Several simulations are conducted using short- and large-scale electrical power systems to find out the performance of both corrective and preventive SCOPF formulations. To formulate the optimization problem, Python [
20] has been used. Moreover, Gurobi [
21] is used as a commercial solver on a computer with the following characteristics: Intel Core i7 3930 (3.20 GHz) six core with RAM 32 GB.
3.1. SCOPF Formulation Applied to an Example Power System (6-Bus)
The first security-constrained simulation takes into account the 6-bus power system presented by Wood and Wollenberg [
8]. Transmission network data can be seen in [
8] or in MatPower [
22].
In comparison with the original generation system, three new power units are located at each load bus (
,
and
).
Figure 1 shows the electrical power system.
Technical data for the power generation system are given in
Table 1. Besides, ramp-up and ramp-down characteristics are incorporated in the last two columns for each power unit.
Solving the classical OPF problem, the optimal cost is
Ctotal = 3003.17
$/h. The power generation solution is
P1 = 50.0 MW,
P2 = 37.5 MW,
P3 = 45.0 MW,
P4 = 5.0 MW,
P5 = 67.5 MW and
P6 = 5.0 MW.
Figure 2 shows the power flow solution. According to the power flow solution, there is no congestion in transmission elements. PyPower [
23] is used to validate the solution and both solutions are the same.
To realize possible transmission effects, the maximum transmission limits are changed to 40 MW in lines 1–4, 2–4 and 3–6. With this new capacity, the OPF problem also obtains the same solution (operational cost, power unit generation and transmission power flows). This new transmission capacity is used to develop the N−1 power system security.
DC-Based and SF-Based SCOPF Formulations
In the first analysis, both classical (CL) and introduced (SF) problems are implemented to model the SCOPF. These formulations are subject to one pre- and eleven post-contingency constraints.
Corrective SCOPF problem—for this analysis, power generation data are incorporated in the formulation assuming that the ISO has 10 min to eliminate overloading transmission effects and to recover the steady-state power system security. The main advantage of the corrective formulation is related to the operator clearly knows the post-contingency economic dispatch. This solution considers not only technical generation constraints but the variable fuel cost of each power unit. Indeed, the new power generation setting will achieve a safety and robust security-constrained N−1 solution no matter which transmission element failure.
Two cases are conducted to determine effects when ramps constraints are added in the SCOPF problem. These optimization problems have been solved using the SF-based formulation. Results are the following:
- Case 1
Without ramp constraints—the operational cost is 3003.17 $/h for the pre-contingency condition and 3487.87 $/h for the post-contingency condition and the optimal total cost is Ctotal = 6491.04 $/h. Actually, the pre-contingency cost is the same than the traditional OPF problem (3003.17 $/h).
- Case 2
Including ramp-up and -down constraints—the cost is 3146.35 $/h for the pre-contingency condition and 3487.87 $/h for the post-contingency condition and the total cost is Ctotal = 6634.22 $/h. For this case, the number of decision variables is 12 and the number of constraints is 280.
Table 2 shows the results, operational cost and power dispatch for each SCOPF solution. Positive values imply that the power unit generation must decrease its dispatch. With this information, the ISO rapidly decides (10 or 20 min) which generator would economically change the power unit setting to optimally eliminate overloading conditions.
Because PyPower does not formulated the security-constrained analysis, it is not possible to contrast this power generation solution.
Because of ramp constraints, the ISO will be not able to achieve a safe post-contingency steady-state with the solution obtained in Case 1). Therefore, the system operator will need to decrease the load of the customers or/and turn-on fast reserve power units.
Notice that the ramp-down constraint of power unit 5 (44.14 − 24.14 = 20 MW) is active with a shadow price of 6.36 $/MWh. Therefore, the pre-contingency cost is higher (143.18 $/h) than the traditional OPF problem. Additionally, the power generation solution for the post-contingency condition is the same for both cases.
In comparison with the SF-based formulation, simulation results obtained by the CL-based formulation are the same. Therefore, both formulations achieve the same optimal solution; that is, these optimization problems are equivalent.
Incorporating the ramp constraints in the CL SCOPF problem, the number of decision variables is 205 and the number of constraints is 483. Comparing with the SF formulation, there is a very significant reduction of 94.1% and 42.0%, respectively.
Figure 3 displays the pre- and the post-contingency solutions when line 1–2 is outage. Both solutions could be compared to realize redispatch effects in the transmission network for generation two, four, five and six (grey color).
Using the post-contingency power generation solution (
Table 2), eleven N−1 power flows are computed. For detailed information,
Table 3 shows the power flow solution for each outage.
Results showed there are two post-contingency congestions in the following lines: (a) transmission line 2–4 when line 1–4 is out; and (b) transmission line 3–6 when line 2–6 is out. We have highlighted these values with bold. Based on the OPF solution, shadow prices (Lagrangian multipliers) for each congestion are the following: (a) for line 2–4 is 28.60 $/MWh; and (b) for line 3–6 is 41.98 $/MWh. The OPF overcost and the bigger Lagrangian multipliers are produced by transmission congestion and ramp-down constraint of unit 5.
Using the SF-based formulation, the number of decision variables is 6 and the number of constraints is 280. The optimal cost obtained by Gurobi is Ctotal = 3487.87 $/h which was previously obtained. In the solution, there is no congestion in the pre-contingency condition. However, there is congestion in transmission line 2–4 (when line 1–4 is out) and line 3–6 (when line 2–6 is out).
Gurobi achieves the same solution solving the CL-based SCOPF formulation. Nevertheless, the number of decision variables is 199 and the number of constraints is 459.
To compare corrective and preventive SCOPF solutions, the power dispatch obtained in the pre-contingency condition must be compared for both solutions. The preventive overcost is 342.52 $/h (9.82%). Therefore, there is an important operational saving cost obtained by the corrective formulation.
3.2. Corrective SCOPF Using a Ranking of Contingencies
In real power systems, all contingencies do not result in a post-contingency alert condition [
8]. To figure out the SCOPF problem, several authors have used as security criterion the most severe contingency [
6].
Regarding the N−1 power flow solution obtained previously by the initial OPF (3003.17 $/h), we solves eleven power flows for each N−1 post-contingency event. Simulations have shown there are five overloading states: (a) when line 1–4 is out, the power flow in line 2–4 is 131.32%; (b) when line 2–4 is out, the power flow in line 1–4 is 117.11%; (c) when line 2–6 is out, the power flow in line 3–6 is 124.41%; (d) when line 3–5 is out, the power flow in line 3–6 is 103.70%; and (e) when line 5–6 is out, the power flow in line 3–6 is 109.42%. Indeed, the ranking of overloading contingencies is the following: (1) line 1–4, (2) line 2–6, (3) line 2–4, (4) line 5–6 and (5) line 3–6.
The outage of these transmission elements produce a risky operational condition and probably technical problems to supply adequately the load of the customers. In the next analysis, each line of this ranking will be added in the SCOPF problem.
Table 4 displays the number of decision variables and constraints and the objective function solving the SCOPF problem.
For the first case, the pre-contingency and the worst contingency (line 1–4) constraints are simultaneously incorporated in the SCOPF problem.
Solving the SCOPF problem, the post-contingency power unit solution is used to validate electrical system security using eleven N−1 power flow problems. Simulation results show there are three overloading conditions: (a) when transmission line 2–6 is out, the power flow in line 3–6 is 51.18 MW; (b) when transmission line 3–5 is out, the power flow in line 3–6 is 42.74 MW; and (c) when transmission line 5–6 is out, the power flow in line 3–6 is 42.11 MW. Notice that the maximum power flow in line 3–6 is 40 MW. Therefore, a SCOPF problem based on the worst contingency does not guarantee a safe post-contingency operational point.
In the second case, the SCOPF problem includes the outage of the transmission line 2–6.
Reviewing the power flow solution, there are two overloading conditions: (a) when transmission line 1–4 is out, the power flow in line 2–4 is 54.34 MW; and (b) when transmission line 2–4 is out, the power flow in line 1–4 is 47.72 MW. Notice that the maximum power flow in line 1–4 and line 2–4 is 40 MW. Even though the operational cost is bigger than the previous case, this solution does not guarantee a safe power system as well. Additionally, this solution obtains the most overloading condition in line 2–4 (136%).
In the third case, failure of transmission line 2–4 is added in the optimization problem.
Simulating the N−1 post-contingency power flow method, the overloading conditions are the following: (a) when transmission line 1–4 is out, the power flow in line 2–4 is 43.97 MW; (b) when transmission line 2–6 is out, the power flow in line 3–6 is 50.73 MW; (c) when transmission line 3–6 is out, the power flow in line 2–6 is 41.62 MW; and (d) when transmission line 5–6 is out, the power flow in line 3–6 is 42.63 MW. Furthermore, this is an unacceptable post-contingency overloading condition.
For the fourth case, transmission constraints for the outage of transmission line 5–6 is included in the N−1 optimization problem.
Simulating the post-contingency power flow method, the overloading conditions are the following: (a) when transmission line 1–4 is out, the power flow in line 2–4 is 53.20 MW; (b) when transmission line 2–4 is out, the power flow in line 1–4 is 47.17 MW; (c) when transmission line 2–6 is out, the power flow in line 3–6 is 46.11 MW; and (d) when transmission line 3–5 is out, the power flow in line 3–6 is 40.16 MW.
We do not show the last contingency because power flow results also display an overloading condition.
Finally, the post-contingency analysis including two N−1 candidate lines is developed. The SCOPF problem incorporates two major contingencies: line 1–4 and line 2–6. This framework adds in the optimization problem simultaneously one set of pre-contingency constraints and two sets of post-contingency constraints. For this optimization problem, the achieved solution is the optimal (the same solution considering the outage of all lines).
Table 4 also shows a reduction of 35.7% in constraints with respect to the original problem. In addition, both problems have the same number of decision variables.
As a result, the SCOPF problem should be modelled not only with the worst contingency but also with two or more contingencies to avoid risky operational conditions and technical problems to supply the load of the customers.
3.3. SCOPF Methodology Applied to Different-Scale Power Systems
We employ different electrical power test systems to determine the performance of both corrective and preventive SCOPF formulations using the SF-based as well as the CL-based frameworks: 14-bus system (five generators and twenty transmission lines); 57-bus system (seven generators and eighty transmission lines); 118-bus system (fifty-four generators and one hundred eighty-six transmission lines); and 300-bus system (sixty-nine generators and four hundred eleven transmission lines). PyPower shows technical information.
3.3.1. Corrective and Preventive Formulations
Because PyPower does not include ramp generation constraints, we have selected these limits using random numbers provided by a normal distribution function with a mean of 7 MW/min and a variance of 1.
Table 5 shows simulation results for both corrective and preventive analyses applying the CL and the SF formulations. For each SC problem, we have included the objective function (
), the number of decision variables (
n), constraints (
) and the average simulation time considering 200 trials.
For both 14-bus and 57-bus power systems, the security-constrained problem models the outage of all transmission lines.
Considering the higher simulation time obtained by a complete N−1 analysis, a reduced number of contingencies should be used to solve the security problem applied to large-scale power systems. In the 118-bus and the 300-bus systems, we have modelled the N−1 analysis using 176 transmission and 100 transmission candidate lines, respectively. This assumption is accomplished not only by the bigger number of variables and constraints but also by the higher simulation time.
For all power systems, there is an important improvement in the simulation time. Comparing both classical and SF-based formulations applied to the 300-bus system, SF-based formulation achieves the best performance with an efficiency of 6.97 times using the corrective formulation and 7.02 times using the preventive formulation.
Simulation results have shown the corrective formulation obtains the lower in comparison with the preventive formulation.
Notice that the reactance bus matrix needs an inverse method.Therefore, the main disadvantage of the proposed methodology is the time computing to obtain the SF matrix. However, the security-constrained analysis could be carried out using an off-line SF matrix; that is, the SF matrix does not need to compute each N−1 time. Therefore, an external file could be used to save these matrixes improving the formulation time.
The LODF matrix could be computed using basic mathematical operations—Equation (
37). Therefore, the computation time is not significant.
3.3.2. Another Classical DC-Based Preventive Formulation Applied to the 300-Bus Power System
In the following analysis, we use a different preventive SCOPF formulation. For this approach, power flow variables are eliminated from the optimization problem using two admittance bus matrixes ( and ). Consequently, there are only two sets of nodal balance constraints for the pre- and post-contingency analyses. Besides, the primitive-admittance incidence matrix for the pre-contingency and the post-contingency conditions is substitute to determine transmission power flows.
Simulating 200 trials, the average simulation time is 0.7648 s for the preventive SCOPF problem which is 19.22% lower than the original CL-based formulation. For this mathematical problem, the number of decision variables is 30,369 and the number of constraints is 113,361. Therefore, the new classical formulation causes a reduction of 57.69% in the number of decision variables and a reduction of 26.76% in the number of equality and inequality constraints. Although there is an improvement in the simulation performance, this time is still higher (5.67 times) than the SF-based formulation.
Based on simulations, the introduced SF-based formulation achieves the best simulation time to carry out the corrective formulation and the preventive formulation applied to the SCOPF problem. This improvement is obtained by the lower number of decision variables as well as by the lower number of equality and inequality constraints. Therefore, this framework could be recommended to solve very large-scale power systems.
3.4. Corrective and Preventive SCOPF Methodology Applied to the Chilean Electrical Power System
The Chilean Interconnected Power System (in Spanish, SEN: Sistema Eléctrico Nacional) is used to validate both SCOPF formulations. Concerning the lower simulation time accomplished previously, the SF-based methodology is only applied to the SCOPF problem.
In January, 2019, the SEN generation capacity was 21,189.95 MW. In 2019, the peak load was 9382.7 MW. Review the Regulatory Company webpage (CNE:
www.cne.cl) or the ISO webpage (CEN:
www.coordinador.cl) to obtain more technical information.
The electrical power system contains one hundred fifty-nine buses, three hundred twenty-one transmission lines from 110 to 500 kV and two hundred sixty-seven generation power units (thermal, hydro and renewable energy). The webpage link
https://infotecnica.coordinador.cl displays detailed technical and economic information about the Chilean study case.
For the power generation system, there are one hundred three renewable power units. For hydro, wind and solar, a capacity factor of 20%, 20% and 30% are selected, respectively. For thermal units, random numbers model ramp generation constraints for each power unit. Additionally, one hundred transmission lines are randomly selected to formulate the N−1 security-constrained analysis.
Corrective SCOPF problem—this optimization problem has 534 decision variables and 22,082 constraints. The cost is 326,948.31 $/h for the pre-contingency condition and 327,087.37 $/h for the post-contingency condition. The simulation time is 1.0218 s using 100 trials.
There is congestion in Cardones-Maintencillo 220 kV in the pre-contingency solution. On the contrary, there is post-contingency congestion in the following transmission elements—(1) Andes-Nueva Zaldivar 220 kV when Antucoya-Antucoya aux. 220 kV is out; (2) Antofagasta-Desalant 110 kV when Atacama-Esmeralda 220 kV is out; (3) Capricornio-Mantos Blancos 220 kV when CD Arica-Tap Quiani 66 kV is out; (4) Carrera Pinto-Carrera Pinto aux. 220 kV when Charrúa-Lagunillas 220 kV is out; and (5) CD Arica-Arica 66 kV when Charrúa-Mulchen 220 kV is out.
Preventive SCOPF problem—this optimization problem has 267 decision variables and 21,013 constraints. The optimal cost is 329,981.43 $/h and the simulation time is 0.6085 s. For the pre-contingency condition, there is congestion in the same line (Cardones-Maintencillo 220 kV). Furthermore, there are seven lines with congestion in the post-contingency condition: (1) Andes-Nueva Zaldivar 220 kV; (2) Antofagasta-Desalant 110 kV; (3) Capricornio-Mantos Blancos 220 kV; (4) Carrera Pinto-Carrera Pinto aux. 220 kV; (5) CD Arica-Arica 66 kV; (6) Charrúa-Hualpen 220 kV (when Colbún-Ancoa 220 kV is out); and (7) Crucero-Nueva Crucero/Encuentro 220 kV (when Don Goyo-Don Goyo aux. 220 kV is out).
Even though the security cost is similar, the OPF formulation only considers one load time-period. Moreover, the unit commitment solution obtained by the ISO should be incorporated in the OPF problem to realize real saving costs.