Q4Economic Operation of Power System

Given Data

• Loss coefficients: B11=0.001 MW^-1, B12=-0.0005 MW^-1, B22=0.0024 MW^-1
• Incremental production costs: dC1/dP1=0.08P1+16 Rs/MW-hr, dC2/dP2=0.08P2+12 Rs/MW-hr
• System lambda (incremental cost) = 20 Rs/MW-hr

Step 1: Set Up the Loss-Coordinated Economic Dispatch (Coordination) Equations

The transmission loss for a two-unit system, using the B-coefficient (loss-coefficient) method, is given by:

The coordination equations for minimum-cost dispatch, accounting for transmission losses, require each unit's incremental cost to equal the system lambda multiplied by its penalty factor (1 minus the incremental transmission loss with respect to that unit's own output):

where the incremental transmission losses are:

Step 2: Substitute Given Values and Form Linear Equations

For unit 1: substituting dC1/dP1=0.08P1+16, lambda=20, B11=0.001, B12=-0.0005:

Rearranging: 0.08P1+0.04P1-0.02P2=20-16, giving the first linear equation:

For unit 2: substituting dC2/dP2=0.08P2+12, lambda=20, B22=0.0024, B12=-0.0005:

Rearranging: 0.08P2+0.096P2-0.02P1=20-12, giving the second linear equation:

Step 3: Solve the Simultaneous Linear Equations

Solving equations (i) and (ii) simultaneously (using standard linear algebra, such as Cramer's rule or substitution):

Step 4: Compute Transmission Loss and Received Power

Substituting P1 and P2 into the loss formula:

The received (delivered/consumed) power is the total generation minus the transmission loss:

Result: for lambda=20 Rs/MW-hr, the economically optimal generation is P1≈41.70 MW and P2≈50.19 MW, with resulting transmission loss≈5.69 MW and received power delivered to the load≈86.20 MW. This calculation illustrates the essential difference the B-coefficient loss model introduces compared to a simplified loss-free economic dispatch: rather than simply equalizing the two units' raw incremental costs, the loss-penalty-factor-adjusted coordination equations account for each unit's differing contribution to system transmission losses (reflected in the differing B-coefficients), correctly incentivizing greater output from whichever unit is, in this loss-model sense, more favorably located relative to the system's load center and transmission network, exactly the loss-aware optimal dispatch principle that distinguishes practical, loss-inclusive economic dispatch from the simpler loss-free equal-incremental-cost rule.

It is instructive to verify the physical reasonableness of the computed dispatch: the total generation P1+P2 is approximately 91.89 MW, of which the transmission loss PL is approximately 5.69 MW (about 6.2 percent of total generation), leaving a received (delivered to load) power of approximately 86.20 MW - this loss percentage is a plausible figure for a real transmission network of moderate length and loading, lending confidence to the B-coefficient values assumed in the problem, and confirming that the coordination-equation approach used here (equating the incremental cost of each unit, augmented by its incremental-loss-penalty term, to the common system lambda) has produced a physically sensible, energy-balanced solution: generation exactly equals load demand plus transmission loss, as required by the fundamental power-balance constraint underlying the entire economic dispatch formulation.

It is also worth noting how the individual unit loadings compare to what a naive, loss-ignoring equal-incremental-cost dispatch would have produced: solving dC1/dP1=dC2/dP2=20 without any loss consideration would give P1=(20-16)/0.08=50MW and P2=(20-12)/0.08=100MW, a substantially different allocation (favoring unit 2 far more heavily) than the loss-inclusive result obtained here (P1 about 41.70MW, P2 about 50.19MW) - this large discrepancy illustrates just how significant the loss-penalty effect can be when a unit's location results in a strongly self-loss-penalizing B-coefficient (here, unit 2's large B22=0.0024 sharply curtails its loss-inclusive optimal dispatch relative to the loss-free calculation), underscoring why real economic dispatch calculations in meshed transmission networks cannot safely ignore transmission losses even as a first approximation.

This two-unit problem also usefully illustrates the general iterative solution procedure that must be used for larger systems (more than two or three units) where the coordination equations, once loss terms are included, no longer reduce to a small linear system solvable in closed form: for an assumed value of lambda, the loss-inclusive coordination equation for each unit individually (a linear or occasionally quadratic equation in that unit's own power, given all other units' powers from the previous iteration) is solved for each unit's tentative output, the resulting total generation is compared against the required total demand plus losses (the losses themselves recomputed from the latest generation values via the B-coefficient loss formula), and lambda is adjusted up or down depending on whether total generation exceeds or falls short of the requirement, with this process repeated until convergence - a method directly analogous in spirit to the iterative approach used elsewhere in power system analysis, such as the Gauss-Seidel load-flow iteration, and one that generalizes gracefully to systems with many more than two generating units, which is why, despite the two-unit example here being solvable in exactly one linear-algebra step, real utility-scale economic dispatch software implements this general iterative lambda-search procedure rather than a closed-form solution.

Finally, it is worth noting the practical significance of the negative cross-coefficient B12 in this problem: a negative B12 physically indicates that, holding total demand fixed, transferring generation from unit 1 to unit 2 (or vice versa) reduces total transmission loss up to a point, reflecting a beneficial power-flow cancellation effect between the two units' contributions to line currents somewhere in the network - this is a common occurrence when two generators are located such that their respective power flows to the load center partially cancel on a shared transmission corridor, and it is precisely this beneficial interaction, captured mathematically by the negative B12 term, that allows the loss-inclusive optimal dispatch to differ so substantially from the naive loss-free dispatch calculated for comparison above.

In a real utility control centre, this economic dispatch calculation would not be performed once and then held fixed, but rather re-solved every few minutes (or continuously, in an automatic generation control context) as the system load, and hence the required total generation, changes throughout the day - the specific numerical solution obtained here (for a fixed demand corresponding to a received power of approximately 86.20MW) therefore represents only a single snapshot of what is, in practice, a continuously evolving dispatch calculation, with lambda itself rising and falling throughout the day as system demand rises toward its daily peak and falls back toward its overnight minimum, tracing out what is sometimes called the system lambda curve over a 24-hour period.

The penalty factor concept, defined for each unit as 1/(1 - dPL/dPi), offers an alternative but equivalent way of viewing this same coordination condition: rather than working with the loss-augmented incremental cost equations directly, the equal-incremental-cost criterion can be restated as requiring each unit's incremental production cost multiplied by its own penalty factor to equal the common system lambda, and units with a larger penalty factor (indicating that their generation contributes more heavily to marginal system loss) are correspondingly dispatched to a lower output relative to what a penalty-free equal-incremental-cost solution would assign them, exactly the qualitative effect already observed for unit 2 in this problem given its comparatively large self-loss coefficient B22.