Robust transmission expansion planning model considering multiple uncertainties and active load

0 Introduction

The main purpose of transmission expansion planning(TEP)is to determine the appropriate time and location for investing in new transmission lines[1].TEP is the basis for the secure and economic operation of power systems and has become one of the most important crucial strategic decisions regarding power systems[2-3].

Recently,with the rapid development of renewable energy sources(RESs),uncertainty has become one of the most important aspects that must be addressed in TEP.Numerous studies that address various uncertainties in TEP models have been conducted.In terms of the time scale,uncertainties can be generally categorized into long- and short-term uncertainties[4].Long-term uncertainties are those that will have an impact in the long run,for example,the uncertainties related to load growth,the future share of RESs in power systems,and future fuel costs[5].By contrast,short-term uncertainties are those that have an impact at a daily,hourly,or even shorter time scale,such as the uncertainty of RES output and contingency events related to equipment outage.

To address the uncertainties caused by RES output and load,the uncertainties can be explicitly expressed,for example,by discretized scenarios,uncertainty intervals,and sets of probability distribution functions,which correspond to stochastic optimization(SO)[6],robust optimization(RO)[7],and distributionally robust optimization(DRO),respectively.Reference[8]proposed a probabilistic TEP model considering load and wind power uncertainties and applied a Monte Carlo simulation to capture the randomness property.Reference[9]proposed a chance-constrained formulation to address the uncertainties of load and wind power in a TEP model and converted the probabilistic nature into a deterministic one.Reference[10]considered the uncertainties of future load demand and wind power production and proposed a two-stage adaptive robust TEP model.Reference[11]established a DRO model for TEP in which both long- and short-term uncertainties of the RES output and load are considered.

Many existing studies have focused on uncertainties caused by RESs or load,while the uncertainty related to contingency events caused by equipment outages has not been well analyzed.To address the uncertainty caused by contingency events,traditionally deterministic and probabilistic approaches have been adopted in TEP models.In the deterministic approach,the N-k security criterion is often applied in the form of constraints[12].Reference[13]considered the uncertainty of RESs and N-1 security criteria in TEP and proposed a min-max regret method.The N-k criteria method is easy to understand,but the result is potentially conservative.The probabilistic approach determines the optimal planning results by balancing the investment cost and pre- and post-contingency costs simultaneously[14],and it strikes a balance between security and cost-effectiveness.The contingency cost is usually computed as the summation of the products of the contingency probability and associated cost.When the contingency probability is considered,a long-term statistical average value is usually applied,which is obtained based on historical data,statistical methods,and practical experience.However,it is difficult to obtain the exact value of the contingency probability because of the insufficient quality and quantity of historical data[15],and the law of large numbers is not applicable[16].The contingency probability would significantly deviate from the real value,leading to a suboptimal solution to the TEP problem.The accuracy and rationality of using an estimated contingency probability need to be studied further.

The uncertainties of contingency probability have been considered in various scheduling models.Reference[17]modeled the contingency probability using an ambiguity set and proposed a DRO unit commitment model.The objective function minimizes the worst-case expected total costs with respect to all contingency probability distributions in the ambiguity set of contingency probability.Reference[18]extended the ambiguity set proposed in[17]to the TEP model and proposed a two-stage DRO TEP model.However,considering only the uncertainty of contingency probability might not be appropriate because the effect of uncertainties caused by RESs is also significant compared with the effect caused by the uncertainty of contingency probability.

Based on previous research,a robust TEP model that considers multiple uncertainties and the active load is proposed in this paper.The model considers the uncertainties of wind power and contingency probability simultaneously.The uncertainty caused by the RESs is described by scenarios,whereas that caused by contingency probability is described by intervals.In addition,the active load is considered as a flexibility source.The proposed model entails a three-level optimization problem and is transformed based on the duality theory.Finally,the model can be solved using the existing state-of-the-art mixed integer linear programming(MILP)solver.The main contributions of this study are as follows.

1)A robust TEP model is proposed.The model considers the uncertainties of both wind power and contingency probability.

2)The proposed three-level model is transformed to a bi-level problem,and the Benders decomposition algorithm is applied to solve the model.

The remainder of this paper is organized as follows.Section 1 introduces the mathematical model and uncertainty set.Section 2 introduces the process for solving the model.Section 3 presents the study case.Finally,Section 4 presents the conclusions.

1 Mathematical Formulation of the Proposed Model

1.1 Mathematical Model

The objective function attempts to minimize the total cost,which includes the investment,operation,and reliability costs.The objective function is expressed as follows:

where CI, CN,and CRare the investment cost,operation cost,and reliability cost,respectively.l,g, ω, w,i,and sare indexes of transmission lines,generators,wind power scenarios,wind farms,nodes,and contingency scenarios,respectively.is the set of candidate transmission lines.NW,NΩ,NI,and NSare sets of w,ω,t,i,and s,respectively.ulis a binary variable that represents whether transmission line lwill be constructed.Pω,gand C(Pω,g)are the output power and quadratic cost function of generator gunder the wind power scenario ω. lols,ω,iis the loss of load due to involuntary load shedding.Δds,ω,iis the load adjustment amount due to voluntary load adjustment.ΔWs,ω,wis the wind power spillage of wind farm wunder contingency scenario sand wind power scenario ω. πWand πLare the prices of the wind spillage and active load adjustment,respectively. is the probability of contingency s.probωis the probability of wind power scenario ω.In the proposed model,the uncertainty of the wind power output is described using discretized scenarios.The uncertainty of contingency is described by the interval.Therefore,in the model,is a random parameter.

Expression(2)calculates the total investment cost.Expression(3)calculates the expected operation cost under normal operating condition and under different wind power scenarios.Expression(4)calculates the expected reliability cost,including involuntary load shedding cost,wind spillage cost,and voluntary load adjustment cost under the worstcase contingency probability and different wind power scenarios.

The constraints of the model are listed as follows:

In(5)-(15),NLand NGare sets of existing transmission lines and generators,respectively.Wwis the wind power output of wind farm w.flis the power flow on transmission line l.Xlis the reactance of transmission line l.θiis the voltage phase angle on bus i.is the capacity of transmission line l. and are the minimum and maximum output of generator g,respectively.μis the proportion of the load that can be adjusted.diis the load on bus i.The variables with the prime symbol(′)in(12)-(18)correspond to contingency scenarios,and their meanings are similar to those of the corresponding variables used in normal operating condition.

Expression(5)represents the nodal power balance constraint.Expressions(6)and(7)calculate the power flow on the existing and candidate transmission lines,respectively.Expression(8)forces the voltage phase angle on the reference node to be 0.Expressions(9)and(10)limit the power flow on the existing and candidate transmission lines,respectively.Expression(11)limits the output of the generators.

Expressions(12)-(18)are the constraints corresponding to contingency scenarios.In this study,only the outages of transmission lines are considered.The power flow on the outage transmission lines are set to zero.Expressions(12)-(18)have similar meanings to those in(5)-(15).Expression(19)limits the wind spillage amount in contingency s.Expression(20)limits the maximum adjustment amount of active load in contingency s.Expression(21)limits the loss of load in contingency s.

The objective function and constraints are very similar to those in the traditional TEP model,except that is a random parameter;consequently,the model cannot be solved directly.Therefore,the random parameter needs to be eliminated,and the model needs to be transformed into a deterministic form.

1.2 Uncertainty Set

Both the uncertainties of wind power and contingency probability are considered in the proposed model.The output of wind power usually follows a normal distribution.If necessary,other distributions can also be applied.The distribution is first curtailed to a double-bounded distribution,and then the double-bounded distribution is discretized into several intervals[14].Each interval corresponds to a scenario.For each scenario,a wind power output value and the corresponding probability are generated.If five intervals are applied,the wind power output for scenario ωwill be equal to the forecast wind power plus the standard deviation of the wind power output multiplied by -2,-1,0,1,and 2,respectively.The probabilities probωfor scenario ω are 0.061,0.242,0.382,0.242,and 0.061[14],respectively.

Presently,the uncertainty of equipment outage probability is commonly described by the interval[19-21].The probability of a system-wide contingency can be analytically expressed by the probability of the equipment outage rate[14].When the equipment outage rate varies within an interval,the contingency probability also varies within an interval.Therefore,the uncertainty of the contingency probability can be described using a box uncertainty set.An auxiliary variable αsis introduced to transform the random contingency probability into a deterministic form,and its uncertainty set can be expressed as follows:

where Γ is a conservativeness parameter that represents

the number of contingency scenarios that reach the worst case at the same time.Expressions(23)-(25)consist of the uncertainty set of contingency probability.They can be expressed in the form of sets,as follows:

1.3 Deterministic Form of the Model

The uncertainty set(26)uses three constraints to describe the random parameter Furthermore,the proposed model can be transformed into a deterministic form as follows.

The objective function of the deterministic form is shown as follows:

The objective function is subject to constraints(5)-(11),(12)-(21),and(26),which correspond to the constraints related to normal operating conditions,contingency conditions,and constraints related to uncertainty sets.

2 Solution Methodology

2.1 Linearization of the Model

Three types of nonlinear terms are included in the model.The first is the quadratic cost function,which can be easily linearized piecewise.The second type is the product of continuous variables and binary variables in(7)and(14).They can be linearized using the big M method[16].After linearization,expressions(7)and(14)are as follows:

The third type of non-linear term is the absolute sign in(25).Because |αs| exists in(25)and finally exists in the objective function of a minimization problem,it can be linearized as follows:

2.2 Model Transformation

The proposed model is a min-max-min model that cannot be solved directly.As the model has three levels,it can be transformed into a bi-level form and solved by the Benders decomposition algorithm.The model can be expressed in a compact format as follows.

Master problem(MP):

The decision variables x in the MP are ul.The MP corresponds to an integer programming model that can be directly solved by a commercial solver.

Sub-problem(SP):

The decision variables in the SP are ΔWs,ω,w,lols,ω,i,, and In the SP,y represents the loss of load,wind spillage,and active load adjustment.s represents the other variables related to the operation stage.v represents the variables that describe the uncertainty of the contingency probability.In the SP,b,c,B,e,C,D,F,G,f,H,g,I,and h are all parameter matrixes.

The SP is a max-min model and cannot be solved directly.It needs to be transformed into a single-level model.The inner minimizing problem can be transformed into a maximizing problem using the duality theory.Then,the single-level SP can be expressed as follows:

where π,φ,and η are dual variables.

2.3 Benders Decomposition

The Benders decomposition algorithm runs as follows[22-24]:

(1)Set UB = +∞ and LB = -∞.Solve the relaxed MP and obtain the initial solution.

(2)Substitute the solution obtained in the MP into SPs and solve the SPs individually.

(3)Verify the solution for each SP:

①If the SP is infeasible,the algorithm terminates.

②If the SP is feasible and has a bounded solution,form and add the optimal cut to the MP.

③If the SP is feasible and has an unbounded solution,form and add the feasibility cut to the MP.

If every SP is feasible and the sum of the optimal solution is less than the UB,the UB is updated.

(4)Solve the MP with the added Benders cut and update LB.If the difference between the UB and LB is less than the given convergence tolerance,stop the iteration;otherwise,return to(2).

3 Case Study

The proposed model is tested on the IEEE-RTS system and the IEEE 118-node system.The RTS system comprises 24 buses,26 existing generators,and 38 existing transmission lines,and the corresponding data can be found in[26].The peak load is 4150 MW.Five wind farms are established on buses 2,8,15,18,and 23,and the capacity of each wind farm is 400 MW.The wind power output follows a normal distribution and is divided into five scenarios.The power deviations of the five scenarios are -2,-1,0,1,and 2 times the standard deviation of the wind power distribution,respectively[26].The standard deviation is set to 20% of the forecast value.A new line can be established in each transmission corridor[26].The maximum capacity of each line has been reduced to half of its original value.The investment cost of each candidate line is given in[26].In the cases,only the outage of the transmission line is considered,and the simultaneous outage events of three and more equipment components are not considered.The outage rate of the equipment is taken from similar equipment in[25]and[26].VOLL takes the value of 5000 $/MWh,and the price of the active load is 1000 $/MWh.The price of wind spillage is 100 $/MWh.The conservativeness parameter Γ was set to 10.The convergence tolerance for the Bender decomposition is 0.01%.The proposed model is coded on the GAMS platform and solved using the commercial solver CPLEX[27].The optimization is performed on a computer with the Windows 10 operating system,3.2 GHz AMD Ryzen 7 5800 processors,and 16 GB of RAM.

3.1 Comparison of Results under Different Conditions

This subsection elucidates the influence of the uncertainties of contingency probability and wind turbine output and the active load.As shown in Table 1,five conditions,which are described as follows,are considered:

Table 1 Comparison of uncertainty and active load

*The unit cost in the table is M$.

ConditionsC1C2C3C4C5 Total cost815.79 856.46 876.85 971.17 890.56 Investment cost314.55 351.04 345.63 363.60 356.56 Operating cost489.77 489.77 517.60 516.30 516.50 Reliability cost11.47 15.66 13.63 91.28 17.50 Number of new constructed lines7108109

C1:Neither the active load nor the uncertainties are considered.This condition is taken as the benchmark.

C2:Only the uncertainty of contingency probability is considered.Neither the uncertainty of wind power nor the active load is considered.

C3:Only the uncertainty of wind power is considered.Neither the uncertainty of contingency probability nor the active load is considered.

C4:Both uncertainties are considered,but the active load is not considered.

C5:The uncertainties and active load are all considered.

From Table 1,it can be found that when uncertainties and the active load are not considered,the total cost of condition C1 is 815.79 M$ and seven new lines are constructed.When only considering the uncertainty of contingency probability in C2,the total cost raises to 856.46 M$ and three more new lines are constructed compared with C1,which shows that considering the uncertainty of contingency probability will lead to a more reliable construction plan and using a fixed contingency probability would overestimate the reliability of the power system.When only considering the uncertainty of wind power output in C3,the total cost also increases,reaching 876.85 M$,and the difference between C1 and C3 is larger than that between C1 and C2,which shows that the uncertainty of wind power might lead to a more obvious influence on total cost compared with the uncertainty of contingency probability.When considering both uncertainties,but not considering the active load in C4,the total cost is the largest.When considering the uncertainties and active load in C5,both the total cost and the number of newly constructed lines decrease compared with C4,which shows that considering the active load provides a system with a better capability to cope with wind power deviations and the uncertainty of contingency probability.

3.2 Sensitive Analysis with Respect to VOLL,
Interval Width,Γ,and μ

To show the effect of the width of the interval of the contingency probability,a parameter,WP,is introduced to describe the interval width.The interval of the contingency probability can be analytically described as follows:

where γmin and γmax are the lower and upper bounds of the contingency probability interval,respectively,and γis the fixed estimated contingency probability derived from the historical data.Table 2 lists the total cost under different VOLL and interval widths.

Table 2 Total cost under different VOLL and interval widths

*The unit cost in the table is M$.

VOLL/($/MWh)10002000500010000 WP= 1863.82 864.40 866.17 869.12 WP= 2864.50 865.79 870.34 876.39 WP= 3865.97 868.73 877.35 885.85 WP= 4868.08 872.90 884.28 896.63 WP= 5870.77 878.76 890.56 911.64

Figs.1 and 2 show the change in total cost when Γ and μchange,respectively.

From Table 2,it can be found that the total cost increases when VOLL increases and the interval becomes wider.From Fig.1,it can be observed that when Γ increases,the total cost also increases.This is because a larger Γ indicates a lower probability,but more conservative situations are considered.The increase in total cost gradually slows.The variation in the total cost with respect to μis shown in Fig.2.When μincreases,the system has more flexible resources to cope with uncertainties;therefore,the total cost decreases.However,finally,the decrease in total also becomes saturated.

Fig.1 Relationship between total cost and

Fig.2 Relationship between total cost and μ

3.3 Case Studies of the IEEE 118-node System

The proposed model is also tested on the IEEE 118-node system.The system data can be found in[28].The IEEE 118-node system contains 118 buses,54 generators,and 186 transmission lines,which can be found in[28].The peak load is 6550 MW.Five wind farms are established on buses 8,34,45,76,and 102,and the capacity of each wind farm is 400 MW.The wind power output follows a normal distribution and is divided into five scenarios,which are the same as those in Subsection 3.1.A new line is allowed to be established in each transmission corridor.The maximum capacity of each line has been reduced to half of its original value.The investment cost of each candidate line is given in[26].In this case,a single outage of the transmission line and simultaneous outage of the two transmission lines are considered.The outage rate data of the equipment are taken from[25]and[26].VOLL takes the value of 10,000 $/MWh,and the price of the active load is 1000 $/MWh.The wind spillage cost is 100 $/MWh.The conservativeness parameter is set to 10.The convergence tolerance of the Bender decomposition is set to 0.5%.

Four conditions are considered,which are described as follows.The corresponding results are presented in Table 3.

Table 3 Comparison of results of cases 6,7,8,and 9

*The unit cost in the table is M$.

ConditionsC6C7C8C9 Total cost2414.54 2473.022582.852601.68 Investment cost108.20 152.40140.30 150.70 Operating cost2249.522250.322380.202370.50 Reliability cost56.82 70.3062.3580.48 Number of new constructed lines10141214 Run time8 h 41 min 8 h 55 min 17 h 43 min 20 h 58 min

C6:Neither the active load nor the uncertainties of contingency probability and wind power are considered.This condition is taken as the benchmark.

C7:The uncertainty of contingency probability and the active load are considered.The uncertainty of wind power is not considered.

C8:The uncertainty of wind power and the active load are considered.The uncertainty of contingency probability is not considered.

C9:The uncertainties of wind power and contingency probability are considered,and the active load is also considered.

From Table 3,it can be found that when neither the active load nor the uncertainties are considered in C6,the total cost is 2414.54 M$,and ten new lines are constructed.When only the uncertainty of contingency probability and the active load are considered in C7,the total cost raises to 2473.02 M$ and four more new lines are constructed compared with C6,which shows that considering the uncertainty of contingency probability will lead to a more reliable construction plan.Using a fixed contingency probability would overestimate the reliability of the power system.When only considering the uncertainty of wind power output and the active load in C8,the total cost is 2582.85 M$,which is significantly higher than that in C6.When considering the uncertainties of contingency probability and wind power and the active load in C9,the total cost increases compared with that in C6 and C7,which shows that the wind power deviation and the uncertainty of contingency probability have similar impacts on the secure and economic operation of power systems.

The proposed model combines the characteristics of robust and stochastic optimization.Many scenarios are involved,and a multilevel optimization model is established.Extensive simulation shows that even for the small IEEERTS system,if the Benders decomposition algorithm is not applied,the optimization will run out of memory.When the Benders decomposition algorithm is applied,for the IEEERTS system,the computation time is approximately 2 h and 35 min.For the IEEE 118-node system,the computation time is reported in the last row of Table 3.

4 Conclusion

In this paper,a robust TEP model that considers multiple uncertainties and the active load is proposed.The uncertainty of wind power is described by scenarios,and the uncertainty of the contingency probability is described by the box uncertainty set.The proposed model involves random parameters and can be transformed into a bi-level model by the dual theory,which can be solved by the Benders decomposition method.The case studies show that the cost and construction plan are significantly influenced by the uncertainties of wind power and contingency probability and the active load.Considering the uncertainties in the system would lead to a more reliable result.The active load will improve the system’s ability to cope with uncertainties and improve the economy and reliability of construction plans.

Acknowledgement

This work was supported by a project of the State Grid Shandong Electric Power Company(52062520000Q)and the National Key Research and Development Program of China(2019YFE0118400).

Declaration of Competing Interest

We have no conflict of interest to declare.