Computational Economics

, Volume 53, Issue 1, pp 1–26 | Cite as

Extracting Appropriate Nodal Marginal Prices for All Types of Committed Reserve

  • Paria Akbary
  • Mohammad Ghiasi
  • Mohammad Reza Rezaie Pourkheranjani
  • Hamidreza Alipour
  • Noradin GhadimiEmail author


This paper proposes a framework to extract appropriate locational marginal prices for each type of reserve (up-/down-going reserves at both generation- and demand-sides). The proposed reserve pricing scheme accounts for the lost opportunity of selling the convertible products (energy and reserve). The fair prices can be obtained for capacity reserves applying this framework, since this framework assigns the same prices to the same services provided at the same location. The proposed reserve pricing scheme provides all the market participants with the appropriate signals to modify their offers according to the system operator requirements. The pricing problem is decomposed to different hourly sub-problems considering the bounding constraints. To show the effectiveness of the proposed algorithm, it is applied to the IEEE reliability test system and the results are discussed.


Marginal pricing Security constraint unit commitment (SCUC) Up-/down-going demand-/generation-side reserves 

List of symbols

Functions and variables

\(C(\ )\)

Generalized objective function

\(com(\ )\)

Reserve commitment indicator; 1 means the regarding reserve is committed and 0 means not committed

\(DT(\ )\)

Shutdown time counter

\(GMP(\ )\)

Generation marginal price


Index for unit

\(ILS(\ )\)

Involuntary load shedding


Index for bus


Index for segment in linearized cost function

\(outg(\ )\)

Unit outage indicator

\(p(\ )\)

Generation of each segment in cost function

\(pg(\ )\)

Generation of a unit

\(pd(\ )\)

Demand at a bus

\(R_d^{dn} (\ )\)

Demand-side down-going reserve

\(R_d^{up} (\ )\)

Demand-side up-going reserve

\(R_g^{dn} (\ )\)

Generation-side down-going reserve

\(R_g^{up} (\ )\)

Generation-side up-going reserve


Index for time

\(u(\ )\)

Unit status indicator; 1 means on and 0 means off


Reactance of a line

\(y(\ )\)

Start-up indicator

\(z(\ )\)

Shut-down indicator

\(\lambda _g^{up} (\ )\)

Lagrange multiplier of maximum available generation-side up-going reserve constraint

\(\lambda _d^{up} (\ )\)

Lagrange multiplier of maximum available demand-side up-going reserve constraint

\(\lambda _g^{dn} (\ )\)

Lagrange multiplier of maximum available generation-side down-going reserve constraint

\(\lambda _d^{dn} (\ )\)

Lagrange multiplier of maximum available demand-side down-going reserve constraint

\(\gamma (\ )\)

Lagrange multiplier of pre-contingency load-generation balance constraint

\(\gamma k (\ )\)

Lagrange multiplier of post-contingency load-generation balance constraint

\(\mu ^{\max }(\ )\)

Lagrange multiplier of maximum output limit

\(\mu ^{\min }(\ )\)

Lagrange multiplier of minimum output limit


\(F_i^{min} \)

Generation cost at the minimum output of unit i

\(IC(\ )\)

Involuntary load shedding price

\(MSR(\ )\)

Maximum sustainable ramp rate


Number of optimization binding constraints


Number of buses


Number of units


Number of lines


Number of optimization independent variables


Number of optimization control variables


Number of segments in linearized cost curves

\(Q(\ )\)

Offered rate for reserve which takes the same subscripts and superscripts as R

\(R^{of} (\ )\)

Maximum offered reserve which takes the same subscripts and superscripts as R

\(RD(\ )\)

Ramping up limit of a unit

\(RMP(\ )\)

Reserve marginal prices which takes the same subscripts and superscripts as R

\(RU(\ )\)

Ramping down limit of a unit

\(SDC(\ )\)

Shutdown cost

\(sl(\ )\)

Slope of each segment in the linearized cost curve

\(SUCF(\ )\)

Start-up cost function


Number of hours in the time span

Vectors and matrices


Bus-to-unit incidence matrix


Post contingency bus-to-unit incidence matrix

\(EMP(\ )\)

Vector of energy marginal prices


Vector of upper limits of line and transformer flows

\(fK^{max}(\ )\)

Vector of post-contingency upper limits of line and transformer flows


Generation shift factors matrix

\(GSFK(\ )\)

Post-contingency generation shift factors matrix

\(Outg(\ )\)

Unit outage matrix

\(PD(\ )\)

Demand vector

\(PG(\ )\)

Generators real power output vector

\(RD(\ )\)

Demand-side reserve vector

\(RG(\ )\)

Generation-side reserve vector


Inverse of DC-LF matrix

\(\lambda \)

Vector of general Lagrange multipliers

\(\Phi (\ )\)

Vector of Lagrange multipliers for pre-contingency line flow constraints

\(\Phi k(\ )\)

Vector of Lagrange multipliers for post-contingency line flow constraints

\(\delta (\ )\)

Bus voltage angle vector


  1. Bolouck Azari, J., & Ghadimi, N. (2014). Firefly technique based on optimal congestion management in an electricity market. International Journal of Information, Security and Systems Management, 3(2), 333–344.Google Scholar
  2. Bouffard, F., Galiana, F. D., & Conejo, A. J. (2005). Market-clearing with stochastic security part I: Formulation. IEEE Transactions on Power Systems, 20, 1818–1826.CrossRefGoogle Scholar
  3. Chen, J., Thorp, J. S., Thomas, R. J., & Mount, T. D. (2003). Locational pricing and scheduling for an integrated energy-reserve market. In Proceedings of 36th Hawaii international conference on system sciences, Hawaii, (pp. 54–63).Google Scholar
  4. Dideban, M., et al. (2013). Optimal location and sizing of shunt capacitors in distribution systems by considering different load scenarios. Journal of Electrical Engineering and Technology, 8(5), 1012–1020.CrossRefGoogle Scholar
  5. Fu, Y., Shahidehpour, M., & Li, Z. (2005). Security-constrained unit commitment with AC constraints. IEEE Transactions on Power Systems, 20, 1538–1550.CrossRefGoogle Scholar
  6. Ghadimi, N. (2014). MDE with considered different load scenarios for solving optimal location and sizing of shunt capacitors. National Academy Science Letters, 37(5), 447–450.CrossRefGoogle Scholar
  7. Ghadimi, N. (2015). A new hybrid algorithm based on optimal fuzzy controller in multimachine power system. Complexity, 21(1), 78–93.CrossRefGoogle Scholar
  8. Jalili, A., & Ghadimi, N. (2015). Hybrid harmony search algorithm and fuzzy mechanism for solving congestion management problem in an electricity market. Complexity. 21(S1), 90–98.Google Scholar
  9. Jamalzadeh, R., Zhang, F., & Hong, M. (2016). An economic dispatch algorithm incorporating voltage management for active distribution systems using generalized benders decomposition. In IEEE power and energy society general meeting (PESGM) (pp. 1–5). Boston, MA, USA.Google Scholar
  10. Li, T., & Shahidehpour, M. (2007a). Price-based unit commitment: A case of Lagrangian relaxation versus mixed integer programming. IEEE Transactions on Power Systems, 20, 2015–2025.CrossRefGoogle Scholar
  11. Li, T., & Shahidehpour, M. (2007b). Risk-constrained generation asset arbitrage in power systems. IEEE Transactions on Power Systems, 22, 1330–1339.Google Scholar
  12. Momoh, J. A., Yan, X., & Boswell, G. D. (2008). Locational marginal pricing for real and reactive power. In Conversion and delivery of electrical energy in the 21st century, power and energy society general meeting-IEEE, (pp. 1–6).Google Scholar
  13. Nouri, A., Afkousi-Paqaleh, M., & Hosseini, S. H. (2013). Probabilistic assessment and sensitivity analysis of marginal price of different services in power markets. IEEE Systems Journal, 7, 873–880.CrossRefGoogle Scholar
  14. Nouri, A., & Hosseini, S. H. (2015). Comparison of LMPs’ sensitivity under payment cost minimization and offer cost minimization mechanisms. IEEE Systems Journal, 9, 1507–1518.CrossRefGoogle Scholar
  15. PJM Manual 06, 11, 12: Scheduling Operations.
  16. Shahidehpour, M., Yamin, H., & Li, Z. (2002). Market operations in electric power systems: forecasting, scheduling, and risk management. New York: Wiley.CrossRefGoogle Scholar
  17. Wang, J., Encinas Redondo, N., & Galiana, F. D. (2003). Demand-side reserve offers in joint energy/reserve electricity markets. IEEE Transactions on Power System, 18, 1300–1306.CrossRefGoogle Scholar
  18. Wong, S., & Fuller, J. D. (2007). Pricing energy and reserve using stochastic optimization in an alternative electricity market. IEEE Transactions on Power Systems, 22, 631–638.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Paria Akbary
    • 1
  • Mohammad Ghiasi
    • 2
    • 3
  • Mohammad Reza Rezaie Pourkheranjani
    • 4
  • Hamidreza Alipour
    • 5
  • Noradin Ghadimi
    • 6
    Email author
  1. 1.Faculty of Marine SciencesChabahar Maritime UniversityChabaharIran
  2. 2.Department of Electrical Engineering, Sciences and Research BranchIslamic Azad UniversityTehranIran
  3. 3.Tehran Metro Operation CompanyTehranIran
  4. 4.Depaertment of Electrical EngineeringFasa Branch, Islamic Azad UniversityFasaIran
  5. 5.Department of Management & EconomicRasht Branch, Islamic Azad UniversityRashtIran
  6. 6.Young Researchers and Elite Club, Ardabil BranchIslamic Azad UniversityArdabilIran

Personalised recommendations