Case I: No Network Congestion

In this case, the thermal limits of lines are considered to be high so that no congestion appears. Therefore, the prices do not change across nodes

Since the driving factor of this chapter is the cost-recovery conditions of producers, we present the market outcomes starting with producer profits.

U76 U50 U155 U50 U155 U197 U50 U400

0 10 20 30 40 50 Unit Day − ahead Profit [K$] Con CR AR SR

Figure 3.4 – Day-ahead Profit (RTS no congestion case).

Fig. 3.4 shows the day-ahead profit obtained from the different approaches. Under the conventional approach without uplift (white bars), units U76and U197incur losses (negative

3.7. Case Studies

Using the proposed approaches, unit U197has a positive day-ahead profit, and thus, does not

need uplift. However, unit U76still incurs losses under the approaches with cost-recovery

constraints in expectation and per scenario, as these approaches do not enforce cost-recovery conditions at the day-ahead market. The day-ahead profit of other units increases using the proposed approaches. Among them, the day-ahead cost-recovery approach results in the largest day-ahead profits. This is caused by the day-ahead prices, as described in the following.

2 4 6 8 10 12 14 16 18 20 22 24 5 10 15 20 25 Day − ahead Prices [$/MWh] Time periods CR AR SR Con

Figure 3.5 – Day-ahead prices at node 2 under different approaches (RTS no congestion case).

Fig. 3.5 depicts the day-ahead prices at node 2, where unit U76is located, over the 24-hour

horizon. Since there is no congestion, the prices are the same at all nodes. Apart from period

t7(morning peak) and period t18(evening peak), the prices obtained from the approaches

with cost-recovery constraints are either equal or slightly higher than the prices obtained by the conventional method. In period t18, the day-ahead prices obtained from the approaches

with cost-recovery constraints are the same and higher than the conventional marginal price. However, a different behavior is observed in period t7. The approach with day-ahead cost-

recovery constraints results in a higher day-ahead price than those from the other approaches. This consequently leads to a higher day-ahead profit from this approach, as observed in Fig. 3.4.

Fig. 3.6 shows the expected profit of the producers. In this figure, we distinguish the producer profits with and without uplift by U and Con, respectively. One can observe that unit U76

cannot recover its expected cost under the prices obtained from the conventional method without uplift (white bars). However, it attains a non-negative expected profit from the approaches with cost-recovery constraints. Other units achieve higher profits under the pricing approaches proposed than under the conventional method with uplift.

Model

U76 U50 U155 U50 U155 U197 U50 U400

0 10 20 30 40 50 Expected profit [K$] Unit CR AR SR U Con

Figure 3.6 – Expected Profit (RTS no congestion case).

Therefore, the proposed pricing approaches demonstrate the performance desired: the ap- proach with day-ahead cost-recovery constraints guarantees non-negative day-ahead profit, and the approaches with cost-recovery constraints in expectation and per scenario result in non-negative expected profits. However, it is important to assess at what expenses these outcomes are obtained. For this purpose, we provide the expected cost, consumer payment as well as the duality gap in Table 3.8. The pricing methodologies with cost-recovery constraints result in slightly higher expected cost than the one from the conventional method.

The approaches with cost-recovery conditions result in higher consumer payments than the method with uplift. The increase in the consumer payments obtained from the proposed approaches as a percentage of the consumer payment from the method with uplift is also listed in Table 3.8. Note that consumer payment comprises day-ahead costs and day-ahead profits. Higher day-ahead profits from the proposed approaches, described above, derive higher consumer payments in these approaches as compared to those from the method with uplift.

Finally, one can observe that the social welfare gaps in the last row are small in comparison with the expected optimal social welfare from primal problem (3.4); they are of order of 0.3% of the optimal expected cost of $127,066.

We should note that these differences are small, but also, they get smaller using a realistic power system, as in a power system with a high penetration of renewable energy, gas units are dominated technology among the conventional units due to their fast-ramping ability. Gas units have small start-up costs and minimum production limits as compared to coal units.

3.7. Case Studies

Table 3.8 – Expected cost, consumer payment and duality gap for the RTS system [$] (RTS no congestion case) CR AR SR U Expected cost 127,066 127,169 127,153 127,066 Consumers payment 2.42×105 2.34×105 2.38×105 2.17×105 (11.5%) (7.8%) (9.7%) – Gap 288.24 (0.2%) 384.80 (0.3%) 371.10 (0.3%) –

Therefore, we obtain a set of uniform prices under which cost-recovery conditions of producers are guaranteed without a need of uplift at the expense of deviating about 0.3% from the optimal social welfare.

For these simulation, we use CPLEX 12.1 under MATLAB on a computer Intel(R) Xeon(R) with two processors clocking at 2.2 GHz and 512 GB of RAM. The sizes of the proposed models in terms of number of variables and constraints as well as the computation time are provided in Table 3.9. We elaborate on computation time in section 3.7.5.

Table 3.9 – Size of the proposed models

CR AR SR U

No. of continuous variables 93368 96968 96968 27600 No. of integer variables 1560 5160 5160 1176

No. of total variables 94928 102128 102128 28776 No. of constraints 95361 108561 108585 65384 Computation time (s) 22705 14231 1624 57