Economical Dispatch Optimal Power Flow and Locational Marginal Pricing - Power Systems Exercise Ariel Villalon M

DURHAM UNIVERSITY - SCHOOL OF ENGINEERING - ENERGY MARKETS AND RISKS

PowerWorld Simulation Exercise

Economic Dispatch, Optimal Power Flow and Locational Marginal Pricing

Ariel Villalón Monsalve

18/03/2011

Ariel Villalón M. – MSc NRE 2010/2011 Page 2

Table of Contents

PART 1 – OPTIMAL DISPATCH ........................................................................................................ 3

I.1 CONSTANT MARGINAL COSTS .............................................................................................. 3

I.2 VARIABLE MARGINAL COSTS ................................................................................................ 9

PART II: LOCATIONAL MARGINAL PRICING (LMP)....................................................................... 21

II.1 BASE CASE – UNIFORM PRICING (TRANSMISSION CONGESTION NEGLECTED)........... 21

II.2 COMPETITION IN QUANTITIES............................................................................................ 25

II.3 COMPETITION IN PRICES ................................................................................................... 33

II.4 LOCATIONAL MARGINAL PRICES: congestion included ...................................................... 37

References ...................................................................................................................................... 42


PART 1 – OPTIMAL DISPATCH I.1 CONSTANT MARGINAL COSTS According to table 1, the generation costs are the followings: Table 1. Generation Costs

Generator 1 6 2 4 7

Incremental Cost (£/MWh) 2 4 6 10 12

Min MW 0 0 0 0 0

Max MW 400 500 500 200 600

They are assumed to be constants. The seven-bus system is shown in the following figure: Figure 1. Configuration of the seven-bus system

For the configuration shown in the figure 1, the output of each of the generators is: Table 2. Output of the generators

Generator 1 6 2 4 7

Output MW 102 200 170 95 888

The total cost per hour for the seven-bus system is 17,092 £/h.

Merit Order – manual action Manually, all the generators but 7 were set to their maximum output, in order to minimize the total cost of generation per hour in the seven-bus system. Having done that, the results are summarized in table 3 and figure 2: Table 3. Outputs of the generators

Generator 1 6 2 4 7

Output MW 400 500 500 200 -51


Figure 2. Seven-bus system. Merit Order – manual action

The total cost per hour for the seven-bus system is 10,645 £/h. Increasing the generation for cheaper generators will decrease the total cost per hour for the whole system. The latter mentioned fact does not assure an optimum solution from the economic point of view, considering the whole seven-bus system. Just the optimum may be found following the equimarginal costs principle. Economic Dispatch with Transmission Losses Neglected – optimisation by PowerWorld Table 4. Economic dispatch with transmission losses neglected

Generator 1 6 2 4 7

Output MW 400 500 500 134 0

Figure 3. Economic dispatch with transmission losses neglected for the seven-bus system


1. Check the economic dispatch calculated by the program and compare it with your manual economic dispatch. Are there any differences? By diminishing the power output of some generators, in this case, the generator 4, that has a marginal cost of 10 £/MWh, the economic dispatch allows to achieve a lower total cost per hour for generation in the whole seven-bus system. In this case, the total cost per hour is 10,597 £/h, for a generation of 1,534 MW, and a demand of 1,406 MW (128 MW of losses). 2. Which generator is at the margin (i.e. is partly loaded and supplies the last MW of demand) and why? What value has the system lambda? Why? The generator 4 is the marginal generator because is partly-loaded (output 134 MW of maximum output of 200 MW), mainly due to its marginal cost which is the highest of the loaded generators. For this case the system lambda corresponds to the marginal cost of the marginal generator, being 10 £/MWh. This system lambda is the same for the whole system because there are no constraints for transmission lines. Economic Dispatch with Transmission Losses Included

Table 5. Economic dispatch with transmission losses included

Generator 1 6 2 4 7

Output MW 400 500 500 148 3

Figure 4. Economic dispatch with transmission losses included for the seven-bus system

3. How have the generation cost and transmission losses changed? Compare with the case when transmission losses were neglected. Considering the net injection at node k, Ik as:

Where Pk and Dk are the generation and demand at the node k, respectively.


Thus, we have the following minimization of the total cost of producing energy in a network of infinite capacity with losses included (Kirschen, 2004):

With Ck = cost of generation at node k (£/h); Ik = net injection of power at node k (MW); L = losses of power in the branches of the network (MW); n = slack bus of the network

Thus, the Lagrangian function of the optimization problem is:

The conditions for optimality are:

Rearranging the first and the second condition, we get:

So, the Lagrange multiplier, represents the marginal cost of marginal benefit of an injection of power

at the slack bus, and the Marginal Cost of Injection of power at bus k is

.

From the economic dispatch calculated in PowerWorld, lambda is 11.86, so the generation costs have

to be considered as nodal prices for each of the nodes.

Additionally, can be said that the penalty factor for bus k, Pfk, is:

So, in table 6 we have the corresponding nodal prices:


Table 6. Nodal prices for each of the buses for the seven-bus system

Bus Marginal Loss Factor for bus

k

Marginal Cost of Injection of Power at

Bus k (£/MWh)

Penalty Factor at Bus k (£/MWh)

1 0.353 7.67 1.546

2 0.281 8.52 1.392

3 0.186 9.65 1.229

4 0.185 9.67 1.226

5 0.044 11.34 1.046

6 0.225 9.19 1.291

7 0.000 11.86 1.000

4. Which generators have changed their outputs and why? Comparing the output in table 4 with the outputs from table 5, the generators that changed their output were number 4 and 7, from 134 to 148 MW to generator 4, and from 0 to 3 MW to generator 7. In this case, the penalty factor for the generator connected to the bus 4 is the smallest penalty factor of the generators (generators 1, 2, 4, 6, and 7), without considering that the generator connected to the bus 7 corresponds to the marginal generator (marginal cost equal to the system lambda). Thus, can be said that the generator 4 has the smallest loss for generating the demanded power, so, it changed its power output. 5. Which generators are at the margin, i.e. operate at equal generalised lambda = (incremental cost) x penalty factor)?

The generator 7, which is connected to the slack bus, works as the marginal generator, being the

nodal price for bus 7, 11.86, the value for the seven-bus system lambda as the incremental cost is

defined by:


Optimal power flow

Figure 5. Optimal Power Flow for the seven-bus system

Table 7. Nodal prices for the optimal power flow and marginal cost of generation, output and power demand for the seven-bus system

Bus Nodal

Price Bus k (£/MWh)

Marginal Cost

Generator (£/MWh)

Output Generator

(MW)

Demand at bus k (MW)

1 2.00 2.00 136 0

2 6.00 6.00 143 74

3 13.86 - 0 204

4 12.66 10.00 200 148

5 14.55 - 0 241

6 8.12 4.00 500 370

7 12.00 12.00 446 370

The difference in nodal prices compared with pure economic dispatch can be seen in the following table:

Table 8. Comparison of nodal prices between pure economic dispatch and optimal power flow

Bus

Economic dispatch - losses included Marginal Cost of

Injection of Power at Bus k (£/MWh)

Optimal Power Flow - Losses included Nodal Price

Bus k (£/MWh)

1 7.67 2.00

2 8.52 6.00

3 9.65 13.86

4 9.67 12.66

5 11.34 14.55

6 9.19 8.12

7 11.86 12.00


I.2 VARIABLE MARGINAL COSTS 6. Note the values of coefficients of cost function. The general form of the cost function is cubic but what is the actual form of the generator input-output costs and incremental costs taking into account the actual values of coefficients? Table 9. Values of coefficients of cost function for the generators

Coefficients of Cost Function

Generator 1

Generator 6

Generator 2

Generator 4

Generator 7

A 761.94 831.92 831.84 530.03 500.08

B 7.62 7.57 7.52 7.84 7.77

C 0.0013 0.00131 0.00136 0.00134 0.00194

D 0 0 0 0 0

Unit Fuel Cost (£/MBtu)

2.04 2.139 2.061 2.093 2.574

In the simulator of the software PowerWorld, the following cubic relationship is used to determine the cost associated with operating a generator:

Ci(Pgi) = ai + ( bi Pgi + ci (Pgi)2 + di (Pgi)

3 ) * unit fuel cost £/Hour

where Pgi is the output of the generator at bus i in MW. The values ai, bi, ci, and di are used to model the generator’s input-output (I/O) curve. This curve specifies the relationship between how much heat must be input to the generator (expressed in MBtu per hour) and its resulting MW output (PowerWorld Corporation, 2009). So, for the generators, the total cost and the marginal cost respectively are:


Figure 6. Total cost curve and marginal cost curve, generator 1

Figure 7.Total cost curve and marginal cost curve, generator 2





Figure 11. Supply curve for the seven-bus system

Economic dispatch without including loss penalty factors into the dispatch

Figure 12. Economic dispatch, transmission losses neglected


7. Make sense out of the results. Check the values of generators incremental costs IC, to see if they indeed satisfy the condition of economic dispatch. Are all the incremental costs the same? The outputs of the generators and the corresponding marginal costs are presented in the following table: Table 10. Power output and marginal cost of the generators

Generator Power output

(MW)

Gen MW Marg.

Cost (£/MWh)

1 276.75 17.01

2 270.07 17.01

4 107.6 17.01

6 146.4 17.01

7 1.89 20.02

As can be seen in table 10 and figure 13, all the generators but the generator 7 that is connected to the slack bus, produce power until they equal the marginal cost, that for this case corresponds to 17.01 £/MWh. This equimarginal condition for the economic dispatch is given by the power outputs that can be seen in the table 10, or alternatively, in figure 13. Thus, in this case, the ED is accomplished minimizing the cost of generation constrained to a certain balance between generation and demand by the following relationship:

I.e. when the incremental costs of all generators are the same and equal to λ.

In this case the total cost of economic dispatch per hour is 16,618 £/h. Figure 13. Marginal costs for the generators


8. Without doing simulations, how can you predict what the total cost of economic dispatch will be if demand increases by 10 MW? Validate your prediction by increasing demand in node 2 by 10 MW and checking the resulting increase in the costs. To the total cost of economic dispatch presented in figure 12, 16,618 £/h, it has to be summed the corresponding increment in demand, considering a marginal cost of 17.01 £/MWh. In that way, we obtain:

Figure 14. Increasing demand in 10 MW in node 2

Can be seen that increasing the demand in node 2 in the software PowerWorld, we obtain a total cost of the economic dispatch of 16,787 £/h.

9. Check the values of penalty factors. Currently the influence of losses is not taken into account but later on, you will include transmission losses in the dispatch. Make a prediction about which generator you expect to increase its output and which decrease when the influence of losses is included.

The values of penalty factors for all the buses (and the generators connected to them) are equal to 1,

as losses are not included.

It is more likely that generator 1 diminishes its output due to its marginal cost function has a bigger

slope, though all the generators are similar in this aspect.

Additionally, it is likely that generator 6 increases its output, mainly due to its closeness to bigger

loads, as generator 1 has to deliver its energy just using the branches 1-3 and 1-2, aspect that will be

reflected in the penalty factors when the losses during transmission are considered.

Additionally, can be mentioned that there is a total generation output of 802.72 MW for a demand of

760.00 MW, and the losses reach a value of 42.72 MW.


Economic dispatch with transmission losses included

Figure 15. Economic Dispatch with transmission losses included

10. Why have results changed? Make sense out of them, i.e. check the values of generation outputs and the overall cost of dispatch. Check the validity of your prediction about generation changes made before. As was mentioned in question 9, the biggest penalty factor is for the generator connected to bus 1 (table 11), because it has to dispatch energy through branches 1-3 and 1-2, because there is no feeder connected to the bus 1, and the impedances in the branches, eventually, are higher. In this way, there are more transmission losses, aspect that is reflected in a bigger incremental loss factor. The generator 6 increased its output from 146.4 MW for the situation with ED without losses to 248.9 MW for the situation of ED including losses to minimize the overall generation cost. As can be seen in table 11, the penalty factor is the second smaller of the whole seven-bus system. Additionally, there is an overall generation of 788.04 MW for a demand of 760 MW, fact that makes to have losses of 28.04 MW, value that is smaller than the value of losses for the ED when transmission losses were not considered.


11. Check how the operating points moved. How and why has the overall generation cost changed? Calculate the savings in both dispatch and losses. Check the values of incremental costs, penalty factors and lambdas for each generator. Check that the equal lambda criterion, modified by loss factors, still holds. Figure 16. Incremental costs for all generators in Economic dispatch, transmission losses included

Table 11. Power output, penalty factor, marginal loss factor, and nodal prices for each bus of the seven-bus system

Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 7

Power output (MW) 140.38 248.59 - 150.19 - 248.88 0

Penalty factor 1.2152 1.1719 1.1372 1.1475 1.0316 1.1256 1

Loss MW Sens 0.1771 0.1467 0.1207 0.1285 0.0306 0.1116 0

Nodal Price (£/MWh) 16.29 16.90 17.41 17.26 19.19 17.59 19.80

The lambda, λ, for the whole system is 19.80 £/MWh and is related to the nodal prices for the following relationship

1:

With Ck = cost of generation at node k (£/h);

Ik = net injection of power at node k (MW); L = losses of power in the branches of the network (MW); n = slack bus of the network The overall generation cost changed from 16,618 £/h for the ED without losses, to 16,447 £/h for the ED considering losses. The overall cost diminished, because now the seven-bus system has different nodal prices which reflect more accurately the real situation, making some generators increase their output and others, decrease their output, according to its location in the network, taking into account the impedances of the transmission lines, physics phenomena that may make or not, more competitive in marginal costs a generator, pursuing a more competitive electric market. The fact of consider the penalty factors for the generators, related to each of the buses, makes that the economic dispatch minimize the overall cost of generation by considering the transmission losses as well, as they were not considered in the ED without considering transmission losses, leading to save losses and maximize the social welfare for the whole electrical market considered (Kirschen, 2004).

1 Check question 3 for more detailed explanation of the relationship


Optimal power flow Figure 17. Optimal Power Flow

12. Compare the results and costs in ED and OPF. Make sense out of them. How was the overload in line 2-5 relieved? Rationalise that in terms of which generator outputs have increased and which decreased and what effect that had on the flow in line 2-5? Table 12. Comparison of the flows in the lines, between Optimum Power Flow, OPF, and Economic Dispatch with transmission losses, ED

Lines Limit MVA

Optimal Power Flow Economic dispatch including

transmission losses

MVA

from

Congestion of

the line (%)

MVA Marginal

Cost

MVA from Congestion of the line

(%)

1-2 150 125.7 83.8% 0 96.1 64.1%

1-3 65 38.0 58.5% 0 46.4 71.4%

2-3 80 23.8 29.8% 0 33.6 42.0%

2-4 100 22.1 22.1% 0 26.9 26.9%

2-5 100 100.0 100.0% 12.00 152.1 152.1%

2-6 200 78.5 39.3% 0 101.9 51.0%

3-4 100 66.3 66.3% 0 39.2 39.2%

4-5 60 53.3 88.8% 0 58.4 97.3%

5-7 200 65.5 32.8% 0 110.6 55.3%

6-7(1)

200 38.8 19.4% 0 74.6 37.3%

6-7(2)

200 38.8 19.4% 0 74.6 37.3%

Table 13. Comparison of the Power Output of the generators, between Optimum Power Flow, OPF, and Economic Dispatch with transmission losses, ED

Generator OPF - Output (MW) ED - Output (MW)

1 160.0 140.4

2 99.6 248.6

4 200.0 150.2

6 200.0 248.9

7 115.5 0.0


Table 14. Comparison of Nodal Prices at the Buses, between Optimum Power Flow, OPF, and Economic Dispatch with transmission losses, ED

Bus OPF - Nodal Price (£/MWh) ED - Nodal Price (£/MWh)

1 16.23 16.29

2 15.78 16.90

3 17.02 17.41

4 17.42 17.26

5 22.30 19.19

6 17.33 17.59

7 20.60 19.80

The line 2-5 has a rating of 100 MVA, so for the ED the generator that is connected to bus 2 was generating 248.6 MW, but considering the optimum power flow (OPF) where the ratings of the branches are considered, the generator 2 is set to produce 99.6 MW that is feasible to be dispatched through the line 2-5. In terms of apparent power, the line 2-5 is congested, meaning that there is no more transmission capacity through this line. The overall cost of generation for the OPF is 16,703 £/h, value that is bigger than the value obtained for ED (16,447 £/h). The overall cost of generation is bigger because the network has a finite capacity, making it more expensive. In table 14, the nodal prices for OPF are presented compared to nodal prices for ED. For the OPF, the total generation is 775.1 MW, a power demand of 760 MW and losses of 15.1 MW for transmission. 13. Click OPF Case Info/OPF Lines and Transformers. Column MVA Marg cost gives the values of Lagrange multipliers connected with a constraint on a maximum value of given line flow. Why are all the values equal to zero apart from one? As can be seen in table 12, in column MVA Marginal cost for Optimal Power Flow, just the line 2-5 has the value different from zero, being 12.1. This value represents the Lagrange multiplier for the line 2-5 that is fully loaded (Lagrange multiplier is nonzero only if the flow on the branch between nodes i and j is equal to its limit), that is the marginal cost of this constraint in the network. Then, for the line 2-5 the Lagrange multiplier λ25, is 12.1 £/MWh, and represents the saving that would accrue each hour if the flow in branch 2-5 could be increased by 1 MW (Kirschen, 2004). The Lagrange multiplier can be expressed mathematically as (Kirschen, 2004):

Where C

min = the value of the cost at the optimum

= the rating of a line ij

= the Lagrange multiplier for the constrained line ij


14. Power system operation must be secure not only for a normal state of operation but also following a contingency, i.e. when any one system element trips to e.g. an unexpected fault. This is referred to as (N-1) security criterion. The transmission line 2-5 was tripped. Figure 18. The seven-bus system when the transmission line 2-5 is tripped

As can be seen in figure above, once the line 2-5 is tripped, the line 4-5 will be overloaded by 157.3% of its thermal capacity. Table 15. Comparison of power flows through transmission lines with the system in normal state and under a contingency (line 2-5 tripped)

Lines Limit MVA

Optimal Power Flow System with line 2-5 tripped

MVA from

Congestion of the line (%)

MVA from


1-2 150 125.7 83.8% 118.2 78.8%

1-3 65 38.0 58.5% 46.8 72.0%

2-3 80 23.8 29.8% 29.4 36.8%

2-4 100 22.1 22.1% 22.5 22.5%

2-5 100 100.0 100.0% - -

2-6 200 78.5 39.3% 142.4 71.2%

3-4 100 66.3 66.3% 44.0 44.0%

4-5 60 53.3 88.8% 94.4 157.3%

5-7 200 65.5 32.8% 86.2 43.1%

6-7(1) 200 38.8 19.4% 70.1 35.1%

6-7(2) 200 38.8 19.4% 70.1 35.1%

The spare capacity of transmission is used, as the rest of the transmission lines were not constrained until the fault. But, since the line 4-5 is overloaded, the generation output should be changed, this meaning, a new OPF should be carried out to restore the normal state of the seven-bus system.

The fundamental principle of security in a power system considers that a power system should always be operated in such a way that no credible contingency could trigger cascading outages or another form of instability. Because all the power systems are routinely affected by unpredictable faults and failures, they have to be operated with a sufficient security margin. There must be enough reserve generation capacity to make up for the loss of a generating unit and enough transmission capacity to


handle the power flows displaced by the outage of a line. Securing the system only calls against all credible contingencies as it is impossible secure it against all possible contingencies. What credible contingencies are assumed to be it is that the probability of two or more independent faults or failures taking place simultaneously is too low to be considered credible. So the N-1 accounts for the feasibility to lose any one of the power system’s N components and continue operating (Kirschen, 2003).

15. Investigate the resulting power flows when you trip a number of different lines (initially one at a time and then some combinations of trips). Try to make sense out of the resulting power flows by thinking how a power that was flowing through a tripped line has to be shared by the remaining lines. Figure 19. The seven-bus system when the transmission lines 2-5 and 4-5 are tripped

Figure 20. The seven-bus system when the transmission lines 2-5, 4-5, and 4-3 are tripped


Table 16. The seven-bus system with different contingencies. Transmission lines tripped.

Lines Limit

MVA

System with line 2-5 tripped

System with lines 2-5 and 4-5 tripped

System with lines 2-5, 4-5, and 4-3 tripped

MVA from


MVA from


MVA from


1-2 150 118.2 78.8% 142.5 95.0% 106.5 71.0%

1-3 65 46.8 72.0% 30.0 46.2% 66.5 102.3%

2-3 80 29.4 36.8% 29.6 37.0% 58.6 73.3%

2-4 100 22.5 22.5% 39.2 39.2% 132.5 132.5%

2-5 100 - - - - - -

2-6 200 142.4 71.2% 244.3 122.2% 238.4 119.2%

3-4 100 44.0 44.0% 93.6 93.6% - -

4-5 60 94.4 157.3% - - - -

5-7 200 86.2 43.1% 141.8 70.9% 141.8 70.9%

6-7(1)

200 70.1 35.1% 118.3 59.2% 116.8 58.4%

6-7(2)

200 70.1 35.1% 118.3 59.2% 116.8 58.4%

In table 16 several contingencies in the power system are presented. First, the line 2-5 is tripped, then, to line 2-5, the line 4-5 is under fault, therefore, tripped, and after, to those two transmission lines tripped, line 4-3 is tripped. These states can be seen in figures 18, 19, and 20, respectively. Table 17. Generation under the different contingencies considered.

Generator

System with line 2-5 tripped

System with lines 2-5 and 4-5 tripped

System with lines 2-5, 4-5, and 4-3 tripped

Power Output (MW) Power Output (MW) Power Output (MW)

1 162.0 166.6 168.9

2 102.0 106.2 108.5

4 200.0 200.0 200.0

6 202.0 206.6 208.9

7 118.0 122.6 125.1

Total Generation 784.0 802.0 811.4

As can be seen, when a transmission line is tripped, the power flow, at first moment, will take the branches in the nearby to make the dispatch to the loads. The power flows don’t take into account by themselves thermal ratings, so the transmission lines capacity, in many cases, will be exceeded. Eventually, this fact may lead to more faults in the system, so more transmission lines may be tripped, as the relays for the different zones respond to the contingency to protect the power system.

Additionally, can be seen from table 17 that to supply the demand, as the branches of the network are decreasing (thus, there will be more losses in the lines), the total generation rises, fact that eventually, may lead to more stability problems during the operation of the power system. The overall cost of generation in the system, like the losses, rises.


PART II: LOCATIONAL MARGINAL PRICING (LMP) II.1 BASE CASE – UNIFORM PRICING (TRANSMISSION CONGESTION NEGLECTED) The generators connected to the respective busbars are set as appear in table 18. Table 18. Generation Costs

Generator 1 6 2 4 7


Min MW 0 0 0 0 0

Max MW 400 500 500 200 600

Figure 21. Supply characteristics (with load multiplied by 1.5 times)

Figure 22. Economic Dispatch with transmission losses neglected and loads increased 1.5 times


16. Why are all LMPs the same? Which generator is at the margin (i.e. is partly loaded and supplies the last MW of demand) and why? By multiplying the load by 1.5, the demand is now of 1,140 MW. All the LMPs are the same because transmission losses are not included in the ED, and therefore, the lambda of the system, λ, is equal to the marginal cost of the generator that supplies the last MW of demand, being for this case, the generator 2. Table 19. Power output of the generators with ED with transmission losses neglected and loads increased 1.5 times

Generator Power Output (MW)

1 400.0

2 335.5

4 0.0

6 500.0

7 0.0

Total Generation 1,235.5

As can be seen in table 18, generator 2 has a marginal cost of 6.00 £/MWh and a maximum capacity of generation of 500 MW, so, it is partly loaded. In general, the following relationship is accomplished; making the nodal price is the same for all the buses in the seven-bus system:

17. Increase the Load Multiplier to 1.8. Answer again the question made in number 16. Figure 23. Economic Dispatch with transmission losses neglected and loads increased 1.8 times


Table 20. Power output of the generators with ED with transmission losses neglected and loads increased 1.8 times


1 400.0

2 500.0

4 93.3

6 500.0

7 0.0


By multiplying the load by 1.8, the demand is now of 1,368 MW. All the LMPs are the same because transmission losses are not included in the ED, and therefore, the lambda of the system, λ, is equal to the marginal cost of the generator that supplies the last MW of demand, being for this case, the generator 4, which has a marginal cost of 10 £/MWh. So, the marginal generator is the one connected to bus 4. Additionally, the maximum capacity of generation of generator connected to bus 4 is 200 MW, so it is partly loaded. 18. Assume that we have a fully competitive market so that the prices the generators bid to the Pool are equal to their marginal costs. Calculate the profit per MW made by each generator and the congestion surplus. For the condition of the seven-bus system with the load multiplied by 1.5, the system marginal price, SMP is 6.00 £/MWh, so producers are paid this price for the output they sell, and consumers pay the same price for the energy they buy. The profits per MW, the producers’ revenues and total consumer payments for the whole system are presented in table 21. Table 21. Producer Profits per MW generated, producer revenues, and consumer payments in the seven-bus system with ED with transmission losses neglected and loads increased 1.5 times


Marginal Cost

(£/MWh)

Profits per MW

(£/MWh)

Producers Revenues

(£/h)

Consumers payments

(£/h)

1 400.0 2.00 4.00 2,400.00 2,400.00

2 335.5 6.00 0.00 2,013.24 2,013.24

4 0.0 10.00 -4.00 0.00 0.00

6 500.0 4.00 2.00 3,000.00 3,000.00

7 0.0 20.00 -14.00 0.00 0.00


7,413.24 7,413.24

As there is only one lambda for the whole system (SMP), due transmission losses are not considered

in the economic dispatch, the Congestion Surplus will be:


For the condition of the seven-bus system with the load multiplied by 1.8, the system marginal price, SMP is 10.00 £/MWh, so producers are paid this price for the output they sell, and consumers pay the same price for the energy they buy. The profits per MW, the producers’ revenues and total consumer payments for the whole system are presented in table 22. Table 22. Producer Profits per MW generated, producer revenues, and consumer payments in the seven-bus system with ED with transmission losses neglected and loads increased 1.8 times

Generator

Power

Output (MW)

Marginal

Cost (£/MWh)

Profits per

MW (£/MWh)

Producer

Revenues (£/h)

Consumer

payments (£/h)

1 400.0 2 8.00 4,000.00 4,000.00

2 500.0 6 4.00 5,000.00 5,000.00

4 93.3 10 0.00 933.30 933.30

6 500.0 4 6.00 5,000.00 5,000.00

7 0.0 20 -10.00 0.00 0.00


14,933.30 14,933.30

As there is only one lambda for the whole system (SMP), due transmission losses are not considered

in the economic dispatch, the Congestion Surplus will be:


II.2 COMPETITION IN QUANTITIES

19. Assume that you are trader working for inframarginal generator 2. Assume that your maximum capacity is as declared so far (500 MW). Try to increase your net profit by reducing your declared capacity to 400 MW, 300 MW, 200 MW or 100 MW. Plot the resulting profits as the reduction of capacity. What are general conclusions about optimal bidding strategy for an inframarginal generator? According to table 23, the generation costs are the followings: Table 23. Generation Costs

Generator 1 6 2 4 7


Min MW 0 0 0 0 0

Max MW 400 500 500 200 600

The load was kept multiplied by 1.5, so the total demand is 1,140 MW.

Figure 24. Generator 2 – 500 MW max output







The system clearing prices (SMP) for each of the situations where the generator 2 withdraws capacity, its power output, and the profits that the producer, owner of the generator 2, would perceive, are presented in table 24.


Table 24. SMPs, power outputs, revenues, costs and profits for each of the situations where generator 2 withdraws an additional unit generation (100 MW)

Withdrawn Capacity

(MW)

Power Output

(MW)

System Marginal

Price (£/MWh)

Revenues (£/h) Costs

(£/h) Profits (£/h)

0 500 10.00 5,000.00 3,000.00 2,000.00

100 400 10.00 4,000.00 2,400.00 1,600.00

200 300 12.00 3,600.00 1,800.00 1,800.00

300 200 12.00 2,400.00 1,200.00 1,200.00

400 100 12.00 1,200.00 600.00 600.00

Figure 29. Graph of the profits obtained by generator 2 by withdrawing capacity in the spot market

As can be seen in the graph of figure 29, the highest profits are obtained when the generator is working at its maximum capacity, without withdrawing any unity capacity. As the generator 2 has a marginal cost of 6.00 £/MWh, when it declares that its maximum capacity for that hour is 500 MW (its maximum power output possible) and, therefore, an inframarginal generator that is under all the situations of withdrawn capacity, fully loaded. It is always in merit as the market clearing price is between 10 and 12 £/MWh, which is determined by the marginal generator, In general, the optimal bidding strategy for an inframarginal generator can be presented as follows: It is considered that all the generators in the pool bid the true marginal costs of running their plants and the system clearing price at demand D is p. The benefit and the cost when one generator withdraws a unity of capacity Δq will be pΔq, the cost of gaming, and qΔp, the increased profit for the remaining in-merit generating capacity. Additionally, the effect of withdrawing a unity of capacity Δq is the increasing of the market clearing price Δp, due to the shifting to the left of the part of supply curve which stars from the gaming generator (Bialek, 2002). The avoidable cost of a generator is given by (Bialek, 2002):

Where a, b, and c are constants, the marginal cost is:

0

500

1,000

1,500

2,000

0100

200300

400

2,000

1,6001,800

1,200

600

£/h

Withdrawn Generation Capacity (MW)

Profits (£/h) as function of withdrawn generation capacity


Then the benefit B is: Δ Δ Δ Δ Δ Δ

with Δc = the saving of withdrawing a unity of generation So, the benefit-cost ratio, G, is:

Δ

Δ

Where

= the market share of the remaining in-merit plants of the generator

= the price sensitivity of supply, i.e., relative increase in the total generation

as a response to the relative increase in the system clearing price. The benefit-cost ratio of the gaming allows quantifying the gain that can be earned. It confirms that for

the gaming to be profitable, a portfolio generator should have a large market share s(D-) and/or the

supply curve should be steep, as then eg(D) is small (Bialek, 2002). In general, the plant to be withdrawn should be the most expensive in-merit plant in the generator’s portfolio as then c’ q is the highest (Bialek, 2002). If we consider a base-load power plant with small marginal cost, the ratio c’ q p can be neglected and the gain is (Bialek, 2002)

So, the gaming is profitable when Gain > 1, i.e., when (Bialek, 2002)

For this sort of power plants, gaming can be profitable if the gaming generator (the owner of the generation portfolio) has large enough market share (Bialek, 2002). 20. Repeat the investigation assuming that you were working now for the marginal generator 4. Draw general conclusions The load was kept multiplied by 1.5, so the total demand is 1,140 MW. Figure 30. Generator 4 – 200 MW max output



The system clearing prices (SMP) for each of the situations where the generator 4 withdraws capacity, its power output, and the profits that the producer, owner of the generator 4, would perceive, are presented in table 25. Table 25. SMPs, power outputs, revenues, costs and profits for each of the situations where generator 4 withdraws an additional unit generation (100 MW)

Withdrawn

Capacity (MW)

Power

Output (MW)

System Marginal

Price (£/MWh)

Revenues (£/h) Costs

(£/h) Profits (£/h)

0 93 10.00 930.00 930.00 0.00

100 93 10.00 930.00 930.00 0.00

As can be seen on table 25, there are no profits, generation for the generator 4 as it withdraws generating capacity. The maximum real capacity for the generator 4 is 200 MW, so as we are considering unities of generation of 100 MW, there is not much space for gaming, and, additionally, its marginal cost is 10.00 £/MWh, fact that makes the generator 4 become marginal generator (figures 30 and 31), as the market clearing price is 10.00 £/MWh. In general, the gaming for a marginal generator that for this case was displaced in the merit order by another plant belonging to another owner, so the marginal gain can be presented as (Bialek, 2002):

assuming that the system clearing price p was set by the marginal cost of the price-setting generator, c’(q+Δq). The latter equation shows that in this case the gain is always greater or equal than 1

providing that the slope of the supply curve is positive, meaning that at the margin, the price obtained is equal to the cost so there can be no profit lost due to withdrawing marginal capacity (Bialek, 2002). Additionally, the demand for energy is supplied by the rest of generators for the seven-bus power system but the last MWh is supplied by generator 4 as it is the marginal generator that sets the System Marginal Price, SMP. In other words, and as will be seen in question 22, the generator 4 has very few market share, so it is not able to be successful when gaming as its auction (its high marginal cost of production and its reduced declared generation capacity) just sets the price for all the power system.


21. Now assume that you are trading for a portfolio generator, i.e. a company owning both generator 2 and generator 4. Try to increase your net profit by reducing the declared capacity of either, or both, of the generators. Generally, what is the best strategy for a portfolio generator? What are the risks? In general, in a Cournot model it is considered that each firm decides the quantity, Pf, that it wants to produce (Kirschen, 2004):

The price, π, is then determined by the inverse market demand function, which expresses the market price as a function of the total amount of energy traded

If firm f assumes that its competitors will not adjust the amount of energy they produce, its revenue is given by

Its marginal revenue is therefore

The Cournot model suggests that firms should be able to sustain prices that are higher the marginal cost of production, with the difference determined by the price elasticity of the demand. For a commodity like electrical energy that has a very low elasticity, the equilibrium price calculated tends to be higher than the prices observed in the actual market (Kirschen, 2004). In the case considering two producers, each of them trying to maximize their profits, they manage to maintain a price that is much higher than the marginal cost of production. In the pool, their interactions will settle at the Nash Equilibrium where neither firm can increase its profit through its own actions (Kirschen, 2004). This equilibrium can be expressed considering the profits earned by each firm as follows

Where π(D) represents the inverse demand curve. Each firm tries to maximize its profit. As both firms share the same market, the supply must be equal to the demand. Therefore we must also have (Kirschen, 2004)

So, the conditions of optimality are (Kirschen, 2004):

Specifically, for the generator 2 and generator 4, the different combinations of withdrawn capacity were found, in order to maximize the profit in the pool for the seven-bus system. The table 26 has the following organization for each cell:

Demands for generator 2 and 4 summed Profit Generator 2

Profit generator 4 SMP (£/MWh)


Table 26. Illustration of Cournot competition between the generators 2 and 4, owned by the same company, in the seven-bus system

Maximum Declared

Capacity (MW)

Generator 2

400 300 200 100

Gen

era

tor

4 200

583 1,600 500 1,800 400 1,200 300 600

366 10 400 12 400 12 400 12

Total profit

(£/h) 1,966

Total profit

(£/h) 2,200

Total profit

(£/h) 1,600

Total profit

(£/h) 1,000

100

500 2,400 400 1,800 300 1,200 200 600

200 12 200 12 200 12 200 12

Total profit (£/h) 2,600



Total profit (£/h) 800

0

400 2,400 300 1,800 200 1,200 100 600

0 12 0 12 0 12 0 12




Total profit (£/h) 600

As can be seen in table 26, the combination that maximizes the profit for the company is generator 2 with a maximum output declared of 400 MW (100 MW withdrawn) and the generator 4 with a maximum output declared of 100 MW (100 MW withdrawn). After of executing the OPF in the software PowerWorld, the output for the generators 2 and 4 are 400 MW and 100 MW, respectively. For this situation, both generators work at its maximum declared capacity, constraining the total capacity of generation for the whole pool, making the SMP be 12.00 £/MWh, becoming both into inframarginal generators (in-merit generators) as the marginal generator is the one connected to the slack bus 7. This generator 7 sets the market clearing price at 12.00 £/MWh. The main risk for the power company that owns both generators is that its most expensive generator turns out to be out of merit as its marginal cost of production may be higher than the market clearing price. As a general rule, can be said that the generator to be withdrawn should be the most expensive in-merit plant in the generator’s portfolio as then the marginal cost of production is the highest (Bialek, 2002). 22. Generally, is it more profitable to reduce capacity of a base load generator or a generator near the top of the merit order? In general, it is more profitable to reduce capacity of a generator near the top of the merit order, though this will depend of the market share of this generator, so, even using base-load plants can be successful. Figure 32. Shares and gain: (a) required market share for withdrawing zero-marginal cost plant, (b) marginal gain due to withdrawing zero-marginal cost plant, (c) marginal gain due to withdrawing marginal plant when demand is inelastic, and (d) marginal gain due to withdrawing marginal plant when the price sensitivity of supply is -0.05

Source: (Bialek, 2002)


In figure 32 can be seen that gaming can be successful even using base-load plants, providing that the gaming generator has large enough market share (curve a). As can be seen, for the demand of 40-75% of the system capacity, the required market share is about 20-40%. In the region of 75-85% of the system capacity, the supply curve is quite flat, making not profitable for a base-load generator at all to withdraw capacity as the required market share would have to be greater than 1, fact that is impossible. As can be seen, as the marginal cost of production of a base-load plant is very low (it can be considered as 0), the only moment when it is worth to withdraw capacity is when the supply curve is very steep, being in the case shown in the curve b of the figure 32, when demand is greater than 90% of the maximum load (Bialek, 2002). For a generator near the top of the merit order, is definitively more profitable to withdraw capacity, compared with a base-load generator. As can be seen for the curve c, the marginal gain is between 1 and 2 until the system load reaches 90%. Then the supply curve becomes very steep and the gain increases reaching the peak value of 7 (Bialek, 2002). II.3 COMPETITION IN PRICES 23. Assume that you are a trader working for inframarginal generator 2. Try to increase your profit by changing your bid price so that it is different than your marginal cost given in table 1. Plot the results. Can you increase your profitability by doing that? What are the conclusions? The results are presented in table 27.

Table 27. SMPs, power outputs, revenues, costs and profits for each of the situations where generator 2 raises its bid price

Price bidden

(£/MWh)

Generation

(MW)

SMP

(£/MWh)

Revenues

(£/h)

Real Marginal

Cost (£/MWh)

Expenses

(£/h)

Profits

(£/h)

6 500 10 5,000 6 3,000 2,000

7 500 10 5,000 6 3,000 2,000

8 500 10 5,000 6 3,000 2,000

10 500 10 5,000 6 3,000 2,000

12 382 12 4,584 6 2,292 2,292

13 0 12 0 6 0 0

14 0 12 0 6 0 0

16 0 12 0 6 0 0

18 0 12 0 6 0 0

20 0 12 0 6 0 0

Figure 33. Graph of the profits obtained by generator 2 by raising the bid price in the spot market

Assuming that the generator 2 bids its true marginal cost (6 £/MWh), the profit that can get is 2,000 £/h. By raising the bid price to 12 £/MWh, the profit turns out to be higher, reaching 2,292 £/h, as can be seen in table 27 and figure 33. This inframarginal generator, by raising the bid price to 12 £/MWh,

0

500

1,000

1,500

2,000

2,500

6 7 8 10 12 13 14 16 18 20

2,000 2,000 2,000 2,000

2,292

00 0

0 0

Pro

fit

(£/h

)

Bid Price (£/MWh)

Profits (£/h) as function of raising of the bid price


becomes into the marginal generator, setting the market clearing price, and getting partly loaded as its power output is 382 MW. For the inframarginal generator 2, there is no effect on the profits meanwhile the bidding price does not reach the value of the market clearing price. 24. Assume that you are a trader working for the marginal generator 4. Try to increase your profit by changing your bid price so that it is different than your marginal cost given in table 1. Plot the results. Can you increase your profit by doing that? To what extent? What are the conclusions? What is the danger in that strategy?

Table 28. SMPs, power outputs, revenues, costs and profits for each of the situations where generator 4 raises its bid price

Price

bidden (£/MWh)

Generation

(MW)

SMP

(£/MWh)

Revenues

(£/h)

Real Marginal

Cost (£/MWh)

Expenses

(£/h)

Profits

(£/h)

10 93 10 930 10 930 0

11 93 11 1,023 10 930 93

12 93 12 1,116 10 930 186

13 0 12 0 10 0 0

14 0 12 0 10 0 0

15 0 12 0 10 0 0

16 0 12 0 10 0 0

Figure 34. Graph of the profits obtained by generator 4 by raising the bid price in the spot market

As can be seen in table 28, the marginal generator 4 increases its profit from 0 £/h to 186 £/h by increasing the bidding price from the real marginal cost until 12 £/MWh, that corresponds to the value of the marginal cost for the generator 7. By bidding prices further this value, the generator 4 just becomes out-of-merit, and therefore, there will not be any power output neither profit. In general for a marginal generator, the outcome of raising the bid price by Δp will depend of the size

of Δp. If it is small then the marginal plant will not be replaced in the merit-order by another plant.

The expected value of the gain for the marginal generator is (Bialek, 2002)

Where P is the probability of the marginal plant is not replaced in the merit-order by another plant, and if Δp is large, then the gaming plant will be replaced in the merit-order by the next cheapest plant

0

50

100

150

200

1011

1213

1415

16

0

93

186

00

00

Pro

fit

(£/h

)

Bid Price (£/MWh)

Profits (£/h) as function of raising of the bid price


which has its bid price p1 lower than the inflated bid price of the gaming plant, i.e., p1 = (p+Δp1) < (p+Δp). The probability of such event is (1-P).

The expected gain can be expressed as well as (Bialek, 2002)

The latter equation shows that the marginal gain is always greater than 1 providing that the slope of the supply curve is positive. P can be considered as the probability of predicting accurately the bids of other generators. Thus accurate prediction of the competitors’ bids has the effect of increasing the gain, as it allows choosing such a value of Δp that the gaming generator is not pushed out of the merit-order (Bialek, 2002). As the outcome of raising the bid price by Δp depends of the size of Δp , the danger in this strategy is

to turn the generator out of merit, as there is always uncertainty in prediction of the competitors’ bids that may lead to the undesirable outcome of having an out-of-merit generator, and therefore, no profits at all. Additionally, the probability of the marginal generator is not replaced in the merit-order by another generator may increase as there is repetitiveness of the auction coupled with the increasingly limited number of competitors as the load increases, especially at the peak load (Bialek, 2002). This fact helps to diminish the uncertainty for gaming with bid prices and to improve the accurate of prediction of the competitors’ bids. Moreover, one would not expect aggressive bidding from the competitors which would have the effect of lowering the probability of keeping the generator in-merit (P). As inflating the market clearing price benefits all the generators, not only the gaming one, it is not in their interest to outbid each other, making tacit collusion, and thus high P, very likely (Bialek, 2002).

25. Now assume that you are trading for a company owning both generator 2 and generator 4. Try to increase your profit by manipulating bid prices. What is the best strategy? Table 29. Illustration of Bertrand competition between the generators 2 and 4, owned by the same company, in the seven-bus system

Price Bidden (£/MWh) Generation (MW) Real Marginal Cost

(£/MWh) SMP (£/MWh)

Revenues (£/h)

Expenses (£/h)

Profits (£/h) Generator

2 Generator

4 Generator

2 Generator

4 Generator

2 Generator

4

6 10 500 93 6 10 10 5,930 3,930 2,000

7 11 500 93 6 10 11 6,523 3,930 2,593

8 12 500 93 6 10 12 7,116 3,930 3,186

10 13 500 0 6 10 12 6,000 3,000 3,000

12 14 500 0 6 10 12 6,000 3,000 3,000

14 15 0 0 6 10 12 0 0 0

As can be seen in table 29, the bidding prices for generators 2 and 4 that have the highest profit , are 8 £/MWh and 12 £/MWh, respectively. The base-load generator 2 is fully loaded and the generator 4 is the marginal generator, setting the market clearing price. For Bertrand competition the optimal bidding strategy, according to (Hao, 2000) can be expressed in a simplified way as follows: For a bidder who participates in a clearing price auction, three outcomes exist: winning below and on the margin, or losing. Corresponding to these outcomes, the bid price is either the same as the winning price, or below the winning price, or above the winning price. The bidding strategy is described by B(ci) for a Bidder i. The probability that a bidder who wins the auction, but not on the margin is R(B-1(bi)) and H(B-1(bi)) the probability that Bidder i wins the auction on the margin.


The common objective for bidders is to maximize their expected payoffs. Assuming the winning price is w, the payoff π for bidder i is (w-ci) if bi wins the auction but not on the margin, and is (bi-ci) if bid bi is on the margin. Thus the expected payoff function π bi) is the sum of above two terms weighted by their probabilities of occurrence:

Increasing bi will increase the bidder’s payoff if Bidder i continues to set the marginal clearing price. However increasing bi will also decrease the probability of winning. The objective for a bidder is to select bi to balance the gain from winning against the increased probability of losing so that the expected payoff is maximized. So, the optimal solution for the bid bi is:

That can be arranged as:

And the latter equation simplified:

And rearranging and integrating by part formula and boundary condition, the general bidding strategy can be expressed as follows:

So, the latter equation describes the general bidding strategy given an estimate of the expected winning price. This result shows that a bidder´s optimal bid is determined by three components: its cost of production, the probability of winning below or on the margin and the difference between its cost and the expected winning price (Hao, 2000).


II.4 LOCATIONAL MARGINAL PRICES: congestion included The network would be according to the next figure: Figure 35. Optimal Power Flow including congestion of transmission

26. Compare the bus marginal costs (i.e. LMPs) with generator prices. Try to make sense of LMPs. Why are they equal to marginal cost for some local generators but not for all of the? What is the regularity? As can be seen in figure 35, the marginal costs of the buses are: Table 30. Marginal costs and load on the buses

Bus Marginal Cost (£/MWh) Load (MW)

1 2.00 0

2 2.68 60

3 9.25 165

4 10.00 120

5 25.77 195

6 8.46 300

7 20.00 300

And the marginal costs of the generators are: Table 31. Marginal costs of the generators

Generator Marginal Cost (£/MWh)

1 2.00

2 6.00

4 10.00

6 4.00

7 20.00

Table 32. Maximum and actual power output of the generators

Generator 1 6 2 4 7

Maximum Output (MW) 400 500 500 200 600

Actual Output (MW) 205.6 500.0 0.0 178.1 255.9


The generators 1, 4, and 7 are the marginal generators as the bus marginal costs are the same that the marginal costs of generation for each of them (table 30 and 31). Additionally, these generators are not generating at their maximum capacity, as can be seen in table 32. They are called part-loaded generators, and each of them set the marginal price at the bus where they are connected. In general, if there are m transmission constraints in the system, there will be m + 1 marginal generators (Kirschen, 2004). Table 33. Transmissions lines capacities, actual power flow, and congestion

Lines Limit (MVA) Flow (MW) Congestion of the line (%)

1-2 150 140.6 93.7%

1-3 65 65.0 100.0%

2-3 80 39.8 49.8%

2-4 100 29.8 29.8%

2-5 100 100.0 100.0%

2-6 200 89.0 44.5%

3-4 100 60.2 60.2%

4-5 60 27.7 46.1%

5-7 200 67.0 33.5%

6-7(1) 200 55.5 27.8%

6-7(2) 200 55.5 27.8%

So, as can be seen in table 33, there are two transmission constraints (100% of congestion of the lines), and that fact does make sense with the regularity of m + 1 marginal generators as generators 1, 4, and 7 are. 27. For those nodal LMPs which are not equal to the marginal costs of local generators, verify experimentally that LMP is equal to the minimum cost of supplying extra MW of demand without violating security constraints. Do it for node 5 as its LMP is higher than any generation cost. To do that increase the local demand by 1 MW, see where the power to supply that extra demand comes from and calculate the nodal LMP as the weighted average of costs of generators supplying this extra 1 MW. Figure 36. Optimal Power Flow including congestion of transmission with load in bus 5 increased in 1 MW

As can be seen in figure 36, the total cost of the total generation per hour, with the load in bus 5 incremented in 5 MW (load of 196 MW) is 12,793.00 £/h, value that is £ 25.77 bigger than the


situation presented in figure 35, where the total cost per hour of the generation is 12,767.00 £/h for a load of 195 MW in the feeder connected to the bus 5. Considering that the nodal price is defined as the cost of supplying an additional MW of demand at that node by the cheapest means, without violating the security-constrained dispatch, therefore, can be said that the definition is accomplished as the total price is incremented in £ 25.77 as the demand at node 5 is 1 MW bigger (table 34) than the situation presented for question 26 (figure 35).

Table 34. Marginal costs and load on the buses

Bus Locational Marginal Price (£/MWh) Load (MW)

1 2.00 0

2 2.68 60

3 9.25 165

4 10.00 120

5 25.77 196

6 8.46 300

7 20.00 300

Table 35. Power output of the generators

Generator 1 6 2 4 7

Actual Output (MW) 205.5 500.0 0.0 177.8 257.4

Table 36. Transmissions lines capacities, actual power flow, and congestion

Lines Limit (MVA) Flow (MW) Congestion of the line (%)

1-2 150 140.4 93.6%

1-3 65 65.0 100.0%

2-3 80 39.9 49.9%

2-4 100 29.9 29.9%

2-5 100 100.0 100.0%

2-6 200 89.4 44.7%

3-4 100 60.1 60.1%

4-5 60 27.6 46.0%

5-7 200 68.1 34.0%

6-7(1) 200 55.3 27.7%

6-7(2) 200 55.3 27.7%

As can be seen in tables 35 and 36, the extra power comes from the generator 7, mainly, and flows through transmission line 5-7.

28. Shadow costs of the lines (i.e. values of Lagrange multipliers associated with a constraint). Why are there only two lines with their shadow costs different than zero? What is the general rule? Table 37. Transmissions lines records

From Bus Number

To Bus Number

Max MVA

% of MVA Limit (Max)

Lim MVA

MVA Marg. Cost (λ), £/MWh

Constraint Status

1 2 140.6 93.7 150

1 3 65 100 65 10 Binding

2 3 39.8 49.8 80

2 4 29.8 29.8 100

2 5 100 100 100 42.5 Binding

2 6 89 44.5 200

3 4 60.2 60.2 100

4 5 27.7 46.1 60

7 5 67 33.5 200

6 7 55.5 27.7 200

6 7 55.5 27.7 200


As can be seen in table 37, just two lines are occupied until their complete capacity (100%), as they are line 1-3 and 2-5. In this case, the Lagrange multiplier, λ, for all the transmission lines but 1-3 and 2-5, are equal to zero as they still have capacity for transmission, so the constraint is not active. For the line 1-3 the Lagrange multiplier is 10 £/MWh, that means that the saving that would accrue each hour if the flow in branch 1-3 could be increased by 1 MW. Equivalently, for the line 2-5, the Lagrange multiplier is 42.5 £/MWh, amount that could be saved each hour by increasing the flow in 1 MW. 29. What is the physical meaning of a Lagrangian multiplier related to a power flow constraint? Figure 37. Optimal Power Flow with the rating of branch 2-5 incremented in 1 MW (from 100 to 101 MVA)

As can be seen in figure 37, the total cost of generation in the system is now 12,725 £/h, value that is smaller in 42.5 £/h than the total cost considered with a rating of 100 MW for the branch 2-5. In this case, and as can be seen in table 37, the Lagrangian multiplier for this branch is 42.5 £/MWh. So, the total cost per hour was reduced in the value of the Lagrangian multiplier for the mentioned branch.


30. Calculate and compare the congestion surplus in each of three ways:

a. The difference between the payments by the loads and the generators revenues Table 38. Economic operation of the seven-bus system: congestion surplus

Bus 1 Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 7

Total System

Consumption (MW) 0.0 60.0 165.0 120.0 195.0 300.0 300.0 1,140

Production (MW) 205.6 0.0 0.0 178.1 0.0 500.0 255.9 1,140

Nodal Marginal Price (£/MWh)

2.00 2.68 9.25 10.00 25.77 8.46 20.00 78.16

Consumer payments (£/h)

0.00 160.8

0 1,526.2

5 1,200.0

0 5,025.1

5 2,538.0

0 6,000.0

0 16,450.20

Producers revenues (£/h)

411.18 0.00 0.00 1,780.9

8 0.00

4,230.00

5,118.74

11,540.90

Merchandising surplus (£/h) 4,909.30

b. The difference between in LMPs between terminal loads in each line Table 39. Contribution of each branch to the merchandising surplus of the seven-bus system

Lines Flow (MW) From price (£/MWh) To price (£/MWh) Surplus (£/h)

1-2 140.6 2 2.7 95.60

1-3 65.0 2 9.3 471.25

2-3 39.8 2.68 9.3 261.49

2-4 29.8 2.68 10.0 217.92

2-5 100.0 2.68 25.8 2,309.00

2-6 89.0 8.46 2.7 -514.30

3-4 60.2 10 9.3 -45.15

4-5 27.7 10 25.8 436.36

5-7 67.0 20 25.8 386.30

6-7(1) 55.5 8.46 20.0 640.59

6-7(2) 55.5 8.46 20.0 640.59

Total 4,899.63

c. Calculation of the congestion surplus utilizing the shadow cost of the transmission lines

As can be seen on table 37, the shadow cost of the each of the branches (column MVA Marg. Cost (λ)) is zero, except for those transmission lines that are fully loaded. Those branches are 1-3 and 2-5, with 10 £/MWh and 42.5 £/MWh, respectively. So, the Congestion Surplus can be found:

Where: So, the congestion surplus is given by:


References

Bialek J.W. Gaming the Uniform-Price Spot Market: Quantitative Analysis [Journal] = Bialek: Gaming

the Uniform // IEEE Transactions on Power Systems / ed. IEEE. - August 2002. - 3 : Vol. 17. - pp.

768-773.

Hao S. A Study of Basic Bidding Strategy in Clearing Price Auctions [Journal] = Hao: A Study of Basic

Bidding Strategy // IEEE Transactions on Power Systems / ed. IEEE. - August 2000. - 3 : Vol. 15. -

pp. 975-980.

Kirschen D., Strbac, G. Fundamentals of Power System Economics [Book]. - [s.l.] : John Wiley &

Sons Ltd., 2004. - p. 284. - ISBN-0-470-84572-4.

Kirschen D., Strbac, G. Why investments do not prevent blackouts [Journal]. - Manchester : [s.n.], 27

August 2003. - UMIST.

PowerWorld Corporation User's Guide. Simulator Version 14 [Book]. - Champaign : [s.n.], 2009. - p.

1517.

Documents

Economical Dispatch Optimal Power Flow and Locational Marginal Pricing - Power Systems Exercise Ariel Villalon M