Generation expansion planning considering efficient linear EENS formulation

Nomenclature

A Indices and Sets

e Index of candidate ESSs

g, i, j, k Index of units,including existing units and candidate units

s Index of contingency events

t Index of optimization periods

N+ESet of candidate ESSs

NG Set of existing units

N+G Set of candidate units

Set of units,including existing units and candidate units

NK Set of intervals of the discretized Gauss distribution

NT Set of time periods

St Set of all possible contingency events during period t

B Parameters

ag,bg,cg Coefficients of unit operation cost function of unit g

ce,t Coefficient of ESS operation cost of ESS e during period t

CEe Construction cost of ESS e

CRe Reserve price of ESS e

CGg Construction cost of candidate unit g

CRg Reserve price of unit g

Cinv Investment cost of candidate units and candidate ESSs

Cope Operation cost of units and candidate ESSs

Crel Reliability cost

Cres Reserve cost

CDt Adjustable load price during period t

dt Forecast load during period t

Dyear Days of a typical year

Start energy stored in ESS e

End energy stored in ESS e

Minimum energy allowed to be stored in ESS e

Maximum energy allowed to be stored in ESS e

ps,t Probability when event s occurs during period t

Probabilitywhen all online units are available during period t

Probability when unit i occurs outage during period t

Probability when units i and j occur synchronous outage during period t

Probability of WT and load falls in the kth interval of the entire uncertainty distribution function during period t

WT and load forecast error when uncertainties fall in interval k during period t

Maximum charging/discharging power of ESS e

Minimum charging/discharging power of ESS e

Maximum output power of unit g

Minimum output power of unit g

Ramp up rate of unit g

VOLL Value of lost load

C Variables

A binary variable to represent whether WT and load uncertainties cause loss of load or not during period t

A binary variable to represent whether WT and load uncertainties and the single outage of unit i cause loss of load or not during period t

A binary variable to represent whether WT and load uncertainties and the simultaneous outage of unit i and j cause loss of load or not during period t

bs,t A binary variable to represent whether loss of load occurs due to event s during period t

Adjustable load amount during period t

Ee,t Energy stored in ESS e during period t

Pg,t Power output of unit g during period t

Pe,t Power output of ESS e during period t

Discharging power of ESS e during period t

Predicted WT power during period t.

Psum-max Summation of the maximum output power in the system

Rg,t Reserve amount provided by unit g during period t

Re,t Reserve amount provided by ESS e during period t

SSRt Spinning reserve provided by all units and ESSs during period t

uGg,uEe Binary variables to represent whether the candidate unit g and the candidate ESS e is constructed or not

Ug,t Binary variable to represent the on/off status of unit g during period t

ΔPs,t Power curtailment under event s during period t

ΔRs,t Reserve curtailment under event s during period t

0 Introduction

Generation expansion planning (GEP)is an important component of power system planning [1].To meet the requirements of future load growth and grid development,GEP is used to determine time,place,and capacity of units required [2].The results of GEP will affect the reliability,economy,power quality,and grid structure of the power system in the future.

Traditional GEP mainly minimizes investment and operation costs.In reference [3],an optimal GEP model using single-stage and two-stage robust optimization (RO)was proposed,which made a balance between investment and operation costs in terms of distribution-free bounded intervals.In reference [4],a new efficient methodology called Genetic Algorithm-Benders’ decomposition to solve the GEP problem was proposed,which evaluated the most economical investment cost and operation cost.However,the reliability of the system was not well understood.In reference [5],[6],a compromise between the operation cost and investment cost was determined.The power system reliability is also considered,and the reliability of the power system was modeled using loss of load (LOL).In reference[7],a power-based GEP-UC model that improved the existing models was proposed.It attempted to minimize the investment and operation costs,and optimized investment decisions on variable renewable energy sources (vRES),energy storage system (ESS),and thermal technologies.

Moreover,the uncertainties of load and large-scale wind turbines (WTs)pose significant challenges to power system planning [8],and the outage of equipment is a problem that should never be ignored in power system planning.Many previous studies on power system planning considered the uncertainties of the load and WT.In reference [9],a twolayer model of source network joint planning is established to minimize the investment,operation and wind curtailment costs,and it can effectively improve the consumption of wind power and reduce the system investment and operation cost.But the wind power output is calculated by wind speed and wind power uncertainty is not well expressed.In reference [10],a GEP model integrated with multiple types of renewable energy resources (RESs),specifically WT and photovoltaic (PV)was proposed,and their joint uncertainty and fluctuation characteristics were evaluated.The uncertainty of contingencies caused by equipment outage events can be considered in deterministic or probabilistic approaches [11].For the deterministic approach,the N-k security criterion is generally applied [12]-[16].However,the contingency probability is ignored in N-k security criterion,and the result may be conservative or radical.

To quantify the effect of uncertainties,many reliability indices including loss of load probability (LOLP),expected energy not supplied (EENS)and conditional value at risk (CVaR)et al are applied in probabilistic methods.In reference [17],a new effective arithmetic including maximum LOLP constrained unit commitment (UC)of high penetration wind power system,and the maximum LOLPconstrained UC aimed to prepare appropriate spinning reserve (SR)to maintain the LOLP under a forecast upper limit.In reference [18],[19],only wind abandonment and lost of load (LOL)were used to express the reliability index,which has not taken equipment failures into account so that it led to an overoptimistic result.EENS is used to quantify consumer losses due to power outages and service interruptions,and is widely used in unit commitment,spinning reserve optimization,and sometimes used in GEP and transmission expansion planning (TEP)problems [20]-[21].Reference [22] proposed an economic dispatching model of the power system with thermal storage station and wind power,in which CVaR was introduced to measure the wind risk loss caused by the uncertainties of the dispatching operation.In reference [23],the EENS formulation was associated with forecast errors about load and WT,and it was illustrated that the reliability of power system is significantly reduced with high penetration of WT.In reference [24],the reliability was estimated using EENS,and the EENS cost was compared with operation cost and investment cost in order to strike a good balance between reliability and economy.In reference [25],an efficient mathematic method was proposed for calculating the EENS by taking time-varying load into consideration.However,the computation of EENS incurs a computation efficiency problem in their models.

In this paper,a GEP model considering the ESS and adjustable load is proposed.The proposed model attempts to minimize the total cost,which includes the investment,operation,reserve,and reliability costs,and the optimal planning results can be achieved by striking a good balance between economy and reliability.The reliability cost is computed by multiplying value of lost load (VOLL)with EENS.For the entire optimization model,the calculation of the EENS was the major impediment.To overcome the tremendous computational burden introduced by EENS,this paper proposes an efficient GEP-based linear EENS formulation,which is applied in a multi-step GEP model.The proposed GEP model is recast as a mixed integer linear programming (MILP)problem,and it can be solved by state-of-art commercial solver.The proposed EENS formulation was compared with the existing linear EENS formulations.The efficiency and validity of the proposed model were verified using an IEEE-RTS system.

The major contributions of this paper can be listed as follows.

(1)A GEP model considering ESS and adjustable load is proposed.The optimal planning results is obtained by striking a good balance between economy and reliability.

(2)The proposed model is recast as a MILP model and it can be efficiently solved by commercial solver.

(3)A new efficient GEP based on a linear EENS formulation is proposed,and it is applied in a multi-step GEP model.The computational efficiency of the proposed model can be significantly improved,while the solution accuracy is desirable.

The remainder of this paper is organized as follows.Section 2 presents the proposed GEP model.Section 3 presents the proposed GEP based on a linear EENS formulation and multi-step method.Section 4 presents the case studies,and the relevant conclusions are presented in Section 5.

1 Model formulation

1.1 Objective function

The objective function of the proposed model is shown as below:

The objective function tries to minimize the total cost,which includes yearly generation and energy storage investment cost Cinv,operation cost Cope,reserve cost Cres,and reliability cost Crel.Here,a quadratic operation cost function is applied and it is piecewise linearized [26].The planning model was implemented based on a typical day.Therefore,the operation cost,reserve cost,and reliability cost are multiplied by Dyear to represent the yearly cost.These costs can be expressed explicitly as follows:

where EENSt represents the EENS during period t.EENSt can be explicitly formulated based on every possible event [27].

where dT is the duration of each period.In this study,dT was assumed to be 1 h.

In this study,the uncertainties caused by the WT and load were combined and represented by a Gaussian distribution [28],and the distribution was further discretized into seven scenarios.Outage events can be classified according to outage orders.Zero order outage implies no unit undergoes outage; first order outage implies outage of a single unit; second-order outage implies synchronous outage of two units,and so on.EENS of different outage order can be indicated by superscripts ‘0’,‘1’,‘2’,etc.,respectively.For simplicity,EENS caused by third and higher orders of outage events is not shown here.

The specific Equation of EENS is formulated as below.

In general,a two-state is used to represent unit availability.Multiple de-rated states are ignored.The outage probabilities p0t, and can be formulated as follows equation [27].

where ui is the outage replacement rate (ORR),and γi is the failure rate of unit i.

The binary variablesand the combined WT and load forecast errorcan be formulated as follows equation [29].

where σtWTL is the standard deviation of the combined WT and load uncertainties during period t.

1.2 Constraints

The objective function is subject to the following constraints.

(1)Power balance constraint

where dt represents the system load during period t.Equation (13)gives the power balance constraint of the total system.

(2)Unit output constraints

Equations (14)and (15)give the output power limits of the existing and candidate units,respectively.

(3)Constraints of ESS

Equation (16)describes the output power constraint of the ESS,and equation (17)describes the constraints of energy stored in ESS [30].

where μd represents the efficiency of discharging stage.

(4)Unit operating constraints

The block constraint of equation (18)generally includes minimum-up and down-time constraints,initial condition constraints,ramp-up and ramp-down constraints [31].These are all considered in this study.

(5)Spinning reserve constraints

Here,only the up-spinning reserve was considered.The SR provided by a unit is constrained by its power output and ramp rate,as follows:

where τ is the reserve deployment time required to deliver the reserve.The ESS reserve is similar to equation (19)and is shown in equation (20).It is constrained by the upper limit of the output power and energy stored in the ESS during period t.

And SSRt can be expressed as below

(6)Adjustable load constraint

Adjustable load is one of the flexibility resources of power system,which plays an important role in dealing with the uncertainties of load and WT fluctuation.Constraint (22)introduces the lower and upper limits of the adjustable load.

where dtadjmax is the upper limit of adjustable load.

2 Linearization of the model

In the model,some nonlinear terms exist in equations(8),(9),(11),and (15).They all belong to the product of some binary variables or the product of a binary variable and a bounded continuous variable,and they can be equally linearized [29].In particular,the product of some binary variables can be linearized as follows:

where z is the product of some binary variables,bi is a binary variable,and n is the number of binary variables.The product of a binary variable and a bounded continuous variable can be linearized as follows:

where b is a binary variable,and y is a bounded continuous variable.

(1)Linearizing EENS based on traditional method

Equation (25)introduces a summation of the maximum power of the system.

Subsequently,equation (11),which includes three binary variables,can be linearized as follows [29]:

Finally, EENS0,EENS1,and EENS2 were linearized.After EENS is linearized,the entire planning problem belongs to an MILP model,which can be solved by a stateof-art MILP commercial solver.

(2)Linearizing EENS based on Epigraph Reformulation

The EENS equation (8)can be equivalently expressed as follows,based on Epigraph Reformulation [32].

After the simplification of Equation (29),the binary variables , ,and are eliminated in EENS formulation,the number of binary variables in the entire optimization model is significantly reduced,and the computational efficiency can be significantly improved.Because the absolute sign is minimized in the objective function,it can be equivalently expressed as follows:

Then,equation (29)can be equivalently expressed as equation (31)with the introduction of auxiliary variables

The constraints related to the auxiliary variables are shown as below.

After the terms of absolute values are linearized,the entire planning problem belongs to a MILP model,which can be solved by a state-of-art MILP commercial solver.

(3)Linearizing EENS based on capacity outage probability table (COPT)

Compared with the original EENS equation,the simplified EENS equation based on Epigraph Reformulation improved the calculation efficiency.However,the computational efficiency is still not desirable because a large number of variables are involved in EENS equation (31).In equation (31),the EENS is still computed by considering individual scenario,and a tremendous number of scenarios would make the computation intractable.Extensive simulations show that the computational efficiency of using equation (31)improves less than ten times compared to that with using equation (8).

Inspired by the simplified EENS calculation proposed in the power system scheduling problem [33],this study proposed an EENS simplification model used in the GEP problem.The procedure of applying the GEP based on the linear EENS formulation is as follows: First,a base GEP model was implemented in which the EENS is not considered.After optimization,the optimization results were obtained,and the corresponding COPT was established.Then,the piecewise linear equation of EENS was established as shown below:

where Nout(t),ΔCi,t,and pi,t are all parameters derived from COPT. Nout(t)is the number of outage events in COPT,ΔCi,t is the outage committed capacity of the ith outage event,and pi,t is the probability corresponding to the ith outage event.The binary variable bi,t satisfies

The planning model is implemented again using a simplified piecewise linear EENS equation.Then,a new COPT can be established based on the new results.The iteration process terminates until the EENS value does not change.Compared with equation (8)and equation (31),the formulation of EENS is considerably simplified,and the computational efficiency is considerably enhanced.Equation (34)can be linearized as follows:

Finally,the EENS has been indicated as a summation of the products of some binary variables and a bounded continuous variable,which can be linearized [29],[34].After EENS is linearized,the entire planning problem belongs to a MILP model,which can be solved by state-ofart MILP commercial solver.

The flowchart of the proposed multi-step method is shown in Fig.1.In the proposed multi-step method,a GEP model is implemented in Step 1,and EENS is not considered in this step.In Step 2,the EENS,which is used in reliability cost,is computed based on the COPT.The SR is scheduled to make the system more reliable.According to the results in Step 2,the value of EENS is updated in Step 3,and more units are scheduled and more available capacity is committed to the scheduled units.The value of EENS used in Step 3 slightly increased compared with that in Step 2.More SR is scheduled,and the reliability of the system is fully guaranteed.As the iteration proceeds,the difference between the EENS computed by equation (33)during optimization and the value of EENS computed using COPT after optimization becomes increasingly smaller.When this difference is smaller than a given threshold,the iteration process stops.

Fig.1 Flowchart of the proposed multi-step method

3 Case study

The proposed model was tested on an IEEE-RTS system without hydro-generation [35].This system consists of 26 units.The generation and load data were obtained from reference [35].The 24-hour period is a typical day,and it is multiplied by 365 to represent a year.The original load was expanded by a factor of 2.0.For simplicity,the uncertainties caused by the WT and load are convolved and considered together.It assumes that the WT uncertainty and load uncertainty both follow a Gaussian distribution.The combined uncertainty follows a Gaussian distribution.The variance of the combined uncertainty was taken from reference [28].

and ce,t can be obtained from references [36]-[38].The VOLL was set as 5000 $/MWh.μd is set as 1.2.The upper limit of the adjustable load was set as 3% of the load level.

The parameters of the candidate generators are listed in Table 1 [35].The types of units are represented by A1,B1,C1,and D1,respectively.The maximum number of each types of generators was set as 4.The parameters of the candidate ESS are listed in Table 2.The types of ESS are represented by A2,B2,C2 and D2.For each type of ESS,only one ESS can be constructed.The construction cost of candidate units is 100000 $/MW,and the construction cost of the ESS is 200000 $/MW [39].

Table 1 Parameters of candidate units

Parameter A1 B1 C1 D1 Pmin/MW 4.00 15.20 54.25 68.95 Pmax/MW ICg Tong/h Toffg/h URg/MW/h DRg/MW/h P0/MW a0/$/MWh2 20 76 155 197-111 332 553-454 55.0 99 0 0.00259 b0/$/MWh c0/$30.5 70 0 0.01020 38.5 80 15.2 0.00932 55.0 78 124.9 0.00463 37.551 117.755 13.407 81.626 10.694 142.7238 23.000 259.131

Table 2 Parameters of candidate ESS

Parameter A2 B2 C2 D2 Pmin/MW 0 0 0 0 Pmax/MW Emine/MWh Emaxe/MWh Estarte/MWh 60 30 300 150 80 40 400 200 100 50 500 250 120 60 600 300

The case studies are coded on a general algebraic modeling system (GAMS)[40] platform and solved using the MILP solver of CPLEX [41].The optimization is implemented on a computer with Win10 system,Intel Core i5-6400 processors at 2.7 GHz and 8 GB of RAM using a duality gap of 0.1%.

3.1 GEP considering the first order outage events

In this subsection,only the first-order outage events are considered,and the performances of the three linearization techniques are compared.Methods I,II,and III correspond to EENS linearization techniques (1),(2),and (3),respectively.After optimization,the construction plans of the new generators and ESSs are listed in Table 3.The numbers in the table represent the number of units/ESSs constructed.The various costs are listed in Table 4.The computation times are presented in Table 5.

Table 3 Construction plans under three linearization methods

Type Method A1 B1 C1 D1 A2 B2 C2 D2 I 0 4 4 4 1 1 1 1Ⅱ 0 4 4 4 1 1 1 1Ⅲ 0 4 4 4 1 1 1 1

Table 4 Various costs under three linearization methods

Method Total cost/M$Investment cost/M$Operation cost/M$Reserve cost/M$Expected interrupt cost/M$I 460.734 251.200 183.789 10.040 15.705Ⅱ 460.748 251.200 183.795 10.050 15.703Ⅲ 460.842 251.200 183.945 9.970 15.728

Table 5 Computation times under three linearization methods

MethodIⅡⅢTime/second 12011.40 1898.37 420.63

From Tables 3 and 4,it can be seen that the construction plans are the same when different EENS equations are applied.The costs of the three techniques were nearly the same.The validity of using Method III is illustrated.

From Table 5,it can be seen that method III possesses the highest computation efficiency,the computation efficiency of Method II is in the middle,while Method I has the lowest computation efficiency.This is because method III replaces the combinatorial and nonlinear nature of EENS by a piecewise linear method,and the computation efficiency is greatly improved.

3.2 EENS considering the first and second order outage events

In this subsection,the first- and second-order outage events are considered,and the three linearization techniques are compared.After optimization,the construction plans of the new generators and new ESSs are listed in Table 6.The various costs are listed in Table 7.The computation times are presented in Table 8.From Table 6,it can be found that when the secondorder outage events are considered,the GEP with the original EENS equation will run out of memory.The calculation of EENS makes the major impediment to the entire optimization model,and it should be well addressed.The construction plans of methods II and III are the same,which shows the validity of methods II and III.From Table 7,it can be seen that various costs are nearly the same for methods II and III.It should be noted that method II does not perform any simplification or approximation,it belongs to an equivalent linearization technique,and the results are credible.Because the relative error between the solution of method II and that of method III is small,it shows that the solution accuracy of method III is desirable.From Table 8,it can be seen that even when only the first- and second-order outage events are considered,method I runs out of memory.Method II obtains the result,but the computation time is long.The computation time of method III is improved by dozens of times compared with that of method II.The good performance of method III is due to the fact that the number of scenarios considered is significantly reduced.The number of scenarios considered in method III corresponds to Nout(t)used in equation (33).

Table 6 Construction plans under three linearization methods

Type Method A1 B1 C1 D1 A2 B2 C2 D2 I out of memoryⅡ 0 4 4 4 1 1 1 1Ⅲ 0 4 4 4 1 1 1 1

Table 7 Various costs under three linearization methods

Method Total cost/M$Investment cost/M$Operation cost/M$Reserve cost/M$Expected interrupt cost/M$I out of memoryⅡ 461.148 251.200 184.978 10.108 14.862Ⅲ 461.583 251.200 185.759 10.033 14.591

Table 8 Computation times under three linearization methods

MethodIⅡⅢTime/second out of emory 7086.651 632.800

3.3 EENS considering the first,second and third order outage events

In this subsection,the first-,second-,and third-order outage events are considered,and the three linearization techniques are compared.After optimization,the construction plans of the new generators and new ESSs are listed in Table 9.Various costs are listed in Table 10.The computation times are presented in Table 11.

Table 9 Construction plans under three linearization methods

Type Method A1 B1 C1 D1 A2 B2 C2 D2 out of memoryⅡout of memoryⅢ 0 4 4 4 1 1 1 1 I

Table 10 Various costs under three linearization methods

Method Total cost/M$Investment cost/M$Operation cost/M$Reserve cost/M$Expected interrupt cost/M$I out of memoryⅡout of memoryⅢ 461.157 251.200 184.299 10.100 15.558

Table 11 Computation times under three linearization methods

MethodIⅡⅢTime/econd out of emory out of memory 663.666

From Table 9,it can be seen that when the third-order outage events are considered,methods I and II will run out of memory.Method III yields the results,and the results are the same as those in Table 6.From Table 10,it can be seen that the various costs of method III are nearly the same as those in Table 7,which illustrates that the third-order outage events do not have a significant effect on the final results,mainly because the probabilities of the third-order outage events are extremely small.

Table 11 shows that methods I and II both run out of memory,which indicates that methods I and II cannot cope with the rapidly increasing number of outage events.Method III yields results within a reasonable run time.The run time was comparable to that in Table 8.This is because the number of scenarios was significantly reduced in method III and it was robust in terms of computational efficiency.

3.4 Impacts of load level and VOLL

To analyze the effect of increasing load demand and VOLL,the GEP using linearization method III was implemented under different load and VOLL values.After optimization,the results are reported in Tables 12-14.Table 12 lists the construction plans for various load levels.Table 13 lists the construction plans for various VOLL.Table 14 lists the various costs under different load levels.

Table 12 Construction plans under various load level

Type Level A1 B1 C1 D1 A2 B2 C2 D2 1.0 0 0 0 0 1 1 1 1 1.2 4 0 0 0 1 1 1 1 1.4 4 0 4 0 1 1 1 1 1.6 4 0 4 4 1 1 1 1 1.8 0 4 4 4 1 1 1 1 2.0 0 4 4 4 1 1 1 1

Table 13 Construction plans under various VOLL

Type VOLL A1 B1 C1 D1 A2 B2 C2 D2 1000 0 0 4 4 1 1 1 1 2000 0 0 4 4 1 1 1 1 5000 0 4 4 4 1 1 1 1 10000 0 4 4 4 1 1 1 1 15000 0 4 4 4 1 1 1 1

Table 14 Costs under different load level

Level 1.0 1.2 1.4 1.6 1.8 2.0 Total cost/M$ 205.6 224.4 296.8 340.6 432.1 461.6 Investment cost/M$ 72.0 80.0 134.0 164.4 243.2 251.2 Operation cost/M$ 124.3 130.4 150.9 164.3 172.3 185.8 Reserve cost/M$ 6.6 9.8 7.2 7.7 9.0 10.0 Expected interrupt cost/M$2.7 4.2 4.7 5.2 7.6 14.6

From Tables 12 and 13,it can be found that with the increase in the load level,more large-capacity candidate units will be established.Similarly,with the increase in VOLL,the GEP model also tends to establish more units.However,the effect of the load level is larger than that of VOLL.The construction plan of new units depends on the balance among the investment cost,operation cost,reserve cost,and expected interrupt cost.From Table 14,it can be seen that all costs tend to increase when the load level increases.

4 Conclusions

In this paper,a GEP model that considers the ESS and adjustable load is proposed.The optimal results are determined by minimizing the cost incurred on investment,operation,reserve,and reliability.The reliability cost is computed by multiplying VOLL with EENS.The proposed model involves a large number of binary and continuous variables.To overcome the tremendous computational burden introduced by EENS,this study proposes a GEP based on the EENS linearization formulation.Case studies show that the computational efficiency of the proposed model can be significantly improved,while the solution accuracy is still at a desirable level.

Acknowledgement

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

Declaration of Competing Interest

We declare that we have no conflict of interest.