Assignment-1 Problems on ELD and Hydro-Thermal Coordination

ECE666: Power Systems Operation: Winter 2024 Assignment-1 Last Date of Submission: Friday, February 16, 2024, 4:30 PM Note: a. All submissions should be in the Drop-Box, on LEARN. b. We follow the Honour System, expecting that you will solve the assignment independently without consulting others or referring to any past year solutions. 1. The cost function of a generator connected at bus-1 is: 𝐶𝐶 1 ( 𝑃𝑃 1 ) = 0.01 𝑃𝑃 1 2 + 5.2 𝑃𝑃 1 + 50 � $ ℎ𝑟𝑟 � � When the system marginal cost is 6.3 $/MWh, which includes the effect of losses, the dispatch of generator at bus-1 is P 1 = 50 MW. i. Find the Incremental Loss Factor at bus-1. ii. If losses were neglected, and the system marginal cost remained the same, what would be the dispatch of generator-1? 2. The optimal solution of an ELD for a system with two thermal generators results in both units being dispatched equally, to supply the 300 MW demand. The cost functions of the units are: 𝐶𝐶 1 ( 𝑃𝑃 1 ) = 𝑎𝑎𝑃𝑃 1 2 + 𝑏𝑏𝑃𝑃 1 + 𝑐𝑐 ($ ℎ ⁄ ) 𝐶𝐶 2 ( 𝑃𝑃 2 ) = 0.03 𝑃𝑃 2 2 + 3 𝑃𝑃 2 + 100 ($ ℎ ⁄ ) Because of this dispatch, generator 1 operates at 75% of its nominal capacity, with a heat rate of 5700 BTU/kWh. The price of the fuel is 1.81712 $/MBTU. For the current dispatch, determine: a. The system marginal cost. b. If the incremental cost of generator 1 at full load is 14 $/MWh, find the coefficients of the cost function C 1 (P 1 ). c. Find total operating cost of the system when serving a 300 MW load. 3. A two-bus system is connected by a transmission line; Generator-1 is at Bus-1, and Generator- 2 at Bus-2. There is a load connected at Bus-2. The cost characteristics of the generators are: ( ) ( ) 40 33 03 . 0 20 18 015 . 0 2 2 2 2 2 1 2 1 1 1 + + = + + = G G G G G G P P P C P P P C The general form of the transmission loss function is: 2 2 3 2 1 2 2 1 1 G G G G Loss P P P P P α α α + + = Given that, when 120 MW power is transmitted from Generator-1 to the load, a total loss of 16.425 MW is incurred. Determine the optimal dispatch of the two generators, and the total load served, if the marginal cost is 36 $/MWh. 4. Consider three generators whose cost characteristics and capacity limits are given below. Unit Cost Characteristics Limits, MW 1 𝐶𝐶 1 ( 𝑃𝑃 1 ) = 25 + 3.5 𝑃𝑃 1 100 ≤ P 1 ≤ 650 2 𝐶𝐶 2 ( 𝑃𝑃 2 ) = 48 + 6.2 𝑃𝑃 2 60 ≤ P 2 ≤ 400 3 𝐶𝐶 3 ( 𝑃𝑃 3 ) = 33 + 4.35 𝑃𝑃 3 100 ≤ P 2 ≤ 500 The system demand is 1000 MW . The system loss function is expressed as,

𝑃𝑃 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 = 0.00015 𝑃𝑃 1 2 + 0.00008 𝑃𝑃 2 2 + 0.0001 𝑃𝑃 3 2 Find the following: a) The optimal generation schedule that minimizes the total generation cost while satisfying all generation limits, and ignoring transmission losses. Find the system marginal cost. b) The optimal generation schedule that minimizes the total system losses while satisfying all generation limits. Find the system marginal loss. c) Examine the solutions obtained in (a) and (b) and comment what would be the appropriate objective for the economic operation of this particular system. 5. Consider two electric utilities whose composite CO 2 emission characteristics, E i (P i ) , and the total generating capacity limits are provided below. Utility Emission Characteristics Limits 1 hr kg P P P E / 0275 . 0 5 . 3 25 ) ( 2 1 1 1 1 + + = MW 650 MW 50 1 ≤ ≤ P 2 , / 065 . 0 2 . 6 48 ) ( 2 2 2 2 2 hr kg P P P E + + = MW 400 MW 30 2 ≤ ≤ P Utility-1 has a system demand of 575 MW and Utility-2 demand is 325 MW. Find the optimal generation schedule that minimizes the total system emissions while satisfying all generation limits. Hence, find the value of system λ and explain its significance. Also find the total reduction in system emissions because of joint dispatch, as compared to the system emissions when the two utilities operate independently. 6. A system comprising two generating units, has the following incremental cost functions: 8 008 . 0 1 1 1 + = G G P dP dC ; and 9 012 . 0 2 2 2 + = G G P dP dC The system is operating on economic dispatch with P G1 = P G2 = 500 MW. It is known that the incremental loss factor of Generator-2 is 0.2. Find the Penalty Factor of Generator-1. 7. Consider two generators whose cost characteristic are given below, ignore their limits. 𝐶𝐶 1 ( 𝑃𝑃 1 ) = 0.00253 𝑃𝑃 1 2 + 3.19 𝑃𝑃 1 + 850 $ ℎ𝑟𝑟 𝐶𝐶 2 ( 𝑃𝑃 2 ) = 0.00325 𝑃𝑃 2 2 + 5.11 𝑃𝑃 2 + 1687 $ ℎ𝑟𝑟 The total system loss can be expressed as follows: 𝑃𝑃 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 ( 𝑃𝑃 1 , 𝑃𝑃 2 ) = 12 × 10 −5 𝑃𝑃 1 2 + 7 × 10 −5 𝑃𝑃 2 2 𝑀𝑀𝑀𝑀 Given that the system demand is 900 MW, find the economic load dispatch: a) To minimize the total system cost, ignoring losses b) To minimize the total system cost, considering losses c) To minimize the total system losses. d) Discuss which of the above solutions is the most acceptable ones by the system operator, and why.

d) If the maximum drawdown from the reservoir is restricted to 180,000 acre-ft over the 168- hour period, how long should the thermal unit operate now? 11. The system load is supplied by a hydro and a steam generator with characteristics below: Steam system: 𝐶𝐶 ( 𝑃𝑃 𝑆𝑆 ) = 0.0027 𝑃𝑃 𝑆𝑆 2 + 9 𝑃𝑃 𝑆𝑆 + 200 $ ℎ𝑟𝑟 30 𝑀𝑀𝑀𝑀 ≤ 𝑃𝑃 𝑆𝑆 ≤ 400 𝑀𝑀𝑀𝑀 Hydro Plant: 𝑞𝑞 ( 𝑃𝑃 𝐻𝐻 ) = 0.0075 𝑃𝑃 𝐻𝐻 2 + 25 𝑃𝑃 𝐻𝐻 + 300 𝑎𝑎𝑎𝑎𝑟𝑟𝑎𝑎−𝑓𝑓𝑓𝑓 ℎ𝑟𝑟 0 ≤ 𝑃𝑃 𝐻𝐻 ≤ 600 𝑀𝑀𝑀𝑀 The system load profile is given below: Hour 1 – Hour 4: 725 MW Hour 5 – Hour 8: 615 MW The hydro reservoir is limited to a total drawdown capacity of 75,000 acre-ft over the 8- hour period. Inflow to the reservoir is to be neglected. a. A hydro-thermal generation schedule is to be drawn up so as to minimize the total cost. Formulate the appropriate Lagrangian function and develop the KKT conditions for the optimum. b. Use an iterative approach, and present three full iterations of the outer loop to solve the KKT equations. (Use starting guess of λ 1 = λ 2 =40 $/MWh, γ = 0.35 $/acre-ft).