06_HW_EEE598

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Apr 3, 2024

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Homework 06 EEE 598 – Electric Energy Markets Professor Kory Hedman Homework 6: Due Thursday, April 4 by 5pm AZ time. You are always required to show your work. AMPL can be used in this assignment (or another solver that is capable of handling the mathematical optimization problems, i.e., determining optimal solutions). 1. (50 points) For this DC OPF (with unit commitment) model, we will assume that the consumers submit bids as well as the generators. That is, we will not assume that the demand is perfectly inelastic but rather that the load at each bus is a variable and it will submit a bid (the maximum price) that it is willing to pay to consume. Since the consumers are submitting bids for this model, the motivation of the ISO is then to maximize the bid surplus. The bid surplus is defined as the amount the load is willing to pay to consume minus the cost to satisfy that load (note that bid surplus is used instead of social welfare since we do not know if the bid is a true reflection of the value to the participant). The formulation is shown below. Maximize: ∑ t ∑ i b it L it − ∑ t ∑ g ( c g P gt + c g NL u gt + c g SU v gt ) s.t. ∑ g ( i ) P gt − Pinj i,t R − L i,t = 0 , ∀ i,t − P k max ≤ ∑ i ( PTDF k ,i R Pinj i,t R ) ≤ P k max , ∀ k ,t ∑ i Pinj i ,t R = 0 , ∀ t P g min u gt ≤ P gt ≤ P g max u gt , ∀ g,t 0 ≤ L it ≤ Load it , ∀ i,t v gt ≥ u gt − u g ,t − 1 , ∀ g ,t ∈ { 2 , .. ,T } v g 1 ≥ u g 1 − Uinitial g , ∀ g,t = 1 0 ≤ v gt ≤ 1 , ∀ g,t u gt ∈ { 0,1 } , ∀ g,t Nomenclature Indices and Sets g : Generator. g ( i ): Set of generators at bus i . k : Transmission element (line or transformer). i : Buses (nodes). R : Reference bus. T : Total number of periods. Page 1 of 6

Homework 06 Parameters c g : Production cost for generator g . c NL g : No load cost for generator g . c SU g : Startup cost for generator g . Load it : Maximum consumption of the Load at bus i in period t . P max g , P min g : Max and min capacity of generator g . P max k , P min k : Max and min capacity rating of transmission element k . Generally, P min k = – P max k . Uinitial g : Initial status (1 for on, 0 for off) of generator g (before period 1). b it : Bid offer of Load at bus i to consume in period t. Variables P gt : Real power supply from generator g at node i for period t . Pinj R it : Net injection at bus i in period t (reference bus R ). L it : Consumption of Load at bus i in period t . u gt : Unit commitment binary variable for generator g in period t (offline: 0, online: 1). v gt : Startup variable for generator g in period t (startup in period t : 1, otherwise: 0). Below is the 4-bus diagram that this problem is based on. We will assume that z1 = z2 = z3 = z4. All of this information is contained within the datasets that are provided on canvas. Line Information: Pmin: Pmax: P1 -25 25 P2 -40 40 P3 -40 40 P4 -40 40 Page 2 of 6 A B D C z1 z2 z4 z3 Ga Gd Gb Gc Ld Lb Lc La P 4 P 1 P 3 P 2

Homework 06 Load Information: Period 1: Period 2: Period 3: La 10 30 80 Lb 20 40 140 Lc 30 100 190 Ld 40 70 160 Generator Information: Pmin: Pmax: Startup Cost: No Load Cost: Operating Cost: Initial Status: Gen A 10 70 $1000 $100 30 $/MWh Off Gen B 20 150 $500 $120 40 $/MWh Off Gen C 30 150 $500 $120 70 $/MWh Off Gen D 20 300 $1000 $90 60 $/MWh Off PTDF k,i r : Power Transfer Distribution Factor for line k for the net injection from bus i to the reference bus r . The PTDFs are shown below. Line k: PTDF k,A A PTDF k,B A PTDF k,C A PTDF k,D A K = 1 0 -0.75 -0.25 -0.5 K = 2 0 -0.25 -0.75 -0.5 K = 3 0 0.25 -0.25 -0.5 K = 4 0 -0.25 0.25 -0.5 Bids from the Loads: Period 1: Period 2: Period 3: La $32/MWh $32/MWh $67/MWh Lb $45/MWh $45/MWh $65/MWh Lc $65/MWh $75/MWh $75/MWh Ld $55/MWh $65/MWh $75/MWh I have posted the code and the datasets for the above optimization program on canvas. a) (20 points) Solve the above program using AMPL (or GUROBI or CPLEX directly if you prefer) and fill in the following tables. You are allowed to use AMPL to calculate the LMPs. Note that this is a maximization problem as compared to a minimization. Period 1: Period 2: Period 3: Gen A Gen B Gen C Gen D U1 U2 Page 3 of 6

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Homework 06 U3 U4 P1 P2 P3 P4 Total Profit for all periods: Gen A Gen B Gen C Gen D Period 1: Period 2: Period 3: LMP A LMP B LMP C LMP D Optimal Bid Surplus for all periods: b) (15 points) What is the total required uplift for each generator and what is the total uplift (for the entire system)? Calculate the uplift payment by looking at the sum of the gen’s profit over all three periods. Total Uplift for all periods: Gen A Gen B Gen C Gen D Total Uplift for all Gens: c) (3 points) Assume that the ISO will cover this uplift payment by taking the total required uplift, your answer in part c above, and dividing it by the total load for all hours (add up each load for each bus for each hour). This would then give a $/MWh calculation. What is this $/MWh value? Page 4 of 6

Homework 06 d) (12 points) Add your answer from part d above to the LMPs so that the load pays the original LMP plus this additional cost to cover the uplift. Period 1: Period 2: Period 3: LMP A + uplift LMP B + uplift LMP C + uplift LMP D + uplift Compare the values that you list in the above table (what the consumers have to pay: LMP + uplift) with the consumers’ bids in the earlier table. Is there any problem (if so, identify the problems)? Discuss. Type your answer below. 2. (50 Points) Load Information: Period 1: Period 2: Period 3: Load: 50 70 140 Generator Information: Pmin: Pmax: Ramp Rate: Marginal Cost to Operate: Gen A 0 70 40 30 $/MWh Gen B 0 100 10 40 $/MWh Gen C 0 100 20 70 $/MWh a) (20 points) Formulate and solve a basic economic dispatch problem with ramp rate constraints . You are allowed to use AMPL. You must turn in your code. You must model all three periods together, in one program. Impose ramp rate constraints from Period 1 to Period 2 and Period 2 to Period 3. Do not impose ramp rate constraints from Period 0 to Period 1 (because there is no period 0), i.e., there is no restriction on Period 1’s outputs based on period 0 (no initial conditions). Generator Information: Period 1 Output: Period 2 Output: Period 3 Output: Gen A Gen B Gen C Period 1 Profit: Period 2 Profit: Period 3 Profit: Total Profit: Gen A Page 5 of 6

Homework 06 Gen B Gen C Optimal Objective: $ b) (5 points) What are the LMPs (for each period) as given by AMPL? Period 1 LMP: Period 2 LMP: Period 3 LMP: c) (5 points) Take the solution you have from part A. For this part, calculate what the LMP would be for each period if you ignore ramp rate constraints and simply buy the next 1MW from the cheapest unit (not at Pmax). This is meant to represent what we did in class when I was teaching LMPs and ramp rates – and asked the class to define the LMP for the example and the class answer was based on the definition of the cost to deliver 1 more MW (while forgetting about ramp rates). Note that these LMPs are not correct. Part B is correct. This is to show you, here in part C, how there can be misunderstandings on LMPs by showing you how some people may interpret the LMPs (here) versus how they should come out (part B). Period 1 LMP: Period 2 LMP: Period 3 LMP: d) (20 points) With the LMPs from part B, are any of the generators required an uplift payment? If so, which ones and how much? Gen 1 Uplift Gen 2 Uplift Gen 3 Uplift With the LMPs from Part C, are any of the generators required an uplift payment? If so, which ones and how much? Gen 1 Uplift Gen 2 Uplift Gen 3 Uplift Give a short discussion on the results. Type your answer below. Page 6 of 6

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