1. LMP calculations Congestion evaluation
Rob Burchett
January, 2005
LMP calculations Congestion evaluation
Rob’s contact information.
Rob Burchett
(518)

2. 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

3. 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

4. Economic Dispatch (No Transmission)
Rob Burchett
January, 2005
Economic Dispatch (No Transmission)
Load: 600 MW (make believe)
G1 = 400 MW $40)
G5 = 200 MW $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. 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
Bus
Bus 3 0
Bus
Bus
Coal:
400
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
100 MW load
300 MW load
9/20/2018
Rob Burchett

6. 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

7. 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

8. 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
Bus
Bus 3 0
Bus
Bus
400
Flow (from load)
Flow (from generation)
Flow on d from load:
-200 *
-100 * 0 +
-300 * = 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 *
200 * = mw
200
9/20/2018
Rob Burchett

9. 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 mw component as a constant.
The “controllable” part is the component due to generation – in this case, mw from generator 1, and 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 = G G5 =
= = Megawatts total flow
9/20/2018
Rob Burchett

10. Transmission Constraint
Rob Burchett
January, 2005
Transmission Constraint
If dmax = 60 MW , the dispatch is not acceptable!!
Flow: d = G G5 = 60
Load Balance: G 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

11. 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 = G G5 = 61
Load Balance: G G5 = 600
G1 = 271 G5 = 329 Cost = $33,870
Shadow price = 33, ,200 = -330 $/mw
9/20/2018
Rob Burchett

12. 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

13. 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

14. 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 g1 at bus 1, and g5 at bus 5.
A flow of exactly 60 mw on branch D 5 g5 3 4
1 mw
9/20/2018
Rob Burchett

15. 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 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 g g (-1) = 0
9/20/2018
Rob Burchett

16. Incremental Equations
Rob Burchett
January, 2005
Incremental Equations
Flow “d” : g g =
Load Balance: g g =
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 =
g5 = (wow !)
9/20/2018
Rob Burchett

17. Marginal Cost at Bus 4... A surprising result!
Rob Burchett
January, 2005
Marginal Cost at Bus 4...
lmp = -3 x $ x $70 = $/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

18. 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 g g (-1) = 0
g g = 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 x 70 = $/mw (!)
9/20/2018
Rob Burchett

19. 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 g g (-1) = 0
g g = 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

20. 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

21. 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

22. 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

23. 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

24. 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
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
100 mw load
300 mw load
9/20/2018
Rob Burchett

25. 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

26. 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

27. 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