Stochastic Unit Commitment

Abstract

Unit commitment (UC) is a process of minimizing operating cost of generation set in given situations and constraints. The traditional deterministic UC considered simple constraints such as demand constraint, reserve constraint, and start-up constraints. However, as the penetration of renewable energy resources grows nowadays, there is an increasing demand to improve UC with better uncertainty management. Stochastic optimization approach to unit commitment problem can be a better way to manage factors of uncertainties. In stochastic unit commitment (SUC), an optimal solution is sought for accommodating diverse scenarios. The process is composed of two stages : the first stage and the second stage. Day-ahead decisions are made in the first stage, and real time decisions are made in the second stage. The day-ahead decisions are unique over all the scenarios, while real-time decisions are scenario-dependent. The solution of stochastic unit commitment problem is an optimum point which satisfies all the constraints for all possible scenarios. The second stage of stochastic unit commitment problem is similar to traditional deterministic UC problem, which is a composition of a security-constrained unit commitment (SCUC) problem and a security-constrained economic dispatch (SCED) problem. This stage is a time-consuming step. Some mathematical methods can be applied to simplify the second stage and reduce the iteration time. Widely used methods are Benders decomposition (BD) and Lagrangian Relaxation (LR).

1. Introduction
Unit Commitment (UC) is an optimization problem that generates commitment status and generation dispatch of various generating units in a way that minimizes the operating cost, including several operating constraints such as generator minimum / maximum generation limits, ramping limits, minimum up/down time constraints, time-dependent start-up costs, and transmission capacity limits. The system operator makes commitment and dispatch decisions on generation units in order to satisfy the demand and reliability requirements. The system operator should take into account several factors that cause uncertainties such as load forecast error, changes of system interchange schedule, and unexpected transmission and generation outages.

2. Deterministic Unit Commitment vs Stochastic Unit Commitment
There are two approaches in solving the unit commitment problem that differ in addressing the uncertainty. Deterministic unit commitment formulation is a traditional solution where the load is modeled with one forecast, while the next day situation including uncertainties is assumed to be fixed, and handled by imposing deterministic reserve requirements. On the other hand, stochastic approaches consider uncertainties for the situations in the scheduling time interval.

There are two types of modeling and solution approaches for stochastic unit commitment which are distinct in handling the uncertainties. Robust unit commitment (RUC) models manage uncertainty based only on the information of the range of the uncertainty without any information of probability distributions. RUC produces conservative solutions by optimizing the cost of the worst case among all the scenarios generated from the uncertainties. Stochastic unit commitment(SUC) models probabilistic scenario based on uncertainty representation, in which a large number of scenarios are generated with probability weighted on each cases according to the information of probability distributions. Due to increasing penetration of renewable energy resources, addressing uncertainties rigorously is getting more important, which leads to imposing larger amount of reserve requirement, thus resulting in higher operation cost in deterministic unit commitment. Stochastic programing is advantageous because it can minimize total expected operation cost while satisfying the reliability improvement. This paper focuses on two-stage models and algorithms associated with stochastic unit commitment and the various methods that can help find the optimal solution for this type of problems.

3. The struture of Stochastic Unit Commitment
Stochastic UC is formulated as a two-stage problem that determines the generation schedule that minimizes the expected cost over all of the scenarios respecting their probabilities. In a two-stage SUC model, two parts of modeling can be discussed. The first part is modeling uncertainties by generating scenarios and the second part is modeling decisions in unit commitment problem.

Scenarios are generated to represent uncertainty realized depending on time and other conditions. It is quite computationally demanding to include a large number of generated scenarios while solving the problem. Therefore, scenarios with ignorable probabilities can be reduced by scenario reduction methods. Similar scenarios in their probability, hourly magnitude, or the resulting cost can be aggregated so that the computational burden is significantly decreased.

UC decision part is divided into two parts : day-ahead decisions (first stage) and real-time decisions (second stage). The day-ahead decisions are unique over all the scenarios, while real-time decisions are scenario-dependent. This means that UC solution is the solution that can manage all the possible scenarios. In the first stage, the model makes commitment decisions for all units, especially traditional units. Several factors that should be determined ahead of scheduling time are included for optimization, such as start-up/shut-down cost of conventional generators and contract cost of demand-side resources. In the second stage, factors that should be considered for real-time operation are included for optimization, such as generation cost where dispatches and reserves of multiple periods are included, deployment cost of demand-side reserve, and cost related to the involuntary load shedding.

4. Solution Algorithm for Stochastic Unit Commmitment(SUC)
Since SUC includes large number of scenarios, SUC problem is a large-scale mixed-integer non-linear problem complicated by a small set of side constraints. The size of the problem increases as the number of simulated scenarios increases which leads to computational intractability. At the very first step of development of unit commitment, solution for large systems could not be obtained; heuristic approaches were the alternatives. Some great mathematical methods have given new approaches for the solution of unit commitment problems for large sized systems. There are several different methods of simplifying the big problem, so that the computational demand is reduced. Two widely used methods of simplifying an optimization problem are Benders Decomposition and Lagragian Relaxation.

4.1. Benders Decomposition
The SUC problem is a large-scale, mixed-integer, and non-linear problem. The Benders decomposition method is applied to decompose such an optimization problem into one relaxed master problem(pure integer programming problem) and several sub-problems(dual linear programming problem) that can check the feasibility of solution of the master problem. The master unit commitment, which is composed of the objective function and constraints associated with commitment and dispatch, provides a commitment and dispatch solution that minimizes the operating cost. Either Lagrangian relaxation or mixed-integer programming method can be applied to solve the unit commitment problem. The feasibility check subproblem checks whether the current commitment and dispatch solution of the master problem can accommodate the several constraints of the sub-problems(e.g. security constraints, transmission constraints, and other physical limitations, etc.) If the given solution violates constraints of the sub-problems, Benders cuts, which is linear constraints, will be generated and added to the master problem, narrowing down search region while preserving the original feasible region. The master problem produces a new solution that satisfies the sub-problem constraints by changing the commitment and/or dispatch. As described above, Benders decomposition is a cutting plane method due to adding a constraint to the problem at each iteration, and makes it progress towards a solution. There is no doubt that Benders decomposition showed a lot of successful implementations in the unit commitment related problems, especially the two-stage stochastic problems without discrete variables in the second stage. However, in order to accomodate flexibility, quick-start resources can be called up in real-time operation, which leads to addition of start-up/shut-down cost of quick-start units to the second stage problem. This newly added integer vaiables in the second stage brings difficulty to apply Bender Decomposition. Therefore, linear relaxation thorough convex hull formulation should be applied to the sub-problem, so that the sub-problem can simply be treated as a linear programming problem.

The convergence of classical Benders Decomposition algorithm is somewhat limited and slow. In order to improve the convergence, accelerated Benders Decompositions are studied. An optimization method by generating multiple strong Benders cuts for accelerating the convergence of Benders Decomposition when solving the network-constrained generation UC problem is suggested. Accelarated Benders Decomposition succesfully reduces the total iteration number and the overall computing time.

4.2. Lagrangian Relaxation
Lagrangian Relaxation is another method used to solve a UC problem efficiently. By applying Lagrangian Relaxation, the coupling constraints and the objective function of the unit commitment problem can be relaxed into less strict optimization problem. Lagrangian relaxation can be applied to solving large-scale, mixed-integer, and non-linear problem. In the relaxation process, the original big problem is broken down to smaller subproblems, coupling constraints between scenarios being dualized.

Lagrangian Relaxation has been developed for getting solutions efficiently. It derived its name from the well-known mathematical technique of using Lagrange multipliers for solving constrained optimization problems, but is really a decomposition technique for the solution of large scale mathematical programming problems.

Usually, an objective function of an UC problem is summation of fuel and operation cost, maintenance cost, start-up cost, and shut down cost for each unit. Moreover, the constraints of a typical UC problem are the demand, capacity, unit limits, system reserve, area reserve, minimum uptime, minimum downtime, must-run units, CO2 emmision, and transmission security constraints. This provides the main idea of LR. To elaborate, LR is a process of relaxing the coupling constraints to decompose the opmization problem into N independent subproblems, where N is the number of units. By dividing one big problem into N small subproblems, the solving process of large-sized optimization can be much faster and efficient. The Lagrangian multiplier (usually noted as λ) is defined, and by adding penalty function in the original objective function, the optimization problem with an equality constraint is relaxed into a one optimization problem without an equality constraint.

The explanation above can be illustrated in a simple example. The following is a simple LR process written in formulae.

min $$\sum_{T=1}^{T} \sum_{i=1}^{N} [F_{i}(P_{i}^t) + start-up cost_{i,t}]U_{i}^t$$

= min $$\sum_{T=1}^{T} \sum_{i=1}^{N} F_{i}(P_{i}^t,U_{i}^t)$$

where $$U_{i}^t$$ is 0 if the ith unit is off at time t, and 1 if on.

F is fuel cost and t is the period of the time.

Two main constraints to consider are Loading constraints and unit limits.

Loading constraints : $$P_{load}^t - \sum_{i=1}^{N} P_{i}^t U_{i}^t = 0 (t=1,2,3, ... T)$$

Unit limits : $$U_{i}^t P_{i}^{min} \le  P_{i}^t  \le  U_{i}^t P_{i}^{max} (i=1,2,3, ... N, t=1,2,3, ... T) $$

The problem can be relaxed into a new problem including lagrangian function; The equality constratint can be included in the objective function by defining Lagrangian multiplier $$ \lambda^t $$.

$$ L(P,U,\lambda) = F(P_{i}^t, U_{i}^t) + \sum_{t=1}^{T} \lambda^t (P_{load} - \sum_{i=1}^N, P_i^t U_i^t)$$

Here, $$\lambda^t$$ could be set high enough to satisfy the previous equality constraint. For instance, when the $$\lambda^t$$ is large enough, if the equality constraint does not hold, the objective function will be differed significantly. The optimum of the new objective function will be found at the point where the previous equality constraint holds.

The minimum of the Lagrangian is found by solving for the minimum for each generating unit over all time periods.

$$\sum_{i=1}^{N} \sum_{t=1}^T ([F_i(P_i^t) + start-up cost_{i,t}]U_i^t - \lambda^t P_i^t U_i^t)$$

$$ U_i^t P_i^{min} \le F_i(P_i) - \lambda^t P_i^t \le U_i^t P_i^{max} for t=1,2,3, ... T$$

at the state where $$ U_i^t = 1 $$, the new objective function can be simplified as follows;

$$ [F_i(P_i) + \lambda^t P_i^t] $$

The minimum of this function can be found by taking the first derivative.

$$ \frac {d}{dP_i^t} [F_i (P_i) - \lambda^t P_i^t] = \frac {d}{dP_i^t}[F_i(P_i^t)] - \lambda^t $$

So, for the optimum value, $$ \frac {d}{dP_i^t} [F_i (P_i^{opt}) = \lambda^t $$ holds.

If the optimum found by this process does not fit the inequality constraints, some adjustment should be given.

If $$ P_i^{opt} \le P_i^{min}, min [F_i(P_i) - \lambda^t P_i^t] = F_i(P_i^{min}) - \lambda^t P_i^{min} $$.

If $$ P_i^{opt} \ge P_i^{max}, min [F_i(P_i) - \lambda^t P_i^t] = F_i(P_i^{max}) - \lambda^t P_i^{max} $$.

If $$ P_i^{min} \le P_i^{opt} \le P_i^{max}, min [F_i(P_i) - \lambda^t P_i^t] = F_i(P_i^{opt}) - \lambda^t P_i^{opt} $$.

5. Conclusion
As the penetration of renewable energy resources grows, there is an increasing demand to improve UC with better uncertainty management. Traditional deterministic UC, which simply uses reserve constraints, has shown limitations on accomodating increasing uncertainties. Stochastic optimization approach can be a better alternative to handle factors of uncertainties by generating scenarios and by ensuring that resulting UC solution meet the demand of each cases appropriately. Robust Optimization methods give solutions towards the worst-case, ensuring the system reliability in a conservative way, whereas Stochastic Unit Commitment gives solutions towards efficiency of system operation, minimizing total expected cost and meeting the system reliability requirement simultaneously. However, in order to ensure the reliability, it is required to have exact information of probability distributions to assign probabilities for scenarios. In computation perspective, SUC is computationally demanding for large numbers of scenarios. The two-stage stochastic UC model has two decision categories, the first stage and the second stage, and different variables and constraints are considered in each stage problems according to the appropriate operating time frames. Many constraints and discrete variables can aggravate the complexity of the problem. Various algorithms and methods are suggested to mitigate given difficulties. Two main methods, which are Benders Decomposition and Lagrangian Relaxation, are discussed in this paper. With these methods, large mixed-integer UC problem can be decomposed into several problems, which can significantly reduce the complexity and computatation time.