Big-M based MILP method for SCUC considering allowable wind power output interval and its adjustable conservativeness

0 Introduction

The increasing energy and environmental concerns and sustainable development strategies have resulted in the rapid growth of wind power generation in many countries and regions.The volatility, limited predictability, nondispatch ability, and anti-peaking regulation inherent in wind power generation have brought significant challenges to power system scheduling and control with high wind power penetration [1,2].Unit commitment (UC) and economic dispatch (ED), which are crucial processes of power system scheduling, have evolved from traditional deterministic formulation into uncertainty formulation to address uncertain wind power generation and simultaneously assess the wind power accommodation capability from the viewpoint of security and economics of system operation [2-10].

Uncertainty optimization methods have been applied to solve UC and ED problems with uncertain wind power output, including stochastic optimization [3,11], chanceconstrained programming [12], and robust optimization [13,14].Among them, the robust optimization method is an alternative method that can guarantee operational security under all possible scenarios given a prescribed uncertainty set for variable wind power generation; thus, it is an effective approach for solving optimization problems with uncertainties.

Currently, robust optimization [15-18] has become a research hotspot.By adopting an uncertainty set to describe the stochastic nature of wind power generation, a two-stage adaptive robust UC model with an adjustable uncertainty set was proposed in [15-17] to reduce the conservativeness of traditional robust optimization.The demand response uncertainty was also considered in [18], and a multi-stage robust UC model based on [15] was developed by assuming that the price elastic demand curve varies within a given range.In addition, the base-case cost [19], worst-case cost [20], recourse cost [21], and min-max regret [22] were utilized in various objective functions of adaptive robust UC with a similar two-stage framework and constraints.

However, the aforementioned robust optimization methods mainly focus on the construction of uncertainty sets and the adjustment of their conservativeness, which may be confronted with significant challenges discussed below.These robust UC models have the premise that power systems can completely accommodate wind power within a predefined uncertainty set regardless of the amount of wind power output and its variation.In other words, wind power curtailment is rarely considered, which may cause a lack of feasible solutions for robust UC models when the worstcase scenario of wind power generation is beyond the redispatch capacity of flexible resources.By contrast, even if the available flexible resources are sufficient, conventional units may provide ramp-up/down capacity at relatively expensive prices to manage drastic wind power fluctuations.

Therefore, for the sake of more accurate analysis of the impact of wind power accommodation on spinning reserve requirements and on/off schedules of conventional units, and to satisfy the transmission capacity limits from the viewpoint of security and economics of system operation, the robust UC model needs to treat the boundaries of the uncertainty set for wind power as the optimizable decision variables to determine the allowable wind power output interval.

In [23] and [24] a robust optimization model was proposed to obtain the allowable wind power output interval and calculate look-ahead ED solutions for conventional units to mitigate the uncertainty inherent to wind power generation.In [25] three alternative solution methodologies were developed to identify the maximum renewable generation ranges, called do-not-exceed limits, that can be accommodated without sacrificing power system reliability.However, UC was not considered in [23-25], which reduced the operational flexibility of systems to a certain extent and was not suitable for the case of large load variation, large uncertainty of wind power generation, and requirements for the start-up and shut-down of units.In [26], the flexible uncertainty set for wind power in security-constrained UC (SCUC) was considered rather than the predefined deterministic one to consider wind power curtailments.In [27] a risk-based admissibility assessment approach was proposed for UC to quantify the extent to which wind power generation can be accommodated.These robust UC problems [26,27] with consideration of wind power curtailments were mathematically formulated as a two-stage robust optimization model, which needed to be solved by a complex column and constraint generation (CC&G)-based iterative algorithm.

In contrast to most existing studies on robust UC, this study proposes a novel big-M-based mixed-integer linear programming (MILP) method to solve the SCUC problem considering the allowable wind power output interval and its adjustable conservativeness.In general, the big-M method is used to linearize the nonlinear constraints formulated as the sum of products of continuous and 0-1 binary variables by introducing a series of linear inequalities with auxiliary binary variables and a large real number.Therefore, the nonlinear optimization problem can be transformed into a mixed-integer linear optimization problem.

First, by employing the big-M method and adding the auxiliary 0-1 binary variables to describe the allowable wind power output interval, a bilinear programming problem meeting the security constraints of system operation is presented.Second, an adjustable confidence level was introduced into the proposed robust optimization model to decrease the level of conservatism of the robust solutions.This can establish a trade-off between the economy and security of system operation.Third, to develop an MILP problem that can be solved by commercial solvers such as CPLEX, the big-M method is utilized again to represent the bi-linear formulation as a series of linear inequality constraints and approximately address the nonlinear formulation caused by the adjustable conservativeness.

The contributions of this study are as follows: 1) it provides a way of combining the allowable wind power output interval with its adjustable conservativeness to establish a trade-off between robustness and economy of system operation; 2) the proposed big-M-based MILP method with significantly reduced problem complexity compared to the existing two-stage robust optimization models allows an easier solution for the robust SCUC problem to obtain the allowable wind power output interval and the adjustable uncertainty set for variable wind power generation.

The remainder of this paper is organized as follows.Section 1 describes the infeasible regions of the conventional robust SCUC model.The proposed robust SCUC model and its big-M-based MILP solution method are presented in Section 2.In Section 3, case studies and simulation results analysis are presented.Finally, in Section 4, the main conclusions are summarized.

1 Infeasible regions of conventional robust SCUC model

1.1 Predicted wind power output interval

Interval prediction is a wind power forecast method that considers uncertainty and thereby can reduce the risk brought about by wind power forecast uncertainty.In the day-ahead UC, the predicted wind power output interval can be estimated with a certain confidence level from the probability density function of the wind power forecast error or the positive and negative deviations calculated via a proportion of the predicted wind power output.As shown in Fig.1, the capacity Wm,t of wind farm m was set to 600 MW, is the predicted wind power output during period t, and are the upper and lower limits of the predicted wind power output interval, respectively.

1.2 Formulation of conventional robust SCUC model

1) Objective Function

The objective function of a conventional UC usually only considers the dispatch cost of thermal units containing the generation, start-up, shut-down, and spinning reserve costs over all scheduling periods.This dispatch cost is defined as

where T is the number of time periods in the scheduling horizon, N is the number of thermal units, is the power output of unit n during period t, are the up-and down-spinning reserve amounts of unit n during period t, respectively; In,t is a binary variable denoting the on/off status of unit n during period t; an, bn, and cn are coefficients of the polynomial approximation of the quadratic generation cost function of unit n; and are the up- and downspinning reserve cost coefficients of unit n, respectively; and and are the start-up and shut-down cost coefficients of unit n, respectively.

2) Power Balance Constraints

where M is the number of wind farms, Q is the number of load demands, Wm,t within the predicted wind power output interval is the most economic power output for wind farm m during period t, and is the predicted amount of load demand q during period t.

3) Transmission Line Power Flow Limits

The transmission line power flow limits are approximated by direct current power flow, which can be formulated as follows:

where J is the number of buses, is the capacity limit of line l, n j∞, m j∞, and q j∞ represent the unit n, wind farm m, and load demand q connected to bus j, respectively; and kj,i is the power transfer distribution factor [28] of bus j to line l.

In robust optimization, the worst-case scenarios for the transmission line power flow limits should be treated as robust solutions, and these scenarios are usually obtained from boundaries of uncertain variable Wm t,.Note that boundaries are positive; thus, when kj l, ≥0, Eq.(3) is equivalent to the inequality constraints in Eq.(4), and when kj l, ＜0, Eq.(3) is equivalent to the inequality constraints in Eq.(5)

4) Upper and Lower Limits for the Power Output of Units

where are the maximum and minimum power outputs of unit n, respectively.

5) Ramp-Rate Limits for Units

where RUn and RDn are the ramp-up and ramp-down rates of unit n, respectively.

6) Spinning Reserve Requirements

In SCUC, the units should provide sufficient spinning reserve capacity to manage all uncertain wind power output scenarios in the predicted interval.In other words, the up- and down-spinning reserve amounts of units should cover the lower and upper limits of the predicted wind power output interval, respectively.In addition, the up- and down-spinning reserve amounts of units are limited by their ramp-rate limits and maximum/minimum power output.Therefore, the spinning reserve requirements can be formulated as follows:

where ΔT is the time resolution of per scheduling period.According to Eq.(9), the worst-case scenarios for the spinning reserve requirements should be treated as robust solutions, and these scenarios are also obtained from boundaries of uncertain variable Wm,t.

7) Minimum On and Off-Time Constraints of Units

where are the minimum on- and off-time of unit n, respectively, and are the on- and off-time that have been accumulated up to period t-1, respectively.

1.3 Analysis for infeasible regions of conventional robust SCUC model

Let Θ= denote the feasible solution set of the previous conventional robust SCUC model.Note that if the conventional robust SCUC model has a feasible solution, it must be guaranteed that the solution set Θ of optimal values exists to manage all wind power output scenarios Wm,t within the predicted wind power output interval The feasible regions of the conventional robust SCUC model are assumed to be In other words, the condition for this model ensures that a feasible solution is defined as follows:

The feasible regions are functions of the predicted wind power output interval and have the premise that wind power generation can be completely accommodated regardless of the wind power output and its uncertainty.However it cannot be ensured that Θ always exists with consideration of the transmission line power flow limits expressed in Eqs.(4) or (5) and the spinning reserve requirements in Eq.(9).From the viewpoint of security and stability of system operation, the spinning reserve constraint violation or transmission line power flow limit violation may lead to wind power curtailments, which means that power grids cannot completely accommodate the predicted wind power output interval.

2 Proposed robust SCUC model and its Big-M based MILP solution method

2.1 Allowable wind power output interval

By shrinking the predicted wind power output interval to a certain extent, the conventional robust SCUC model can be ensured to have a feasible solution.In extreme cases, if power grids have no ability to accommodate the predicted interval, this interval will be shortened to a constant value, and the lower boundary of the wind power output is equal to its upper boundary, that is, Consequently, by adding the optimization variable of the wind power output limit to shrink the predicted wind power output interval, the allowable wind power output interval can be calculated to obtain the feasible solution of the SCUC model that satisfies this condition are the upper and lower boundaries of the allowable power output interval of wind farm m, respectively.

The impact of the wind power output limit on the allowable wind power output interval Ωm t, is shown in Fig.2.This interval Ωm t, can be divided into two types: 1) when is within the predicted interval, i.e., when is smaller than the lower boundary of the predicted interval, i.e., In real operation, if the actual wind power output is larger than the limit , the wind power needs to be curtailed to satisfy the allowable interval Ωm t,.The curtailed power of the wind farm m can be calculated as Note that indicates that there is no wind-power curtailment.Note also that for further accommodation of wind power, the curtailed wind power needs to be as small as possible to ensure maximum allowable wind power output interval Ωm t,.

2.2 Adjustable conservativeness of allowable wind power output interval

In the previous model, the worst-case scenarios were obtained from boundaries of the allowable wind power output interval, where Ωm t, could not be adjusted.However, robust dispatch decisions are inevitably conservative to ensure security in worst-case scenarios.The robust SCUC model provides overly conservative solutions with large prediction errors, which may result in unnecessary dynamic response capacity to meet the security constraints for all wind power output scenarios.System operators prefer to establish a trade-off between robustness and economy to achieve the desired level of security.Therefore, in this section, further investigation of methods for reducing conservativeness in robust SCUC is presented.

The first step is to construct an uncertainty set that consists of the allowable wind power output interval Ωm t, and the adjustable wind power output interval with a certain confidence level β.The formulation of is defined as

where condition a is necessarily compared with condition are the upper and lower boundaries of the uncertainty set respectively; are the upper and lower boundaries of the confidence interval with a given confidence level β, i.e., According to Eq.(12), the determination of the adjustable uncertainty set can be divided into four cases, as shown in Fig.3.

Confidence boundaries are the parameters determining the size of confidence interval and adjusting its conservativeness by altering the confidence level β.Specifically,are defined as

where is defined as the budget of uncertainty, which represents the number of wind farms allowing their output during period t to reach their predicted maximum or minimum value and reflects the preference for the security and economy of system operation.If Γt( )β is increased, the optimization solution will be more conservative, the security of system operation will be higher, and accordingly, its economy will be worse; conversely, the optimization solution will be more optimistic.In particular, when Γt( β )= 0, the uncertainty of wind power output is not considered in the scheduling process, and the SCUC model is transformed into a deterministic model; Γt( β) =M indicates that all possible worst-case scenarios are considered in the process of scheduling, and the resultant schedules are most conservative.The value of Γt( β) can be selected as follows [24]:

2.3 Dispatch framework of proposed robust SCUC

The dispatch framework of the proposed robust SCUC is a two-level hierarchical dispatch system that can be summarized as shown in Fig.4.This dispatch system consists of an electric power dispatch and control center, wind farm dispatch systems, and conventional thermal plant dispatch systems.

The predicted wind power output interval is periodically sent from the wind farms to the dispatch-andcontrol center, where the robust SCUC program is executed, and the optimized generation schedules are sent back to wind farms and thermal plants.Concerning wind farms, the optimized generation schedules refer to the uncertainty set of the wind power output interval, which is effective in reducing wind power curtailments and regulation frequency of wind turbines.For thermal plants, the corresponding schedules are set-point values of units rather than interval values.

2.4 Formulation of proposed robust SCUC model

1) Objective Function

To obtain the minimum dispatch cost of thermal units and maximum allowable wind power output interval, the objective function of the proposed SCUC model should consist of two parts: the dispatch cost of thermal units and the penalty cost of wind power curtailments over all scheduling periods.The objective function is defined as follows:

where is the penalty coefficient of the upper limit deviation of the wind power output interval for wind farm m.

2) Power Balance Constraints

In the power balance constraints expressed in Eq.(2) for the conventional robust SCUC, Wm,t is a decision variable within the predicted wind power output interval However, in the proposed robust SCUC, Wm,t is limited by the uncertainty set of the wind power output interval, which is also a decision variable.Thus, the constraints expressed in Eq.(2) for the conventional robust SCUC containing Wm t, needs to be transformed as

3) Transmission Line Power Flow Limits

Likewise, according to the transmission line power flow limits in Eqs.(4) and (5) for the conventional robust SCUC containing Wm,t also need to be transformed as

4) Spinning Reserve Requirements

In the proposed robust SCUC, the up- and downspinning reserve amounts of units should cover the upper, and lower, limits of the uncertainty set respectively.Therefore, except for the up- and downspinning reserve limits in Eq.(8) caused by the ramp-rate limits of units and their maximum/minimum power output, the other spinning reserve requirements expressed in Eq.(9) containing need to be transformed as

5) Allowable Wind Power Output Interval Constraints:

6) Other Constraints

The other constraints, including the upper and lower limits in Eq.(6) for the power output of units, ramp-rate limits in Eq.(7) for units, and minimum on and off-time constraints in Eq.(10) of units remain unchanged.

2.5 Big-M based MILP method for proposed SCUC problem solution

Considering the four cases of the uncertainty set for the wind power output interval shown in Fig.3, we first redefine the mathematical formulation of this set via the following conditional expression:

By employing auxiliary 0-1 binary variables ym,t and zm,t, the nonlinear conditional expression in Eq.(22) can be rewritten as follows:

In Eq.(23), is formulated as the sum of products of a continuous variable and two binary variables, whereas is formulated as the sum of products of a continuous variable and a binary variable.According to the big-Mbased linearization method in [23], in Eq.(23) can be transformed into a series of linear inequalities.

The big-M based linearization process can be described as follows:

Step 1.A binary variable xm t, and two continuous variables, are introduced.Let xm,t be the product of binary variables ym,t and zm,t, let be the product of binary variable ym,t and bounded continuous variable be the product of binary variable ym,t and bounded continuous variable Then, is transformed into the summation of products of a binary variable and a continuous variable, and becomes linearized.can be formulated as

The aforementioned new variables, can be equivalent to the following linear constraints:

where M is a large real number.Note that when ym t, =1, Eq.(27) can be equivalent to the constraint i.e., when ym t, =0, Eq.(27) can be equivalent to the formulation It should be pointed out that, in practice, M should not be selected to be extremely large and only needs to satisfy the complementary constraint conditions ym,tM and (1-ym,t)M.Otherwise, it may cause numerical instability.Consequently, in this study, M is selected to be max

Step 2.Note from Eq.(25) that is a nonlinear formulation consisting of the product of a bounded continuous variable and a binary variable that can be finally equivalent to the following linear constraints by employing the big-M method:

Note that when xm t, =1, Eq.(29) is equivalent to the formulation when xm t, =0, Eq.(29) is equivalent to the formulation In addition, in view of the nonlinear formulationconsisting of the product of two continuous variables in Eq.(13), we first redefine by searching for the set sj , m , t∈{0,0.1,0.2, … , 0.8,0.9,1}, which can be expanded in its actual application to carry out its dynamic selection in the optimization process.A binary variable Mj,m,t is introduced to indicate whether the j-th value in this set is chosen during period t and satisfies the constraint defined as

where J represents the total number of set elements that can be selected.Similarly, continuous variables Wj,m,t were introduced.Let in Eq.(13) becomes linearized.Equation (13) can be formulated as

Wj,m,t can be equivalent to the following linear constraints:

In addition, the dispatch cost of thermal units in the objective function given by Eq.(16) and the minimum on- and off-time constraints of units in Eq.(10) can be linearized according to the method proposed in [29].Finally, the proposed robust SCUC problem is formulated as an MILP model, which can be directly solved using the commercial software CPLEX.This detailed MILP model is summarized as subject to the linear constraints in Eqs.(6), (7), (8), (10), (14), (15), (17)-(21), (24), (25-2), and (26)-(32).

3 Case studies and analysis of simulation results

3.1 Modified IEEE 26-generator reliability test system

This study adopted a modified IEEE 26-generator reliability test system to verify the effectiveness and advantages of the proposed model.This system contains 26 thermal units with a total capacity of 3105 MW.The capacity limits of transmission lines, ramp rates, cost coefficients, and minimum up- and down-time of units were obtained from [30].Table 1 lists the forecast loads.The unit penalty cost of the wind power spillage smd isequal to 10 $/(MW·h).The minimum up- and down-spinning reserve requirements were set to 400 MW.The proposed MILP model was solved with a commercial solver CPLEX in a MATLAB environment on a PC with 8 GB of RAM.

3.2 Analysis of results with consideration of wind power curtailments

To analyze the impact of spinning reserve requirements and transmission line capacity on the wind power accommodation capability and the allowable wind power output interval, one case, irrespective of the adjustable conservativeness, was considered.In this case, one wind farm was added at bus 14, and its capacity was set at 600 MW.The predicted power output intervals used for the testing are shown in Fig.1.

1) Analysis of the Impact of Transmission Line Capacity on Wind Power Accommodation Capability

The capacity limits of transmission lines from buses 11 to 14, and from buses 14 to 16 were decreased to 100 MW during periods 7-8.Wind power curtailments must be considered from the network security constraints of systems.A comparison of the predicted and optimized allowable interval upper and lower bounds of the wind power output is illustrated in Fig.5.Note that during periods 7 and 8, the transmission line capacity limits were all 100 MW between buses 14 and 11, and buses 14 and 16.The insufficient transmission line capacity shortens the allowable wind power output interval to a constant that is less than the predicted wind power output interval in the proposed model.Meanwhile, the conventional robust UC model, irrespective of the wind power output limit, cannot satisfy the network security constraints and has no feasible solution.

2) Analysis of the Impact of Spinning Reserve Requirements on Wind Power Accommodation Capability

To eliminate the influence of network security constraints, the capacity limits of transmission lines from buses 14 to 11, and from buses 14 to 16 were altered to 500 MW.Figs.6, 7, and 8 compare the upper and lower bounds of the wind power output interval, the operational schedules of conventional units, and the up-and downspinning reserves between the proposed and conventional robust UC models, respectively.From these figures, during periods 1-6, 15, 18, and 24, when the wind power output was high and the system load level was low, the power output of conventional units was close to their minimum power output.Given that the conventional units do not have sufficient down-spinning reserves and downward adjustment capacity, the proposed model reduces the allowable wind power output interval to ensure that the system security constraints can be satisfied within the allowable interval, while the conventional robust UC model has to stop units 22 and 23 to meet the system dynamic adjustment requirements.As the allowable wind power output interval is reduced, the downward adjustment capacity requirements of the systems and the down-spinning reserve requirements are reduced as well.During other periods, with an increase in the system load and a decrease in wind power output, the power output of conventional units will increase and have sufficient adjustment capacity and spinning reserves to maximize the wind power accommodation capability; consequently, the wind power output limit becomes equal to the upper limit of the wind power output.

3) Analysis of the Impact of Wind Power Curtailments on Economy Improvement

A comparison of different costs, including fuel cost, start-up and shut-down cost, spinning reserve cost, penalty cost of wind power curtailments, and total cost between the conventional and the proposed robust UC models, is shown in Table 2.Note from this table that in the proposed robust UC model, although the penalty cost of wind power curtailments increases by 4 040.80 $ compared to the conventional robust UC model, inexpensive units 22 and 23 are in operation so that the fuel cost of conventional units decreases 1 476.82 $, and the start-up and shut-down costs of conventional units are reduced by 246.10 $.Moreover, the allowable wind power output interval reduces the spinning reserve cost to 3 380.34 $.Therefore, the total cost is reduced to 1 062.46 $, and its reduced proportion is approximately 0.15%.

3.3 Simulation analysis considering different Γt values

To analyze the impacts of budget Γt of uncertainty on the upper and lower limits of uncertainty set another case was considered.In this case, two wind farms were added at buses 14 and 15, respectively.One wind farm capacity was set to 400 MW.Its predicted power output interval used for testing is shown in Fig.1.Table 3 lists the robust optimization results for different Γt values.

With an increase in the budget of uncertainty, the mutation range of wind power output within a given interval may increase, and the security requirements of system operation may increase as well, which will make the robust solutions more conservative.Meanwhile, the maximum allowable wind power generation gradually increases and the minimum allowable wind power generation gradually decreases, which enlarges the allowable wind power output interval.The improvement of security requirements will alter the optimal schedules of conventional units with the main consideration of operational economy, resulting in a continuous increase in the total cost and deterioration of the economy of system operation.

4 Conclusions

The wind power accommodation capability is usually limited by the spinning reserve requirements and transmission line capacity in power systems with largescale wind power integration.Therefore, by employing the big-M method and adding auxiliary 0-1 binary variables to describe the allowable wind power output interval, a bilinear programming problem meeting the security constraints of system operation is presented.Furthermore, an adjustable confidence level was introduced into the proposed robust optimization model to decrease the level of conservatism of the robust solutions, which can establish a trade-off between economy and security.To develop an MILP problem that can be solved by commercial solvers such as CPLEX, the big-M method was utilized to represent the bilinear formulation as a series of linear inequality constraints and approximately address the nonlinear formulation caused by the adjustable conservativeness.

Simulation studies were conducted on a modified IEEE 26-generator reliability test system connected to a wind farm.According to the simulation results, this model can reduce the wind power output interval to maximize the overall economic efficiency and simultaneously satisfy the security requirements of system operation.By introducing an adjustable confidence level as the basis for adjusting the predicted wind power output interval, the proposed robust SCUC model can obtain the relationship between the confidence level and the uncertainty set for wind power generation and reduce its conservativeness, thereby achieving a trade-off between the economy and security of system operation.

Acknowledgements

This research was supported by State Grid Jiangsu Electric Power Co.,Ltd (JF2020001).National Key Technology R&D Program of China (2017YFB0903300) and State Grid Corporation of China (5210EF17001C).

Declaration of Competing Interest

We have no conflict of interest to declare.