Robust generation expansion planning considering high penetration renewable energies uncertainty

The proper optimal generation expansion planning (GEP) should meet the reliability criteria requirements over a planning horizon under the presence of uncertainties. The intermittent nature of renewable energy sources (RES) introduces an enormous uncertainties impact within the planning model. A simulation model for RES uncertainty is developed using the capacity factor (CF) of the RES historical data. The RES simulation model is handled via the probability density function (PDF). The uncertainty parameter of different RES is described as a flexible polyhedral uncertainty set and incorporated within the proposed GEP model. The influence of different uncertainty scenarios for each RES uncertainty on the GEP model can be analyzed separately. The RES uncertainty scenarios are predefined and incorporated within the proposed GEP model through a proposed parameter named as a confidence level. The proposed confidence level parameter is beneficial to the power system planner to control the degree of robustness. Different GEP results are presented for various RES uncertainty scenarios. Three methods are proposed as appropriate solutions to deal with the RES uncertainty impact. The most economical method among the three proposed methods is determined by developing an objective function tailored to achieve the optimality of the economic factor.

This paper introduces a robust GEP model considering a high share of RES and shows the importance of considering the RES uncertainties within the proposed GEP model. The available historical data of wind and PV are used to predict the uncertainties via developing a representation model. RES uncertainties simulation model is handled by the PDF curve. Based on a predefined confidence level, various scenarios can be obtained from the PDF curve to clarify the influence of the weight of the uncertainties on the planning model. The uncertainties are incorporated into the planning model through a developed flexible polyhedral set. The boundaries of the polyhedral sets can be controlled based on an appropriate predefined parameter. Hence, the solution's robustness is controlled. Analyzing the influence of each uncertainty weight will be beneficial for planners. A confidence level parameter is proposed to control the weight of each RES uncertainty set separately. A single uncertain budget will be investigated which is beneficial for practical implementation purposes and easily controlled. The single uncertain budget approach is predefined only once based on the weight of the uncertainty set. RES overcapacity, reserve margin, and energy storage system (ESS) methods are proposed to cope with the effect of RES uncertainty. The most appropriate method is determined through a tailored economic objective function.
The main contributions of this paper are as follows.
1. An analytical simulation model is developed to handle the estimated future RES uncertainty set accurately. The weight of each renewable uncertainty set can be controlled separately (ie, the influence of different uncertainty scenarios for each uncertainty source on the GEP model can be analyzed separately). 2. Based on the uncertainty weight, a flexible polyhedral uncertainty set is developed to analyze the impact of uncertainty on the proposed robust GEP model. The degree of robustness is controlled through an adjustable robustness parameter that depends on the weights of each uncertainty set. 3. RES overcapacity, reserve margin, and ESS are introduced as effective methods to cope with the RES uncertainty problem. The optimal method is suggested based on technical and economic aspects.
The paper is organized as follows. Section 2 explains the RES uncertainty simulation model. Section 3 introduces the RO approach. The deterministic GEP model is introduced in Section 4. The robust GEP model is proposed in Section 5. Section 6 lists the used data and assumptions. GEP results are discussed in Section 7. Section 8 concludes the paper.

LONG-TERM RES UNCERTAINTY SIMULATION MODEL
The simulation of the long-term RES uncertainty data can be depicted from the renewable historical data. RES historical data 34,35 are used as input to calculate the annual CF for different renewable generation types. The historical data used for the simulation model depend on the renewable generation type, that is, wind speed data is used for wind farms and the solar radiation data is used for PV plants. The historical CF variation range indicates the range of historical uncertainty. Hence, historical uncertainty CF is beneficial to predict uncertainty range. The annual CF during the last 30 years of candidate wind and PV sites in Egypt are utilized to develop the simulation model. CF calculation is discussed in References 36,37. Figure 1 shows the historical calculated annual CF for both wind and PV energy. The resulted statistical mean and SD of the calculated CF for wind energy is 31% and 2.8%, respectively, according to historical data shown in Figure 1. For PV, mean and SD results are 18.5% and 0.62%, respectively. According to the mean and SD information, the corresponding uncertainty model of each renewable energy type is simulated by the PDF curve as shown in Figure 2.

RO APPROACH
RO is one of the most known algorithms dealing with optimization problems under the presence of uncertainties. RO algorithm presents a worst-case optimal solution that is immunized against the uncertain parameters. The degree of the optimal solution robustness due to the uncertainties can be controlled via an adjustable robustness parameter. 38,39 There are several definitions for uncertainty sets, which are defined as box, ellipsoidal, and polyhedral. The box set considers that all parameters will take the worst possible value. 40 Hence, a high level of conservatism is achieved and causing deterioration for the objective function accordingly. To avoid the disadvantage of such over conservatism, polyhedral, and ellipsoidal uncertainty sets are proposed. 38 The polyhedral is simple and less complex than the ellipsoidal. Therefore, RO induced by the polyhedral uncertainty set is utilized in this paper to describe and incorporate the RES uncertainty data into the GEP model. It is worth mentioning that, a flexible bounded-free polyhedral type is implemented in this paper to control the trade-off between robustness and performance.

Flexible polyhedral uncertainty set
Flexible uncertainty set means that a variety of different amounts of each RES uncertainty source weight can be defined and implemented within the GEP model. Therefore, analyzing the influence of each uncertainty source will be beneficial for planners. Moreover, the boundaries of the uncertainty sets can be controlled based on an appropriate predefined parameter. Hence, the solution's robustness is controlled. Figure 3 shows the geometric view of both the typical and flexible polyhedral uncertainty sets. The flexible polyhedral is described in (1) as follows.
where represents the random variable that is under uncertainty and distributed in the range ∈ [−R j , R j ]. J is the cardinality whose corresponding coefficient is subject to uncertainty. R j is the ratio parameter that is defined as the ratio of the uncertainty amounted by a confidence level defined from the PDF of the uncertainty simulation model per the maximum expected value of the uncertainty. The ratio parameter is ranging between 0 and 1 (0 ≤ R j ≤ 1). Figure 4 shows different predefined confidence levels scenarios (50%, 80%, and 99%) from the PDF based on historical calculated CF. It is noticed from Figure 4 that the uncertainty sets are flexible based on the confidence value. Therefore, the bounds of uncertainty become changeable and more independent.

RO counterpart based on flexible polyhedral uncertainty set
Reference 39 proposed the concept of robust counterpart optimization formulation for linear programming problem. Consider the following linear optimization problem as follows.
whereã ij represents the constraint coefficient subject to uncertainties. The uncertain coefficients are described in (4) as follows.ã where a ij represents the mean value of the PDF curve andâ ij denotes the maximum range of uncertainty. So the main constraint (3) can be rewritten in (5) as follows.
where U denotes the uncertainty set. If set U is the flexible polyhedral, then the corresponding RO counterpart is equivalent to the following.

GEP MODEL PROBLEM FORMULATION
The main objective of the proposed GEP problem is to minimize the total system costs over a certain planning horizon. The minimization of the total costs includes thermal resources, renewable energy sources, energy not served, CO 2 emissions penalty, and the salvation value. The proposed objective function is formulated in (9) as follows.
The equations represent each term in the objective function are explained as follows.

Thermal resources cost
The total investment, maintenance, and fuel costs of both candidate and existing thermal plants are defined in (10). The retirement of old inefficient plants is considered as well.

Renewable energy sources cost
Minimizing the total utility costs comprising the investment and maintenance costs of the candidate RES plants is formulated in (11) as follows:

Energy not served cost
Equation (12) aims to minimize the amount of energy expected not to be supplied in a given year.

CO 2 emissions penalty cost
A carbon tax on the CO 2 emissions from thermal plants is considered as described in (13). The total cost of a CO 2 emission tax is counted for both the existing and planned thermal units.

Salvation cost
The salvation value of the investment cost of both candidate thermal and renewable generating units is modeled and described in (14).
The equations describing the various constraints are detailed as follows. The constraint (15) represents the power balancing equation. A predefined value of the reserve margin should be declared to meet the reliability and security requirements of the power system in case of unit outages.
The impact of a high share of RES integration, specifically wind and PV is incorporated in the GEP model as described in (16) and (17) as follows.
The LOLE should be guaranteed as described in (18).
As the reduction of the CO 2 emission has become an environmental mandatory requirement, the annual emission reduction rate is considered. The CO 2 constraints are represented in (19) and (20). The maximum amount of CO 2 is annually reduced as described in (20).

ROBUST GEP MODEL
RES uncertainty creates imbalances between generation and load demand. Therefore, the effect of the RES uncertainty should be incorporated within the planning model. A RO approach is a perfect tool used to deal with problems under the presence of uncertainties. RES uncertainty impact is incorporated into the GEP model through developing a flexible polyhedral-based RO. The solution robustness is controlled based on a predefined confidence level. Renewable energy overcapacity, reserve margin, and ESS are proposed to cope with the RES uncertainty problem. The optimal method among the three proposed methods is determined by developing an economic objective function.

Renewable energy overcapacity method
The power balancing constraint (15) and the high sharing RES integration constraint (16) are modified to represent the RES uncertainty as defined in (21), (22), and (23).
whereCFis the capacity factor under uncertainty. CF is the capacity factor mean value.ĈF max is the maximum deviation of capacity factor (ie, maximum range of uncertainty). is distributed in the range ∈ [−R i , R i ]. R i is the ratio parameter which can be determined based on the confidence level of the CF uncertainty resulted from the PDF curve.
According to the RO counterpart induced by the flexible polyhedral uncertainty set, the constraints (21), (22), and (23) can be rewritten as follows.
When applying the Mixed Integer Linear Programming (MILP) optimization technique on the robust GEP model through modified constraints (24), (25), (26), and (27), it is expected that the number of RES units X RE will be increased to maintain the decrease in CF(i) due to the RES uncertainty based on the preselected confidence level.

Reserve margin method
Reserve capacity is useful to preserve operational flexibility as well as maintain system reliability. Capacity margin is proposed to deal with the RES uncertainty. So that previous power balancing constraint (15) is modified and utilized as (24), (26), and (27). The RES uncertainty is represented in the power balancing constraint (15) only in this method and not represented in the integration of high sharing RES constraint (16). The reason behind that the goal of this method is to deal with the RES uncertainty impact via increasing the number of thermal units X CS only as clarified in (24), (26), and (27). Hence, the reserve margin will be increased.

Electrical ESS
ESS may be a preferable solution used by the grid operators to solve some of the critical characteristics of electricity generation and operation. Using ESS is beneficial for power system reliability and flexibility. An ESS is proposed as a solution to deal with the RES uncertainty problem. Two stages are required to obtain the appropriate energy storage size and type. Firstly, the amount of annual power (P u ) required to cover the RES uncertainty should be determined. The amount of required power (P u ) is determined by calculating the difference between the amount of total power obtained from the GEP results without considering the RES uncertainty impact, in (15) and (16), and total power obtained from the GEP considering the RES uncertainty in (24), (25), (26), and (27). The amount of the required power (P u ) obtained from the first stage will be implemented in the second stage as a constraint. An objective function and technical constraints are developed in the second stage to determine the suitable energy type and the capacity of each type. The objective function is developed in (28) to minimize the total costs including investment and maintenance costs of candidate energy storage plants as follows.
The sum of the planned energy storage plants must be greater than or equal to the electrical uncertainty power (P u ) in each year within the planning period to maintain the adequacy of the electrical power system as constrained in (29).

Determining the optimal method to cope with the RES uncertainty problem
In this section, the optimal economical method to deal with the RES uncertainty impact is determined. The three previously proposed methods are implemented within an objective function and appropriate constraint to achieve the optimality of the economic factor. The proposed objective function and the constraints are formulated in (30) and (31), respectively.

DATA AND ASSUMPTIONS
According to the Egyptian power grid, the available candidate thermal plants for the case study are Combined Cycle Gas Turbine (CCGT), nuclear, and coal-fired. The available candidate renewable plants are PV and wind power plants. The candidate energy storage technologies are represented by Pumped Hydro Station (PHS), Compressed Air Energy Storage (CAES), and Lithium-ion (Li-ion) battery. The technical and economic characteristics of candidate thermal and renewable plants are shown in Table 1 while the energy storage technologies are shown in Table 2.
The list of other assumptions is shown in Table 3. The forecasted peak load and energy demand over the planning period are shown in Figure 5. Figure 6 shows the retirement schedule of the existing plants. The planning horizon is assumed to be from the year 2021 up to 2040. A total of 35% of renewable energy sharing is aimed to be achieved by the end of 2040. The historical data of the wind and PV are attained from the New and Renewable Energy Authority 46 and Renewables ninja. 34,35 The carbon tax rate is varied from $0/ton to $30/ton with an incremental rate of $

RESULTS AND DISCUSSION
Variety case studies are presented in this section based on the above-mentioned data and assumptions. The GEP is studied without considering the RES uncertainty impact and different cases considering the RES uncertainty. The obtained results clarify the impact of the uncertainty on the proposed GEP model. The GEP model is solved by MILP optimization technique.

Total generation without considering uncertainty
The annual results of the capacity plants and the corresponding type over the proposed planning horizon are shown in Figure 7. Various case studies considering the RES uncertainty impact in the proposed robust GEP model are conducted. The GEP results for the different proposed solutions to deal with the RES uncertainty impact are presented. RES uncertainties with different confidence level scenarios (50%, 80%, and 99%) are studied to clarify the influence of the RES uncertainty weight on the proposed robust GEP model. DIgSILENT Power Factory generation adequacy tool is used to verify the LOLE and the EENS of the GEP results. Figure 8 shows the RES overcapacity results for different RES uncertainty weights. The different RES uncertainty weights are based on the degree of confidence level. Increasing the confidence level resulted to increase the RES capacity as shown in Figure 8.

F I G U R E 7
Planning results without considering renewable energy sources uncertainty impact F I G U R E 8 Impact of different renewable energy sources uncertainty weights on renewable capacity F I G U R E 9 Impact of different renewable energy sources uncertainty weights on reserve margin capacity

Reserve margin method results
In this case, reserve margin results for different RES uncertainty weights are shown in Figure 9.

Electrical ESS method results
PHS, CASE, and Li-ion storage types are studied as ESS systems to cope with the RES uncertainty. The proposed objective function and constraints described by (28) and (29) resulted in the annual optimal number of each ESS type. The objective function resulted that PHS is the best economic solution of ESS types. Hence, CASE and Li-ion types are not recommended as a feasible ESS solution. Therefore, both types are not shown in Figure 10.

Proposed optimal solution for RES uncertainty problem
Solving the objective function (30) resulted in using the ESS system is the most economical method to cope with the RES uncertainty. Moreover, from a technical point of view, ESS decreases the RES curtailment and enhances the grid operational flexibility.

CO 2 gas emission reduction
The high tax rate value of CO 2 emission, as well as the high RES sharing, is constrained in the proposed GEP planning model. Therefore, it is observed from the results that the sharing percentage of the CO 2 gas emission is reduced while RES sharing is increased gradually as shown in Figure 11.
F I G U R E 10 Impact of different uncertainty weights on pumped hydro storage

F I G U R E 11
The reduction of CO 2 gas emissions and the renewable energy sources gradually increasing

CONCLUSION
A robust long-term GEP model is proposed in this paper considering the high sharing RES effect. A mathematical simulation model to predict the long-term RES uncertainty is developed to consider the uncertainty effect within the proposed GEP model. CF of the historical RES data is used to build an uncertainty simulation model. The PDF is used to handle the predicted uncertainty data to be described within the GEP model. A confidence level parameter is proposed to control the weight of each RES uncertainty set separately. The RES uncertainty is incorporated within the proposed GEP model through a flexible polyhedral to provide realistic and precise GEP results. The confidence level parameter is used to describe different RES uncertainty scenarios within the GEP model and control the degree of robustness. RES overcapacity, reserve margin, and ESS are proposed as three appropriate methods to deal with the uncertainty problem. The results of the three methods have demonstrated that increasing the confidence level resulted in increasing the RES capacity, the total reserve margin, and the number of ESS. An economic objective function has been developed to determine the best solution among the three proposed methods. The results have shown that the use of PHS is the most economical method among all the proposed methods. The results obtained from the robust GEP model have validated the gradual CO 2 gas emission reduction and increasing the sharing of RES during the proposed planning horizon. The generation adequacy tool using DIgSILENT program has been used to verify the reliability criteria.