Co-optimization of electricity transmission and generation resources for planning and policy analysis: review of concepts and modeling approaches

Abstract

The recognition of transmission’s interaction with other resources has motivated the development of co-optimization methods to optimize transmission investment while simultaneously considering tradeoffs with investments in electricity supply, demand, and storage resources. For a given set of constraints, co-optimized planning models provide solutions that have lower costs than solutions obtained from decoupled optimization (transmission-only, generation-only, or iterations between them). This paper describes co-optimization and provides an overview of approaches to co-optimizing transmission options, supply-side resources, demand-side resources, and natural gas pipelines. In particular, the paper provides an up-to-date assessment of the present and potential capabilities of existing co-optimization tools, and it discusses needs and challenges for developing advanced co-optimization models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. 1.

    In states where the power sector is unbundled, the term “co-optimization” is a slightly misleading characterization of these models, since the transmission owner would not use such models to optimize resource investment, but instead to simulate the decision process of resource owners.

  2. 2.

    Kirchhoff’s Laws include the current and voltage laws. The former says that there is a current balance at any node (bus) in a network, with inflows equaling outflows. The latter says that the net voltage drop around any loop in a network must be zero. In a linearized DC load flow model, the analogies to these laws are, respectively, that the net inflow of power to any bus is zero and that the sum of the products of power times reactance around any loop is also zero. One result of these laws is that power travels in parallel paths between sources and sinks, and another result is that given a set of sources and sinks, the flow over any given line is completely determined and cannot be controlled. More generally, however, phase shifters and other FACTS devices can be introduced into a network, which allows for partial control of flows.

  3. 3.

    A typical power system with multiple transformers utilizes different voltage levels. The per-unit (p.u.) system simplifies the analysis of power systems by choosing a set of voltage and power base values, and then computing a set of current and impedance base values; these four base values are used in normalizing all system impedances, powers, voltages, and currents so that standard electric circuit relations continue to hold among the normalized quantities.

References

  1. 1.

    Sauma, E.E., Oren, S.S.: Proactive planning and valuation of transmission investments in restructured electricity markets. J. Regul. Econ. 30(3), 261–290 (2006). (358–387)

    Article  Google Scholar 

  2. 2.

    Krishnan, V., McCalley, J., Lemos, S., Bushnell, J.: Nation-wide transmission overlay design and benefits assessment for the US. Energy Policy (2013). doi:10.1016/j.enpol.2012.12.051

  3. 3.

    McCalley, J., Krishnan, V., Gkritza, K., Brown, R., Mejia-Giraldo, D.: Planning for long haul- Investment strategies for national energy and transportation infrastructures. IEEE Power Energy Mag. 11(5), 24–35 (2013)

    Article  Google Scholar 

  4. 4.

    Shahidehpour, M.: Investing in expansion: the many issues that cloud electricity planning. IEEE Power Energy Mag. 2, 14–18 (2004)

    Article  Google Scholar 

  5. 5.

    Awad, M., Casey, K.E., Geevarghese, A.S., Miller, J.C., Rahimi, A.F., Sheffrin, A.Y., Zhang, M., Toolson, E., Drayton, G., Hobbs, B.F., Wolak, F.A.: Economic assessment of transmission upgrades: application of the California ISO approach, Ch. 7. In: Zhang, X. (ed.) Restructured Electric Power Systems: Analysis of Electricity Markets with Equilibrium Models, Power Engineering Series, pp. 241–270. J. Wiley & Sons/IEEE Press, New York (2010)

    Chapter  Google Scholar 

  6. 6.

    Gu, Y., McCalley, J.D., Ni, M.: Coordinating large-scale wind integration and transmission planning. IEEE Trans. Sustain. Energy 3(4), 652–659 (2012)

    Article  Google Scholar 

  7. 7.

    McCalley, J., Bushnell, J., Krishnan, V., Cano, S.: Transmission design at the national level: benefits, risks and possible paths forward. In: White Paper to PSERC, The Future Grid to Enable Sustainable Energy Systems. http://www.pserc.wisc.edu/research/FutureGrid/broadanalysis.aspx (2012)

  8. 8.

    Roh, J.H., Shahidehpour, M., Fu, Y.: Market-based coordination of transmission and generation capacity planning. IEEE Trans. Power Syst. 22(4), 1406–1419 (2007)

    Article  Google Scholar 

  9. 9.

    Short, W., et al.: Regional energy deployment system (ReEDS). NREL Technical Report NREL/TP-6A20-46534. http://www.nrel.gov/analysis/reeds/pdfs/reeds_documentation.pdf (2011)

  10. 10.

    van der Weijde, A.H., Hobbs, B.F.: The economics of planning electricity transmission to accommodate renewables: using two-stage optimisation to evaluate flexibility and the cost of disregarding uncertainty. Energy Econ. 34(5), 2089–2101 (2012)

    Article  Google Scholar 

  11. 11.

    Zheng, Q.P., Liu, A.L.: Transmission and generation capacity expansion with unit commitment: a multiscale stochastic model. Presentation at the INFORMS Annual Meeting (2011)

  12. 12.

    Pfeifenberger, J.P., Hou, D.: Transmission’s true value: adding up the benefits of infrastructure investments. Publ. Util. Fortnightly, 44–50. http://www.fortnightly.com/fortnightly/2012/02/ (2012)

  13. 13.

    Chang, J.W., Pfeifenberger, J.P., Hagerty, J.M.: A WIRES report on the benefits of electric transmission: identifying and analyzing the value of investments. http://www.WIRESgroup.com The Brattle Group (2013)

  14. 14.

    Sauma, E., Oren, S.: Economic criteria for planning transmission investment in restructured electricity markets. IEEE Trans. Power Syst. 22(4), 1394–1405 (2007)

    Article  Google Scholar 

  15. 15.

    Pozo, D., Contreras, J., Sauma, E.: If you build it, he will come: anticipative power transmission planning. Energy Econ. 36, 135–146 (2013)

    Article  Google Scholar 

  16. 16.

    Hobbs, B.F.: Regional energy facility location models for power system planning and policy analysis. In: Lev, B., Murphy, F., Bloom, J., Gleit, A. (eds.) Analytic Techniques for Energy Planning, pp. 53–66. North-Holland Press, Amsterdam (1984)

    Google Scholar 

  17. 17.

    Stoll, H.: Least-Cost Electric Utility Planning. John Wiley, New York (1989)

    Google Scholar 

  18. 18.

    International Atomic Energy Agency: Expansion Planning for Electrical Generating Systems: A Guidebook (1984)

  19. 19.

    Wang, X., McDonald, J.: Modern Power System Planning. McGraw Hill Book Company, London (1994)

    Google Scholar 

  20. 20.

    Ventosa, M., Baíllo, Á., Ramos, A., Rivier, M.: Electricity markets modeling trends. Energy Policy 33(7), 897–913 (2005)

    Article  Google Scholar 

  21. 21.

    Madrigal, M., Stoft, S.: Transmission Expansion for Renewable Energy Scale-Up: Emerging Lessons and Recommendations. World Bank, Washington, DC (2012)

    Book  Google Scholar 

  22. 22.

    Areiza, J.M., Latorre, G., Cruz, R.D., Villegas, A.: Classification of publications and models on transmission expansion planning. IEEE Trans. Power Syst. 18(02), 938–946 (2003)

    Article  Google Scholar 

  23. 23.

    Liu, A., Zheng, Q., Ho, J., Krishnan, V., Hobbs, B., Shahidehpour, M., McCalley, J.: Co-optimization of Transmission and Other Supply Resources, NARUC Project No. 3316T5, prepared for the Eastern Interconnection States Planning Council. Available at: http://www.naruc.org/grants/Documents/Co-optimization-White-paper_Final_rv1.pdf (2013). Accessed 1 Sep 2013

  24. 24.

    Khodaei, A., Shahidehpour, M.: Microgrid-based co-optimization of generation and transmission planning in power systems. IEEE Trans. Power Syst. 28(2), 1582–1590 (2013)

    Article  Google Scholar 

  25. 25.

    Turvey, R., Anderson, D.: Electricity Economics: Essays and Case Studies. Johns Hopkins University Press, Baltimore (1977)

    Google Scholar 

  26. 26.

    Hobbs, B.F., Hu, M., Chen, Y., Ellis, J.H., Paul, A., Burtraw, D., Palmer, K.L.: From regions to stacks: spatial and temporal downscaling of future pollution scenarios for the power sector. IEEE Trans. Power Syst. 25(2), 1179–1189 (2010)

    Article  Google Scholar 

  27. 27.

    ICF Inc, Integrated Planning Model. http://www.icfi.com/insights/products-and-tools/ipm Fairfax (2013)

  28. 28.

    Sawey, R., Zinn, C.: A mathematical model for long range expansion of generation and transmission in electric utility systems. IEEE Trans. Power Apparatus Syst. 96(2), 657–666 (1977)

    Article  Google Scholar 

  29. 29.

    Pereira, M., Pinto, L., Cunha, S., Oliveira, G.: A decomposition approach to automated generation/transmission expansion planning. IEEE Trans. Power Apparatus Syst. 104(11), 3074–3083 (1985)

    Article  Google Scholar 

  30. 30.

    Li, W., Billinton, R.: A minimum cost assessment method for composite generation and transmission system expansion planning. IEEE Trans. Power Syst. 8(2), 628–635 (1993)

    Google Scholar 

  31. 31.

    Alizadeh, B., Jadid, S.: Reliability constrained coordination of generation and transmission expansion planning in power systems using mixed integer programming. IET Gener. Transm. Distrib. 5(9), 948–960 (2011)

    Article  Google Scholar 

  32. 32.

    Motamedi, A., Zareipour, H., Buygi, M.O., Rosehart, W.D.: A transmission planning framework considering future generation expansions in electricity markets. IEEE Trans. Power Syst. 25(4), 1987–1995 (2010)

    Article  Google Scholar 

  33. 33.

    Murugan, P., Kannan, S., Baskar, S.: Application of NSGA-II algorithm to single objective transmission constrained generation expansion planning. IEEE Trans. Power Syst. 24(4), 1790–1797 (2009)

    Article  Google Scholar 

  34. 34.

    Sepasian, M., Seifi, H., Foroud, A., Hatami, A.: A multiyear security constrained hybrid generation-transmission expansion planning algorithm including fuel supply costs. IEEE Trans. Power Syst. 24(3), 1609–1618 (2009)

    Article  Google Scholar 

  35. 35.

    Baringo, L., Conejo, A.J.: Transmission and Wind Power Investment. IEEE Trans. Power Syst. 27(2), 885–893 (2012)

    Article  Google Scholar 

  36. 36.

    Tor, O., Guven, A., Shahidehpour, M.: Congestion-driven transmission planning considering the impact of generator expansion. IEEE Trans. Power Syst. 23(2), 781–790 (2008)

    Article  Google Scholar 

  37. 37.

    Tor, O., Guven, A., Shahidehpour, M.: Promoting the investment on IPPs for optimal grid planning. IEEE Trans. Power Syst. 25(3), 1743–1750 (2010)

    Article  Google Scholar 

  38. 38.

    Head, W.J., Nguyen, H.V., Kahle, R.L., Bachman, P.A., Jensen, A.A., Watry, S.J.: The procedure used to assess the long range generation and transmission resources in the Mid-Continent Area Power Pool. IEEE Trans. Power Syst. 5(4), 1137–1145 (1990)

    Article  Google Scholar 

  39. 39.

    Castillo, A., O’Neill, R.P.: Computational performance of solution techniques applied to the ACOPF, Optimal power flow paper-5, FERC staff paper (2013)

  40. 40.

    Model Types, General Algebraic Modeling System (GAMS). http://www.gams.com/modtype/index.htm (2014). Accessed 07 March 2014

  41. 41.

    Zhang, H., Heydt, G.T., Vittal, V., Mittelman, H.D.: Transmission Expansion Planning Using an AC Model: Formulations and Possible Relaxations IEEE PES General Meeting (2012)

  42. 42.

    Jabr, R.: Optimization of AC transmission system planning. IEEE Trans. Power Syst. 28(3), 2779–2787 (2013)

    Article  Google Scholar 

  43. 43.

    Taylor, J., Hover, F.: Conic AC transmission system planning. IEEE Trans. Power Syst. 28(2), 952–959 (2013)

    Article  Google Scholar 

  44. 44.

    Bent, R., Coffrin, C., Gumucio, R., van Hentenryck, P.: Transmission Network Expansion Planning: Bridging the Gap between AC Heuristics and DC Approximations. PSCC (2014)

  45. 45.

    Krishnan, V., Liu, H., McCalley, J.D.: Coordinated reactive power planning against power system voltage instability. In: Proceedings of IEEE/PES Power Systems Conference and Expo. (2009)

  46. 46.

    Li, Y., McCalley, J.: Design of a high capacity inter-regional transmission overlay for the U.S. IEEE Trans. Power Syst. 30(1), 513–521 (2015)

    Article  Google Scholar 

  47. 47.

    Gutman, R., Marchenko, P.P., Dunlop, R.D.: Analytical development of loadability characteristics for EHV and UHV transmission lines. IEEE Trans. Power Apparatus Syst. PAS–98(2), 606–617 (1979)

    Article  Google Scholar 

  48. 48.

    Quelhas, A.M., Gil, E., McCalley, J.D.: A multiperiod generalized network flow model of the U.S. integrated energy system: Part I-model description. IEEE Trans. Power Syst. 22, 829–836 (2007)

    Article  Google Scholar 

  49. 49.

    Bertsekas, D.P., Polymenakos, L.C., Tseng, P.: Epsilon-relaxation method for separable convex cost network flow problems. SIAM J. Optim. 7, 853–870 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Ding, J., Somani, A.: Parallel computing solution for capacity expansion network flow optimization problems. J. Comput. 4(7), (2012)

  51. 51.

    McCalley, J., Krishnan, V.: Survey of transmission technologies for planning long distance bulk transmission overlay in US. Int. J. Electr. Power Energy Syst. 54, 559–568 (2014)

    Article  Google Scholar 

  52. 52.

    Mohitpour, M., Golshan, H., Murray, A.: Pipeline Design and Construction: A Practical Approach, 3rd edn. American Society of Mechanical Engineers (2007)

  53. 53.

    Lamont, A.: User’s guide to the META-Net economic modeling system; version 1.2, Lawrence Livermore National Laboratory, UCRL-ID-122511 (1994)

  54. 54.

    Gabriel, S.A., Conejo, A.J., Fuller, J.D., Hobbs, B.F., Ruiz, C.: Complementarity Modeling in Energy Markets. Springer-Verlag, Berlin (2012)

    MATH  Google Scholar 

  55. 55.

    Lund, H., Kempton, W.: Integration of renewable energy into the transport and electricity sectors through V2G. Energy Policy 36(9), 3578–3587 (2008)

    Article  Google Scholar 

  56. 56.

    Krishnan, V., Gonzalez-Marciaga, L., McCalley, J.: A planning model to assess hydrogen as an alternative fuel for national light-duty vehicle portfolio. Energy 73(14), 943–957 (2014)

    Article  Google Scholar 

  57. 57.

    Connolly, D., Lund, H., Mathiesen, B.V., Leahy, M.: A review of computer tools for analysing the integration of renewable energy into various energy systems. Appl. Energy 87(4), 1059–1082 (2010)

    Article  Google Scholar 

  58. 58.

    Lund, H., Werner, S., Wiltshire, R., Svendsen, S., Thorsen, J.E., Hvelplund, F., Mathiesen, B.V.: 4th Generation District Heating (4GDH). Integrating smart thermal grids into future sustainable energy systems. Energy 68, 1–11 (2014)

    Article  Google Scholar 

  59. 59.

    Krishnan, V., Das, T., McCalley, J.D.: Impact of short-term storage on frequency response under increasing wind penetration. J. Power Sources (2014)

  60. 60.

    Krishnan, V., Das, T.: Optimal allocation of energy storage in a co-optimized electricity market: Benefits assessment and deriving indicators for economic storage ventures. Energy 81, 175–188 (2015)

    Article  Google Scholar 

  61. 61.

    Das, T.: Performance and economic evaluation of storage technologies. Ph.D. Dissertation, Iowa State University (2013)

  62. 62.

    Das, T., Krishnan, V., McCalley, J.D.: Incorporating cycling costs in generation dispatch program: an economic value stream for energy storage. Int. J. Energy Res. 38(12), 1551–1561 (2014)

    Article  Google Scholar 

  63. 63.

    Navid, N., Rosenwald, G.: Market solutions for managing ramp flexibility with high penetration of renewable resource. Sustain. Energy IEEE Trans. 3(4), 784–790 (2012)

    Article  Google Scholar 

  64. 64.

    Das, T., Krishnan, V., McCalley, J.: High-fidelity dispatch model of storage technologies for production costing studies. IEEE Trans. Sustain. Energy 5(4), 1242–1252 (2014)

    Article  Google Scholar 

  65. 65.

    Das, T., Krishnan, V., McCalley, J.D.: Assessing the benefits and economics of bulk energy storage technologies in the power grid. Appl. Energy 139, 104–118 (2015)

    Article  Google Scholar 

  66. 66.

    Krishnan, V., Das, T., Ibanez, E., Lopez, C.A., McCalley, J.D.: Modeling operational effects of wind generation within national long-term infrastructure planning software. IEEE Trans. Power Syst. 28(2), 1308–1317 (2013)

    Article  Google Scholar 

  67. 67.

    Walawalkar, R., Apt, J., Mancini, R.: Economics of electric energy storage for energy arbitrage and regulation in New York. Energy Policy 35(4), 2558–2568 (2007)

    Article  Google Scholar 

  68. 68.

    Hedman, K.W., Oren, S.S., O’Neill, R.P.: A review of transmission switching and network topology optimization. In: IEEE Power and Energy Society General Meeting (2011)

  69. 69.

    Grinold, R.C.: Model building techniques for the correction of end effects in multistage convex programs. Oper. Res. 31(3), 407–431 (1983)

    Article  MATH  Google Scholar 

  70. 70.

    Krishnan, V., McCalley, J.D.: Building foresight in long-term infrastructure planning using end-effect mitigation models. IEEE Syst. J. PP(99), 1–12 (2015)

  71. 71.

    MISO Transmission Expansion Plan 2012: Appendix E2 EGEAS, Assumptions Document

  72. 72.

    De Jonghe, C., Hobbs, B.F., Belmans, R.: Optimal generation mix with short-term demand response and wind penetration. IEEE Trans. Power Syst. 27(2), 830–839 (2012)

    Article  Google Scholar 

  73. 73.

    López, J.A., Ponnambalam, K., Quintana, V.H.: Generation and transmission expansion under risk using stochastic programming. IEEE Trans. Power Syst. 22(3), 1369–1378 (2007)

    Article  Google Scholar 

  74. 74.

    Roh, J.H., Shahidehpour, M., Wu, L.: Market-based generation and transmission planning with uncertainties. IEEE Trans. Power Syst. 24(3), 1587–1598 (2009)

    Article  Google Scholar 

  75. 75.

    Mejia-Giraldo, D.: Robust and flexible planning of power system generation capacity. Graduate Theses and Dissertations. Paper 13225. http://lib.dr.iastate.edu/etd/13225 (2013)

  76. 76.

    Pozo, D., Sauma, E., Contreras, J.: A three-level static MILP model for generation and transmission expansion planning. IEEE Trans. Power Syst. 28(1), 202–210 (2013)

    Article  Google Scholar 

  77. 77.

    Denholm, P., Drury, E., Margolis, R.: The solar deployment system (SolarDS) model: documentation and sample results. Technical Report, NREL/TP-6A2-45832, Sep 2009

  78. 78.

    Krishnan, V., Kastrouni, E., Pyrialakou, D., Gkritza, K., McCalley, J.: An optimization model of energy and transportation systems: assessing the high-speed rail impact in the United States. Transp. Res. Part C: Emerg. Technol. 54, 131–156 (2015)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to acknowledge the National Association of Regulatory Utility Commissioners (NARUC) for supporting our efforts in writing the whitepaper on co-optimization [23]. The authors are also grateful to Bob Pauley, Doug Gotham, Stan Hadley, and Patrick Sullivan for their comments. Opinions expressed in this paper, however, as well as any errors or omissions, are the authors’ alone.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Venkat Krishnan.

Appendices

Appendix A.1: Full ACOPF-GTEP model

This appendix provides a detailed description of a general ACOPF based multiyear Generation-Transmission Expansion Problem (ACOPF-GTEP), which is a minimization problem defined by the following Eqs. (1)–(12). The objective function is the present worth of operation and annualized investment costs over several years t. The investment variables are the number of new lines that are built in each corridor (integer variables) and new generation capacity (modelled as being continuous). More detailed formulations can include unit commitment costs, maintenance expenses, and environmental damages in the objective; pollution limits, ramping limits, and other operational constraints in the constraint set; and integer variables for generation investments and start-ups. Constraints (2)–(3) model the nodal real and reactive power balances, which are affected by integer transmission expansion variables of candidate corridors. Constraints (4) – (5) model non-linear AC power flow relations across line ij. Constraints (6)–(8) model the network security limits for voltage magnitude, voltage angle and apparent line power flow, the latter bounds being a function of line expansion decisions. Constraints (9)–(10) model generation power bounds, which depend on generation expansion variables. Constraints (11) and (12) model the bounds on the number of investments in transmission lines and generation size respectively within a given investment time period.

$$\begin{aligned} {\mathbf{Minimize}}&T_t \sum \limits _t {\sum \limits _{(i,j)} {C_i (t)P_{gi} (t)} +\sum \limits _t {\sum \limits _i {I_i (t)PI_{gi} (t)} +\sum \limits _t {\sum \limits _{(i,j)} {I_{(i,j)} (t)z_{(i,j)} (t)} } } }\nonumber \\\end{aligned}$$
(1)
$$\begin{aligned} {\mathbf{subject}}\,\mathrm{\mathbf{to}}&P_{gi} (t)-P_{di} (t)=\sum \limits _j {P_{(i,j)} ( {V,\theta ,t})\left( {z_{(i,j)} (0)+\sum \limits _{\tau =start}^t {z_{(i,j)} (\tau )} }\right) ,\forall i}\end{aligned}$$
(2)
$$\begin{aligned}&Q_{gi} (t)-Q_{di} (t)=\sum \limits _j {Q_{(i,j)} ( {V,\theta ,t})\,\left( {z_{(i,j)} (0)+\sum \limits _{\tau =start}^t {z_{(i,j)} (\tau )} }\right) ,\;\forall i}\nonumber \\\end{aligned}$$
(3)
$$\begin{aligned}&P_{(i,j)} (V,\theta ,t)=V_i^2 (t)G_{(i,i)} +\sum \limits _{j=1,j\ne 1}^N {\vert V_i (t)\vert \left| {V_j (t)} \right| } ( G_{(i,j)} cos \theta _{ij}\nonumber \\&\quad \quad \quad \quad \quad \quad \quad \quad +B_{(i,j)} sin \theta _{ij} )\end{aligned}$$
(4)
$$\begin{aligned}&Q_{(i,j)} (V,\theta ,t)\;=\;-V_i ^2( t){}{}B_{(i,i)} +\sum \limits _{j=1,j\ne i}^N {\left| {V_i ( t)} \right| } \left| {V_j ( t)} \right| ( G_{(i,j)} sin\theta _{ij}\nonumber \\&\qquad \qquad \qquad \qquad \qquad -B_{(i,j)} cos\theta _{ij} )\end{aligned}$$
(5)
$$\begin{aligned}&V_i^{min} \le V_i (t)\le V_i^{max}\end{aligned}$$
(6)
$$\begin{aligned}&-\pi \le \theta _i (t)\le +\pi \end{aligned}$$
(7)
$$\begin{aligned}&0\;\le \;P^2_{(i,j)} (t)+Q^2_{(i,j)} (t)\;\le \;S^2_{(i,j)} {}^{max}\,\left( {z_{(i,j)} (0)+\sum \limits _{\tau =start}^t {z_{(i,j)} (\tau )} }\right) \nonumber \\\end{aligned}$$
(8)
$$\begin{aligned}&P_{gi} ^{min}\;\le \;P_{gi} (t)\;\le \;P_{gi}^{max}+\sum \limits _{\tau =start}^t {PI_{gi} (\tau )}\end{aligned}$$
(9)
$$\begin{aligned}&Q_{gi} ^{min}\;\le \;Q_{gi} (t)\;\le \;Q_{gi}^{max}+\beta \sum \limits _{\tau =start}^t {PI_{gi} (\tau )}\end{aligned}$$
(10)
$$\begin{aligned}&0\le z_{(i,j)} (t)\le n_{(i,j)}^{max}\end{aligned}$$
(11)
$$\begin{aligned}&0\le PI_{gi} (t)\le PI_{gi}^{max} \end{aligned}$$
(12)

In this model, t and \(\tau \) are time periods, start is the first year in which investments can be made, \(T_{t}\) is the number of hours in an individual operating time interval, \(C_{i}\) is the operational cost of generation in $/MWh, \(I_{i}\) is the investment cost of generation in $/MW, \(I_{(i,j)}\) is the investment cost of transmission line in $/MW, \(P_{gi}\) is the real power generation in MW (decision variable), \(Q_{gi}\) is the reactive power generation in MVar (decision variable), \(P_{gi}^{min}\) and \(P_{gi}^{max}\) are the minimum and maximum real power generation in MW, \(Q_{gi}^{min }\) and \(Q_{gi}^{max}\) are the minimum and maximum reactive power generation in MVar, \(PI_{gi}\) is the generation capacity investment (continuous decision variable), \(PI_{gi}^{max}\) is the maximum allowable generation capacity investment, \(z_{(i,j)}(0)\) is number of existing transmission lines in a corridor (ij),\( z_{(i,j)}\) is the transmission investment (integer decision variable), \(n_{(i,j)}^{max}\) is the maximum allowable transmission lines across a corridor, \(P_{(i,j)}\) is the real power flow across line ij (decision variable), \(Q_{(i,j)}\) is the reactive power flow across line ij (decision variable), \(S_{(i,j)}^{max}\) is the maximum allowed apparent power flow across line ij, \(P_{di}\) is the real power demand, \(Q_{di}\) is the reactive power demand, \(V_{i}\) is the bus voltage magnitude (decision variable), \(V_{i}^{max}\) is the maximum limit on bus voltage magnitude, \(\theta _{i}\) is the bus voltage angle (decision variable), \(\theta _{ij}\) is the bus voltage angle difference, \(G_{(i,j)}\) is the line conductance (Y-bus element), \(B_{(i,j)}\) is the line susceptance (Y-bus element), and \(\beta \) is the maximum real to reactive power conversion constant at rated voltage based on generator capability curve.

Relaxed ACOPF-GTEP model using binary variables

Instead of the integer decision variable in (11) for optimizing the total number of lines to be built across a transmission corridor, a binary decision variable can be used to decide if a certain candidate line across a corridor should be built at any time (\(s_{b}(t))\) or not? This replaces the integer variable with multi-stage to a variable with two stages (0 or 1), thereby reducing the problem complexity. In such a case, the existing arc (by Eq. (15)) and a candidate arc (by Eq. (16)) are represented individually in the network, and Eqs. (1318) are used in the model instead of Eqs. (211). The presented formulation can be further improvised by giving it the ability to consider investments in multiple lines across a corridor. This can be modeled by designing many candidate arcs across that corridor, each having its own binary decision variable. Though this MINLP formulation will have a larger number of constraints and variables due to the inclusion of many candidate arcs, it will have reduced problem solving complexity due to the removal of multi-stage integer variables.

$$\begin{aligned}&P_{gi} (t)-P_{di} (t)\;=\;\sum \limits _{j} {P_{(i,j)} ( {V,\theta ,t})\;z_{(i,j)} (0)\;+\;P_{(i,j)} ( {V,\theta ,t})\,s_{b_{(i,j)}} (t)}\end{aligned}$$
(13)
$$\begin{aligned}&{Q_{gi}}(t) - {Q_{di}}(t)\; = \;\sum \limits _j {{Q_{(i,j)}}\left( {V,\theta ,t} \right) \;{z_{(i,j)}}(0)\; + \;{Q_{(i,j)}}\left( {V,\theta ,t} \right) \,{s_b}_{(i,j)}(t)}\qquad \quad \end{aligned}$$
(14)
$$\begin{aligned}&0\le P^2_{(i,j)} (t)+Q^2_{(i,j)} (t)\le z_{(i,j)} (0) S^2_{(i,j)}{}^{max}\end{aligned}$$
(15)
$$\begin{aligned}&0\le P^2_{(i,j)} (t)+Q^2_{(i,j)} (t)\le s_{b(i,j)} (t)S^2_{(i,j)}{} ^{max}\end{aligned}$$
(16)
$$\begin{aligned}&0\le z_{(i,j)} (t)\le 1\end{aligned}$$
(17)
$$\begin{aligned}&0\;\le \;s_{b _{(i,j)}} (t)=\sum \limits _{\tau =start}^t {z_{(i,j)} (\tau )} \;\le \;1 \end{aligned}$$
(18)

where, \(s_{b(i,j)}(t)\) is a binary decision variable used to decide if a certain candidate line across the corridor (ij) should be built at any time or not. In the above model, Eqs. (13) and (14), which have binary variables multiplying non-linear power flow equations, can be further relaxed by considering a disjunctive formulation of MINLP problem using the big “M” method [41]. The MINLP or MILP formulation can be further relaxed to NLP or LP problem by assuming the transmission investment variable as continuous, instead of binary. The continuous variable can be constrained close to discrete 0 or 1 value by using a binding constraint relaxed using \(\varepsilon \), as shown in (19).

$$\begin{aligned} z_{(i,j)} (t)( {1-z_{(i,j)} (t)})\le \varepsilon \end{aligned}$$
(19)

The AC formulation also allows shunt devices such as MSCs (Mechanically switched capacitors) and SVCs (Static Var Compensators) as investment options. Their influence can be accounted within Eq. (5), which has bus shunt susceptance \(b_{i}(t)\) as shown in (20).

$$\begin{aligned} B_{(i,i)} (t)=\sum \limits _j {b_{(i,j)} (t)} +b_i (t) \end{aligned}$$
(20)

where \(b_{(i,j)}\) is the line susceptance and \(b_{i}\) is the bus shunt susceptance.

Appendix A.2: DC optimal power flow based generation-transmission expansion planning model

The DCOPF formulation is based on the following simplifications to ACOPF model:

  1. 1.

    R\(<<<\)X: The resistance of transmission circuits is significantly less than the reactance.

  2. 2.

    Voltage angle differences very small: For typical operating conditions, the difference in voltage angles for two buses is very low (at the max, 10\(^{\circ }\)–15\(^{\circ }\)). For smaller angle differences, the cosine function approaches 1.0 and the sine function is the angle difference itself (expressed in radians).

  3. 3.

    Voltage magnitudes are assumed to be 1.0 in the per-unit system.Footnote 3

The resulting power flow model has two equations, a real power flow Eq. (21) which is directly proportional to angle difference (in radians) and reactive power flow Eq. (22) which is directly proportional to bus voltage difference.

$$\begin{aligned}&P_{(i,j)} (\theta ,t)=B_{(i,j)} ( {\theta _i (t)-\theta _j (t)})\end{aligned}$$
(21)
$$\begin{aligned}&Q_{(i,j)} (V,t)=-b_i +\sum \limits _{j=1,j\ne 1}^N {\left| {b_{(i,j)} } \right| \left( {\left| {V_i (t)} \right| -\left| {V_j (t)} \right| }\right) } \end{aligned}$$
(22)

The DCOPF-GTEP problem has the following constraints as shown in Eqs. (2329) and Eqs. (7, 9, and 12) in Appendix Sect. A.1. It should be noted that in the formulation of (2329), the transmission investment decision variable is binary, and hence the network expansion problem is formulated using arcs representing existing (by Eqs. (24) and (25)) and candidate lines (by Eqs. (26) and (27)) individually, so if \(z_{(i,j)}(t) = \)1, this indicates that a certain candidate line across a corridor is to be built at time t.

$$\begin{aligned}&\sum \limits _i {P_{gi} ( t)=\sum \limits _i {P_{di} ( t)} }\end{aligned}$$
(23)
$$\begin{aligned}&P_{(i,j)} (t)=B_{(i,j)} ( {\theta _i (t)-\theta _j (t)})z_{(i,j)} (0)\end{aligned}$$
(24)
$$\begin{aligned}&-z_{(i,j)} (0)P_{(i,j)} ^{max}\le P_{(i,j)} (t)\le P_{(i,j)} ^{max}z_{(i,j)} (0)\end{aligned}$$
(25)
$$\begin{aligned}&P_{(i,j)} (t)=B_{(i,j)} ( {\theta _i (t)-\theta _j (t)})s_{b_{(i,j)} } (t)\end{aligned}$$
(26)
$$\begin{aligned}&-s_{b_{(i,j)} } (t)P_{(i,j)} ^{max}\le P_{(i,j)} (t)\le P_{(i,j)} {}^{max}s_{b_{(i,j)} } (t)\end{aligned}$$
(27)
$$\begin{aligned}&0\le z_{(i,j)} (t)\le 1\end{aligned}$$
(28)
$$\begin{aligned}&0\;\le \;s_{b _{(i,j)}} (t)=\sum \limits _{\tau =start}^t {z_{(i,j)} (\tau )} \;\le \;1 \end{aligned}$$
(29)

The above MINLP model can be relaxed to a MILP using a disjunctive formulation, sometimes referred to as the big “M” method, for candidate branches as shown in (3032), instead of (26).

$$\begin{aligned}&P_{(i,j)} (t)=B_{(i,j)} ( {\theta _i (t)-\theta _j (t)})+( {s_b (t)-1})M+U_b (t)\end{aligned}$$
(30)
$$\begin{aligned}&U_b (t)\le 2( {1-s_b (t)})M\end{aligned}$$
(31)
$$\begin{aligned}&U_b (t)>0 \end{aligned}$$
(32)

Appendix A.3: network flow based generation-transmission expansion planning model

The network flow model-based linear programming cost minimization formulation is shown in Eqs. (3336), where the operational arc flows and investments are minimized in (33). Since, both generation and transmission are represented as arcs (refer Fig. 2), i.e., as transportation pipelines (with different properties), the only equation that governs this model is (34), the nodal power flow balance equation. The efficiency term \(\eta _{(i,j)}\) in Eq. (34) for a generation arc represents its capacity factor and for a transmission arc its losses. Equation (35) represents the capacity constraint for both generation and transmission arcs.

$$\begin{aligned} {\mathbf{Minimize}}&\sum \limits _t {\sum \limits _{(i,j)} {C_{(i,j)} } (t)P_{(i,j)} (t)+\sum \limits _i {\sum \limits _{(i,j)} {I_{(i,j)} (t)} PI_{(i,j)} } (t)}\end{aligned}$$
(33)
$$\begin{aligned} {\mathbf{Subject}\, \mathbf{to}}&\sum \limits _i {\eta _{(i,j)} (t)P_{(i,j)} (t)-\sum \limits _k {P_{(j,k)} (t)} =d_j (t)}\end{aligned}$$
(34)
$$\begin{aligned}&P_{(i,j)} ^{min}\;\le \;P_{(i,j)} (t)\;\le \;P_{(i,j)} ^{max}+\sum \limits _{\tau =start}^t {PI_{(i,j)} (\tau )}\end{aligned}$$
(35)
$$\begin{aligned}&0\le PI_{(i,j)} (t)\le PI_{(i,j)} ^{max} \end{aligned}$$
(36)

DC lines are modeled as real power injections (positive and negative) at both the ends of the lines, which effectively translate to modeling it as a transportation pipeline. Equation (37) shows the inclusion of power injection from a DC line into nodal real power balance equation. To consider DC lines among the transmission investment options, candidate arcs for DC lines are created separately from AC lines with appropriate cost and operational characteristics. The cost may also include the power electronics component costs at both the terminals.

$$\begin{aligned} P_{gi} (t)-P_{di} (t)\;=\;P_{(i,j)} ( {V,\theta ,t})\;z_{(i,j)} (0)\;+P_{(i,j)} ( {V,\theta ,t})\,s_{b _{(i,j)}} (t)+P_{(i,j)} ^{HVDC}(t) \end{aligned}$$
(37)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Krishnan, V., Ho, J., Hobbs, B.F. et al. Co-optimization of electricity transmission and generation resources for planning and policy analysis: review of concepts and modeling approaches. Energy Syst 7, 297–332 (2016). https://doi.org/10.1007/s12667-015-0158-4

Download citation

Keywords

  • Co-optimization
  • Transmission expansion planning
  • Generation expansion planning
  • Model fidelity
  • Energy storage
  • Demand response
  • Integrated network uncertainty
  • Long-term planning
  • AC and DC power flow