Q1Power System Engineering

(a) Economic Distribution of Load Neglecting Transmission Losses

Consider n generating units supplying a total system load PD, with each unit i having its own fuel cost function Ci(Pi). The economic dispatch problem is to determine the individual generation Pi for each unit so as to minimize total system fuel cost CT=sum(Ci(Pi)), subject to the equality constraint that the sum of all generations equals the total demand: sum(Pi)=PD (transmission losses neglected here).

This constrained optimization is solved using the method of Lagrange multipliers: form the Lagrangian function L = CT + lambda*(PD - sum(Pi)), where lambda is the Lagrange multiplier. Setting the partial derivative of L with respect to each Pi to zero gives the necessary condition for a minimum:

This is the classical equal incremental cost criterion: for minimum total system fuel cost (neglecting transmission losses), all online generating units must be loaded so that their individual incremental fuel costs (dCi/dPi, the additional cost of generating one more MW from that unit) are all equal to each other, and equal to a common value lambda, the system incremental cost (also interpreted as the system marginal cost of supplying one additional MW of demand).

The economic justification for this equal-incremental-cost condition follows directly by contradiction: suppose two units, i and j, were operating at unequal incremental costs, with dCi/dPi less than dCj/dPj. Then transferring a small amount of generation from unit j to unit i (increasing Pi slightly and decreasing Pj by the same amount, keeping total generation unchanged) would decrease total cost, since the cost saved by reducing unit j's output (at its higher incremental cost) exceeds the cost added by increasing unit i's output (at its lower incremental cost) - this demonstrates that any dispatch with unequal incremental costs cannot be the true cost-minimizing solution, and the true minimum-cost dispatch is reached only when no further such beneficial transfer is possible, which occurs precisely when all incremental costs are equal.

(b) Economic Loading with Transmission Loss (Two-Plant System)

Plant 2 is directly connected at the same bus as the load, so all power generated by plant 2 is delivered directly to the load with no transmission loss associated with it, and plant 2's coordination equation is simply its ordinary equal-incremental-cost condition dC2/dP2=lambda (no penalty factor, since plant 2 contributes no loss).

The transmission loss is caused entirely by the power flowing from plant 1 to plant 2, so it can be modeled with a single loss coefficient B11 using the simple relation PL=B11*P1^2. From the given data (100 MW flow causes 15 MW loss):

Plant 1's coordination equation must include the loss-penalty factor, since increasing P1 increases the transmission loss: dC1/dP1 = lambda(1-dPL/dP1), where dPL/dP1=2B11*P1.

Verifying: dC1/dP1 = 0.20(105.26)+20 = 41.05 Rs/MWh; the penalty factor is 1/(1-20.0015105.26) = 1/(1-0.3158) = 1.4615; and 41.05*1.4615=60.0, confirming the coordination equation is satisfied.

So the economic loading is P1=105.26 MW and P2=200 MW, the transmission loss is 16.62 MW, and the total demand (load) supplied is 288.64 MW. It is worth noting that plant 1, despite having a lower incremental cost at low loading (a lower base cost coefficient of 20 versus plant 2's 30), is dispatched to a lower absolute output than plant 2 specifically because of the loss penalty associated with its remote location relative to the load - this illustrates the general principle that loss-penalty-factor-based coordination systematically favors generation located electrically closer to the load center, even when that generation's intrinsic (loss-free) incremental cost characteristic might otherwise favor a more distant unit.

It is worth noting an important practical qualification to the equal-incremental-cost criterion derived in part (a): this simple condition applies only when transmission losses are neglected and all units are unconstrained by any minimum or maximum generation limit - in a real system, individual units also have physical maximum and minimum stable generation limits (Pi,min and Pi,max), and if the unconstrained equal-incremental-cost solution would require any unit's output to fall outside its permissible range, that unit is instead held fixed at its violated limit (min or max, whichever was exceeded), while the equal-incremental-cost condition is reapplied to the remaining, still-unconstrained units to redistribute the remaining load among them, a refinement necessary for the criterion to be directly usable in practical economic dispatch software.

The two-plant loss-penalty calculation in part (b) also illustrates why simply comparing each plant's base incremental-cost coefficients in isolation (plant 1's cost intercept of 20 being lower than plant 2's intercept of 30) does not, by itself, determine which plant should be loaded more heavily once transmission losses are taken into account - despite plant 1's inherently cheaper cost characteristic at low output levels, its more remote electrical location (reflected in its nonzero loss coefficient B11) means increasing its output imposes a real, quantifiable additional system cost through increased transmission loss, and it is only by combining both the intrinsic cost characteristic and this loss-penalty effect through the coordination equation that the true economically optimal dispatch, P1=105.26 MW and P2=200 MW, can be correctly determined.

It is also worth noting the broader generalization of the two-plant loss formula used here to a system with more than two generating plants: for n plants, the full B-coefficient loss formula (discussed further in relation to the second question of this examination) requires a full n-by-n matrix of loss coefficients Bij, and the corresponding coordination equations for economic dispatch with losses become a system of n simultaneous nonlinear equations requiring iterative numerical solution (typically via a lambda-iteration or gradient-based method) rather than the simple direct algebraic solution possible in this specific two-plant, single-loss-coefficient example.

In summary, this problem demonstrates the complete practical workflow for two-plant economic dispatch with transmission losses: deriving the plant-specific loss coefficient from given operating data, formulating each plant's loss-penalty-adjusted coordination equation, and solving the resulting system algebraically to obtain the true cost-minimizing generation schedule together with the associated system loss and total demand.

This complete two-plant worked example, from the equal-incremental-cost derivation through the loss-inclusive numerical solution, serves as a compact but representative illustration of the economic dispatch methodology used, in substantially more elaborate multi-plant form, by real utility control centers on a continuous, near-real-time basis.