Q4Power System 2

Question

Q.4. The fuel-cost function in $/h of two thermal plants are:

C_{1} = 320 + 6.2 P_{1} + 0.004 P_{1}^{2}

C_{2} = 200 + 6.0 P_{2} + 0.003 P_{2}^{2}

where P1 and P2 are in MW. Plants outputs are subject to the following limits (in MW): 50<=P1<=250, 50<=P2<=350. The per-unit system real power loss with generation expressed in per unit on a 100-MVA base is given by:

P_{L (p u)} = 0.0125 P_{1 (p u)}^{2} + 0.00625 P_{2 (p u)}^{2}

The total load is 412.35 MW. Determine the optimal dispatch of generation.

Answer 1

Given Data

C1 = 320 + 6.2P1 + 0.004P1^2 ($/h, P1 in MW), with 50<=P1<=250
C2 = 200 + 6.0P2 + 0.003P2^2 ($/h, P2 in MW), with 50<=P2<=350
PL(pu) = 0.0125 P1(pu)^2 + 0.00625 P2(pu)^2, on 100 MVA base
Total load = 412.35 MW

Step 1: Convert Loss Formula to MW

Converting the given per-unit loss formula (with P1, P2 expressed in per unit of the 100 MVA base, i.e., P(pu)=P(MW)/100) to a direct MW-based expression:

P_{L} (MW) = 100 [0.0125 (\frac{P _{1}}{100})^{2} + 0.00625 (\frac{P _{2}}{100})^{2}] = 0.000125 P_{1}^{2} + 0.0000625 P_{2}^{2}

Step 2: Set Up the Coordination (Lagrangian) Equations

Minimizing total system cost C1+C2 subject to the power balance constraint P1+P2-PL=PD (demand), using the Lagrangian method, gives the classical coordination equations requiring each plant's incremental cost, adjusted by its penalty factor, to equal the common system incremental cost lambda:

\frac{d C _{1}}{d P _{1}} = λ (1 - \frac{\partial P _{L}}{\partial P _{1}}), \frac{d C _{2}}{d P _{2}} = λ (1 - \frac{\partial P _{L}}{\partial P _{2}})

Computing the required derivatives from the given cost and loss functions:

\frac{d C _{1}}{d P _{1}} = 6.2 + 0.008 P_{1}, \frac{\partial P _{L}}{\partial P _{1}} = 0.00025 P_{1}

\frac{d C _{2}}{d P _{2}} = 6.0 + 0.006 P_{2}, \frac{\partial P _{L}}{\partial P _{2}} = 0.000125 P_{2}

Substituting into the coordination equations and rearranging to express P1 and P2 explicitly in terms of lambda:

6.2 + 0.008 P_{1} = λ (1 - 0.00025 P_{1}) \Rightarrow P_{1} = \frac{λ - 6.2}{0.008 + 0.00025 λ}

6.0 + 0.006 P_{2} = λ (1 - 0.000125 P_{2}) \Rightarrow P_{2} = \frac{λ - 6.0}{0.006 + 0.000125 λ}

Step 3: Solve Iteratively for Lambda to Satisfy the Power Balance

The correct value of lambda must satisfy the overall power balance equation P1+P2-PL(P1,P2)=412.35 MW. Solving this non-linear equation iteratively (for example, via bisection or Newton-Raphson search on lambda) converges to:

λ \approx 7.804 $/ MWh

Substituting this converged lambda value back into the expressions for P1 and P2:

P_{1} = \frac{7.804 - 6.2}{0.008 + 0.00025 \times 7.804} = \frac{1.604}{0.009951} \approx 161.18 MW

P_{2} = \frac{7.804 - 6.0}{0.006 + 0.000125 \times 7.804} = \frac{1.804}{0.006976} \approx 258.60 MW

Step 4: Verify Power Balance and Operating Limits

Computing the transmission loss at this operating point:

P_{L} = 0.000125 (161.18)^{2} + 0.0000625 (258.60)^{2} \approx 3.248 + 4.181 \approx 7.43 MW

Check: P1+P2-PL = 161.18+258.60-7.43 = 412.35 MW, exactly matching the specified total load, confirming the solution's consistency. Checking against the specified generation limits: P1=161.18 MW lies within 50<=P1<=250 (satisfied), and P2=258.60 MW lies within 50<=P2<=350 (satisfied), so neither unit's limit is binding, and the unconstrained coordination-equation solution obtained above is indeed the optimal economic dispatch.

Result: the optimal (loss-inclusive) economic dispatch is P1 approximately 161.18 MW and P2 approximately 258.60 MW, with total transmission loss approximately 7.43 MW and system incremental cost (lambda) approximately 7.804 $/MWh. This solution illustrates the essential effect of transmission losses on economic dispatch: rather than simply equalizing the two plants' raw incremental costs (dC1/dP1=dC2/dP2), the loss-penalty-factor-adjusted coordination equations used above cause the dispatch to favor the plant whose incremental transmission loss contribution is smaller for a given output level, which in this case is plant 2 (with the smaller loss coefficient, 0.0000625 versus plant 1's 0.000125), explaining why P2 is dispatched to a notably higher output than P1 despite plant 2's cost function being only modestly cheaper than plant 1's at comparable output levels — a good illustration of why real-world economic dispatch, unlike the simplified loss-free equal-incremental-cost rule, must properly account for each plant's specific location-dependent contribution to system transmission losses.

It is worth noting the practical procedure by which the lambda value above would typically be found in an actual computation: starting from an initial guess of lambda (for instance, based on the average of the two plants' no-load incremental costs), the corresponding P1 and P2 are computed from the coordination equations, the resulting total generation minus losses is compared against the specified load, and lambda is then adjusted iteratively (increased if generation is insufficient, decreased if generation exceeds the required load-plus-losses) until the power balance equation converges to the specified tolerance — this iterative lambda-search procedure is the standard practical technique used in economic dispatch software to solve the loss-coordination equations, since the presence of the loss term (itself a function of the unknown generation outputs) makes a direct, closed-form analytical solution for lambda generally unavailable except in the simplest loss-free case.