Abstract
Currently, there is a national push for a smarter electric grid, one that is more controllable and flexible. Only limited control and flexibility of electric assets is currently built into electric network optimization models. Optimal transmission switching is a low cost way to leverage grid controllability: to make better use of the existing system and meet growing demand with existing infrastructure. Such control and flexibility can be categorized as a “smart grid application” where there is a co-optimization of both generators or loads and transmission topology. In this paper we form the dual problem and examine the multi-period N-1 reliable unit commitment and transmission switching problem with integer variables fixed to their optimal values. Results including LMPs and marginal cost distributions are presented for the IEEE RTS 96 test problem. The applications of this analysis in improving the efficiency of ISO and RTO markets are discussed.
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Abbreviations
- c::
-
Operating state; c=0 indicates the no contingency state (steady-state); c>0 is a single contingency state, i.e. c∈C, C=CT∪CG.
- e::
-
Generator, load, or transmission element.
- g::
-
Generator or load; for generators, g∈G; for load, g∈D.
- g(n)::
-
Set of generators or load at node n.
- k::
-
Transmission element (line or transformer).
- k(n,.)::
-
Set of transmission assets with n as the ‘from’ node.
- k(.,n)::
-
Set of transmission assets with n as the ‘to’ node.
- m,n::
-
Nodes.
- n(g)::
-
Generator g located at bus n.
- t::
-
Time period; t=1,…,T.
- B k ::
-
Electrical susceptance of transmission element k.
- c g ::
-
Production cost for generator (or value of load) g; generally c g >0.
- CG::
-
Set of generator contingencies.
- \(cr^{+}_{g},cr^{-}_{g}\) ::
-
Ramp rate cost in the up and down direction for generator (or load) g.
- CT::
-
Set of transmission contingencies.
- d n ::
-
Real power load (fixed) at bus n.
- N1 ec ::
-
Binary parameter that is 0 when element e is the contingency and is 1 otherwise.
- \(P^{+}_{gc},P^{-}_{gc}\) ::
-
Max and min capacity of generator (or load) g in state c; for \(g\in D,P^{+}_{gc}=P^{-}_{gc}\) .
- \(P^{+}_{kc},P^{-}_{kc}\) ::
-
Max and min rating of transmission element k in state c; for lines \(P^{+}_{kc}=-P^{-}_{kc}\) .
- \(R^{+}_{gt},R^{-}_{gt}\) ::
-
Max ramp rate in the up and down direction for generator (or load) g at node n in period t except in the startup period.
- \(R^{c+}_{gt},R^{c-}_{gt}\) ::
-
Max emergency (contingency) ramp rate in the up and down direction for generator (or load) g in period t.
- \(R^{s}_{g}\) ::
-
Max ramp rate for the start up period in the up direction for generator (or load) g at node n.
- SU gt ::
-
Startup cost for generator (or load) g in period t; generally for g∈G,SU gt ≥0.
- T::
-
Number of periods.
- UT g ,DT g ::
-
Min up and down time for generator (or load) g.
- θ+,θ−::
-
Max and min voltage angle; θ +=−θ −.
- ρ c ::
-
Contingency c indicator; ρ c =1 for c=0; ρ c =0, otherwise.
- P gct ::
-
Real power supply from generator g(>0) or demand from load (<0)g (at node n), in state c and period t.
- P kct ::
-
Real power flow from node n to m for transmission element k, in state c and period t.
- \(r^{+}_{gct},r^{-}_{gct}\) ::
-
Ramp rate in the up and down direction for generator g, in state c and period t.
- u gt ::
-
Binary unit commitment variable for generator (or load) g in period t (0 down, 1 operational).
- v gt ::
-
Startup variable for generator (or load) g in period t (1 for startup, 0 otherwise).
- w gt ::
-
Shutdown variable for generator (or load) g in period t (1 for shutdown, 0 otherwise).
- z kt ::
-
Binary variable for transmission element k in period t (0 open/not in service, 1 closed/in service).
- \(\alpha^{-}_{nct},\alpha^{+}_{nct}\) ::
-
Marginal value of lowering (raising) the min (max) phase angle at node n, in state c and period t.
- \(\beta^{-}_{gct},\beta^{+}_{gct}\) ::
-
Marginal value of reducing (increasing) the min (max) level of generator (or load) g, in state c and period t.
- γ g ::
-
Uplift or additional profit for generator g.
- δ kt ::
-
Marginal value of switching transmission element k in period t.
- \(\eta^{-}_{kct},\eta^{+}_{kct}\) ::
-
Marginal value of reducing (increasing) the lower (upper) limit for transmission element k, in state c and period t.
- θ nct ::
-
Bus voltage angle at node n, in state c and period t.
- λ nct ::
-
Marginal value of a unit of generation or load at node n, in state c and period t.
- μ kct ::
-
Marginal susceptance value of transmission element k, in state c and period t.
- σ gt ::
-
Marginal value of enforcing the startup value for generator g in period t.
- τ gt ::
-
Marginal value of enforcing the relationship between startup, shutdown, and unit commitment variables for generator g in period t.
- \(\chi^{+}_{gct},\chi^{-}_{gct}\) ::
-
Marginal value of increasing the up (down) ramp rate for generator g, in state c and period t.
- \(\chi^{c+}_{gct},\chi^{c-}_{gct}\) ::
-
Marginal value of increasing the up (down) emergency ramp rate for generator g, in state c and period t.
- ψ gt ::
-
Marginal value of enforcing the shutdown value for generator g in period t.
- \(\omega^{+}_{gct},\omega^{-}_{gct}\) ::
-
Marginal value of increasing the up (down) ramp rate constraint for generator g, in state c and period t.
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O’Neill, R.P., Hedman, K.W., Krall, E.A. et al. Economic analysis of the N-1 reliable unit commitment and transmission switching problem using duality concepts. Energy Syst 1, 165–195 (2010). https://doi.org/10.1007/s12667-009-0005-6
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DOI: https://doi.org/10.1007/s12667-009-0005-6