Weak-Grid Wind Plant (2D)

Why a weak grid is a problem for wind and solar plants

A wind or solar plant connects through power-electronic inverters at the point of common coupling (PCC). Seen from the PCC, the rest of the power system is a voltage source E behind an impedance Zg = R + jX. The short-circuit ratio is SCR = Ssc / Sn = 1 / |Zg| (per unit on the plant rating): a strong grid has a large SCR, a weak grid (SCR below about 3, very weak below 2) has a large impedance, so the voltage at the PCC moves a lot when the plant changes its output.

The exact PCC voltage

The plant injects S = P + jQ. With a = R P + X Q and b = X P - R Q the circuit equation reduces to E^2 V^2 = (V^2 - a)^2 + b^2, a quadratic in V^2:

V^2 = [ (2a + E^2) + sqrt( (2a + E^2)^2 - 4 (a^2 + b^2) ) ] / 2

The PCC leads the grid by theta = atan2( b / V, V - a / V ). For small injections this becomes the familiar estimate dV ~ (R P + X Q) / E: active power raises the voltage in proportion to R, reactive power in proportion to X (X dominates in a high-voltage network). Because the discriminant can become negative, there is a maximum transferable power (the nose of the P-V curve). For unity power factor: Pmax = E^2 (R + Z) / (2 X^2), which for R = 0 is E^2 / (2X) = SCR E^2 / 2: a plant on a SCR 1.4 grid can never push more than about 0.7 of its rating.

Reactive-power control

The inverter can inject or absorb reactive power within its capability P^2 + Q^2 <= S^2 (P has priority, so the headroom shrinks at full output; a STATCOM adds fixed extra capability). Three modes: unity power factor (Q = 0), fixed Q, and Q-V droop: Q = kq (V0 - V), which absorbs reactive power when the voltage is high. The steady state solves V = V(P, Q(V) + Qc V^2), where Qc is the charging of an export cable. A droop that is too stiff on a weak grid, combined with the measurement delay and the controller time constant, is the textbook route to voltage oscillations: the loop gain is roughly kq X and a delay turns gain above one into growing oscillation until the inverter saturates.

Numerical checks on this page

(1) The closed-form voltage is compared with an independent Gauss iteration of the complex equation V = E + Z conj(S / V). (2) The maximum power found by bisection on the existence of a solution is compared with the closed form above. (3) A time-domain run with a stable loop settles to the same voltage as the static solution.

Honest simplifications

This is a one-bus, per-unit teaching model. The grid is a Thevenin equivalent with a constant E; there is no transformer tap changer, no real-time electromagnetic transient behaviour of the phase-locked loop or current controller (the weak-grid PLL stability that real studies examine is only hinted at by the loop-gain argument), no fault ride-through, the wind farm is one lumped power curve with a simple smoothing, and the cable is one shunt capacitance. Real interconnection studies use full network models. Use the SCR labels as rules of thumb, not as grid-code values.

• Load Strong grid (SCR 8) and then Weak grid (SCR 2.2): the same plant output raises the PCC voltage much more.
• Load Weak grid + Q-V droop: the inverter absorbs reactive power and pulls the voltage back inside the band (graph 4, green curve).
• Load Very weak grid (SCR 1.4) and raise P: the curve on graph 1 ends at the nose and the plant collapses the voltage.
• Load Droop gain too high with delay: the voltage oscillates; lower the gain or the delay to calm it.
• Load Long export cable at low output: the cable charging pushes the voltage up (Ferranti-like rise).
• Add a STATCOM at SCR 1.6 and watch the headroom tile and the lamps on the STATCOM containers.