Long-term voltage stability : load aspects

Transcription

1 ELEC Power system dynamics, control and stability Thierry Van Cutsem t.vancutsem@ulg.ac.be December / 15

2 Table of contents Voltage instability results from the inability of the combined transmission and generation system to provide the power requested by loads Transmission aspects Generation aspects Load aspects 2 / 15

3 Load power restoration If loads behaved as constant admittances, no voltage instability would occur (low but steady voltages would be experienced in severe cases) voltage instability is largely caused by the trend of loads to restore their pre-disturbance power after a disturbance this may take place in several time scales : component time scale internal variable equilibrium condition induction motor 1 second motor speed mechan. torque = electrom. torque load tap changer few minutes transformer controlled voltage ratio within deadband thermostatically few minutes amount of temperature controlled load - tens of min. connected load within deadband other devices/processes: distribution voltage regulators, consumer reaction to voltage drop 3 / 15

4 Load power restoration in induction motor Steady-state characteristics of motor large industrial motor X s = 0.067, X m = 3.800, X r = 0.17, R s = 0.013, R r = pu 4 / 15

5 Load tap changers Also referred to as on-load tap changers or under-load tap changers. Widely used to control voltages in networks of lower nominal voltage HV sub-transmission and MV distribution networks where no longer power plants are connected (replaced by more powerful ones connected to transmission network) to compensate for voltage deviations in the EHV transmission network and serve the consumers under correct voltage. Main way of controlling voltages in MV distribution grids. Other ways available at distribution level: switch on/off shunt capacitors (but this is mainly for power factor correction) ajust the active and/or reactive production of distributed generation units not much used yet, but likely to be required in the future, with the expected deployment of renewable energy sources 5 / 15

6 Automatic load tap changer: adjusts r to keep V 2 into a deadband [V o 2 ɛ V o 2 + ɛ] r [ ] r % r < 2ɛ Voltage setpoint V2 o : standard MV distribution systems importing active power: V2 o higher than nominal voltage to counteract the voltage drop in MV grid in some cases, the tap changer controls a downstream voltage V 2 Z c Ī Ī : see figure Z c : compensation impedance MV distribution systems hosting distributed generation sources and exporting active power: V2 o lower than nominal voltage to avoid overvoltages at MV buses 6 / 15

7 Load tap changers are rather slow devices long-term dynamics Delay between two tap changes: minimum delay T m of mechanical origin 5 seconds intentional additional delay: from a few seconds up to 1 2 minutes to let network transients die out before reacting (avoid unnecessary wear) fixed or variable e.g. inverse-time characteristic: the larger the deviation V 2 V2 o, the faster the reaction delay before first tap change ( seconds) usually larger than delay between subsequent tap changes ( 10 seconds) if several levels of tap changers in cascade: the higher the voltage level, the faster the reaction (otherwise risk of oscillations between tap changers) 7 / 15

8 Load power restoration through LTCs Assume the load is represented by an exponential model: P 2 (V 2 ) = P o ( V 2 V2 o ) α Q 2 (V 2 ) = Q o ( V 2 V2 o ) β For simplicity, the reference voltage V2 o is taken equal to the LTC set-point. The power balance equations at bus 2 are: P o ( V 2 V2 o ) α = V 1V 2 sin θ (1) r X Q o ( V 2 V2 o ) β BV2 2 = V 2 2 X + V 1V 2 cos θ r X (2) 8 / 15

9 For given values of V 1 and r, Eqs. (1,2) can be solved numerically with respect to θ and V 2 (using Newton method for instance) from which the power leaving the transmission network is obtained as: P 1 = V 1V 2 r X sin θ (= P 2) Q 1 = V 1 2 r 2 X V 1V 2 r X cos θ repeating this operation for various values of V 1 and r yields the curves shown on the next slide. Numerical example transformer: 30 MVA, X = 0.14 pu, V2 o = 1 pu load: α = 1.5, β = 2.4, P 2 = 20 MW under V 2 = 1 pu, cos φ u = 0.90 (lagging) under V 2 = 1 pu with the compensation capacitor: cos φ c = 0.96 (lagging) under V 2 = 1 pu On the S B = 100 MVA base: X = 0.14(100/30) = pu V o 2 = 1 pu P o = 0.20 pu Q o = P o tan φ u = = pu B.1 2 = Q o P o tan φ c B = = pu 9 / 15

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11 Initial operating point: A, where V 1 = 1 pu, r = 0.97 pu/pu, and V 2 = V o 2 = 1 pu Response to a 0.05 pu drop of voltage V 1 : in the short term, r does not change; the oper. point changes from A to B at point B, V 2 < V2 o ɛ = 0.99 pu hence, the LTC makes the ratio decrease by three positions, until V 2 > V2 o ɛ and the operating point changes from B to C. Neglecting the deadband 2ɛ: the V 2 voltage is restored to the setpoint value V o 2 hence, the P 2 and Q 2 powers are restored to their pre-disturbance values the same holds true for the P 1 and Q 1 powers. This was to be expected since: P 1 = P 2 (V 2 ) Q 1 = Q 2 (V 2 ) BV XI 2 2 = Q 2 (V 2 ) BV X P2 2 (V 2) + Q 2 2 (V 2) V 2 2 hence, the load seen by the transmission system behaves in the long-term (i.e. after the tap changer has acted) as a constant power. This is true as long as the tap changer does not hit a limit. 11 / 15

12 Power recovery of thermostatic loads Heating resistors are switched on/off by thermostats so that the mean power consumed over a cycle = power required to keep the temperature = P req t on t on + t off GV 2 = P req (3) If V drops, P = GV 2 drops t on increases until (3) is satisfied For a large number n of identical thermostatically-controlled resistors: ( ) n P(t) = f i (t)gv 2 1 n = n f i (t) GV 2 t on n GV 2 n t on + t off i=1 i=1 where f i (t) = 1 if the i-th resistor is on at time t f i (t) = 0 if it is off. 12 / 15

13 t on P(t) n GV 2 t on + t off following a voltage drop, ngv 2 decreases t on but after some time, increases until P(t) recovers to np req t on + t off thus, the load behaves as constant admittance in the short term and as constant power in steady state thermostatically controlled loads are also referred to as constant-energy loads. However, if the voltage drop is too pronounced, all resistors stay connected (t off = 0) but P req cannot be obtained. Then, the load behaves as constant admittance. 13 / 15

14 Generic model of load power restoration Power consumed by the load at any time t: P(t) = z P P o ( V V o ) αt Q(t) = z Q Q o ( V V o ) βt (4) z P, z Q : dimensionless state variables associated with load dynamics α t, β t : short-term (or transient) load exponents In steady-state, the load obeys: ( ) αs ( ) βs V V P s = P o Q s = Q o (5) V o V o α s, β s : steady-state (or long-term) load exponents; usually α s < α t, β s < β t. The load dynamics are given by: ( ) αs ( ) αt ( ) βs ( ) βt V V V V T ż P = z P T ż Q = z Q (6) V o V o V o V o with z min P z P z max P and z min Q z Q zq max. T several minutes 14 / 15

15 Response of active power P to a step decrease of voltage V α t P t/p o V /V o α s P s/p o V /V o Similar expressions hold true for reactive power 15 / 15