Abstract
In this paper, a new formulation for modeling the problem of stochastic security-constrained unit commitment along with optimal charging and discharging of large-scale electric vehicles, energy storage systems, and flexible loads with renewable energy resources is presented. The uncertainty of renewable energy resources is considered as a scenario-based model. In this paper, a multi-objective function which considers the reduction of operation cost, no-load and startup/shutdown costs, unserved load cost, load shifting, carbon emission, optimal charging and discharging of energy storage systems, and power curtailment of renewable energy resources is considered. The proposed formulation is a mixed-integer linear programming (MILP) model, of which the optimal global solution is guaranteed by commercial solvers. To validate the proposed formulation, several cases and networks are considered for analysis, and the results demonstrate the efficiency.
THE problem of unit commitment is a fundamental optimization task in the management of power systems to determine the optimal set of on/off status of existing units, which minimizes the total operation cost and at the same time meets certain limitations in the short term. The main challenges faced by the problem of unit commitment can be summarized as follows.
One challenge is created by the high penetration of renewable energy resources for power system operators in the unit commitment. Another challenge is the use of grid-connected electric vehicles (EVs) as movable energy storage units with intelligent charging and discharging schedules.
Two basic approaches are to use demand-side management programs and large-scale energy storage systems to increase grid flexibility in the presence of renewable energy resources and grid-connected EVs.
Given the above, the main objectives for power system operators are to provide a mathematical model for unit commitment with security constraints and model the uncertainty of renewable energy resources by modeling EVs and energy storage systems under flexible loads that can lead to finding optimal global solutions.
Reference [
In [
Reference | Solver | Model | EV | Security | Renewable | Multi-objective | Stochastic | Flexible load | Battery | Emission |
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This paper | Gurobi | MILP | √ | √ | √ | √ | √ | √ | √ | √ |
[ | Gurobi | MILP | √ | |||||||
[ | Cplex | MISOCP | √ | √ | √ | |||||
[ | GBO | MINLP | ||||||||
[ | Cplex | MILP | √ | √ | √ | |||||
[ | Gurobi | MILP | ||||||||
[ | Cplex | MILP | √ | √ | √ | |||||
[ | Cplex | MIP | √ | √ | ||||||
[ | Xpress | MILP | √ | √ | ||||||
[ | Gurobi | MILP | √ | √ | √ | |||||
[ | Cplex | MILP | √ | √ | √ | √ | ||||
[ | Gurobi | MILP | √ | √ | √ | |||||
[ | Cplex | MILP | √ | |||||||
[ | Gurobi | MILP | √ | √ | ||||||
[ | Gurobi | MILP | √ | √ | ||||||
[ | Cplex | MILP | √ | |||||||
[ | DICOPT | MILP | √ | √ | √ | √ | √ | |||
[ | Gurobi | MILP | √ | √ | ||||||
[ | Gurobi | MISOCP | √ | |||||||
[ | ADMM | MINLP | √ | √ | ||||||
[ | Cplex | MILP | √ | √ | √ | √ |
According to the aforementioned literature review and the comparison of previous models, it can be found that in most of the previous studies, there was no single or complete model considering the modeling of the demand-side management problem, multi-objective functions, stochastic model, EVs, security constraints, and renewable energy resources in the problem of unit commitment. Thus, in this study, we try to present a complete model considering all these issues.
The main innovations and contributions of this paper can be summarized as follows.
1) A new formulation is introduced for the problem of unit commitment considering the optimal charging and discharging of large-scale EVs, energy storage systems, and flexible loads with renewable energy resources.
2) The uncertainty of renewable energy resources is considered as a scenario-based model.
3) A framework in the form of MILP model with an optimal global solution guaranteed is proposed.
4) A multi-objective function is considered in the proposed formulation, which includes the reduction of operation cost, no-load and startup/shutdown costs, unserved load cost, load shifting, carbon emission, optimal charging and discharging of energy storage systems, and power curtailment of renewable energy resources.
The remainder of this paper is organized as follows. The proposed formulation is presented and fully explained in Section II. The results obtained from the simulation are analyzed in Section III. Finally, the conclusions and some suggestions are presented in Section IV.
The modeling of the problem for stochastic security-constrained unit commitment is presented in this section.
(1) |
where , , and are the sets of bus, time, and scenario, respectively; , , and are the indices of bus, time, and scenario, respectively; is the power of the units on bus b at time t; is the unit production cost on bus b; is the state of commitment of units on bus b at time t; is the no-load cost of each unit on bus b; and are the on and off statuses of the units on bus b at time t, respectively, and their related costs are and ; is the load curtailment on bus b at time t; is the cost related to the load interruption; and are the charging and discharging power of the battery on bus b at time t, respectively; is the cost of charging and discharging of the battery on bus b; is the initial load on bus b at time t; is the shifted load after the demand-side management program; is the cost associated with the change or load shedding on bus b in the demand-side management program; is the emission cost of the units in bus b; is the probability of each scenario; is the actual production of renewable energy resources on bus b at time t in scenario s; is the operation power from renewable energy resources on bus b at time t; and is the cost of a power outage of renewable energy resources on bus b.
The problem of security-constrained unit commitment is also shown by (2)-(14).
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
where is the power flow from bus i to j; is the power flow from bus j to i; is the injected or received power of the EVs on bus b at time t; and are the lower and upper limits of the power of the units on bus b, respectively; is the set of grid lines; is the index of line; is the state of lines, if it is equal to 1, the line is connected, and if it is equal to 0, the line is disconnected; is the maximum power flow through line ; is the susceptance of line ; is the voltage angle of bus i at time t; k is the number of lines disconnected from the network; ref is the set of reference buses; is the real power flow; and are the minimum and maximum ramp rates, respectively; is the minimum up time; and is the minimum down time.
Relationships (15)-(26) show the modeling of EVs in the problem of security-constrained unit commitment. The net charging and discharging energy of the EV battery is shown in (15) according to the efficiency of each battery.
(15) |
(16) |
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
(23) |
(24) |
(25) |
(26) |
where is the net charging and discharging energy of the EV battery on bus b at time t; and are the discharging and charging power of EV batteries on bus b at time t, respectively; is the charging efficiency of the EV battery; and are the charging and discharging statuses of EVs on bus b at time t, respectively; is the state of connection to the network of EVs, if it is equal to 1, the EV is connected to the grid, otherwise, it is disconnected; and are the minimum and maximum charging power of EVs, respectively; and are the minimum and maximum discharging power of EVs, respectively; is the energy status of the EV battery; is the energy demand of EVs on bus b at time t; and are the minimum and maximum energies of the EV battery, respectively; is the operation cost of EVs; is the slope of the
Formulas (
(27) |
(28) |
(29) |
(30) |
(31) |
(32) |
where is the battery status; is the maximum battery capacity on bus b; is the energy in the battery on bus b at time t; is the battery efficiency; and is the amount of initial energy in the battery on bus b.
Relationships (33)-(35) represent the demand-side management model in the problem of unit commitment.
(33) |
(34) |
(35) |
where is the shifted demand; is the initial demand; and is the percentage of load changes.
To analyze the proposed formulation, two test networks (6-bus and 24-bus) are considered for analysis. The simulation is implemented using Julia programming language and solved with the Gurobi solver. A computer system with a 1.8 GHz CPU and 6 GB RAM is used.
In this subsection, the results related to the 6-bus network are presented [

Fig. 1 Schematic of 6-bus network.

Fig. 2 Generated scenarios for renewable energy resources. (a) Scenarios for PV. (b) Scenarios for wind power.
To better analyze the proposed formulation, four different case studies are considered as follows.
Case 1: considering a maximum of 0% load shifting and the number of outage lines is equal to 1.
Case 2: considering a maximum of 10% load shifting and the number of outage lines is equal to 1.
Case 3: considering a maximum of 0% load shifting and the number of outage lines is equal to 3.
Case 4: considering a maximum of 10% load shifting and the number of outage lines is equal to 3.
The results related to different cases in the 6-bus network are presented in
Case | Objective function value ($) | No. of disconnected line |
---|---|---|
Case 1 | 535887 | 9 |
Case 2 | 529545 | 9 |
Case 3 | 535890 | 2, 8, 9 |
Case 4 | 529738 | 4, 8, 9 |

Fig. 3 Status of unit commitment in 6-bus network in case 1.

Fig. 4 Status of unit commitment in 6-bus network in case 2.

Fig. 5 Status of unit commitment in 6-bus network in case 3.

Fig. 6 Status of unit commitment in 6-bus network in case 4.

Fig. 7 Comparison of charging and discharging status of EV parking in 6-bus network. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

Fig. 8 Comparison of SOE of EVs in cases 1-4 in 6-bus network.

Fig. 9 Optimal charging and discharging statuses of energy storage system in 6-bus network. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

Fig. 10 Status of energy remaining in energy storage system in 6-bus network. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

Fig. 11 Comparison of load shifting in different cases in 6-bus network.

Fig. 12 Comparison of production of units in different cases in day-ahead operation in 6-bus network.
Load shifting (%) | Number of outage lines | No. of disconnected line | Objective function value ($) | Solution time (s) |
---|---|---|---|---|
0 | 0 | 535756 | 5 | |
1 | 9 | 535887 | 6 | |
2 | 4, 9 | 535889 | 16 | |
3 | 2, 8, 9 | 535890 | 22 | |
4 | 1, 4, 8, 9 | 544464 | 57 | |
10 | 0 | 529537 | 5 | |
1 | 9 | 529545 | 6 | |
2 | 2, 9 | 529638 | 8 | |
3 | 4, 8, 9 | 529738 | 19 | |
4 | 1, 4, 8, 9 | 534143 | 40 |
Load shifting (%) | Objective function value ($) | Load shifting (%) | Objective function value ($) |
---|---|---|---|
0 | 535887 | 15 | 526232 |
5 | 532671 | 20 | 525171 |
10 | 529545 | 30 | 524341 |
In this subsection, the results related to the IEEE RTS 24-bus network shown in

Fig. 13 Schematic of IEEE RTS 24-bus network.
To better analyze the proposed formulation, four different case studies have been considered as follows.
Case 5: considering a maximum of 0% load shifting and the number of outage lines is equal to 1.
Case 6: considering a maximum of 10% load shifting and the number of outage lines is equal to 1.
Case 7: considering a maximum of 0% load shifting and the number of outage lines is equal to 5.
Case 8: considering a maximum of 10% load shifting and the number of outage lines is equal to 5.
The simulation results related to different cases in the 24-bus network are presented in
Case | Objective function value ($) | No. of outage lines |
---|---|---|
Case 5 | 2727219 | 8 |
Case 6 | 2719348 | 34 |
Case 7 | 2727226 | 1, 2, 7, 8, 12 |
Case 8 | 2719383 | 1, 6, 9, 12, 21 |
Load shifting (%) | Objective function value ($) | Load shifting (%) | Objective function value ($) |
---|---|---|---|
0 | 2727210 | 15 | 2717550 |
5 | 2721800 | 20 | 2716250 |
10 | 2719340 | 30 | 2715340 |
Figures 14-17 show the status of unit commitment of cases 5-8 in the 24-bus network for the day-ahead operation (1-24 hours), which indicate the modeling and accuracy of the unit commitment problem in each case. The numbers 1 and 0 indicate that the unit is on and off, respectively. It can be observed that with the change of load shifting and the number of outage lines, the status of unit commitment changes. As can be observed from Figs.

Fig. 14 Status of unit commitment in 24-bus network in case 5.

Fig. 15 Status of unit commitment in 24-bus network in case 6.

Fig. 16 Status of unit commitment in 24-bus network in case 7.

Fig. 17 Status of unit commitment in 24-bus network in case 8.

Fig. 18 Comparison of charging and discharging statuses of EVs in 24-bus network. (a) Case 5. (b) Case 6. (c) Case 7. (d) Case 8.

Fig. 19 Comparison of energy status of EV parking in cases 5-8 in 24-bus network. (a) Case 5. (b) Case 6. (c) Case 7. (d) Case 8.

Fig. 20 Comparison of load shifting in different cases in 24-bus network.
Besides, with the increase of the number of outage lines, the peak load increases in case 8 compared with case 6.

Fig. 21 Comparison of production of units in different cases in day-ahead in 24-bus network.
By examining the results obtained from the 6- and 24-bus networks, it can be found that the proposed formulation, in addition to being efficient, can be easily implemented and implemented for any system with any dimension.
In this study, a formulation for modeling the problem of stochastic security-constrained unit commitment is presented. The proposed formulation is considered as an MILP model, which includes modeling of charging and discharging of large-scale EVs, optimal charging and discharging of energy storage systems, and demand-side management along with renewable energy resources. A multi-objective function is considered to minimize different objectives in order to get the maximum efficiency from the proposed formulation. The 6-bus and 24-bus networks are considered for analysis with different cases. The results demonstrate that by managing the demand and optimal charging and discharging of batteries and EVs in the unit commitment problem, it is possible to reduce the effect of line outages on the performance of the objective function. For example, the objective function value only differs by 0.015% for the cases with 1 outage line and 5 outage lines. Besides, the proposed formulation can be well implemented in different transmission networks. Also, various cases considered by changing the parameters of the problem to investigate the demand-side management performance in the problem of unit commitment show that the proposed formulation has good scalability. However, as we know, by modeling very large transmission networks, the numbers of variables and parameters of the proposed problem increase dramatically. Therefore, one of the leading limitations of this work is to solve it by solvers.
To continue this paper, the following items are suggested for future research.
1) Modeling distribution networks in each bus of the transmission network, so as to analyze the proposed formulation of the transmission network in distribution networks.
2) Modeling the natural gas network along with the transmission network, so as to analyze the impact of natural gas reduction on gas-fired power plants in the power grid.
3) Presenting a bi-level optimization model with the aim of maximizing the resilience of the transmission network to the proposed formulation in times of natural disasters.
4) Adding the discussion of network resilience to the proposed formulation and modeling natural disasters and cyber-attacks on the network.
Appendix
Power levels of renewable energy resources for each scenario can be expressed as:
(A1) |
where is the forecast value of renewable power; and is the forecast error of the renewable power. Since each scenario has a probability of occurrence, the normalized probability of each generated scenario is obtained using the following equation.
(A2) |
where is the probability of the th renewable power interval at time ; and is a binary parameter indicating whether the th renewable power interval of renewable energy resource on bus is selected in scenario () or not () at time . The full description of the scenario generation method can be found in [
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