Q2Power System Engineering

(a) Function of Transmission Losses for a Two-Generator System

For a system with two generators supplying a common load through a transmission network, the total transmission loss PL can be expressed as a quadratic function of the individual generator power outputs using the loss-coefficient (B-coefficient) formula, derived originally by Kron from a detailed load-flow-based analysis of the network but expressed in the simplified, widely used form:

where B11 and B22 are the self-loss coefficients of generators 1 and 2 respectively (representing each generator's own individual contribution to system loss, always positive), and B12 is the mutual (cross) loss coefficient between the two generators (which can be positive or negative depending on the specific network topology and relative electrical location of the two generators, since a negative B12 reflects a beneficial power-flow-cancellation interaction between the two generators' contributions to the network's line currents).

This B-coefficient loss formula is derived under the assumption that the ratio of each load's current to the total system current remains approximately constant as generation levels vary around a given base-case operating condition (obtained from a detailed load-flow study), and that all loads have essentially constant power factor - under these assumptions, the total transmission loss (which is fundamentally the sum of I-squared-R losses in every line of the network) can be shown, after a moderately involved derivation combining the network's individual line resistances with this fixed-current-distribution-ratio assumption, to reduce to the quadratic B-coefficient form shown above, expressed purely in terms of the total generation from each of the system's generating plants rather than requiring a full network load-flow solution to be repeated for every trial dispatch during the economic dispatch optimization process.

The B-coefficients themselves are computed once (as constants) from a base-case load-flow solution of the actual network, and are then treated as fixed constants during the iterative economic dispatch calculation - this is a considerable practical simplification, since it avoids needing to re-solve the full nonlinear network load-flow equations at every single trial dispatch point during the economic dispatch iteration, at the cost of the loss formula's accuracy gradually degrading for large deviations in generation pattern away from the original base case used to derive the B-coefficients (motivating periodic recalculation of the B-coefficients from an updated base-case load-flow as actual system operating conditions drift over time).

(b) Economic Loading and Savings Comparison

Given: two 250 MW units with dC1/dP1=0.20P1+30, dC2/dP2=0.15P2+40, total load 225 MW, transmission losses neglected in this problem.

For economic (equal-incremental-cost) loading, setting dC1/dP1=dC2/dP2 and using P1+P2=225:

The common incremental cost (system lambda) is dC1/dP1 = 0.20(125)+30 = 55 Rs/MWh (verified: dC2/dP2 = 0.15(100)+40 = 55 Rs/MWh, confirming both units are at the same incremental cost).

The total fuel cost for economic loading is found by integrating each unit's incremental cost function to obtain its total cost function (C1(P)=0.1P^2+30P, C2(P)=0.075P^2+40P, ignoring any fixed no-load cost constant since it is identical and cancels in the comparison performed here):

For equal load division (each unit supplying half the total load, P1=P2=112.5 MW):

Assuming the same total load of 225 MW is maintained continuously (throughout the year, as stated in the problem), the annual saving is obtained by multiplying the hourly saving by the total number of hours in a year:

So the economic loading is P1=125 MW, P2=100 MW (compared to 112.5 MW each for equal division), and adopting economic loading instead of simple equal load division saves approximately Rs 27.34 per hour, or approximately Rs 2,39,531 per year, assuming this exact 225 MW load level were sustained continuously - in a real system where load varies considerably over the day and year, the actual annual saving would be computed by integrating this same economic-versus-equal-division cost comparison across the full range of actual load levels experienced, using the system's actual load-duration curve, but the fixed-load calculation shown here illustrates the underlying economic dispatch benefit clearly and is the standard simplified approach used for this type of textbook-style demonstration problem.

It is also worth noting an important limitation of the B-coefficient loss formula derived and used in these problems: the B-coefficients themselves are strictly valid only in the vicinity of the specific base-case operating condition (generation pattern and network configuration) from which they were originally computed via a detailed load-flow study, and using them for economic dispatch calculations under significantly different operating conditions (a substantially different total system load, a materially different generation pattern, or following a major network topology change such as a line outage) can introduce meaningful inaccuracy, since the underlying assumption of a fixed, unchanging current-distribution-ratio pattern across the network becomes progressively less valid the further the actual operating condition drifts from the original base case.

The comparison between economic dispatch and equal load division performed in part (b) also illustrates a more general principle applicable well beyond just these two specific 250 MW units: whenever two or more generating units have different marginal (incremental) cost characteristics, any dispatch strategy other than the equal-incremental-cost economic dispatch necessarily incurs some avoidable additional cost, and the magnitude of this avoidable cost (as quantified by the saving calculation performed above) grows larger the more the units' individual cost characteristics differ from one another and the further the total system load moves away from the specific load level at which equal division happens to coincide with economic dispatch (which would only occur, for these two units, at the one specific total load level where the equal-incremental-cost condition is exactly satisfied by equal individual outputs).

In summary, this problem set demonstrates both the theoretical basis of the B-coefficient transmission loss formula and its direct, quantifiable economic consequence when comparing optimal economic dispatch against the simpler, non-optimal equal-load-division alternative, reinforcing why utilities invest in economic dispatch computation despite its greater complexity relative to simpler heuristic loading strategies.

This worked comparison between economic and equal-division loading, quantifying a concrete annual cost saving from adopting proper economic dispatch, reflects the same underlying financial motivation that has driven the electric utility industry's decades-long investment in increasingly sophisticated economic dispatch and unit commitment optimization software.

Ultimately, this saving calculation, however modest it may appear on a per-hour basis, compounds into a substantial annual figure precisely because power plants operate continuously, illustrating why utilities dedicate considerable engineering and computational resources to economic dispatch optimization even for comparatively small percentage cost improvements.