LMP calculations Congestion evaluation Rob Burchett January, 2005 LMP calculations Congestion evaluation Rob’s contact information. Rob Burchett (518) 384-0872 burchett@nycap.rr.com www.PowerAnalyticsSoftware.com

What’s coming today… Understanding constrained dispatching. Rob Burchett January, 2005 What’s coming today… Understanding constrained dispatching. The Important role of Shift Factors. What is a shadow price, really? How are LMP’s computed? How are congestion charges computed? 9/20/2018 Rob Burchett

Generation Cost Oil Unit Coal Unit MW Output MW Output 600 Inc Cost Rob Burchett January, 2005 Generation Cost Oil Unit Coal Unit 70 $/mw Our 5 bus system has only two generators. 400 mw of $40 power at bus 1 600 mw of $70 power at bus 5 40 $/mw Inc Cost Inc Cost 600 400 MW Output MW Output 9/20/2018 Rob Burchett

Economic Dispatch (No Transmission) Rob Burchett January, 2005 Economic Dispatch (No Transmission) Load: 600 MW (make believe) G1 = 400 MW (max @ $40) G5 = 200 MW (Marginal @ $70) Cost = 400 x $40 + 200 x $70 = $30,000 The economic dispatch without transmission is simple. Since the load is 600mw, the best dispatch is to run the cheapest generation first, And the expensive unit last. 9/20/2018 Rob Burchett

5 Bus Power Network 1 a 2 b d f 5 e c 4 3 Coal: 400 mw @$40 Rob Burchett January, 2005 5 Bus Power Network 1 a 2 Shift Factors for “d” Bus 1 0.1818 Bus 2 0.3636 Bus 3 0 Bus 4 -0.1818 Bus 5 0.0909 Coal: 400 mw @$40 200 MW load b The dispatch induces flows on the transmission system, according to electrical laws. These laws are nonlinear, involving sines and cosines, and quadratic expressions. Since we are only concerned about mw flow, we may use the “DC” approximation, Relating mw flow in branches to generation and load using a matrix of numbers called “Shift Factors” These constant shift factors indicate how much of a bus’ mw injection flows through each transmission branch. They are efficiently computed with sparse matrix algebra, but the resulting matrix is dense, so we cannot store all of them. We only store the shift factors for those branches we wish to constrain. We use special methods to compute any shift factors that are associated with constraints under a branch outage. Line “d” is special, because we wish to constrain it, while ignoring the flow on other branches. Shift Factors tell us the portion of bus injection that flows on our branch. Bus 3 absorbs the injection inserted at any bus. They are calculated from the branch topology and impedance data. d f 5 c e 3 4 Oil: 200 mw @$70 100 MW load 300 MW load 9/20/2018 Rob Burchett

Shift Factor Concepts 1 2 1Ω a 1Ω b 1Ω 1Ω 5 3 1Ω 4 1Ω 9/20/2018 Rob Burchett January, 2005 Shift Factor Concepts 1 2 1Ω a 1Ω b 1Ω In this network, everything is measured with respect to the previous optimal solution. The values shown are changes from that solution. The only difference is the 1mw of load added at bus 4. 1Ω 5 3 1Ω 4 1Ω 9/20/2018 Rob Burchett

Reference Bus Shift factors measure the flow Rob Burchett January, 2005 Reference Bus Shift factors measure the flow resulting from an injection a single bus and withdrawn from another bus. The point of the withdrawal is common. This is the reference bus. All shift factors are computed relative to the reference (or slack) bus. 9/20/2018 Rob Burchett

5 Bus Power Network 2 1 5 4 3 200 mw 300 mw 100 mw 400 mw @$40 Rob Burchett January, 2005 5 Bus Power Network 2 4 3 200 mw 300 mw 100 mw 1 Shift Factors Bus 1 0.1818 Bus 2 0.3636 Bus 3 0 Bus 4 -0.1818 Bus 5 0.0909 400 mw @$40 Flow (from load) Flow (from generation) Flow on d from load: -200 * 0.3636 + -100 * 0 + -300 * -0.1818 = -18.18 mw Here we compute the flow on branch “d” using shift factors. Multiply each bus’ injection (generation is positive, load is negative) by the branch d’s shift factors. It is convenient to split the component due to generation from the component due to load. 5 Flow on d from gen: 400 * 0.1818 + 200 * 0.0909 = +90.91 mw 200 mw @$70 9/20/2018 Rob Burchett

Flow “d” Components Load Component Shift Factors Rob Burchett January, 2005 Flow “d” Components Load Component Shift Factors Since the load does not change during any particular hour in Promod, we may treat the -18.18 mw component as a constant. The “controllable” part is the component due to generation – in this case, 72.73 mw from generator 1, and 18.18 mw from generator 5. Note that twice as many mw from G1 (per mw) wind up on “d” than from G5. Flow on Line d = -18.18 + .1818 G1 + .0909 G5 = -18.18 + .1818 400 + .0909 200 = -18.18 + 72.73 + 18.18 = 72.73 Megawatts total flow 9/20/2018 Rob Burchett

Transmission Constraint Rob Burchett January, 2005 Transmission Constraint If dmax = 60 MW , the dispatch is not acceptable!! Flow: d = -18.18 +.1818 G1 + .0909 G5 = 60 Load Balance: G1 + G5 = 600 Here is where the transmission constraint shows its effect. If the flow limit is 60 mw – how do we adjust G1 and G5 to obey the limit, while at the same time having the lowest production cost? Solution: Solve a system of equations with two unknowns. The first equation is the flow constraint, using the shift factor terms. The second equation is the original load balance, making sure there is just enough generation to meet the load. G1 must ramp down 140 mw, and G5 must ramp up 140 mw – contrary to economics. The resulting dispatch cost is $34,200, an increase of $4,200. G1 = 260 (was 400) G5 = 340 (was 200) Cost = $34,200 Both Marginal ! 9/20/2018 Rob Burchett

Constraint Shadow Price Rob Burchett January, 2005 Constraint Shadow Price What if the constraint were 61 mw? The increase in cost is the shadow price The “shadow price” is the sensitivity of the production cost with respect to the flow constraint. If the flow were relaxed to 61 mw, the production cost would decrease (hence the negative sign) by $330. Flow: d = -18.18 +.1818 G1 + .0909 G5 = 61 Load Balance: G1 + G5 = 600 G1 = 271 G5 = 329 Cost = $33,870 Shadow price = 33,870 - 34,200 = -330 $/mw 9/20/2018 Rob Burchett

Bus Marginal (Locational) Prices Rob Burchett January, 2005 Bus Marginal (Locational) Prices How much will the next mw of load cost? Usually, it is simply the marginal unit cost With flow constraints, it’s not so simple The LMP measures the cost of serving the next mw of load if we are forced to move the generation to obey transmission constraint. We are not free to simply ramp up the cheapest generator, as this will cause the flow on “d” to violate its limits. For LMP calculations, we only pay attention to those constraints that are currently at a limit (“binding”). All other constraints do not affect this incremental calculation. 9/20/2018 Rob Burchett

Marginal Cost at Bus 4 Compute the margin cost at load bus 4 Rob Burchett January, 2005 Marginal Cost at Bus 4 Compute the margin cost at load bus 4 Get equations for a 1 MW added load Derive an expression for the margin cost We are going to add 1mw of load at bus 4 (this is conceptual only, as the real calculations are performed algebraically) 9/20/2018 Rob Burchett

Incremental Network 1 2 g1 5 g5 4 3 0 mw 1 mw 9/20/2018 Rob Burchett January, 2005 Incremental Network 1 2 g1 0 mw In this network, everything is measured with respect to the previous optimal solution. The values shown are changes from that solution. The only difference is the 1mw of load added at bus 4. We zero everything out so the math is simpler. Since the network is linear (or at least we can pretend it is), the solution to this incremental network may be added to the original network to achieve what we want: A load of 601 mw, increased by one at bus 4, A net dispatch of 260 + g1 at bus 1, and 340 + g5 at bus 5. A flow of exactly 60 mw on branch D 5 g5 3 4 1 mw 9/20/2018 Rob Burchett

The Incremental Flow Equation Rob Burchett January, 2005 The Incremental Flow Equation The change in generation must result in zero flow change Power changes at the marginal units, and at the load bus The sum of generation change must be 1 mw Here we use the shift factors for branch “d” again. Only now, we also need the factors for bus 4, since it has an injection change. Previously, we were only concerned with the factors at the generation buses. The 18.18 mw due to load is not present in these equations, because we set the base load to zero. Remember – we are solving for the change in dispatch. Bus 4: Add 1 MW Demand. New flow equation... .1818 g1 + .0909 g5 - .1818 (-1) = 0 9/20/2018 Rob Burchett

Incremental Equations Rob Burchett January, 2005 Incremental Equations Flow “d” : .1818 g1 + .0909 g5 = - .1818 Load Balance: g1 + g5 = 1 We have the same two equations, only now they are incremental. The first equation states that the generation must adjust to equal the change in load. The second equation makes sure that the adjustments at the three buses will not change the line flow on “d”. The result is surprising. The generation adds to 1.0, but it is not what we would expect. Solution: g1 = - 3 g5 = + 4 (wow !) 9/20/2018 Rob Burchett

Marginal Cost at Bus 4... A surprising result! Rob Burchett January, 2005 Marginal Cost at Bus 4... lmp = -3 x $40 + 4 x $70 = 160 $/mw One mw of load at bus 4 costs $160 to supply, if we are to keep the transmission constraint happy. This is quite a surcharge for congestion, considering there is $40 power on the margin. A surprising result! 9/20/2018 Rob Burchett

Price at Bus 2 g5 = -2 lmp = 3 x 40 - 2 x 70 = - 20 $/mw (!) Rob Burchett January, 2005 Price at Bus 2 Bus 2: Add 1 MW Demand. New equations... .1818 g1 + .0909 g5 + .3636 (-1) = 0 g1 + g5 = 1 Now do the same calculation, only at bus 2. The equations are exactly the same, except now we use the shift factor of bus 2 for branch “d” Here, adding load at bus 2 actually helps the constraint (the shift factor is positive). The more load, the more the constraint is relaxed. Thus we are actually lessening the congestion by adding load at this bus. Thus the price is negative. The cost of producing more mw is more than offset by the congestion relief provided. Solution: g1 = 3 g5 = -2 lmp = 3 x 40 - 2 x 70 = - 20 $/mw (!) 9/20/2018 Rob Burchett

What about Bus 3 (slack bus)? Rob Burchett January, 2005 What about Bus 3 (slack bus)? Bus 3: Add 1 MW Demand. New equations... .1818 g1 + .0909 g5 + .0000 (-1) = 0 g1 + g5 = 1 Bus 3 is special, because it is our slack bus. All shift factors are zero at this bus. Note: The system “lambda” – the cost of serving one more mw of pool load, is always the LMP at the slack bus. Changing the slack bus does not affect the LMP at any bus – but it will affect the congestion components, as well as the pool “lambda” Solution: g1 = -1 g5 = +2 lmp = (-1*40) + (2*70) = 100 $/mw 9/20/2018 Rob Burchett

Buses 1 and 5? These are the “price setting buses” Rob Burchett January, 2005 Buses 1 and 5? These are the “price setting buses” LMP at these buses will be the gen price LMP1 = $40 LMP5 = $70 The price is always the margin price at the marginal buses, of course. 9/20/2018 Rob Burchett

Comparison... 9/20/2018 Rob Burchett January, 2005 Comparison... The first column contains the five prices we just calculated. The 2nd column is the vector of shift factors for branch “d”. The 3rd column is the Shift Factor column multiplied by -330 (shadow price for branch D) The 4th column is the third, with $100 added (price at the reference node). Note, these are exactly equal to the prices. Note: LMPi = LMP (energy component) + shift factor (for branch d at bus i) x shadow price of branch d. The energy component is the price of electricity at the reference node. In a real ISO, a virtual bus is used as the reference bus, such as the RTO load zone. The math is the same, only the shift factors reflect this different reference point. 9/20/2018 Rob Burchett

Very Important Concept !! Rob Burchett January, 2005 Very Important Concept !! Electricity prices around the system Are flat when no transmission is congested Vary from bus-to-bus when split. The price at each bus is directly proportional to that bus’ effect on the binding constraints (shift factors). With the shadow prices, we can compute LMP everywhere. (see spreadsheet) 9/20/2018 Rob Burchett

Congestion charges Use our computed LMP to compute congestion charges Rob Burchett January, 2005 Congestion charges Use our computed LMP to compute congestion charges 9/20/2018 Rob Burchett

Single Area Model 1 2 $40 -$20 60 mw 5 $100 $160 $70 3 4 260mw @$40 Rob Burchett January, 2005 Single Area Model 1 2 $40 -$20 260mw @$40 200 mw load The dispatch induces flows on the transmission system, according to electrical laws. These laws are nonlinear, involving sines and cosines, and quadratic expressions. Since we are only concerned about mw flow, we may use the “DC” approximation, Relating mw flow in branches to generation and load using a matrix of numbers called “Shift Factors” These constant shift factors indicate how much of a bus’ mw injection flows through each transmission branch. They are efficiently computed with sparse matrix algebra, but the resulting matrix is dense, so we cannot store all of them. We only store the shift factors for those branches we wish to constrain. We use special methods to compute any shift factors that are associated with constraints under a branch outage. Line “d” is special, because we wish to constrain it, while ignoring the flow on other branches. 60 mw 5 $100 $160 $70 3 4 340 mw @$70 100 mw load 300 mw load 9/20/2018 Rob Burchett

Individual Flow Method Rob Burchett January, 2005 Individual Flow Method branch congestion = From bus To bus Flow Lower Upper Shadow LMP src LMP sink CC 1 2 120 -250 250 40 -20 -7,200 3 140 -260 260 100 8,400 4 60 -100 -330 160 10,800 -40 2,400 5 -200 70 18,000 -140 -300 300 -12,600   Congestion charge: 19,800 9/20/2018 Rob Burchett

Congested Flow Method Total Congestion = -Shadow Price x Flow Rob Burchett January, 2005 Congested Flow Method Total Congestion = -Shadow Price x Flow CC = 330 $/mw x 60 mw = $19,800 9/20/2018 Rob Burchett

Total Congestion = Demand $ - Gen $ Rob Burchett January, 2005 Single Area Method Total Congestion = Demand $ - Gen $ Bus Generation Demand LMP Gen $ Demand $ CC 1 260   40 $10,400 2 200 -20 -$4,000 3 100 $10,000 4 300 160 $48,000 5 340 70 $23,800 Totals 600 $34,200 $54,000 $19,800 9/20/2018 Rob Burchett