Dispatch & control formulas

Cost in Rs/h, $P_i$ in MW; incremental cost is in Rs/MWh and is linear in $P_i$.

A unit held at $P_{i,\max }$ has $d{C}_i/d{P}_i\le \lambda$; a unit held at $P_{i,\min }$ has $d{C}_i/d{P}_i\ge \lambda$. $\lambda$ is the system incremental cost in Rs/MWh.

Valid while every unit is inside its limits. The fixed term $a_i$ does not affect the dispatch.

Symmetric $B_{ij}$. For two units: $P_L=B_{11}{P}_1^2+2B_{12}{P}_1{P}_2+B_{22}{P}_2^2$. $\partial P_L/\partial P_i$ is the incremental transmission loss of unit $i$.

Holds for units inside limits. Here $\lambda$ is the incremental cost of power delivered at the load.

Every unit inside limits has the same product of penalty factor and incremental cost. The approximation holds when $\partial P_L/\partial P_i$ is small. A unit with larger $\partial P_L/\partial P_i$ has a larger $L_i$ and so a lower optimal IC.

$R_i$ in per unit on the unit's own rating; $\Delta P_i$ is the change in generator output in steady state with the speed changer setting fixed.

All units settle at one common frequency. A smaller $R$ or a larger rating takes a larger share of a load change, as long as no unit reaches its limit.

Free governor action, no supplementary control. $\Delta P_D$ is the load increase in MW and $D$ is load damping in MW/Hz (set $D=0$ if not given).

For one isolated area with governor action only (no integral control). $1/R_i$ is the governor response of unit $i$ and $D$ the load damping.

Steady state with no supplementary control; $\Delta P_{12}$ is the change in flow from area 1 to area 2. For a load step in area 1 only, $\Delta P_{12}=-{\beta }_2\Delta P_{D1}/({\beta }_1+{\beta }_2)$: area 2 sends part of the step over the tie line.

$\Delta P_{tie}$ is the tie-line flow deviation (positive out of the area) and $B_f>0$ is the frequency bias in MW/Hz, commonly set equal to the area's $\beta$. The integral controller changes the set point as $\Delta P_{ref}=-K\int \mathrm{ACE}dt$: negative ACE raises generation.

Per phase or per unit consistently; $P$ and $Q$ are the real and lagging reactive power received, $V$ the receiving-end voltage; valid for a small angle across the line. In a mainly inductive line ($X\gg R$) the drop depends mostly on $Q$.

Reactive power $\Delta Q_{inj}$ injected (for example by a shunt capacitor) at a bus whose Thevenin source reactance is $X$; $S_{sc}$ is the three-phase short-circuit level at that bus (per-unit form, $X$ dominant).

Tap-changing transformers at both ends of a line of impedance $R+jX$. $V_1$ and $V_2$ are the nominal voltages at the two ends referred to the line side, $P$ and $Q$ are the power received at end 2, and the angle across the line is small. With $V_1=V_2=V$: $t_s^2=1/\left(1-(RP+XQ)/V^2\right)$.

Total three-phase kVAr for total real power $P$ in kW. The rating is the reactive power the bank delivers at its rated voltage.

$Q_c$ is the three-phase VAr, $V_L$ the line voltage in V, $\omega =2\pi f$, and $C$ comes out in F. Note $C_{\Delta }=C_Y/3$.

At constant $P$ and constant load voltage, the source-side current falls when $\cos {\varphi }_2>\cos {\varphi }_1$, and feeder $I^{2}R$ loss falls with its square.

Constant $P$; demand charge $x$ per kVA of maximum demand per year and capacitor annual cost $y$ per kVAr per year (both linear). It minimises $xP/\cos {\varphi }_2+yP(\tan {\varphi }_1-\tan {\varphi }_2)$.