That is the argument for gain scheduling. It is also, as Chapter 3 argues, the argument for a control law that does not measure the angle at all.
State, in terms of R06 and R18, one sentence on why adding more grid-following plant does not raise SCR for the plant already connected there.
SCR is set by the Thevenin impedance of the sources behind the point of connection (Definition 2.5), and a current source contributes no Thevenin voltage source, so it adds no short-circuit power and leaves Xg unchanged.
Expand in the chapter's terms. R06 gives a synchronous machine a restoring coefficient Ks = (E′Vinf/X) cos δ0, which exists because the machine holds an internal emf E′ behind a finite reactance. That emf is what feeds a fault, so it is what sets Ssc. Model 2.6 gives a grid-following converter no internal emf at all; equation (2.7) also caps its fault contribution near 1.1 to 1.3 pu. R18 then shows that each added converter increases the current flowing through the same Xg, which pushes θ0 up and Kpll down for every plant at that node. Adding grid-following plant therefore consumes grid strength; it does not produce it. Corollary 2.10 (R23) is the general statement.
Chapter 1 gave a machine an angle backed by kinetic energy, and read the stability of that angle off two numbers, Ks and H. This chapter gave a converter an angle backed by a measurement, and read the behaviour of that angle off one number, Kpll. The algebra of the two restoring coefficients matches. The energy behind them does not.
The practical statement is short. A grid-following plant on a weak grid loses terminal voltage and export power at the same rate, because both are the same quantity Vg cos θ0 = Kpll in per unit at unity power factor. It loses loop bandwidth and damping ratio at the same moment, as √Kpll (equation (2.19)). At SCR = 1.2 the first pair is down 44.7 % and the second pair 25.7 %. The static ceiling on power is SCR · Vg2/2, which falls below plant rating at SCR = 2. The true stable region is smaller than the static one, by a mechanism this chapter cites and does not prove.
| Local | Global | Statement | Status |
|---|---|---|---|
| Definition 2.1 | R12 | Clarke and Park transforms, amplitude-invariant, d axis on the measured voltage | stated in §2.1 |
| Model 2.2 | R13 | Averaged converter with L filter in dq, with cross-coupling | derived in §2.2 |
| Model 2.3 | R14 | Decoupled current control; first-order closed loop of bandwidth αc | derived in §2.3 |
| Definition 2.4 | R15 | Synchronous-reference-frame phase-locked loop | stated in §2.4 |
| Definition 2.5 | R16 | Short-circuit ratio; Xg = 1/SCR; strength classes | stated in §2.5; classes cited [S19 — verify] |
| Model 2.6 | R17 | Current source behind jXg; four omissions listed | derived in §2.5 |
| Theorem 2.7 | R18 | Existence bound on current; power curve; Kpll = Vg cos θ0 | proved in §2.5 |
| Theorem 2.8 | R19 | ωn,pll = √(Kpllki,pll); ζpll ∝ √Kpll | proved in §2.6 |
| Example 2.1 | R20 | SCR = 1.2: θ0 = 56.44°, Kpll = 0.5528, 7.435 Hz, ζpll = 0.5257 | computed in §2.6.1 |
| Example 2.2 | R21 | Five-row SCR sweep with both power columns | computed in §2.6.2 |
| Cited Result 2.9 | R22 | Phase-locked-loop and current-loop interaction destabilises below the static bound | cited [S07, S08 — verify]; not proved here |
| Corollary 2.10 | R23 | A grid-following fleet supplies neither Ks nor H | proved in §2.6.4 |
Forward. Corollary 2.10 leaves a hole where Ks and H used to be. Chapter 3 defines grid-forming control by terminal behaviour (R24). It shows that four control families reduce to one angle model with an equivalent inertia constant and an equivalent damping coefficient (R32). It recovers Ks. It then bounds the recovered inertia by the direct-current-side energy reserve and by the current limit Imax of equation (2.7) (R33). Those two bounds have no counterpart in Chapter 1.
Rendering convention: the plan writes symbol names in plain text, such as omega_n_pll. This chapter renders the same symbol as ωn,pll. The mapping is one to one: an underscore becomes a subscript, and a spelled-out Greek name becomes the Greek letter. So k_p_pll is kp,pll, zeta_pll is ζpll, alpha_c is αc, and K_pll is Kpll. This chapter never shortens kp,pll to kp.
Cross-chapter references in this chapter use the global result identifiers R01 to R40 of the book plan. Those already drafted in Chapter 1 (R01, R03, R04, R06 to R08) are linked to ch1.html; identifiers for Chapters 3 and 4 (R24, R32, R33, R39, R40) are not yet linked, because those chapters' anchors do not yet exist. Every citation carries the mark "verify" until a human reads the source page.