JP7610003B2 - Power Grid Resource Allocation - Google Patents

Description

TECHNICAL FIELD The present disclosure relates generally to power grids and power grid operation, and in particular embodiments to power grid resource allocation.

Background Independent system operators (ISOs) use unit commitment (UC) to obtain generation resource and demand commitment and dispatch in the power grid. Unit commitment establishes the commitment state and generation levels of all generators, energy storage systems, and price-dependent loads over a scheduling horizon to minimize the total generation cost while meeting all system and area constraints, such as load balance and spinning reserve requirements, transmission network constraints, and individual unit operation constraints. Unit commitment is often formulated as a mixed integer linear programming (MILP) problem.

Since a typical power grid is operated closer and closer to its security margin, security related transmission constraints are included to constrain unit commitment. Thus, a typical power system resource scheduling implies security constrained unit commitment (SCUC), where the security constraint may be, for example, a transmission line thermal capacity constraint for base case and contingency operating conditions. SCUC is used in day-ahead, intraday, and real-time power grid scheduling. In solving for security constrained unit commitment, a minimum-cost operating schedule for generators (which may also be referred to as power plants) is determined over the scheduling horizon. For example, a minimum-cost operating schedule is identified that satisfies the operating constraints of each generator unit, the electrical network constraints in the base case network topology, and various operator-specified contingency scenarios.

SUMMARY In some embodiments, a method of operating a power grid includes generating, by a power management system of the power grid, an intermediate resource allocation schedule, where the intermediate resource allocation schedule provides a tentative schedule of resource allocations for power grid resources operating in the power grid; determining, by the power management system, whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a coupling constraint of a power grid resource allocation profile of the power grid, where the power grid resource allocation profile indicates operation of the power grid constrained by operational information of the power grid; and in response to determining that the intermediate resource allocation schedule is infeasible, repairing, by the power management system, the intermediate resource allocation schedule to generate a feasible resource allocation schedule, where the resource allocation schedule provides a final schedule of resource allocations for the power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible.

In some embodiments, a power management system for a power grid includes one or more processors and a memory storing a program executed within the processor, the program including instructions, the instructions when executed within the one or more processors, for generating an intermediate resource allocation schedule, the intermediate resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating within the power grid, and determining whether the intermediate resource allocation schedule is executable by checking whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid, the power grid being configured to generate an intermediate resource allocation schedule for the power grid resources. and a memory for causing the intermediate resource allocation schedule to generate a feasible resource allocation schedule in response to determining that the intermediate resource allocation schedule is infeasible, the intermediate resource allocation schedule providing a final schedule of resource allocations for the power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule including determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible.

In some embodiments, a system for a power grid is a power management system, the system comprising: generating an intermediate resource allocation schedule, the intermediate resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating in the power grid; determining whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid, the power grid resource allocation profile directing operation of the power grid constrained by the power grid operational information; and in response to determining that the intermediate resource allocation schedule is not feasible, determining whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid. The power management system includes a power grid resource configured to generate a feasible resource allocation schedule by repairing the allocation schedule, the resource allocation schedule providing a final schedule of resource allocation for power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible; the power grid resource configured to be controlled according to the resource allocation schedule; and a transmitter configured to transmit the resource allocation schedule from the power management system to the power grid resource.

BRIEF DESCRIPTION OF THE DRAWINGS The details of one or more embodiments of the present disclosure are set forth in the accompanying drawings and the following description. Other features, objects, and advantages of the present disclosure will become apparent from the description and drawings, and from the claims. In the figures, identical reference characters generally indicate the same component parts throughout the various views, which are generally not re-described in the interest of brevity. For a more complete understanding of the present disclosure, reference is now made to the following description in conjunction with the accompanying drawings.

FIG. 1 illustrates a block diagram of a power grid having a power management system (PMS) in the control loop of the power grid in accordance with some embodiments. 1 is a functional block diagram of a power management system (PMS) in some embodiments. 1 is a functional block diagram of a Security Constrained Unit Commitment (SCUC) solver in accordance with some embodiments. FIG. 2 illustrates a flowchart of a method for solving the Lagrangian-relaxed (LR) dual of a Security Constrained Unit Commitment (SCUC) system in an embodiment. FIG. 2 illustrates a flowchart of a method for repairing an infeasibility of a power resource allocation profile in an embodiment. 1 is a block diagram of a processing system for performing operations of a power management system according to some embodiments. 1 is a flowchart of a method for operating a power grid according to some embodiments.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS The making and using of the presently preferred embodiments is discussed in detail below. However, it should be appreciated that the present disclosure provides many applicable inventive concepts that can be embodied in a wide variety of specific contexts. The specific embodiments discussed are merely illustrative of specific ways to make and use the present disclosure and do not limit the scope of the present disclosure.

Solving a large-scale security constrained unit commitment (SCUC) can require significant time and computing resources due to its large dimension and complexity. An embodiment of the present disclosure describes an SCUC solver that efficiently solves large-scale SCUC. An exemplary SCUC solver first finds a tentative power resource allocation schedule by solving the Lagrangian Relaxation (LR) dual of the power grid resource allocation profile. Then, the SCUC solver checks whether the tentative resource allocation schedule is feasible. If the tentative resource allocation schedule is infeasible, the SCUC solver repairs the infeasibility of the tentative resource allocation schedule to achieve a feasible resource allocation schedule.

1 illustrates a block diagram of a power grid 100 in some embodiments. The power grid 100 (which may also be referred to as a power system) includes multiple power resources 101 (e.g., power generators, or energy storage systems). The power resources 101 may be of different types, such as thermal plants (e.g., coal, natural gas, nuclear plants), hydroelectric plants, renewable resource power plants (e.g., wind farms, solar plants), energy storage systems, and demand response programs. Depending on the power plant type, each of the power resources 101 may be subject to complex technical and business constraints, such as minimum up/down time, ramp up/down rate, regulation/stability (e.g., a unit may not change its production level too many times), and start up/shut down ramp rate (e.g., when starting/stopping, a unit follows a specific power curve, which may depend on how long the plant has been offline/online).

The power grid 100 also includes a transmission grid 103, which is an electrical grid that distributes electricity generated by the power resources 101 (e.g., power plants) to consumers. The transmission grid 103 is an interconnected network that may span a large geographic region (e.g., a country). The complex characteristics of the transmission grid 103, such as network topology, equipment parameters, and line flow limitations, as well as resource response rates, may need to be considered in solving the security constrained unit commitment system.

As illustrated in FIG. 1, the power grid 100 further includes a power management system (PMS) 105, which may be a market management system (MMS) or an emergency management system (EMS). The PMS 105 may include various function blocks for implementing various functions, such as function blocks 200 and 300 of FIG. 3. For example, the PMS 105 may include a clearing engine that runs a market scheduling application to clear the market to establish market clearing prices for energy and ancillary services (such as various reserves and responses) and schedule commitments and dispatches for reliability.

In some embodiments, the clearing engine solves the security constrained unit commitment system to provide a resource allocation schedule for the power grid 100. The resource allocation schedule may include commitment decisions (e.g., whether a power plant is online and capable of producing energy at any time), production (also called dispatch) decisions (e.g., how much energy a power plant is producing at any time), and controllable network element decisions (e.g., power flows on HVDC and AC transmission lines with phase angle regulators). The resource allocation schedule may also be referred to as an operating schedule for the power grid 100. In some embodiments, after being generated, the resource allocation schedule is sent to the power resources 101, for example, by a transmitter (which may be part of the power management system 105), so that the power resources 101 are operated according to the resource allocation schedule.

In some embodiments, the resource allocation schedule is generated by solving a security constrained unit commitment system while taking into account the power grid 100's operational information (e.g., various constraints), such as power resource 101 constraints (e.g., generation capacity and operating margins, including safe operating ranges), transmission grid 103 constraints, and market information (e.g., bidding information, cost information, forecasted demand information, regulatory requirements such as emission targets). The goal of the resource allocation schedule is to clear market offers and bids while maximizing total welfare or minimizing total cost. In one exemplary embodiment, the goal may be to meet energy demand while minimizing energy production costs subject to reliability and emission constraints. In another exemplary embodiment, the goal may be to maximize energy production profits, e.g., the difference between revenue (from selling electricity) and cost (from producing electricity).

FIG. 2 is a functional block diagram of a power management system (PMS) 105 in an embodiment. The PMS 105 in FIG. 2 may be used in the power management system 105 of FIG. 1, for example. For simplicity, not all functional blocks of the PMS 105 are illustrated in FIG. 2.

2, the PMS 105 formulates the SCUC problem as a mixed integer programming (MIP) representation 133 by using the SCUC model 131 and applying market and grid data 132 to the SCUC model 131. Thus, the MIP representation 133 is, in some embodiments, a mathematical model with constraints (e.g., 131). Details of the SCUC model and the formulation of the MIP representation 133 are described below. In the discussion herein, the term mixed integer programming (MIP) may be used interchangeably with mixed integer linear programming (MILP).

The PMS 105 further includes an SCUC solver 134 configured to generate an SCUC solution 135 (e.g., a resource allocation schedule). In some embodiments, the SCUC solver 134 is configured to implement function blocks 200 and 300 of FIG. 3, which are described in more detail below.

In an embodiment, the SCUC model 131 for a security constrained unit commitment (SCUC) system solves the following optimization problem:

This is formulated as follows:

and

where N is the number of power grid resources (e.g., the number of power plants and loads in the power grid), j is a resource index, _xj is a decision vector for resource j, _cj is a cost coefficient vector for resource j, Aj is a constraint matrix for resource j, _{Ac,j is} a coupling constraint matrix for resource j, _bj is a constraint boundary vector for resource j, and _bc is a constraint boundary vector of the coupling constraints. The constraints in equation (1.2) may be referred to as the constraints of each of the power grid resources, and the constraints in equation (1.3) may be referred to as the coupling constraints of the SCUC system. In the discussion herein, the SCUC system (also referred to as the SCUC problem) of equations (1.1)-(1.3) may be referred to as a power grid resource allocation function, or a power grid resource allocation profile. Note that in equation (1.1), the vector x is used to represent all of the decision vectors _xj ,j=1, 2, ..., N. Throughout the discussion herein, unless otherwise specified, the same mathematical symbols in different mathematical formulas (e.g., xj, _cj , _Aj , Ac _,j ) refer to the _same or similar entities.

Therefore, solving the security constrained unit commitment system is equivalent to finding a decision vector x _{j , j} = 1, 2, ..., N, that minimizes the loss function (or cost function) shown in equation (1.1) while satisfying various constraints in equations (1.2) and (1.3), where the decision vector x _j includes information about the resource allocation schedule for the jth power grid resource. For example, each decision vector x _j includes binary information about the jth power resource (e.g., a commitment decision to turn on or off the jth power resource in a particular scheduled time slot). In addition, each decision vector x _j includes non-binary (e.g., continuous value) information about the jth power resource, such as production information (e.g., the amount of energy to produce). The decision vectors x _j , j = 1, 2, ..., N, are used to form a power resource allocation schedule that provides a schedule of resource allocation for the power grid resources operating in the power grid (e.g., whether to turn on or off a power station in a particular time slot and how much power to generate, etc.).

In some embodiments, various constraints of the power grid 100 discussed above (e.g., market and grid data 132) may be included in the constraint matrix _Aj and/or the combined constraint matrix _Ac,j . The cost coefficient vector _cj may include information such as cost information of energy production. In various embodiments, the security constrained unit commitment system is a large-scale mixed integer linear non-convex optimization problem. The security constrained unit commitment system is non-deterministic polynomial-time hardness (NP hard) and scales poorly with parallel MIP solvers using conventional methods.

As illustrated in the example of FIG. 2, a security constrained unit commitment (SCUC) system may be represented as a mixed integer programming (MIP) representation 133. Traditionally, a MIP solver is used to solve the SCUC problem. A MIP solver may be a software package that runs on a processor (e.g., a computer) to solve an MIP problem (e.g., a security constrained unit commitment system). A commercially available general-purpose MIP solver, such as CPLEX or Gurobii, may be used to solve the security constrained unit commitment system. An open source MIP solver, such as CBC, may also be used as a MIP solver.

Typically, a MIP solver searches through a vector space of decision vectors through a series of branch-and-bound (B&B) and/or branch-and-cut (B&C) operations to find integer feasible solutions with increasingly smaller loss functions for the SCUC system, thereby gradually approaching an optimal solution (e.g., the set of decision vectors x _i corresponding to the minimum loss function for the SCUC system). The MIP solver computes and updates values for the decision vectors in multiple iterations/steps such that the loss function decreases over time until the MIP solver reaches a final solution that is an optimal solution for the SCUC system or is sufficiently close to the optimal solution. The path from an initial (e.g., tentative) solution to a final solution (e.g., the vector space traversed by the MIP solver) is said to be the convergence path of the MIP solver (or of the SCUC system), and the MIP solver is said to converge to a final solution along this convergence path.

For example, due to the larger number of variables (e.g., hundreds, thousands, or millions) in the decision vector and the fact that the decision vector contains both integer-valued and real-valued variables (e.g., with continuous values), it may take a long time and intensive computing resources for a MIP solver to find a feasible solution to an SCUC system. This disclosure describes an SCUC solver that uses a two-step process to efficiently find a feasible solution with good optimality (typically with a duality gap <0.5%) for an SCUC system.

FIG. 3 illustrates a functional block diagram of a security constrained unit commitment (SCUC) solver 134 in some embodiments. The SCUC solver 134 may be used in the SCUC solver 134 of FIG. 2. The SCUC solver 134 includes two functional blocks 200 and 300 for a two-step process to find a feasible solution to an SCUC system.

3, in function block 200, the SCUC solver 134 optimizes the Lagrangian Relaxation (LR) dual (also called the LR dual problem) of the SCUC problem of Equations (1.1)-(1.3) to generate an intermediate (e.g., tentative) resource allocation schedule. However, the intermediate resource allocation schedule may not satisfy all the coupling constraints of the SCUC system. The SCUC solver 134 then checks whether the tentative resource allocation schedule is feasible (e.g., satisfies all the constraints of the SCUC system). If the tentative resource allocation schedule is infeasible, the SCUC solver 134 repairs the infeasibility of the tentative resource allocation schedule to achieve a feasible resource allocation schedule in function block 300. Details of function blocks 200 and 300 are discussed below.

FIG. 4 illustrates a flowchart of a method 250 for optimizing the Lagrangian Relaxation (LR) dual of a Security Constrained Unit Commitment (SCUC) system in an embodiment. The process illustrated in FIG. 4 is used in the illustrated embodiment to implement the function block 200 of FIG. 3.

In block 201 of FIG. 4, the LR dual of the SCUC system is set up. In an exemplary embodiment, optimizing the LR dual of the SCUC problem is the following optimization problem:

This is formulated as

where λ is a Lagrange multiplier vector (e.g., a vector containing multiple Lagrange multipliers) and Φ (λ) is

This is the LR dual of SCUC, which is defined as

Receive.
By rewriting (2.3) into (2.4), the LR-dual problem in (2.3) and (2.5) can be broken down into several LR-dual subproblems:

Note that the LR duality subproblems are decomposed into, which is subject to the constraints of equation (2.5) (e.g., A _j x _j >_b _j , j=1, 2, ..., N.). Each of the LR duality subproblems in (2.6) contains only the j-th decision vector x _j and the j-th coupling constraint matrix A _c,j . Thus, the number of variables in each of the LR duality subproblems is much smaller than the number of variables in the original SCUC problem, and therefore can be solved much faster. In addition, each of the LR duality subproblems may be solved by a different solver (e.g., a MIP solver), and therefore, all of the LR duality subproblems may be solved in parallel by multiple solvers (e.g., N MIP solvers) to reduce the amount of time required to solve the LR duality subproblems. In other words, solving the LR duality problem can be achieved by solving multiple LR duality subproblems in parallel. In the discussion herein, a solver for solving the LR dual subproblem may be referred to as an LR dual subproblem solver. As described in more detail below, optimizing the LR dual of the SCUC problem is achieved by solving the LR dual problem and alternately updating the Lagrangian multiplier vector λ in multiple iterations.

Next, in block 203, the LR dual of the SCUC system is solved, for example, by solving multiple LR dual subproblems of Equation (2.6) using multiple LR dual subproblem solvers (e.g., multiple MIP solvers). Note that when solving the LR dual subproblems of Equation (2.6), the variables to be solved (e.g., optimized) are the decision vectors _xj , and the Lagrangian multiplier vector λ has fixed values (e.g., treated as a constant vector). In the first iteration, the Lagrangian multiplier vector λ is assigned an appropriate initial value (e.g., a zero vector). The Lagrangian multiplier vector λ is subsequently updated in each iteration (see blocks 205 and 207). Thus, in subsequent iterations, for the processing of block 203, the Lagrangian multiplier vector λ uses the updated value of the Lagrangian multiplier vector λ obtained from the previous iteration.

Next, in block 205, the partial gradient of the Lagrange multiplier vector λ is calculated using the following formula:

where the decision vector x _j in equation (2.7) is the decision vector x _j obtained from solving the LR dual problem in block 203.

Next, in block 207, the Lagrangian multiplier vector λ is updated using the calculated partial gradient sg from block 205. Those skilled in the art will readily appreciate that different methods for updating the Lagrangian multiplier vector λ are available and that any suitable update method may be used to update the Lagrangian multiplier vector λ. As a non-limiting example, the Lagrangian multiplier vector λ may be expressed as follows:

where Δ is the step size used in updating the Lagrangian multiplier vector λ and k is the iteration index.

Next, in block 209, the convergence of the LR-dual optimization is checked, for example, by checking the difference between the values of Φ(λ) achieved in the current iteration and the last iteration. For example, if the difference between Φ(λ) in the current iteration and the last iteration is smaller than a predefined threshold, it is determined that the convergence of the LR-dual optimization is achieved and processing proceeds to block 211. Another criterion for stopping the iterative processing may be that a predefined maximum number of iterations has been reached. If the convergence of the LR-dual optimization is not achieved and the maximum number of iterations has not been reached, processing returns to block 203 for the next iteration.

In block 211, an intermediate resource allocation schedule is formed based on the decision vector _xj of the solution to the LR dual problem (or LR dual problem solution for short). Since the LR dual problem solution may not be a feasible solution with respect to the relaxed constraints, the intermediate resource allocation schedule provides a tentative resource allocation schedule. Next, the feasibility of the LR dual problem solution is checked. If the decision vector _xj of the LR dual problem solution satisfies all the constraints in equations (1.2) and (1.3), the LR dual problem solution is said to be feasible, the intermediate resource allocation schedule is output by the SCUC solver 134 as the resource allocation schedule, and the processing in function block 300 is skipped.

On the other hand, if the decision vector _xj does not satisfy all the constraints in (1.2) and (1.3), the LR dual problem solution is said to be infeasible or have residual infeasibility. Note that in the illustrated embodiment, the solution to the LR dual subproblem (2.6) must satisfy the constraints in (2.5), which are the same as the constraints in (1.2), and therefore, to determine the feasibility of the LR dual problem solution, only the coupling constraints in (1.3) need to be checked. When any of the coupling constraints in (1.2) are not satisfied, the LR dual problem solution is said to have residual infeasibility due to the coupling constraints.

When the LR dual problem solution is determined to be infeasible, the intermediate resource allocation schedule is repaired by function block 300 (see FIG. 3) to obtain a feasible resource allocation schedule. Thus, the processing of function block 300 may also be referred to as repairing (residual) infeasibility of the intermediate resource allocation schedule, repairing (residual) infeasibility of the LR dual problem solution, or repairing (residual) infeasibility of the coupling constraints.

FIG. 5 illustrates a flowchart of a method 350 for repairing an infeasibility of a power resource allocation profile in an embodiment. The process illustrated in FIG. 5 is used to implement the functional block 300 of FIG. 3 in the illustrated embodiment.

5 is iterative, and therefore an index k is used to indicate the number of iterations of the iterative process. In block 301, the index k is assigned an initial value of zero. Next, in block 303, the dispatch problem (also called the economic dispatch problem) is solved. For the first iteration (e.g., when k=0), the dispatch problem is solved with fixed integer decisions from the LR duality problem solution (obtained in function block 200 of FIG. 3). For subsequent iterations (e.g., when k>0), the dispatch problem is solved with fixed integer decisions from the extended subproblem solution (obtained in block 309 of the last iteration). For ease of discussion, the dispatch problem may be referred to as being solved with fixed integer decisions without identifying the source of the fixed integer decisions (e.g., from the LR duality problem solution or from the extended subproblem solution). The dispatch problem is

This is formulated as

and

Receive.
The dispatch problem in (3.1)-(3.3) is similar to the original SCUC problem in (1.1)-(1.3), but with the addition of a slack variable vector S (containing real-valued slack variables). In other words, the dispatch problem is obtained by modifying the SCUC problem with the slack variables. The dispatch problem also includes a penalty vector M for the slack variable vector S. Note that the dispatch problem is solved with fixed integer decisions, which means that when solving the optimization problem in (3.1)-(3.3), the integer variables in the decision vector _xj (e.g., decisions to turn on or off power grid resources and other variables with integer values, if any) are fixed to the values provided by the LR dual problem solution (when k=0) or by the extended subproblem solution (when k>0). In other words, the optimization of the dispatch problem is performed only for the non-integer variables in the decision vector _xj , and the integer variables in the decision vector _xj are treated as constants. The dispatch problem is easier to solve because the number of variables to solve in the dispatch problem is smaller than that in the original SCUC system, and because the variables to optimize are all real-valued (and therefore may have continuous gradients). A MIP solver may be used to solve the dispatch problem, but in this case the MIP solver only needs to solve the non-integer variables, which results in less time being required to solve the dispatch problem. Alternatively, a simpler solver than a MIP solver may be used to solve the dispatch problem, for example a solver that only optimizes non-integer variables.

The slack variable vector S is added to the dispatch problem to ensure technology feasibility, so that the solver for the dispatch problem can find a solution (e.g., decision vector _xj and slack variable vector S) that satisfies all the constraints in (3.2) and (3.3). In particular, the slack variables help to satisfy the coupling constraints in (3.3). Without the slack variable vector S, the dispatch problem may not have a feasible solution, because the integer variables in the decision vector _xj are fixed to values provided by the LR dual problem solution or the extended subproblem solution, which may not allow a feasible solution. Since the optimization process is likely to avoid the high costs associated with the slack variables, the elements in the penalty vector M are assigned sufficiently large values (e.g., a high cost for the slack variables) to inhibit the solver from relying on the slack variable vector S to satisfy the coupling constraints in (3.3).

Next, in block 305, the feasibility of the solution to the dispatch problem (also called the dispatch solution) is checked. A dispatch solution is feasible when the coupling constraints in (3.3) are satisfied with a zero-valued slack variable vector S, which is equivalent to satisfying the coupling constraints (1.3) of the original SCUC system. Thus, a dispatch solution is considered feasible when the slack variables in the slack variable vector S are zero. A non-zero-valued slack variable vector S means that the dispatch solution is technically feasible (e.g., satisfies (3.3)) relying on the slack variables, but does not satisfy the coupling constraints (1.3) of the SCUC system, and is therefore not feasible. If the dispatch solution is feasible, processing proceeds to block 306, where the SCUC solver 134 generates a resource allocation schedule based on the decision vector _xj of the feasible dispatch solution, and processing stops.

If the dispatch solution is infeasible, method 350 corrects the infeasibility by solving multiple extended subproblems in one or more iterations and solving the dispatch problem with fixed integer decisions from the solutions to the extended subproblems, the details of which are discussed below. For ease of discussion, checking whether the dispatch solution is infeasible may also be referred to as checking the joint constraint infeasibility of the dispatch solution.

To repair the infeasibility of the dispatch solution, processing of the method 350 proceeds to block 307 for the first iteration (e.g., k=0), where multiple extended subproblems are set up and the joint constraint bounds for the extended subproblems are initialized. The extended subproblems are

is defined as, which is

and

where k is the iteration index, λ ^(k) is the multiplier vector (e.g., including multiple multipliers) at the kth iteration, _sj ^(k) is the slack variable vector for the jth extended sub-problem at the kth iteration, _yj ^(k) is the coupling constraint bond vector (e.g., including multiple bounds for the coupling constraints) for the jth extended sub-problem at the kth iteration, and ρ is the penalty factor for the quadratic term in (4.1). The multiplier vector λ ^(k) may also be referred to as the Lagrangian multiplier vector for the extended sub-problem, and the multiple multipliers in the multiplier vector λ ^(k) may also be referred to as Lagrangian multipliers.

The multiplier vector λ ^(k) is initialized to the final value obtained from block 200 of FIG. 2. In subsequent iterations (k>0), the multiplier vector λ ^(k) is updated at each iteration. The penalty coefficient ρ is an input parameter for the extended subproblem and therefore has a fixed value during optimization. The bond constraint bond vector _yj ^(k) is initialized in the first iteration (when k=0) with the assigned infeasibility, if any, plus the value from the dispatch solution. Details are discussed below.

Once the extended subproblems have been set up and the coupling constraint bond vectors y _j ⁽⁰⁾ have been initialized, an appropriate solver, such as a MIP solver, may be used to solve each of the extended subproblems, as illustrated in block 309. To reduce the time required to find solutions to the extended subproblems, multiple solvers may be used to solve the extended subproblems in parallel. In the discussion herein, the solutions to the extended subproblems (e.g., decision vectors x _j , j=1, 2,....N) are also referred to as extended subproblem solutions.

Next, in block 310, the iteration index k is incremented by 1. Processing returns to block 303 for the next iteration of the process.

Then, in block 303, the dispatch problem is solved again with the fixed integer decisions from the extended subproblem solution. The details are the same or similar to those discussed above and therefore are not repeated here.

The dispatch solution is then checked for feasibility in block 305 using the same or similar process as discussed above. If the dispatch solution is feasible, processing proceeds to block 306 where the SCUC solver 134 generates a resource allocation schedule based on the decision vectors _xj of the feasible dispatch solution and processing stops. If the dispatch solution is infeasible, processing proceeds to blocks 311 and 313 where the multiplier vectors λ ^(k) and the coupling constraint boundary vectors _yj ^(k) are updated as discussed below.

Next, updates are calculated for the multiplier vector λ ^(k) and the joint constraint boundary vector _yj ^(k) in block 311. In particular, the update Δλ ^(k) (i) for the i-th element of the multiplier vector λ ^(k) in the k-th iteration is given by

It is calculated by:
The update Δy _j ^(k) (i) for the i-th element of the j-th joint constraint boundary vector y _j ^(k) in the k-th iteration is given by

Next, in block 313, the multiplier vector λ ^(k) and the joint constraint boundary vector _yj ^(k) are updated using the updates calculated in block 311. In particular, the multiplier vector λ ^(k) is given by

and the joint constraint boundary vector y _j ^(k) is updated by

is updated by
Next, in block 309, the extended subproblem is solved using the updated multiplier vector λ ^(k) and the updated joint constraint boundary vector _yj ( ^k) using the same or similar process as discussed above, and therefore the details are not repeated.

Next, in block 310, the iteration index k is incremented by 1 and processing returns to block 303 for the next iteration of the process.

Next, in block 303, the dispatch problem is solved with fixed integer decisions from the extended subproblem solution obtained with the updated multiplier vector λ ^(k) and the updated coupling constraint boundary vector _yj ^(k) . The iterative process continues until a feasible solution is achieved. Note that the process in block 307 is performed only once at the beginning, when k=0. In subsequent iterations (k>0), the process of method 350 follows a path along blocks 311 and 313. By updating the multiplier vector λ ^(k) and the coupling constraint boundary vector _yj ^(k) in each iteration, the process of method 350 reduces the residual infeasibility of the coupling constraints and eventually finds a feasible dispatch solution.

FIG. 6 is a block diagram of a processing system 900 for performing processing of a power management system in some embodiments. For example, the processing system 900 may be used to implement the PMS 105 of FIG. 1.

As shown in FIG. 6, the processing system 900 includes a processor 902, a memory 904, and interfaces 906-910, which may be arranged (or not arranged) as shown in FIG. 6. The processor 902 may be any component or collection of components adapted to perform computing and/or other processing-related tasks, and the memory 904 may be any component or collection of components adapted to store programming and/or instructions for execution by the processor 902. In an embodiment, the memory 904 includes a non-transitory computer-readable medium. The interfaces 906, 908, 910 may be any component or collection of components that allow the processing system 900 to communicate with other devices/components. For example, one or more of the interfaces 906, 908, 910 may be adapted to communicate data, control, or management messages from the processor 902 to applications installed on the host device and/or remote devices. As another example, one or more of the interfaces 906, 908, 910 may be adapted to allow a device (e.g., a personal computer (PC), etc.) to interact/communicate with the processing system 900. The processing system 900 may include additional components not depicted in FIG. 6, such as long-term storage (e.g., non-volatile memory, etc.).

FIG. 7 illustrates a flow chart of a method 1000 of operating a power grid according to some embodiments. It should be understood that the embodiment method shown in FIG. 7 is merely an example of many possible embodiment methods. Those skilled in the art will recognize many variations, substitutions, and modifications. For example, various steps as illustrated in FIG. 7 may be added, removed, substituted, rearranged, or repeated.

7, in block 1010, a power grid resource allocation profile is generated by a power management system of the power grid, the power grid resource allocation profile directing operation of the power grid constrained by the power grid operation information. In block 1020, an intermediate resource allocation schedule is generated by the power management system, the intermediate resource allocation schedule providing a tentative schedule of resource allocation for the power grid resources operating in the power grid. In block 1030, the power management system determines whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies the binding constraints of the power grid resource allocation profile. In block 1040, in response to determining that the intermediate resource allocation schedule is not feasible, the power management system repairs the intermediate resource allocation schedule to generate a feasible resource allocation schedule, the resource allocation schedule providing a final schedule of resource allocation for the power grid resources operating in the power grid.

The embodiments may achieve advantages. For example, it may take a long time for a MIP solver to find a feasible solution for an SCUC system, for example, due to the larger number of variables to optimize, the types of variables to optimize (e.g., integer-valued and real-valued variables), and the NP-hardness of the SCUC system. The present disclosure provides a computationally efficient way to solve the SCUC problem in two steps. In a first step, the SCUC solver solves a Lagrangian Relaxation (LR) dual of the SCUC problem. The LR dual includes multiple LR subproblems that have lower dimensions (e.g., fewer variables) and are suitable for parallel processing. The feasibility of the LR dual solution is checked. If the LR dual solution is feasible, a resource allocation schedule is generated from the decision vector of the LR dual solution. If the LR dual solution is infeasible, the infeasibility of the LR dual solution is repaired to achieve a feasible solution. The repair process involves solving a dispatch problem with fixed integer decisions and solving multiple extended subproblems with lower dimensionality and suitable for parallel processing. The dispatch problem and extended subproblems allow for finding feasible solutions quickly and efficiently.

Variations and modifications to the disclosed embodiments are possible and are fully intended to be included within the scope of the present disclosure. For example, although the disclosed method is described within the framework of solving the SCUC problem, the disclosed method may be readily applied to other MIP problems. As another example, the process for updating the multiplier vector λ ^(k) (e.g., blocks 311 and 313 in FIG. 5) may be performed only for the combined constraints with infeasibility from the dispatch solution. As yet another example, whereas in FIG. 5 the feasibility test is performed on the dispatch solution, it is possible to test feasibility directly on the extended subproblem solution.

Exemplary embodiments of the present disclosure are summarized here. Other embodiments may be understood from the entirety of this specification and the claims filed herein.

Example 1. In an embodiment, a method of operating a power grid includes generating, by a power management system of the power grid, an optimizer for addressing an optimization problem related to power grid resource allocation, the optimizer including an objective, constraints, and a computing model, where a coupling is between at least two of the constraints; generating, by the computing model of the optimizer, a solution to the optimization problem as an intermediate resource allocation schedule, where the intermediate resource allocation schedule provides a tentative schedule of resource allocation for power grid resources operating in the power grid; and determining, by the computing model of the optimizer, whether the solution satisfies the coupling constraints of the optimizer. determining whether the intermediate resource allocation schedule is feasible by checking; and, in response to determining that the intermediate resource allocation schedule is not feasible, repairing the intermediate resource allocation schedule to generate a feasible resource allocation schedule by a computing model of an optimizer, the resource allocation schedule providing a final schedule of resource allocation for power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible.

Example 2. The method of Example 1, further comprising transmitting, by the power management system, the resource allocation schedule to the power grid resources.

Example 3. The method of example 1, wherein the optimizer constraints include information for constraining the operation of the power grid, including security-related transmission constraint information and cost information.

Example 4. The method of Example 1, wherein generating the optimizer includes formulating the optimization problem as a security constrained unit commitment (SCUC) system based on a mixed integer program.

Example 5. The method of Example 1, in which the intermediate resource allocation schedule is generated by solving a Lagrangian Relaxation (LR) dual of the optimization problem, the LR dual of the optimization problem including Lagrangian multipliers and coupling constraints of the optimization problem, the LR dual of the optimization problem including multiple LR subproblems, and generating the intermediate resource allocation schedule includes solving the multiple LR subproblems in parallel using multiple mixed integer programming (MIP) solvers.

Example 6. The method of example 5, wherein generating the intermediate resource allocation schedule further includes computing partial gradients for the Lagrangian multipliers and updating the Lagrangian multipliers using the computed partial gradients.

Example 7. The method of Example 1, wherein determining whether the first dispatch solution is feasible includes forming a dispatch problem by modifying an optimization problem with slack variables, solving the dispatch problem with fixed integer decisions from the intermediate resource allocation schedule to obtain the first dispatch solution, and determining whether there are non-zero slack variables in the first dispatch solution.

Example 8. The method of Example 7, wherein repairing the intermediate resource allocation schedule further includes generating a resource allocation schedule from a commitment decision of the first dispatch solution in response to determining that there are no non-zero slack variables in the first dispatch solution.

Example 9. The method of example 8, wherein in response to determining that there are non-zero slack variables in the first dispatch solution, the first dispatch solution is determined to be infeasible.

Example 10. The method of Example 9, wherein repairing the intermediate resource allocation schedule further includes, in response to determining that the first dispatch solution is infeasible, solving a plurality of extended subproblems to generate a first extended subproblem solution, generating a second dispatch solution by solving the dispatch problem with fixed integer decisions from the first extended subproblem solution, determining whether the second dispatch solution is feasible, and generating a resource allocation schedule from the commitment decisions of the second dispatch solution if the second dispatch solution is feasible.

Example 11. The method of Example 10, wherein repairing the intermediate resource allocation schedule further includes setting up a plurality of extended subproblems prior to solving the plurality of extended subproblems, the plurality of extended subproblems including Lagrangian multipliers and joint constraint bounds for the plurality of extended subproblems, and initializing the joint constraint bounds for the plurality of extended subproblems.

Example 12. The method of Example 11, wherein repairing the intermediate resource allocation schedule further includes updating Lagrangian multipliers and joint constraint bounds of the multiple extended subproblems in response to determining that the second dispatch solution is infeasible, solving the multiple extended subproblems with the updated Lagrangian multipliers and the updated joint constraint bounds to generate a second extended subproblem solution, generating a third dispatch solution by solving the dispatch problem with fixed integer decisions from the second extended subproblem solution, determining whether the third dispatch solution is feasible, and generating a resource allocation schedule from the commitment decisions of the third dispatch solution if the third dispatch solution is feasible.

Example 13. In an embodiment, a power management system for a power grid includes one or more processors and a memory storing a program executing in the processor, the program including instructions, when executed in the one or more processors, for causing the one or more processors to generate an intermediate resource allocation schedule, the intermediate resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating in the power grid, and determining whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid, the power grid resources being allocated to the power grid resources. The resource allocation profile indicates operation of the power grid constrained by the power grid operation information, and in response to determining that the intermediate resource allocation schedule is infeasible, repair the intermediate resource allocation schedule to generate a feasible resource allocation schedule, the resource allocation schedule providing a final schedule of resource allocations for the power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible.

Example 14. The power management system of Example 13, further comprising a transmitter configured to transmit the resource allocation schedule to the power grid resources, the power grid resources being configured to be controlled according to the resource allocation schedule.

Example 15. The power management system of Example 13, wherein the intermediate resource allocation schedule is generated by solving a Lagrangian Relaxation (LR) dual of the power grid resource allocation profile, the LR dual of the power grid resource allocation profile including multiple LR subproblems, and the program includes instructions that cause one or more processors to generate the intermediate resource allocation schedule by solving the multiple LR subproblems in parallel using multiple solvers.

Example 16. The power management system of Example 13, wherein determining whether the first dispatch solution is feasible includes forming a dispatch problem by modifying a power grid resource allocation profile with slack variables, solving the dispatch problem with a fixed integer solution from the intermediate resource allocation schedule to obtain the first dispatch solution, and determining whether there are non-zero slack variables in the first dispatch solution, and repairing the intermediate resource allocation profile further includes generating a resource allocation schedule from a commitment decision of the first dispatch solution in response to determining that there are no non-zero slack variables in the first dispatch solution.

Example 17. The power management system of Example 16, wherein the program further includes instructions for repairing the intermediate resource allocation schedule by having one or more processors solve a plurality of extended subproblems to generate a first extended subproblem solution in response to determining that the first dispatch solution is infeasible, the plurality of extended subproblems including Lagrangian multipliers and coupling constraint bounds for the plurality of extended subproblems, generate a second dispatch solution by solving the dispatch problem with fixed integer decisions from the first extended subproblem solution, determine whether the second dispatch solution is feasible, and in response to determining that the second dispatch solution is feasible, generate a resource allocation schedule from a commitment decision of the second dispatch solution.

Example 18. The power management system of Example 17, wherein the program includes instructions to repair the intermediate resource allocation schedule by having one or more processors, in response to determining that the second dispatch solution is infeasible, update Lagrangian multipliers and coupling constraint bounds of the multiple extended subproblems, solve the multiple extended subproblems with the updated Lagrangian multipliers and updated coupling constraint bounds to generate a second extended subproblem solution, and generate a third dispatch solution by solving the dispatch problem with fixed integer decisions from the second extended subproblem solution.

Example 9. In an embodiment, a system for a power grid is a power management system, the system comprising: generating an intermediate resource allocation schedule, the intermediate resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating in the power grid; determining whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid, the power grid resource allocation profile indicating operation of the power grid constrained by power grid operational information; and in response to determining that the intermediate resource allocation schedule is not feasible, determining whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid. The power management system includes a power grid resource configured to generate a feasible resource allocation schedule by repairing the schedule, the resource allocation schedule providing a final schedule of resource allocation for power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible; the power grid resource configured to be controlled according to the resource allocation schedule; and a transmitter configured to transmit the resource allocation schedule from the power management system to the power grid resource.

Example 20. The system of Example 19, wherein determining whether the first dispatch solution is feasible includes forming a dispatch problem by modifying a power grid resource allocation profile with slack variables, solving the dispatch problem with fixed integer decisions from the intermediate resource allocation schedule to obtain the first dispatch solution, and determining whether there are non-zero slack variables in the first dispatch solution, and repairing the intermediate resource allocation schedule further includes generating a resource allocation schedule from the commitment decisions of the first dispatch solution in response to determining that there are no non-zero slack variables in the first dispatch solution.

While this disclosure has been described with reference to illustrative embodiments, this description is not intended to be construed in a limiting sense. Various modifications and combinations of the illustrative embodiments, as well as other embodiments of the present disclosure, will be apparent to those skilled in the art upon reference to the description. Accordingly, it is intended that the appended claims cover any such modifications or embodiments.

Claims (19)

1. A method of operating a power grid, comprising:
generating an interim resource allocation schedule by a power management system of the power grid, the interim resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating within the power grid;
determining, by the power management system, whether the intermediate resource allocation schedule is executable by checking whether the intermediate resource allocation schedule satisfies a binding constraint of a power grid resource allocation profile of the power grid, the power grid resource allocation profile directing operation of the power grid constrained by operational information of the power grid;
in response to determining that the intermediate resource allocation schedule is infeasible, repairing, by the power management system, the intermediate resource allocation schedule to generate a feasible resource allocation schedule, the resource allocation schedule providing a final schedule of resource allocations for the power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible;
Determining whether the first dispatch solution is feasible includes:
forming the dispatch problem by adding slack variables to the coupling constraints of the power grid resource allocation profile and to an objective function of the power grid resource allocation profile;
solving the dispatch problem with the fixed integer decisions from the intermediate resource allocation schedule to obtain the first dispatch solution;
determining whether there are any non-zero slack variables in the first dispatch solution . The method of claim 1, further comprising transmitting, by the power management system, the resource allocation schedule to the power grid resources. The method of claim 1 or 2, wherein the operational information includes information for constraining the operation of the power grid, including security-related transmission constraint information and cost information. The method of any one of claims 1 to 3, wherein generating the power grid resource allocation profile includes formulating the power grid resource allocation profile as a security constrained unit commitment (SCUC) system based on a mixed integer program. The method of any one of claims 1 to 4, wherein the intermediate resource allocation schedule is generated by solving a Lagrangian Relaxation (LR) dual of the power grid resource allocation profile, the LR dual of the power grid resource allocation profile including Lagrangian multipliers and coupling constraints of the power grid resource allocation profile, the LR dual of the power grid resource allocation profile including multiple LR subproblems, and generating the intermediate resource allocation schedule includes solving the multiple LR subproblems in parallel using multiple mixed integer programming (MIP) solvers. generating the intermediate resource allocation schedule;
Calculating partial gradients with respect to the Lagrange multipliers; and
The method of claim 5 , further comprising: updating the Lagrangian multipliers using the calculated partial gradients. 7. The method of claim 6, wherein repairing the intermediate resource allocation schedule further comprises generating the resource allocation schedule from a commitment decision of the first dispatch solution in response to determining that there are no non-zero slack variables in the first dispatch solution. 8. The method of claim 7 , wherein in response to determining that there are non-zero slack variables in the first dispatch solution, the first dispatch solution is determined to be infeasible. Repairing the intermediate resource allocation schedule
in response to determining that the first dispatch solution is infeasible,
solving a plurality of extended subproblems to generate a first extended subproblem solution;
generating a second dispatch solution by solving the dispatch problem with fixed integer decisions from the first extended subproblem solution;
determining whether the second dispatch solution is feasible; and
10. The method of claim 8 , further comprising: generating the resource allocation schedule from a commitment decision of the second dispatch solution if the second dispatch solution is feasible. Repairing the intermediate resource allocation schedule includes, prior to solving the plurality of extended sub-problems,
setting up the plurality of extended subproblems, the plurality of extended subproblems including Lagrangian multipliers and joint constraint bounds for the plurality of extended subproblems;
The method of claim 9 , further comprising: initializing the joint constraint bounds for the plurality of extended sub-problems. Repairing the intermediate resource allocation schedule
in response to determining that the second dispatch solution is infeasible,
updating the Lagrange multipliers and the joint constraint bounds of the plurality of extended sub-problems;
solving the plurality of extended subproblems with updated Lagrangian multipliers and updated joint constraint bounds to generate a second extended subproblem solution;
generating a third dispatch solution by solving the dispatch problem with fixed integer decisions from the second extended subproblem solution;
determining whether the third dispatch solution is feasible; and
The method of claim 10 , further comprising: generating the resource allocation schedule from a commitment decision of the third dispatching solution if the third dispatching solution is feasible. 1. A power management system for a power grid, comprising:
one or more processors;
a memory storing a program for execution within the processors, the program including instructions that, when executed within the one or more processors, cause the one or more processors to:
generating an interim resource allocation schedule, the interim resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating in the power grid;
determining whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a coupling constraint of a power grid resource allocation profile of the power grid, the power grid resource allocation profile directing operation of the power grid constrained by operational information of the power grid;
and in response to determining that the intermediate resource allocation schedule is infeasible, repairing the intermediate resource allocation schedule to generate a feasible resource allocation schedule, the resource allocation schedule providing a final schedule of resource allocations for the power grid resources operating in the power grid, repairing the intermediate resource allocation schedule including determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible ;
Determining whether the first dispatch solution is feasible includes:
forming the dispatch problem by modifying the power grid resource allocation profile with slack variables;
solving the dispatch problem with a fixed integer solution from the intermediate resource allocation schedule to obtain the first dispatch solution;
determining whether there is a non-zero slack variable in the first dispatch solution . a transmitter configured to transmit the resource allocation schedule to the power grid resources, the power grid resources being configured to be controlled according to the resource allocation schedule.
The power management system of claim 12 . 14. The power management system of claim 12 or 13, wherein the intermediate resource allocation schedule is generated by solving a Lagrangian Relaxation (LR) dual of the power grid resource allocation profile, the LR dual of the power grid resource allocation profile including multiple LR subproblems, and the program includes instructions that cause the one or more processors to generate the intermediate resource allocation schedule by solving the multiple LR subproblems in parallel using multiple solvers. 15. The power management system of claim 12, wherein repairing an intermediate resource allocation profile further comprises generating the resource allocation schedule from a commitment decision of the first dispatch solution in response to determining that there are no non -zero slack variables in the first dispatch solution. the one or more processors;
in response to determining that the first dispatch solution is infeasible,
solving a plurality of extended subproblems to generate a first extended subproblem solution, the plurality of extended subproblems including Lagrangian multipliers and joint constraint bounds for the plurality of extended subproblems;
generating a second dispatch solution by solving the dispatch problem with fixed integer decisions from the first extended subproblem solution;
determining whether the second dispatch solution is feasible; and
16. The power management system of claim 15, further comprising instructions for repairing the intermediate resource allocation schedule by : generating the resource allocation schedule from a commitment decision of the second dispatch solution in response to determining that the second dispatch solution is feasible. the one or more processors;
in response to determining that the second dispatch solution is infeasible,
updating the Lagrange multipliers and the joint constraint bounds of the plurality of extended sub-problems;
solving the plurality of extended subproblems with updated Lagrangian multipliers and updated joint constraint bounds to generate a second extended subproblem solution;
and generating a third dispatch solution by solving the dispatch problem with fixed integer decisions from the second extended subproblem solution. 1. A power grid system comprising:
1. A power management system comprising:
generating an interim resource allocation schedule, the interim resource allocation schedule providing a tentative schedule of resource allocations for power grid resources operating in the power grid;
determining whether the intermediate resource allocation schedule is feasible by checking whether the intermediate resource allocation schedule satisfies a coupling constraint of a power grid resource allocation profile of the power grid, the power grid resource allocation profile directing operation of the power grid constrained by operational information of the power grid;
a power management system configured to: in response to determining that the intermediate resource allocation schedule is infeasible, generate a feasible resource allocation schedule by repairing the intermediate resource allocation schedule, the resource allocation schedule providing a final schedule of resource allocations for the power grid resources operating in the power grid, and repairing the intermediate resource allocation schedule includes determining whether a first dispatch solution obtained by solving a dispatch problem with fixed integer decisions from the intermediate resource allocation schedule is feasible;
a power grid resource configured to be controlled according to the resource allocation schedule;
a transmitter configured to transmit the resource allocation schedule from the power management system to the power grid resources ;
Determining whether the first dispatch solution is feasible includes:
forming the dispatch problem by modifying the power grid resource allocation profile with slack variables;
solving the dispatch problem with the fixed integer decisions from the intermediate resource allocation schedule to obtain the first dispatch solution;
determining whether there are any non-zero slack variables in the first dispatch solution . 20. The system of claim 18, wherein repairing the intermediate resource allocation schedule further comprises generating the resource allocation schedule from a commitment decision of the first dispatch solution in response to determining that there are no non- zero slack variables in the first dispatch solution.

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