Power Systems formula sheet

98 formulas in 6 topics, each with when to use it. Each is explained in the topic notes.

HYDRO POWER OUTPUT

P_{hydro} = ρ g Q H η

$ρ = 1000 kg/m^{3}$ , $g = 9.81 m/s^{2}$ , $Q$ in m $^{3}$ /s, $H$ net head in m, $η$ overall efficiency; result in W.

LOAD FACTOR AND CAPACITY FACTOR

Load factor = \frac{average load}{peak load}, Plant capacity factor = \frac{energy generated}{rated capacity \times time}

Both over the same period; each is at most 1.

Per-unit system, bus admittance matrix and load flow

All 11 formulas

BASE CURRENT AND BASE IMPEDANCE

I_{ba se} = \frac{S _{ba se} \times 1 0 ^{3}}{3 V _{ba se}} A, Z_{ba se} = \frac{V _{ba se}^{2}}{S _{ba se}} Ω

Three-phase MVA base $S_{ba se}$ in MVA and line-to-line kV base $V_{ba se}$ in kV; the impedance is per phase of the equivalent star.

PER-UNIT IMPEDANCE

Z_{p u} = \frac{Z _{Ω}}{Z _{ba se}} = Z_{Ω} \frac{S _{ba se}}{V _{ba se}^{2}}

Per-unit voltage, current and power follow in the same way: $V_{p u} = V / V_{ba se}$ , $I_{p u} = I / I_{ba se}$ , $S_{p u} = S / S_{ba se}$ .

Economic dispatch, voltage and frequency control, power factor correction

All 19 formulas

QUADRATIC COST AND INCREMENTAL COST

C_{i} = a_{i} + b_{i} P_{i} + c_{i} P_{i}^{2}, \frac{d C _{i}}{d P _{i}} = b_{i} + 2 c_{i} P_{i}

Cost in Rs/h, $P_{i}$ in MW; incremental cost is in Rs/MWh and is linear in $P_{i}$ .

EQUAL INCREMENTAL COST CRITERION (NO LOSSES)

\frac{d C _{i}}{d P _{i}} = λ (all units inside limits), i \sum P_{i} = P_{D}

A unit held at $P_{i, m a x}$ has $d C_{i} / d P_{i} \leq λ$ ; a unit held at $P_{i, m i n}$ has $d C_{i} / d P_{i} \geq λ$ . $λ$ is the system incremental cost in Rs/MWh.

Symmetrical components and fault analysis

All 18 formulas

THE A-OPERATOR

a = 1∠12 0^{\circ} = - \frac{1}{2} + j \frac{3}{2}, a^{2} = 1∠24 0^{\circ} = - \frac{1}{2} - j \frac{3}{2}, a^{3} = 1, 1 + a + a^{2} = 0

Multiplying a phasor by $a$ rotates it by $12 0^{\circ}$ anticlockwise without changing its magnitude. Also $a^{2} - a = - j 3$ .

SYNTHESIS (PHASE FROM SEQUENCE)

V_{a} V_{b} V_{c} = 111 1 a^{2} a 1 a a^{2} V_{a 0} V_{a 1} V_{a 2}

Same form for currents. Here $V_{b} = V_{a 0} + a^{2} V_{a 1} + a V_{a 2}$ and $V_{c} = V_{a 0} + a V_{a 1} + a^{2} V_{a 2}$ .

Protection and circuit breakers

All 11 formulas

RELAY (SECONDARY) QUANTITIES

I_{sec} = \frac{I _{pri}}{N _{C T}}, V_{sec} = \frac{V _{pri}}{N _{V T}}, Z_{sec} = Z_{pri} \frac{N _{C T}}{N _{V T}}

N_CT = CT ratio and N_VT = VT ratio, each as primary over secondary. Z_pri is the impedance in primary ohms, from the primary voltage and current.

PLUG SETTING MULTIPLIER

PSM = \frac{relay current}{pickup current} = \frac{I _{fault} / N _{C T}}{PS \times I _{CT,rated sec}}

Pickup current = plug setting (as a fraction) times the CT rated secondary current. The relay operates only if PSM > 1.

Power system stability and the equal area criterion

All 16 formulas

SWING EQUATION (PER UNIT)

\frac{H}{π f} \frac{d ^{2} δ}{d t ^{2}} = P_{m} - P_{e} = P_{a}

$δ$ in electrical radians, $t$ in seconds, $f$ the system frequency in Hz, $P_{m}$ and $P_{e}$ in per unit on the same base as $H$ ; $P_{a}$ is the accelerating power. Equivalent form: $\frac{2 H}{ω _{s}} \ddot{δ} = P_{m} - P_{e}$ with $ω_{s} = 2 π f$ .

SWING EQUATION WITH THE ANGLE IN ELECTRICAL DEGREES

\frac{H}{180 f} \frac{d ^{2} δ}{d t ^{2}} = P_{m} - P_{e}

Same equation with $δ$ in electrical degrees (since $π$ rad $= 18 0^{\circ}$ ). Do not mix the two forms.