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Formulation

Currently, ProxAL is designed to solve contingency-constrained AC Optimal Power Flow (ACOPF) formulations over multiple time periods.

Time coupling

Each time period $t \in T$ involves the solution of an ACOPF with active $p_{dt}$ and reactive $q_{dt}$ load forecasts, which may differ from one time period to the next. In each time period $t \in T$, we must determine the 'base-case' active power generation level of generator $g \in G$, denoted by $p^0_{gt}$. The active power generations in consecutive time periods are constrained by the generator's ramping capacity, which can be modeled as follows:

\[-r_g \leq p^0_{g,t-1} - p^0_{g,t} \leq r_g \qquad \forall g \in G, \; \forall t \in T \setminus \{1\}.\]

Here, $r_g$ denotes the ramping capacity of generator $g$ (per unit of time in which $T$ is defined).

For convenience, ProxAL also provides the functionality to solve the full/"non-decomposed" model using JuMP/Ipopt. In this case, one can switch between the above inequality form, or an equivalent equality form, as well as a quadratic penalty form of the ramping constraint by setting the time_link_constr_type field of ProxAL.ModelParams in Model parameters. When solving the full/"non-decomposed" model with the quadratic penalty form of the ramping constraints, the user must provide a value for the corresponding quadratic penalty parameter by setting θ_t in Algorithm parameters.

Contingency constraints

Each single-period ACOPF problem may itself be constrained further by a set of transmission line contingencies, denoted by $K$. The active and reactive power generations, and bus voltages must satisfy the following constraints in each time period and each contingency:

  1. the power flow equations,
  2. bounds on active and reactive generation and voltage magnitudes, and
  3. line power flow limits.
Note

It is possible that the problem parameters are such that (some of) the above constraints can become infeasible. To model this, ProxAL also allows constraint infeasibility (except on variable bounds) by penalizing them in the objective function with a quadratic penalty. This can be controlled by setting allow_constr_infeas = true and a corresponding value for weight_constr_infeas in Model parameters.

Note

Line power flow limits can be disabled by setting allow_line_limits = false.

The contingencies in each time period are linked together via their active power generations in one of several forms. The choice of the form can be set using the ctgs_link_constr_type field of ProxAL.ModelParams in Model parameters.

  • Preventive mode: active power generation in contingency $k$ must be equal to the base case value. This constraint has one of two forms:

    • Preventive equality: This is the original form of the constraint. For numerical convergence reasons, ProxAL does not allow using this form whenever the decompCtgs field of ProxAL.AlgParams is set to true, see Algorithm parameters.

    \[p_{gt}^k = p_{gt}^0 \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T.\]

    • Preventive penalty: In this form, ProxAL introduces additional continuous variables $z_{g,k,t}$ along with the following constraints. Note that a quadratic penalty term $θ_c \|z_k \|^2$ is also added to the objective function, where the parameter $θ_c$ is controlled within ProxAL whenever the decompCtgs field of ProxAL.AlgParams is set to true. Otherwise, its value can be set using the θ_c field of ProxAL.AlgParams in Algorithm parameters.

    \[p_{gt}^k = p_{gt}^0 + z_{gkt} \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T.\]

  • Corrective mode: active power generation is allowed to deviate from base case by up to $\Delta$ fraction of its ramping capacity. The parameter $\Delta$ can be set using the corr_scale field of ProxAL.ModelParams in Model parameters. This constraint has one of two forms:

    • Corrective inequality: This is the original form of the constraint. For numerical convergence reasons, ProxAL does not allow using this form whenever the decompCtgs field of ProxAL.AlgParams is set to true, see Algorithm parameters.

    \[0.1\, r_g \leq p_{gt}^k - p_{gt}^0 \leq \Delta \, r_g \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T\]

    • Corrective equality: In this form, ProxAL introduces additional continuous variables $s_{g,k,t}$ along with the following constraints. As before, ProxAL does not allow using this form whenever the decompCtgs field of ProxAL.AlgParams is set to true.

    \[\left.\begin{aligned} 0 \leq s_{gkt} \leq 2 \Delta \, r_g \\ p_{gt}^0 - p_{gt}^k + s_{gkt} = \Delta \, r_g \end{aligned}\right\} \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T\]

    • Corrective penalty: In this form, ProxAL introduces additional continuous variables $s_{g,k,t}$ and $z_{g,k,t}$ along with the following constraints. A quadratic penalty term $θ_c \|z_k \|^2$ is also added to the objective function, where the parameter $θ_c$ is controlled within ProxAL whenever the decompCtgs field of ProxAL.AlgParams is set to true. Otherwise, its value can be set using the θ_c field of ProxAL.AlgParams in Algorithm parameters.

    \[\left.\begin{aligned} 0 \leq s_{gkt} \leq 2 \Delta \, r_g \\ p_{gt}^0 - p_{gt}^k + s_{gkt} + z_{gkt} = \Delta \, r_g \end{aligned}\right\} \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T\]

  • Frequency control mode: In this case, ProxAL defines new continuous variables $\omega_{kt}$ which is the (deviation from nominal) system frequency in contingency $k$ of time period $t$, and $\alpha_g$ is the droop control parameter of generator $g$. The objective functions includes an additional term $w_\omega \| \omega \|^2$, where the parameter $w_\omega$ must be set using the weight_freq_ctrl field of ProxAL.ModelParams in Model parameters. This constraint has one of two forms:

    • Frequency equality: This is the original form of the constraint. For numerical convergence reasons, ProxAL does not allow using this form whenever the decompCtgs field of ProxAL.AlgParams is set to true, see Algorithm parameters.

    \[p_{gt}^k = p_{gt}^0 + \alpha_g \omega_{kt} \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T.\]

    • Frequency penalty: In this form, ProxAL introduces additional continuous variables $z_{g,k,t}$ along with the following constraints. Note that a quadratic penalty term $θ_c \|z_k \|^2$ is also added to the objective function, where the parameter $θ_c$ is controlled within ProxAL whenever the decompCtgs field of ProxAL.AlgParams is set to true. Otherwise, its value can be set using the θ_c field of ProxAL.AlgParams in Algorithm parameters.

    \[p_{gt}^k = p_{gt}^0 + \alpha_g \omega_{kt} + z_{gkt} \qquad \forall g \in G, \; \forall k \in K, \; \forall t \in T.\]