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


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.

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  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.


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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.

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Correspondence to Venkat Krishnan.


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.

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

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.

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

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).

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

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).

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

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.

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

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.

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

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).

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

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.

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

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.

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

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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).

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  • 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