Generation Capacity Expansion Problem Considering Different Operation Modes of Combined-Cycle Gas Turbines

The design of future decarbonized power systems is one of the most relevant and challenging problems that power system planners are facing nowadays. It is expected that Combined-Cycle Gas Turbines (CCGTs) play a relevant role in these systems to provide peak power and reserve capacity when intermittent power units be unavailable. However, the large computational size of planning problems has prevented so far from modeling the actual operation of CCGTs accurately. This paper intends to quantify the effect of the modeling of CCGTs in the generation capacity expansion problem. For doing that, the operation of CCGTs is modeled using a mixed-integer linear formulation that considers different operation modes. Afterwards, generation expansion decisions in a realistic case study are analyzed using different accuracy degrees for modeling the operation of CCGTs. The numerical results suggest that the simplification of the modeling of the operation of CCGTs overestimates the flexibility provided by these units, which increases the capacity installed from renewable units between 15 and 25%.

The notation used throughout the rest of this section is included below for quick reference.  The European Commission has defined a long-term strategy aiming to achieve net-zero greenhouse gas emissions by 2050 [1]. In this transformation process, the electricity production coming from open-cycle (OCGTs) and combined-cycles gas turbines (CCGTs) is intended to play a principal role in providing a dispatchable and reliable supply of electricity as the participation of renewable technologies increases [2]. OCGTs burn natural gas or an alternative fuel to rotate one or several gas turbines using the combustion gases. A compressor is used to force the injection of air in the chamber for increasing the efficiency of the generating unit. OCGTs are scalable for different electric power capacities and they are suitable to be used for base or peak load electricity production. On the other hand, CCGTs are equipped with one or several steam turbines that are used VOLUME 10, 2022 to exploit the remaining heat of exhaust gases of the gas turbines. CCGTs have been installed widely in power systems around the world because these units present high energy efficiency, low installation costs, and short construction times. Additionally, CCGTs, as well as OCGTs, allow fast startups and shutdowns and they can change their production very quickly. Unfortunately, the modeling of the operation of CCGTs is more complex than that of OCGTs because of the simultaneous presence of gas and steam turbines result in different operation modes depending if gas and steam turbines are either at operation or not. The modelling of the operation characteristics of CCGTs is key in the formulation of short-term operation problems where generating units must be characterized as detailed as possible. This detail level is difficult to reach in the long-term planning problems, where the computational complexity is much higher. However, the adequate modeling of decision-making problems intended to design future power systems also requires a high enough degree of accuracy in the modelling of generating units if the resulting generation capacity expansion decisions are desired to be implemented in practice.
The modeling of the operational behaviour of CCGT units is currently an active research topic. Reference [3] proposes a unit commitment problem formulation that takes into account the transition processes among the different operation modes of CCGTs and their efficiencies. In [4], the formulation derived in [3] is improved by increasing the computational efficiency and including tighter constraints. The authors of [5] propose a novel tight mixed-integer linear formulation for the modeling of CCGTs. This formulation can be used in either unit commitment problems or in the bidding problems faced by those CCGTs participating in energy markets. Reference [6] proposes an enhanced stochastic-constrained unit commitment model especially tailored for the presence of CCGTs that is tested in the Midcontinent Independent System. In [7], the Surrogate Lagrangian Relaxation technique is applied to solve large unit commitment problems with high presence of CCGTs. Reference [8] develops a node-based formulation for the operation of CCGTs considering different levels of accuracy in the representation of the physical constraints of these units. The authors of [9] analyze different existing CCGT models and propose a hybrid model considering the best features of configuration-base and component-base models. Reference [10] considers the medium-term operation under uncertainty of a CCGT unit that has a gas storage available. Reference [11] proposes a mixed-integer linear formulation that considers the transitions between different operation modes spanning more than one time interval. In [12], the startups and shutdowns of individual turbines of CCGTs are precisely modelled and they are included in the formulation of a unit commitment problem that also considers the provision of spinning reserves. Reference [13] formulates a unit commitment problem that considers simultaneously gas and power network constraints.
Observe that all works mentioned here are focused in the short-term operation of CCGTs and none of them is designed to be implemented in large-scale long-term expansion problems.
Many modeling approaches have been proposed to formulate the generation capacity expansion problem in the last years, [14]- [22]. Table 1 lists the most relevant features of these works regarding the modeling of CCGTs. Observe that none of the analyzed works takes into account the operation modes of CCGTs. Additionally, most of these works do not take into account the binary variable that models the commitment of generating units. This paper seeks to overcome these limitations proposing for the first time a generation capacity expansion formulation considering explicitly the different operation modes of CCGTs. Therefore, the objective of this paper is to analyze the influence of the modeling of CCGT units in the generation expansion decisions. Traditionally, the large computational size of expansion problems has prevented decision-makers of an accurate modeling of the operation of CCGTs. As a consequence of this, minimum power outputs, startup costs and the establishment of specific operation costs for each operation mode of CCGTs are usually neglected. This fact may cause an overestimation of the flexibility of this type of generating units, which may render in the implementation of generation capacity portfolios that are not able to supply the demand in situations with high penetration of intermittent capacity.

A. DESCRIPTION OF CCGTs
A CCGT is a type of power plant specifically fed with natural gas. The operation of these units is based on the combination of two thermodynamic cycles that seeks to improve the efficiency of the plant. The first thermodynamic cycle occurs in the gas turbines, which is composed of a compressor and a combustion chamber. In this Brayton cycle, the energy, which is present in the chemical bonds of fuel, is transformed into work and thermal energy. In the Rankine cycle, the thermal energy of exhaust gases outgoing from the gas turbine is transformed into work through a Heat Recovery Steam Generator (HRSG) and the steam turbine. In both cases, the mechanical work is transformed into electricity by a generator. The configuration of a CCGT could be by single-shaft or multi-shaft. In the first case, gas turbine and steam turbine share shaft and generator. In the second case, each turbine has its own generator.
Considering that the determination of the generation capacity expansion of a power system is a long-term decisionmaking problem, the following assumptions are considered: • The generation capacity expansion is determined for a target year that is represented using a set of characteristic days. Each day is divided into 24 hourly periods.
• Different operation modes are defined for each CCGT depending on the number of gas and steam turbines that are at operation.
• Each operation mode is characterized by maximum and minimum power limits, as well as by different operation costs.
• The uncertainty related to demand growth and fuel costs of thermal units is characterized using a set of scenarios, indexed by ω ∈ .
• The operation cost of the resulting power system is modelled considering the day-ahead energy and reserve capacity markets.

B. MATHEMATICAL FORMULATION OF THE OPERATION MODES OF CCGTs
The mathematical formulation of the different operation modes of CCGTs is presented in this section. For the sake of clarity, the proposed formulation is derived for a typical multi-shaft CCGT with a configuration comprising two gas turbines (2GT), a steam turbine (1ST) and three generators. Fig. 1 represents the schema of this type of unit. In this approach it is assumed that the exhaust gases of the steam turbines are not used for other purposes. Considering a 2GT+1ST configuration, the following five operation modes can be distinguished: • Mode 0: The unit is offline. • Mode 1: One gas turbine is at operation.
• Mode 2: One gas turbine and the steam turbine are at operation.
• Mode 3: Two gas turbines are at operation. • Mode 4: Two gas turbines and the steam turbine are at operation. The different operation modes are denoted by index m ∈ M . In this manner, variable v M gmdtω is equal to 1 if CCGT unit g is at operation in mode m in day d, period t and scenario ω, being equal to 0 otherwise. Considering this, the power output of the CCGT is denoted by p G,D gdtω and it is equal to the sum of the power output in each operation mode, p G,D,M gmdtω . Considering this, the mathematical formulation of the power output of a CCGT unis can be expressed as follows: 0 ≤ v M gmdtω ≤ 1, ∀g, ∀m, ∀d, ∀t, ∀ω (4) Expression (1) computes the power output of the CCGT as the sum of the power generated in each operation mode. The power limits in each operation mode are defined by (2). Then, if v M gmdtω = 1, the power output in mode m is constrained by gdtω are defined to model each specific operation mode. Considering these auxiliary variables, the selection of the operation modes denoted by v M gmdtω , ∀m = 0, · · · , 4, is computed as follows: Constraints (5)-(9) compute the minimum value of variable v M gmdtω for each operation mode m = 0, · · · , 4. Observe that if this minimum value is equal to 1, then constraints (4) VOLUME 10, 2022 should not be further constrained by (5)- (9). Since only five operation modes are considered, constraints (10)-(11) are used to avoid combinations of auxiliary variables not desired. Constraints (12) define the binary nature of auxiliary binary variables.
It is also worth to note that, because of the binary nature of these auxiliary variables and the enforcement of constraints (3), (4) and (5)-(12), the value of variable v M gmdtω is always either 0 or 1. Then, it is not necessary to define v M gmdtω as a binary variable. As Considering that variables v M gmdtω are bounded by 0 and 1 by constraints (4), it is observed that constraints (14)-(17) do not further constrain the value of v M gmdtω for m = 1, · · · , 4. However, constraint (13), together with (14)- (17), enforces Table 2 shows the value of variable v M gmdtω for each operation mode and for all possible combinations of auxiliary variables v M,1 gdtω , v M,2 gdtω and v M,3 gdtω .

C. PARTICIPATION OF CCGTs IN THE RESERVE CAPACITY MARKET
The formulation (1)-(4) can be expanded to consider that CCGTs can participate in the reserve capacity market. If the participation of CCGT unit g in the up-and down-reserve capacity services is modeled by r G,U gdtω and r G,D gdtω , the following constraints must be also considered: Constraints (18) and (19) define the power limits of the operation of CCGTs when they provide up-and down-reserve capacity, respectively. Constraints (20) and (21) establish that the scheduled reserve capacity associated with operating mode m has to be equal to zero if the CCGT is not working in that operating mode. Constraints (22) and (23) enforce that the total scheduled reserve capacity of a CCGT unit has to be equal to the sum of the reserve capacities associated with each operating mode.

D. MODELING OF THE TRANSITION COST BETWEEN OPERATION MODES OF CCGTs
The transition between different operation modes of CCGTs has associated a cost that depends on the number of gas and steam turbines that are started in the transition process.
gmm ω is the cost related to the transition of unit g from operation mode m to m in scenario ω, the resulting transition cost between these operation modes in day d, period t and scenario ω, c TC gmm dtω , is formulated as follows: ∀m <> m , ∀g, ∀d, ∀t, ∀ω c TC gmm dtω ≥ 0, ∀m, ∀m , ∀g, ∀d, ∀t, ∀ω In some CCGT units, the transition between two operation modes in a single time period is not physically possible. For instance, the transition between modes 0 (0GT + 0ST) and 4 (2GT+1ST) may not be performed in a single time period. For modeling this situation, we define parameter FT mm that is equal to 1 if the transition from mode m to m is forbidden, being equal to 0 otherwise. Therefore, the prohibition of the transition between two operation modes is formulated as follows:

IV. GENERATION EXPANSION PROBLEM A. DECISION-MAKING FRAMEWORK
The generation expansion problem consists in determining the power capacity to install in a power system for a given target year from a set of candidate generation technologies minimizing the total cost including investment and operation costs. For the sake of tractability, the target is represented by a set of characteristic days.
In this paper, a static generation expansion model has been considered. In this manner, it is assumed that all investment decisions are made simultaneously at the beginning of the planning horizon. Note that static expansion models are used because they provide an appropriate trade-off between modeling accuracy and computational tractability. Since the objective of this paper is to quantify the effect of modelling the different operation modes of CCGTs, transmission expansion decisions are not considered in the proposed model. Consequently, all transmission expansion plans decided beforehand by the transmission system planner must be incorporated in the proposed model as input data. The power flow in the transmission lines is computed using the DC model. Investments in storage units based on electrochemical batteries are also considered to favor the installation of intermittent generation units. Finally, the operation of the resulting system is considered by modeling the day-ahead and reserve capacity markets.

B. OPTIMIZATION PROBLEM
The mathematical formulation of the problem described above is included below. The proposed generation expansion problem is formulated as a risk-neutral two-stage stochastic programming problem [24], in which investment decisions in generating and storage units are made at the first stage, whereas the operation of the system is determined at the second stage. The objective function of this problem is the total cost, including annualized investment and operation costs.
Subject to: • Investment constraints: 0 ≤ p I,G g ≤ P I,G max,g , ∀g ∈ G I • Operation of generating units: • Operation of CCGT units: Constraints (1) • Energy balance • Reserve-capacity provision • Transmission power flow where is the set of all optimization variables in this problem. The objective function (27) formulates the total costs, which are equal to the summation of the annualized investment costs of generating and storage units, the yearly operation and reserve capacity costs of generating and storage units, startup costs of OCGTs, the transition mode costs of CCGTs and the unserved demand cost. Observe that the operation costs of CCGTs depend on the operation mode m. The investment decisions on new generating and storage units are formulated through constraints (28)-(33). Observe that the power to install from intermittent units, p I,G g , and storage facilities, p I,SP s , are continuous variables because of the modularity and scalability of these technologies. On the other hand, the power to install from OCGTs and CCGTs depends on binary variable v I,G g , which is used to force that the whole capacity of the unit be installed if the unit is built. The investment decisions related to storages comprise two different decisions: 1) maximum power to charge and discharge and 2) the energy capacity of the storage. In practice, both variables are related. This relationship has been modeled through parameter γ S,EP s in constraint (32).

V. NUMERICAL RESULTS
A realistic case study based on the power system of Gran Canaria (Spain) has been solved to test the performance of the proposed formulation. Gran Canaria is an island that has a total population over 866 thousand inhabitants and an annual electricity consumption equal to 3469 GWh in 2019. The objective of this case study is to determine the optimal generation capacity expansion decisions that should be implemented by year 2050. It is assumed that all generating units that are currently at work will be decommissioned prior to this year.

A. INPUT DATA
The main input data describing the Gran Canaria system is provided in reference [25]. This power system comprises 43 candidate generating units that can be installed in 30 buses that are connected by 58 lines. The transmission system considers the future transmission expansion plans announced by the Spanish transmission planner, [25]. Fig. 2 represents the single-line diagram of Gran Canaria transmission network. The technologies of the generating candidate units are CCGT, OCGT and wind and solar PV. Two different types of OCGT units are considered, and they are denoted as OCGT-1 and OCGT-2. The characteristics of these units are provided in Table 3 and they are based on actual OCGT plants installed   in the Spanish power system which are named as Ibiza16 and Ibiza25, respectively, [26].
A single type of candidate CCGTs is considered, which is based on an actual CCGT plant installed in the Spanish power system, which is named as CA's, [26]. This is a CCGT plant with two gas turbines and a single steam turbine. This plant has 5 operation modes. The technical characteristics of this candidate power plant are listed in Table 4. Fig. 3 depicts the thermal consumption of the considered candidate OCGTs and CCGTs units. Slashed lines represent the actual quadratic thermal consumption obtained from [26], whereas continuous lines represent the linearization of these values. Table 5 provides the parameters of the linear approximation of heat consumption (MWh-t) of the different thermal generating units and operation modes represented in Fig. 3.
The thermal consumption in transitions between two operation modes entailing the startup of either a steam or a gas turbine is not negligible. In this manner, the thermal consumption in MWh-t and, in brackets, the fixed cost    in ke of the transitions between different operation modes is included in Table 6. In the same manner, Table 7 lists the thermal consumption and the fixed cost of startups of OCGT units. These values have been obtained from [26]. Transitions between modes 0 and 4 in a single period are not allowed. The minimum up and down times of CCGT units in modes 2 and 4, in which the steam turbine is on, are 2 hours.
Natural gas prices are considered as an uncertain parameter, which is modeled using the data provided by [27]. Two scenarios with maximum and minimum prices equal to 62.63 and 20.14 e/MWh-t are considered for Spain in 2050 assuming current energy policies.
The capital costs of wind and solar PV power units are equal to 1452 and 660 e/kW which are based on the estimations provided by IEA [28]. Capital costs are annualized using a capital recovery factor equal to 10.95%. The technology, location and capacity of each candidate generating unit are included in Table 8.
Additionally, four utility-scale storage units based on Li-on batteries can be installed in buses 13 and 18. The maximum energy capacity that can be installed per bus is 2000 MWh, and a typical relationship energy/power (γ SP i ) equal to 6 has been considered [29]. The investment costs of storage units is equal to 300 e/kWh.
The target year is represented by a set of characteristic days. The hourly values of the system demand, and wind and solar PV availabilities in each day are assigned using real data pertaining to 2019. A set of 6 characteristic days has been selected by using the scenario reduction algorithm described in [30]. The weights of each day, W d , are assigned minimizing the daily square error between the reduced set of days and the original set containing 365 days. Fig. 4 plots the daily square error of wind and solar availabilities and demand for different number of selected days. As Fig. 4 shows, a number of days greater than or equal to 6 is adequate to represent the planning horizon. Fig. 5 represents the expected system demand, and wind and solar PV availabilities for the considered days.
The system demand in 2050 is characterized as a random variable characterized by a set of scenarios. According to [31], two electricity demand scenarios are computed considering demand increases of 1.9 and 2.1% per year between 2018 and 2050.
The values of parameters γ U,D and γ U,I used to establish the minimum requirements of up and down reserve capacities are equal to 0.03 and 0.05, respectively. It is considered that up-reserve requirements are equal to the down-reserve requirements, i.e., γ D,D = γ U,D and γ D,I = γ U,I . Reserve capacity costs are equal to 0.25 times the operating cost for each unit. It is assumed that wind and solar PV units are not qualified to supply reserve capacity. The unserved demand cost is equal to 1000e/MWh.

B. RESULTS
In this section, we analyze the influence of the modelling of the operation of CCGTs in the generation expansion decisions. For the sake of comparison, two different models are tested considering if minimum power outputs and the different operation modes of CCGTs are formulated or not: In order to test the performance of both models, five different cases have been solved for each model: • Th: Only thermal units can be installed • 0Sto: Thermal and renewable units can be installed, but storages are not considered.
• 0.25Sto: Thermal and renewable units can be installed, and only 25% of the potentials of storages.
• BAU : Business-as-usual case where thermal and renewable units can be installed, as well as the full potential of storages.
• 75R: This case is similar to BAU, but it is enforced that 75% of the demand must be provided by CO 2 -free sources. Note that higher limits of non pollutant production lead to large amounts unserved demand in the analyzed system. All simulations are performed with CPLEX 12.6.1 using a server with four 3.0 GHz processors and 250 GB of RAM.   The numbers of constraints and binary and continuous variables are reported in Table 9. As expected, the problem that considers all constraints and variables, F-BAU case, achieves the largest numbers of variables and constraints, as well as the highest solution times. However, note that the obtained times are reasonable for solving a long-term problem. Fig. 6 represents the optimallity gap of F-BAU case with respect to the solution time. It is observed that the time needed to reduce the gap increases as the gap becomes smaller. Table 10 provides the resulting expected cost obtained in each case. Startup costs include startup costs of OCGTs and transition mode costs of CCGTs. Note that these costs are not considered in the modelling of S-cases. It is worth noting that S-cases overestimate the total expected cost with respect to those in F-cases. Observe that F-cases obtain significant lower day-ahead energy costs due to a more accurate modelling of the operation costs of CCGT units. It is also interesting to note the high relevance that the amount of available capacity to install from storages has in the total costs. As expected the BAU case attains the lowest cost for S-and F-cases.
Tables 11 lists the generation and storage capacities installed in each case. In all cases, it is observed that solar  PV is the generation technology most installed, which value increases as the potential storage capacity grows. However, it is worth emphasizing that the investment decisions on renewable units in S-and F-cases are appreciably different. For instance, it is remarkable that the capacity installed from wind and solar PV units ranges between 15 and 25% higher in S-cases with respect F-cases. Then, it is observed that the simplification of the modeling of thermal units in S-cases overestimates the flexibility of these units which favors the installation of wind and solar PV units.
Tables 12 provides the energy produced by technology in each case. The energy produced by storages refers to the energy discharged. In most cases, the technologies that produce the maximum quantity of energy are solar PV and CCGT. Note that the energy produced by thermal units in Fcases is higher than in S-cases. This result is consequence of the reduced power capacity installed from renewable energies in F-cases. Then, the estimation of the total renewable production using S-cases is also overestimated. Despite of the simultaneous charge and discharge of storage units is not explicitly forbidden in the presented formulation, we have verified that storage units do not simultaneously charge and discharge in the analyzed case studies. Table 13 shows the annual scheduled up and down reserve capacities per technology and case. It is noticed that CCGT units provide most of the up reserves, storages schedule down-reserve capacity if storage units are available to be installed. However, the contribution of storages to the up-reserve capacity in F-cases is much more higher than in S-cases. The reason of this result is that the simplification of VOLUME 10, 2022  the operation of CCGTs in S-cases does not allow to estimate adequately the reserve capacity available from thermal units. Table 14 includes some relevant data pertaining to the performance of thermal units in the different analyzed cases. In particular, two items are considered. First, it is provided the percentage of hours in which thermal units are providing up-reserve capacity when they are not participating in the day-ahead energy market. This undesired situation indicates that the provided reserve cannot be considered as spinning reserve, and it may occur in S-cases where the minimum power output of the units is not considered. The second item is the number of operation mode changes of CCGTs with respect to the number of working hours of these units. Note that high values of these parameters are negative since they indicate high cycling costs of thermal units.
In this table we observe that the percentage of hours in which thermal units provide irregular up-reserve in Scases is very high. It should be highlighted that in S-75R case, more than half of the hours in which thermal units are assumed to provide up-reserve they are offline. This table also reveals that the cycling of CCGT units measured as the number of operation mode changes modes is higher in most S-cases. This result is explained by the fact that S-cases do not take into account transition mode costs in their formulations. Fig. 7 represents the energy production for each considered day and the scenario with lowest demand and gas prices. This figure shows that the solar PV supplies most of the demand in the middle of the day. This fact forces to reduce  the power output of OCGT and CCGT units between 10:00 and 20:00 hours. Storages are charged during these hours, and discharged when the production of solar PV is low.
As an example, the operation of the CCGT unit number 13 during the first day of the planning horizon is represented in Fig. 8 in terms of the energy production, reserve capacity scheduling, operation mode and transition cost between operation modes. In this figure, it is observed how the production of the CCGT unit is equal to the maximum capacity (214.5 MW) at hours 22 and 23, in which the system demand is high and there is not renewable production. However, the output of the CCGT unit is equal to the minimum power output (6.39 MW) during the middle part of the day, where the solar PV production is high. During these hours, the CCGT unit is providing up-reserve capacity. The unit is operating at modes 1 and 2 during most part of the day, except in hours 21-23, in which it is operating at mode 4 to provide a higher quantity of power. Observe that the minimum up time constraint forces that the CCGT unit remain in operating mode 2 at least two hours, 19 and 20. Finally, observe that at hours 19 and 21, the CCGT unit changes from mode 1 to mode 2 and from mode 2 to mode 4, which has transition costs equal to 3.6 and 14.4 ke, respectively.

VI. CONCLUSION
This paper presents a generation capacity expansion problem formulation that explicitly accounts different operation modes of CCGTs. The proposed model considers the long-term uncertainty pertaining to the demand growth and natural gas costs. The model is tested on a realistic case study based on the Gran Canaria power system. Based on the numerical results presented in the case study, the following main conclusions can be drawn: • The proposed formulation is able to characterize properly the operation modes and the transitions between different modes of CCGT units in generation capacity expansion problems.
• The computational size of the problem that considers different operation modes of CCGTs is significantly higher than that of the problem in which operation modes are ignored. Specifically, the number of binary variables grows linearly with the number of time periods, scenarios and CCGT units. This fact has a strong impact in solution times.
• The obtained generation capacity expansion depends on the degree of precision of the modeling of CCGT units. The capacity installed of the different candidate generation technologies varies if the technical characteristics of CCGT and OCGT units are relaxed.
• The up-reserve capacity provided by CCGT units is significantly overestimated if the technical characteristics of CCGT units are neglected. The reserve capacity in the simplified model is 10% higher than that obtained with the proposed model.
Future research is underway to determine the optimal upgrade of existing CCGT units to improve their performance in renewable dominated power systems.