06_HW_EEE598

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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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