Energy Systems

, Volume 3, Issue 2, pp 153–179 | Cite as

An energy management system for off-grid power systems

  • Daniel Zelazo
  • Ran Dai
  • Mehran Mesbahi
Original Paper


Next generation power management at all scales will rely on the efficient scheduling and operation of both generating units and loads to maximize efficiency and utility. The ability to schedule and modulate the demand levels of a subset of loads within a power system can lead to more efficient use of the generating units. These methods become increasingly important for systems that operate independently of the main utility, such as microgrid and off-grid systems. This work extends the principles of unit commitment and economic dispatch problems to off-grid power systems where the loads are also schedulable. We propose a general optimization framework for solving the energy management problem in these systems. An important contribution is the description of how a wide range of sources and loads, including those with discrete states, non-convex, and nonlinear cost or utility functions, can be reformulated as a convex optimization problem using, for example, a shortest path description. Once cast in this way, solution are obtainable using a sub-gradient algorithm that also lends itself to a distributed implementation. The methods are demonstrated by a simulation of an off-grid solar powered community.


Off-grid Energy management Lagrangian relaxation Shortest-path algorithm 



number of generating units, number of loads


set of generating units, set of loads


time horizon


set of time indices


index for generating units, loads, and time


power level of generating unit \(i \in \mathcal{G}\) and demand of load \(j \in\mathcal{L}\) at time \(t \in\mathcal{T}\)

\(x_{i}^{g}(t), u_{i}^{g}(t)\)

state and control variables for generating unit \(i \in\mathcal{G}\) at time \(t \in\mathcal{T}\)

\(x_{j}^{l}(t), u_{j}^{l}(t)\)

state and control variables for load \(j\in\mathcal{L}\) at time \(t \in\mathcal{T}\)


dynamic evolution of generating unit variables for unit \(i \in\mathcal{G}\)


dynamic evolution of load variables for load \(j \in\mathcal{L}\)


operating cost of generator unit \(i \in\mathcal{G}\) at time \(t \in\mathcal{T}\)


utility of load \(j \in\mathcal {L}\) at time \(t \in\mathcal{T}\)


abstract constraint set for generator unit \(i \in\mathcal{G}\) and load \(j \in\mathcal{L}\) at time \(t \in \mathcal{T}\)


abstract constraint set for generator load state and control variables at \(t \in\mathcal{T}\)

\(\mathcal{X}_{j}^{l}(t), \,\mathcal{U}_{j}^{l}(t)\)

abstract constraint set for load state and control variables at \(t \in\mathcal{T}\)


Lagrangian and dual functions


Lagrange multiplier and sub-gradient at time \(t \in\mathcal{T}\), step size



The authors would like to thank the associate editor and reviewers for the in-depth comments and discussions regarding this work.


  1. 1.
    US department of energy, smart grid.
  2. 2.
    European commission on energy, smartgrids.
  3. 3.
    Ipakchi, A., Albuyeh, F.: Grid of the future. IEEE Power Energy Mag. 7(2), 52–62 (2009) CrossRefGoogle Scholar
  4. 4.
    Fan, J., Borlase, S.: The evolution of distribution. IEEE Power Energy Mag. 7(2), 63–68 (2009) CrossRefGoogle Scholar
  5. 5.
    Farhangi, H.: The path of the smart grid. IEEE Power Energy Mag. 8(1), 18–28 (2010) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brown, R.E.: Impact of smart grid on distribution system design. In: Power and Energy Society General Meeting—Conversion and Delivery of Electrical Energy in the 21st Century, 2008, pp. 1–4. IEEE Press, New York (2008) CrossRefGoogle Scholar
  7. 7.
    Hatziargyriou, N.: Microgrids [guest editorial]. IEEE Power Energy Mag. 6(3), 26–29 (2008) CrossRefGoogle Scholar
  8. 8.
    Katiraei, F., Iravani, R., Hatziargyriou, N., Dimeas, A.: Microgrids management. IEEE Power Energy Mag. 6(3), 54–65 (2008) CrossRefGoogle Scholar
  9. 9.
    Thornley, V., Kemsley, R., Barbier, C., Nicholson, G.: User perception of demand side management. In: SmartGrids for Distribution IET-CIRED, Frankfurt, Germany (2008) Google Scholar
  10. 10.
    Zhang, D., Papageorgiou, L.G., Samsatli, N.J., Shah, N.: Optimal scheduling of smart homes energy consumption with microgrid. In: Energy 2011: The First International Conference on Smart Grids, Green Communications and IT Energy-aware Technologies, no. c, pp. 70–75 (2011) Google Scholar
  11. 11.
    Misak, S., Prokop, L.: Off-grid power systems. In: 9th International Conference on Environment and Electrical Engineering, 2010, pp. 14–17. IEEE Press, New York (2010) CrossRefGoogle Scholar
  12. 12.
    Muntean, N., Cornea, O., Petrila, D.: A new conversion and control system for a small off—grid wind turbine. In: 12th International Conference on Optimization of Electrical and Electronic Equipment, 2010, pp. 1167–1173. IEEE Press, New York (2010) CrossRefGoogle Scholar
  13. 13.
    Leak, M.H., . Rashid, M.: Feasibility of off-grid residential power. In: CONIELECOMP 2011, 21st International Conference on Electrical Communications and Computers, pp. 14–17. IEEE Press, New York (2011) CrossRefGoogle Scholar
  14. 14.
    Chan, C.C.: The state of the art of electric, hybrid, and fuel cell vehicles. Proc. IEEE 95(4), 704–718 (2007) CrossRefGoogle Scholar
  15. 15.
    Koot, M., Kessels, J., DeJager, B., Heemels, W., VandenBosch, P., Steinbuch, M.: Energy management strategies for vehicular electric power systems. IEEE Trans. Veh. Technol. 54(3), 771–782 (2005) CrossRefGoogle Scholar
  16. 16.
    Davey, K., Longoria, R., Shutt, W., Carroll, J., Nagaraj, K., Park, J., Rosenwinkesl, T., Wu, W., Arapostathis, A.: Reconfiguration in shipboard power systems. Am. Control Conf. 80(6), 064501 (2007) Google Scholar
  17. 17.
    Maldonado, M., Shah, N., Cleek, K., Walia, P., Korba, G.: Power Management and Distribution System for a More-Electric Aircraft (MADMEL)-Program Status. IEEE Press, New York (2004) Google Scholar
  18. 18.
    Cloyd, J.: Status of the United States air force’s more electric aircraft initiative. IEEE Aerosp. Electron. Syst. Mag. 13(4), 17–22 (1998) CrossRefGoogle Scholar
  19. 19.
    Luongo, C.A., Masson, P.J., Nam, T., Mavris, D., Kim, H.D., Brown, G.V., Waters, M., Hall, D.: Next generation more-electric aircraft: a potential application for HTS superconductors. IEEE Trans. Appl. Supercond. 19(3), 1055–1068 (2009) CrossRefGoogle Scholar
  20. 20.
    Mohsenian-Rad, A.H., Leon-Garcia, A.: Optimal residential load control with price prediction in real-time electricity pricing environments. IEEE Trans. Smart Grid 1(2), 120–133 (2010) CrossRefGoogle Scholar
  21. 21.
    Ricquebourg, V., Menga, D., Durand, D., Marhic, B., Delahoche, L., Loge, C.: The smart home concept: our immediate future. In: 1st IEEE International Conference on E-Learning in Industrial Electronics, 2006, pp. 23–28. IEEE Press, New York (2006) CrossRefGoogle Scholar
  22. 22.
    Bertsekas, D.P., Lauer, G.S., Sandell, N.R.J., Posbergh, T.A.: Optimal short-term scheduling of large-scale power systems. IEEE Trans. Autom. Control AC-28(1), 1–11 (1983) CrossRefGoogle Scholar
  23. 23.
    Gruhl, J., Schweppe, F., Ruane, M.: Unit commitment scheduling of electric power systems. In: System Engineering for Power: Status and Prospects, Henniker, NH, pp. 116–128 (1972) Google Scholar
  24. 24.
    Muckstadt, J., Koenig, S.: An application of Lagrangian relaxation to scheduling in power generation systems. Oper. Res. 25(3), 387–403 (1977) CrossRefzbMATHGoogle Scholar
  25. 25.
    Kirchmayer, L.K.: Economic Operation of Power Systems. Wiley, New York (1958) Google Scholar
  26. 26.
    Guan, X., Luh, P., Yan, H., Amalfi, J.: An optimization-based method for unit commitment. Electr. Power Energy Syst. 14(1), 9–17 (1992) CrossRefGoogle Scholar
  27. 27.
    Ongsakul, W., Petcharaks, N.: Unit commitment by enhanced adaptive Lagrangian relaxation. IEEE Trans. Power Syst. 19(1), 620–628 (2004) CrossRefGoogle Scholar
  28. 28.
    Papalexopoulos, A.: Optimization based methods for unit commitment: Lagrangian relaxation versus general mixed integer programming. In: IEEE Power Engineering Society General Meeting (IEEE Cat. No. 03CH37491), 2003, pp. 1095–1100. IEEE Press, New York (2003) Google Scholar
  29. 29.
    Li, T., Shahidehpour, M.: Price-based unit commitment: a case of Lagrangian relaxation versus mixed integer programming. IEEE Trans. Power Syst. 20(4), 2015–2025 (2005) CrossRefGoogle Scholar
  30. 30.
    Joo, J.Y., Ilic, M.D.: A multi-layered adaptive load management (alm) system: Information exchange between market participants for efficient and reliable energy use. In: Transmission and Distribution Conference and Exposition, 2010 IEEE PES, April, pp. 1–7 (2010) Google Scholar
  31. 31.
    Fahrioglu, M., Alvarado, F.: Using utility information to calibrate customer demand management behavior models. In: IEEE Power Engineering Society Winter Meeting. Conference Proceedings (Cat. No. 02CH37309), 2002, vol. 1, p. 26. IEEE Press, New York (2002) CrossRefGoogle Scholar
  32. 32.
    Samadi, P., Mohsenian-Rad, A.-H., Schober, R., Wong, V.W.S., Jatskevich, J.: Optimal real-time pricing algorithm based on utility maximization for smart grid. In: First IEEE International Conference on Smart Grid Communications, 2010, pp. 415–420. IEEE Press, New York (2010) CrossRefGoogle Scholar
  33. 33.
    Pedrasa, M.A.A., Spooner, T.D., MacGill, I.F.: Coordinated scheduling of residential distributed energy resources to optimize smart home energy services. IEEE Trans. Smart Grid 1(2), 134–143 (2010) CrossRefGoogle Scholar
  34. 34.
    Chiang, M., Low, S.H., Calderbank, A.R., Doyle, J.: Layering as optimization decomposition: a mathematical theory of network architectures. Proc. IEEE 95(1), 255–312 (2007) CrossRefGoogle Scholar
  35. 35.
    Nedic, A., Ozdaglar, A.: Subgradient methods in network resource allocation: Rate analysis. In: 42nd Annual Conference on Information Sciences and Systems, 2008, pp. 1189–1194. IEEE Press, New York (2008) CrossRefGoogle Scholar
  36. 36.
    Andrade, L., Tenning, C.: Design of Boeing 777 electric system. IEEE Aerosp. Electron. Syst. Mag. 7(7), 4–11 (1992) CrossRefGoogle Scholar
  37. 37.
    Huneault, M., Galiana, F.D.: A survey of the optimal power flow literature. IEEE Trans. Power Syst. 6(2), 762–770 (1991) CrossRefGoogle Scholar
  38. 38.
    Xu, D., Girgis, A.: Optimal load shedding strategy in power systems with distributed generation. In: IEEE Power Engineering Society Winter Meeting. Conference Proceedings (Cat. No. 01CH37194), no. C, 2001, pp. 788–793. IEEE Press, New York (2001) Google Scholar
  39. 39.
    Bhattacharyya, K., Crow, M.: A fuzzy logic based approach to direct load control. IEEE Trans. Power Syst. 11(2), 708–714 (1996) CrossRefGoogle Scholar
  40. 40.
    Cohen, A., Wang, C.: An optimization method for load management scheduling. IEEE Trans. Power Syst. 3(2), 612–618 (1988) MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ng, K.H., Sheble, G.B.: Direct load control—a profit-based load management using linear programming. IEEE Trans. Power Syst. 13(2), 688–695 (1998) CrossRefGoogle Scholar
  42. 42.
    Ramanathan, B., Vittal, V.: A framework for evaluation of advanced direct load control with minimum disruption. IEEE Trans. Power Syst. 23(4), 1681–1688 (2008) CrossRefGoogle Scholar
  43. 43.
    Widergren, S.E.: Demand or request: Will load behave? In: IEEE Power & Energy Society General Meeting, 2009, pp. 1–5. IEEE Press, New York (2009) CrossRefGoogle Scholar
  44. 44.
    Ruiz, N., Cobelo, I.N., Oyarzabal, J.: A direct load control model for virtual power plant management. IEEE Trans. Power Syst. 24(2), 959–966 (2009) CrossRefGoogle Scholar
  45. 45.
    Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) zbMATHGoogle Scholar
  46. 46.
    Rockafellar, R.T.: Network Flows and Monotropic Optimization. Wiley, New York (1984) zbMATHGoogle Scholar
  47. 47.
    Frangioni, A.: Solving nonlinear single-unit commitment problems with ramping constraints. Oper. Res. 54, 775 (2006) CrossRefGoogle Scholar
  48. 48.
    Ruszczynski, A.: Nonlinear Optimization. Princeton University Press, Princeton (2006) zbMATHGoogle Scholar
  49. 49.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control. Athena Scientific, Belmont (2007) Google Scholar
  50. 50.
    Sherali, H., Choi, G.: Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs. Oper. Res. Lett. 19(3), 105–113 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute for Systems Theory and Automatic ControlUniversität StuttgartStuttgartGermany
  2. 2.Department of Aeronautics and AstronauticsUniversity of WashingtonSeattleUSA

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