# Recourse-based Stochastic Market Clearing Algorithm

## Abstract

Solutions obtained from the deterministic market-clearing problem may be feasible only for those conditions when point forecasts of random variables such as load and renewable sources of energy are within a tight range of accuracy. Unfortunately, point forecasts of renewable sources of energy have a higher error percentage. Under such circumstances, dynamism associated with renewable sources such as wind must be formulated as stochastic formulations which would encompass feasible solutions for a broader spectrum of forecast possibilities. This paper describes stochastic formulation for market clearing using recourse method. This method gives twofold solutions—the first being day-ahead market schedules obtained as here-and-now variables while the second being reserves applicable for different scenarios of wind forecast obtained as wait-and-see variables. This recourse-based stochastic formulation is validated for modified 24-node IEEE reliability test system.

## Keywords

Stochastic market clearing Recourse method Renewable energy Here-and-now variables Wait-and-see variables## Notations

## Indices

*t*Time period

*i*Individual conventional power plant

*j*Individual wind farm

- sc
Individual scenario

*k*Individual load

*r*Individual transmission line

*n*Individual node/bus

*ω*Individual scenario

*Nt*Total time period

*Ng*Total number of conventional power plants

*Nω*Total scenarios considered

*Nj*Total wind farms

*Nl*Total loads in the system

*Nr*Total number of transmission lines in the system

*Ng*_{n}Total number of generators on bus

*n**Nw*_{n}Total number of wind generators on bus

*n**Nr*_{n}Total number of transmission lines on bus

*n**Nl*_{n}Total number of load on bus

*n*

## Here-and-now and wait-and-see variables

- \( C_{it}^{\text{su}} ,C_{it\omega }^{\text{su}} \)
Start-up cost of conventional generator

*i*at time*t*- \( P_{it}^{g} ,P_{it\omega }^{g} \)
Power generated by conventional generator

*i*at time*t*- \( R_{it}^{U} \)
Up reserve of conventional generator

*i*at time*t*- \( R_{it}^{D} \)
Down reserve of conventional generator

*i*at time*t*- \( R_{it}^{\text{NS}} \)
Non-spinning reserve of conventional generator

*i*at time*t*- \( rG_{it\omega } \)
Additional power to be generated by conventional generator

*i*at time*t*under scenario*ω*- \( P_{jt}^{\text{wind}} ,P_{jt\omega }^{\text{wind}} \)
Power generated by wind farm

*j*at time*t*- \( S_{t\omega }^{\text{wind}} \)
Curtailment due to scenario

*ω*in time*t*- \( U_{it} ,U_{it\omega } \)
Unit commitment status binary variable

- \( f_{\omega } (n,r) \)
Transmission line flow between bus

*n*and bus*r*- \( rG_{it\omega }^{U} \)
Up reserve for generator

*i*at time*t*under scenario*ω*- \( rG_{it\omega }^{D} \)
Down reserve for generator

*i*at time*t*under scenario*ω*

## Constants

- \( \lambda_{it} \)
Offer cost of conventional generator

*i*at time*t*- \( \lambda_{it}^{\text{su}} \)
Cost for starting the conventional generator

*i*at time*t*- \( \lambda_{it}^{\text{RU}} \)
Cost for up reserve of conventional generator

*i*at time*t*- \( \lambda_{it}^{\text{RD}} \)
Cost for down reserve of conventional generator

*i*at time*t*- \( \lambda_{it}^{\text{RNS}} \)
Cost for non-spinning reserve of conventional generator

*i*at*t*- \( \pi_{\omega } \)
Probability of scenario

*ω*- \( L_{kt} \)
Demand of load \( k \) at time

*t*- \( P_{{{ \min },i}}^{g} \)
Minimum generation limit of generator

*i*- \( P_{{{\max}, i}}^{g} \)
Maximum generation limit of generator

*i*- \( R^{{{system}}} \)
System reserve

## References

- 1.Conejo A, Carrion M, Morales J (1999) Decision making under uncertainty in electricity markets, 2nd edn. Elsevier, LondonzbMATHGoogle Scholar
- 2.Gazafroudi A (2017) A novel stochastic reserve cost allocation approach of electricity market agents in the restructured power systems, Electr Power Syst Res pp 223–236CrossRefGoogle Scholar
- 3.Wu L, Shahidehpour M, Li Z (2012) Comparison of scenario-based and interval optimization approaches to stochastic SCUC. IEEE Trans Power Syst 27(2):913–921CrossRefGoogle Scholar
- 4.Morales-Espana G, Ramos A, Garcia-Gonzalez J (2014) An MIP formulation for joint market clearing of energy and reserves based ramp-scheduling, IEEE Trans Power Syst 29(1)Google Scholar
- 5.Xiong P, Jirutitijaroen P (2013) A stochastic optimization formulation of unit commitment with reliability constraints, IEEE Trans Smart Grid 4(4)CrossRefGoogle Scholar
- 6.Ye H, Ge Y, Shahidehpour M, Li Z (2017) Uncertainty marginal price, transmission reserve, and day-ahead market clearing with robust unit commitment, IEEE Trans Power Syst 32(3)CrossRefGoogle Scholar
- 7.Ye H, Li Z (2016) Robust security-constrained unit commitment and dispatch with recourse cost requirement, IEEE Trans Power Syst 31(5)CrossRefGoogle Scholar
- 8.Vlachos A, Biskas P (2016) An integrated intuitive exchange- and flow-based market clearing model, IEEE Trans Power syst 31(5)CrossRefGoogle Scholar
- 9.Chatzigiannis D, Biskas P, Dourbois G (2017) European-type electricity market clearing model incorporating PUN orders, IEEE Trans Power Syst 32(1)CrossRefGoogle Scholar
- 10.Abbaspourtorbati F, Conejo A, Wang J, Cherkaoui R (2017) Three- or two-stage stochastic market clearing algorithm, IEEE Trans Power Syst 32(4)CrossRefGoogle Scholar