3.4 Decision-Making Process
3.7.6 Impact of Linearization Step
The impact of linearization steps on the problem outcomes is elaborated in this section. On one hand, a smaller linearization step approximates more precisely the problem, but on the other hand, significantly increases the dimension of the problem and its computational burden.
Computational burden particularly matters when considering small linearization steps for the deployed reserves in the approach with cost-recovery constraints in expectation and the ap- proach with cost-recovery constraints per scenario. The reason is that the number of required decision variables (yi tωmU , yi tDωm, zUi tωm, and zDi tωm,∀i,∀t,∀ω,∀m), and constraints, (con- straints (3.16i)-(3.16l) for the approach with average cost-recovery conditions, and constraints (3.17i)-(3.17l) for the approach with per-scenario cost-recovery conditions) significantly in- crease.
In the simulations, thus far presented, linearization steps of 5 MW and 19 MW have been considered, respectively, for the day-ahead scheduled productions and the deployed reserves. We next consider a linearization step of 2 MW for both the day-ahead productions and de- ployed reserves (a drastic reduction) and provide the outcomes in Table 3.13. Comparing these outcomes with those provided in Table 3.8 (with linearization steps of 5 MW and 19 MW), we conclude that this smaller linearization step results in very similar expected costs (less than 0.03% differences), but smaller gaps for all proposed approaches. It also results in smaller consumer payments for the approach with day-ahead cost-recovery constraints (2.5% reduction) and the approach with average cost-recovery constraints (0.4% reduction), and the same consumer payment for the approach with cost-recovery constraints per scenario. From a computational point of view, the smaller linearization steps of 2 MW increase the
3.7. Case Studies
computation time from 14321 s to 19581 s for the approach with cost-recovery constraints in expectation, and from 1624 s to 10983 s for the approach with cost-recovery constraints per scenario. The computation time for the approach with day-ahead cost-recovery constraints remains the same as the one for the linearization steps of 5 MW and 19 MW.
To summarize, a smaller linearization step leads to slightly more precise results at the expense of higher computation time, while still the same conclusions are derived.
Table 3.13 – Expected cost, consumer payment and duality gap for the RTS system: no conges- tion case and linearization steps of 2MW for both schedules and deployed reserves.
CR AR SR
Expected cost 127055 127122 127131
Consumers payment 2.36×105 2.33×105 2.38×105
Gap 282.8 347.6 359.9
3.7.7 Case Study Conclusion
We propose pricing approaches with cost-recovery conditions at the day-ahead, in expectation, and per scenario for a stochastic non-convex clearing model.
Day-ahead prices obtained from the proposed approaches are higher than conventional marginal prices in some periods. This, consequently, causes higher producer profits, and therefore, higher consumer payments. However, the new prices eliminate the need of uplifts and allow the market to fully rely on these new marginal prices. These conclusions are not affected by network congestion, as well as by considering other sources of non-convexity such as minimum up/down time constraints of units.
The increase in consumer payment varies from 3% to 9% of a payment derived from the method with uplift considering different load profiles and network congestion.
From a social welfare point of view, the proposed approaches with cost-recovery features imply deviating the least possible amount from the optimal value of the expected cost. The approach with day-ahead cost-recovery constraints results in the same expected cost as the original primal two-stage problem for the different load profiles considered. The other proposed approaches also attain optimal expected costs close to the one of the primal problem. The increases in expected costs are of order of 0.5% of the optimal expected cost. In the same vein, the duality gaps are also small. Considering different load profiles, the duality gaps are of order of 0.5% of the optimal social welfare.
Model
From the computational point of view, the models proposed have bi-linear terms, which need to be linearized using integer variables. In this process, a smaller linearization step leads to slightly more precise results at the expense of higher computation burden, but still the same conclusions are derived. Generally, the models proposed are tractable and solvable in a reasonable time using a MILP state-of-the-art solver.
3.8 Summary and Conclusion of the Chapter
Pricing schemes in non-convex electricity markets constitute an active area of research. Re- cently, the growth of renewable generation and the possibility of using of stochastic clearing models have added a new dimension to the traditional pricing problem: uncertainty.
This chapter proposes pricing methodologies in the presence of non-convexity and stochas- ticity in electricity markets. The proposed approaches result in locational marginal prices which guarantee cost-recovery conditions for producers, and therefore, eliminate a need for uplifts. The prices obtained deviate in the least possible manner from conventional marginal prices. This implies that a minimum deviation from the optimal value of social welfare is also guaranteed. Moreover, the new prices preserve the short-term economic efficiency and long-term cost recovery properties of marginal prices.
The proposed pricing methods may be of interest for regulators to replace the existing pricing methods that require uplifts.
4
Economic Impact of Flexible De-
mands
4.1 Introduction
In the previous chapters, we have focused on market-clearing models as tools facilitating a large-scale integration of renewable generation. In this chapter, we change the view from the role of tools to the role of market players, particularly demands.
A high penetration of renewable generation requires a system with sufficient flexibility. Flexi- bility is the operational ability of a generating unit or a demand to be scheduled by the system operator with some degree of freedom. The operational flexibility of demands and units allows the system operator to adjust them in order to absorb renewable production to the largest extent at a minimum cost. While flexibility of a generating unit is reflected in its ramping capability, demand flexibility includes the ability to move consumption across periods, and to change the consumption level per period. Hence, a system with a large amount of renewable generation needs to promote demand flexibility and building fast-ramping units. This implies that a common future power generation mix may consist of comparatively cheap renewable units and comparatively expensive fast-ramping units.
Given that the driving factors behind marginal prices are the production costs of units, a swing between high marginal prices (due to high production costs of fast-ramping units) and low marginal prices (due to small production costs of renewable units) seems likely. But also, high demand flexibility may alter energy prices such that what is known nowadays as peak and off-peak prices may fade by demand flexibility, as it basically shifts energy consumption from peak periods to off-peak periods.
The interaction between energy prices and flexible demands is complex. On one hand, de- mand flexibility is recognized to be beneficial to the system as a whole since such flexibility facilitates the integration of renewable generation with a reduced operation cost, but on the
other hand, shifting demands from peak periods to off-peak periods may influence prices to an extent that affects the willingness of demands to be flexible.
Therefore, in a power system with a high penetration of cheap renewable production and expensive fast-ramping units, a legitimate question is whether being flexible is advantageous for demands.
To address this question, this chapter analyzes the operational and economic impacts of demand flexibility, particularly demand revenues.
Note that the contribution of demands in providing flexibility to assist the integration of renewable generation ([44] and [1]) from the system point of view is discussed in [39], [75], and [74]. However, analyses focusing on the economic impacts of demand flexibility are not common in the literature, particularly the impact of different degrees of demand flexibility on day-ahead prices.