Voon Keck

Rémy Ling

Jan Rigo-Mariani

Maciejowski

Remy Rigo-Mariani
email: remyrm@ntu.edu.sg

Keck Voon

Jan Maciejowski

A Generic Method to Model CO 2 Emissions of Combined Cycle Power Plants for Environmental Power Dispatch

Keywords: combined cycle gas turbine, design, CO2 emissions, environmental unit commitment, power dispatch, optimization

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The work presented in this paper has been led in the framework of a wider study whose objective is to simulate the operation of the Jurong Island power grid in Singapore. The island has four thermal power plants with thirteen identified combined cycle gas turbines (CCGT) [START_REF]Singapore Energy Statistics[END_REF]. The first concern is to model the power generation on site to perform a power dispatch that minimizes the amount of CO2 generated. Such unit commitment (UC) problems have been widely studied in the literature. The traditional objective is to minimize the generation cost while meeting the load and fulfilling system operating constraints [START_REF] Van Den Bergh | A Mixed-Integer Linear Formulation of the Unit Commitment Problem[END_REF]. Here the objective only refers to the minimization of the CO2 emissions. In the literature those emissions are commonly added to the objective function as a linear cost with the generated power as in [START_REF] Deng | What is the Cost of Negative Bidding by Wind? A Unit Commitment Analysis of Cost and Emissions[END_REF][START_REF] Dupuis | Comparing Costs and Generation Mix Alternatives for the Application of the Clean Power Plan in the PJM Electricity Market: Application of Unit Commitment[END_REF]. Another alternative consists in considering a quadratic curve for the CO2 emissionsthe same way it is commonly done for the generation costas well as CO2 start-up costs [START_REF] Yamashita | Trade-off Analysis of CO2 versus Cost by Multi-objective Unit Commitment[END_REF]. Then either a bi-objective (cost/CO2) optimization is performed or the CO2 is simply added to the conventional objective function while introducing carbon penalties [START_REF] Van Den Bergh | A Mixed-Integer Linear Formulation of the Unit Commitment Problem[END_REF]. Cubic functions can also be found in the literature [START_REF] Krishnamurthy | Impact of price penalty factors on the solution of the combined economic emission dispatch problem using cubic criterion functions[END_REF] as well as exponential formula when other types of pollutant considered [START_REF] Gong | Environmental/economic power dispatch using a hybrid multi-objective optimization algorithm[END_REF]. In addition to that variety of models, the attached coefficients are rarely justified. They come from pre-established test cases and can display wide ranges of values even for the same type of unitfrom 336 kg/MWh to 616 kg/MWh for CCGT emissions in [START_REF] Deng | What is the Cost of Negative Bidding by Wind? A Unit Commitment Analysis of Cost and Emissions[END_REF]. Indeed, a great part of such power systems studies mainly focus on validating UC problems formulation and the algorithms used to solve themwhich is still a hot topic in research on power systems. In order to accurately model those emissions for existing thermal units one must pay attention to energy conversion aspects and model calibration. CCGT has received extensive attention, since it offers enhanced performances compared to conventional gas turbines (GT). An increasing number of heavy duty units has been installed in electrical systems across the world in recent decades to replace conventional coal/oil power plants [START_REF] Boyce | Handbook for cogeneration and combined cycle power plants[END_REF]. Most CCGT studies are concerned with optimal design that minimizes the cost for the plant owner as in [START_REF] Zhu | Simplified Performance Model of Gas Turbine Combined Cycle Systems[END_REF][START_REF] Ganjehkaviri | Modelling and optimization of combined cycle power plant based on exergoeconomic and environmental analyses[END_REF]. However, their performance dramatically decreases when deviating from their nominal operating point. Thus off-design operations have to be considered carefully, especially when a power dispatch will require the system to work on its whole operating range [START_REF] Kim | Comparative analysis on the part load performance of combined cycle plants considering design performance and power control strategy[END_REF]. A particular attention also has to be paid to the computational time once the power dispatch problem is written. Some more complex studies such as grid expansion planning with grid constraints, and sizing for renewable or storage units can be considered [START_REF] Alnaser | Optimal sizing and control of energy storage in wind power-rich distribution networks[END_REF]. In such cases prohibitive computational times should be avoided and tight formulations can prove to be of interest [START_REF] Van Den Bergh | A Mixed-Integer Linear Formulation of the Unit Commitment Problem[END_REF]. The main motivation of the present study is then to use a CCGT model in order to get realistic coefficients for carbon emissions in the environmental UC problem. With only on type of unit considered (CCGT here) it is important to discriminate the costs attached to the different technologies. The biggest challenge lies in obtaining consistent values from the sparse information given by the manufacturers for the identified units -usually only the rated power and efficiency are advised. Then major contributions of this work are:

 The cross disciplinary bridge between CCGT modeling and power dispatch strategies.



The development of a systematic comprehensive CCGT model whose granularity is adapted to with the MILP formulation for UC problems -the scalability does not allow too much detailed representation.

 The CO2 start-up costs are not explicitly entered as parameters in the UC. Instead the off design performance is estimated below the minimum operating point and start-up phases are modeled.

 A tight formulation of the environmental UC is proposed. Especially the units are gathered with distinct power plants. It allows the reduction of the number of variables and equations compared to a more complete representation.

The rest of the paper is organized as follows. Section 2 describes the model used to represent the considered units and the obtained results. The general equations are given and a particular attention is attached to the optimal procedures that allow estimating both nominal and off design performance. Section 3 present the obtained results and compare the performance for the different units. The environmental unit commitment is introduced in Section 4 with thirteen independent units distributed among four power plants. Section 5 describes the tight formulation of the power dispatch problem while the conclusions of the work are presented in Section 6.

CCGT MODELING

Fundamentals and Identified Generation on Site

Fig. 1 shows a simplified diagram of a CCGT. The operation of such a unit lies on the combination of two thermodynamic cycles. In the top Brayton cycle the input air mass low ma (in kg/s) at ambient conditions is compressed with a ratio rc before entering the combustion chamber of the gas turbine. The combustion with a fuel mass flow mf (in kg/s) allows the flue gases to reach the turbine inlet temperature TIT (in O C) before their expansion that produces a work WGT (in MW). A nomenclature of all the considered parameters is given on Table 1. Among those both rc and TIT have the greatest influence on the GT performances with highest values around 40 and 1600 O C respectively for the most efficient systems [START_REF] Boyce | Handbook for cogeneration and combined cycle power plants[END_REF]. The main limitations come from the maximum stress that the different materials can endure under operating constraints. In the bottom Rankine cycle, the hot gases at the turbine exhaust temperature TET (in O C) are used in a heat recovery steam generator (HSRG) to produced superheated steam at high temperature for a steam turbine. Note that typical architectures [START_REF] Boyce | Handbook for cogeneration and combined cycle power plants[END_REF]. Such units represent more than half the installed power generation capacity in Singapore and up to 95 % of mix in term of electricity generation [START_REF]Singapore Energy Statistics[END_REF]. Field data [13] and manufacturers websites identify five types of units on Jurong Island with the characteristics advised in Table 2 and distributed among four power plant as follows:

 Keppel Merlimau (KPL): Units A1 and A2, Units B1 and B2.

 Pacific Light Power (PLP): Units C1 and C2.

 SembCorp Cogen (SMB): Unit A1, Units D1 and D2.

 YLT Seraya (SRY): Units C1 and C2, Units E1 and E2.

Model Equations

This subsection describes the steady state equations commonly used in the literature for the modeling of CCGT units. Firstly the temperature at the output of the compressor Tc,out (in O C) is computed as well as the corresponding work Wc (in MW) as in (1) and (2) [START_REF] Ibrahim | Thermal Impact of Operating Conditions on the Performance of a Combined Cycle Gas Turbine[END_REF]. Note that Cpa (in kJ/kg/K) is estimated at the average temperature between Ta and Tc,out and referring to the tables for air properties.

               c c a out c n r T T c c 1 1 1 ,       m a out c pa a c n T T C m W      1000 ,  
A complete combustion of pure methane is considered (3) [START_REF] Khaliq | Thermodynamic performance evaluation of combustion gas turbine cogeneration system with reheat[END_REF]. Similarly to [START_REF] Ganjehkaviri | Modelling and optimization of combined cycle power plant based on exergoeconomic and environmental analyses[END_REF] the combustion equations has to be solved in order to estimate the fuel mass flow mf (in kg/s) required to reach the turbine inlet temperature. The heat capacity of the flue gas Cpg (in kJ/kg/K) is also obtained after solving a second order polynomial equation with the molar air fuel ratio λ (4). Cpg is computed with the molar quantities and specific heat of each component in the flues gas and using the corresponding property tables. Ma, Mf, na, and nf denote the molar masses and quantities of fuel and air.

    2 2 2 2 2 2 4 .O 2 . 2 N . 76 , 3 . . 2 O 2H CO .... .... 3,76.N O . . 2 CH                                                                       . 2
f pg cc f out c pa pg a C C C C C C n n n n C n C n C n C n C n n m m TIT C n LHV m T C TIT C m



The computation of the GT output work WGT (in MW) requires the estimation of the turbine exhaust temperature TET (in O C) as shown in ( 5) and [START_REF] Krishnamurthy | Impact of price penalty factors on the solution of the combined economic emission dispatch problem using cubic criterion functions[END_REF]. The different net output powers are then obtained following [START_REF] Gong | Environmental/economic power dispatch using a hybrid multi-objective optimization algorithm[END_REF] and [START_REF] Boyce | Handbook for cogeneration and combined cycle power plants[END_REF]. Note that the HSRG is not fully modeled with the superheater, the economizer, the deaerator and condenser equations. In this work the steam turbine work output WSTnet, once the heat is extracted from the exhaust gas, is given as an input fixed parameter at design conditions. The specific CO2 emissions SCE (in kg/MWh) consider the CO2 molar mass MCO2 and the fuel molar quantity nf [START_REF] Zhu | Simplified Performance Model of Gas Turbine Combined Cycle Systems[END_REF].

                          e e c gt r n TIT TET   1 1 1 1     1000 TET TIT C m W pg g GT       c gen m GT GTnet W n n W W       STnet GTnet CC W W W     CC CO f W M n SCE     1000 3600 2  
The heat supplied to the gas turbine QGT (in MW) [13] is used to compute the thermal efficiencies for the GT and the overall CC unit as follows:

  1000 ,out c pg g GT T TIT C m Q       GT CC CC GT net GT GT Q W Q W     and  

Off-Design Operation

The previous equations describe the behavior of a typical CCGT at the design operating point. Such a representation with a more detailed model of the HSRG is used in [START_REF] Zhu | Simplified Performance Model of Gas Turbine Combined Cycle Systems[END_REF] to perform the optimal sizing of a CCGT unit with regard to capital and operating costs. In power system studies the units never operate at the nominal point all along the considered time horizon [START_REF] Kim | Comparative analysis on the part load performance of combined cycle plants considering design performance and power control strategy[END_REF]. It becomes more obvious with the totality of a variable load fed by a finite numbers of units as it is the case in the system modeled here. Thus a particular attention has to be paid to the off-design operating points and constraints. Especially the objective is to estimate the carbon emissions on the whole operating range of the units. The first off-design constraint refers to the fulfillment of the schoked conditions of the gas turbine with a constant swallowing capacity [START_REF] Kim | Comparative analysis on the part load performance of combined cycle plants considering design performance and power control strategy[END_REF]. Here the compression ratio is used instead of the compressor output pressure [START_REF] Kim | Comparative analysis on the part load performance of combined cycle plants considering design performance and power control strategy[END_REF].

D sh sh des c des des a c a G G r TIT m r TIT m      , ,

 

Reference [START_REF] Zhang | Thermodynamic performance simulation and concise formulas for triple-pressure reheat HRSG of gas-steam combined cycle under off-design condition[END_REF] establishes semi empirical equations for the behavior of the steam turbine under off-design conditions. The output power has to be corrected regarding the deviations of the flue gas mass flow mg (in kg/s) and TET from their design values while considering ηc, the Carnot cycle efficiency [START_REF] Alnaser | Optimal sizing and control of energy storage in wind power-rich distribution networks[END_REF]. WSTnet then tends to decrease in the same time than QAV the heat available in the GT exhaust gas ((13) and ( 14)). Note that a more accurate representation of the off-design performances should take the deviations of the isometric efficiencies for the compressor and the gas turbine into account. It is not considered here and constant values are used.

TET T

a c   1    Fig. 2.
                  des g g des des AV AV m m TET TET Q Q , 2 ,                            2 , 2 , , , 32872 . 0 64575 . 1 32063 . 0 ... ... des AV AV des AV AV des c c gen m des STnet STnet Q Q Q Q n n W W    

Optimization for Design and Part-Load Operating Points

The model developed for CCGT units displays the architecture shown on Fig. 2 with fixed parameters LHV = 48685 kJ/kg and Ta = 30 O C. The equations are validated with the injection of the parameters found in the literature for the PG9351FA unit [START_REF] Zhang | Thermodynamic analysis of combined cycle under design/off-design conditions for its efficient design and operation[END_REF]. Acceptable values of the error for the output variables are observed (Table 3). The next step consists in calibrating the model. The objective is to optimize all the inputs (i.e. both design and operating variables) in order to minimize the error between the outputs and the values advised by the manufacturers (noted with the superscript . D ). The design objective odj D is then expressed as the sum of the errors regarding the values for TET, WGTnet and ηGT. [START_REF] Khaliq | Thermodynamic performance evaluation of combustion gas turbine cogeneration system with reheat[END_REF]. The CC output power and efficiency are implicitly fit as the ST net power is entered as a fixed parameter here. For the offdesign operations only the three operating variables are optimized. The unit operation consists in the control of the inlet guide vane of the compressor. The operation is based on IVG control up to 30 % air flow reduction with constant TET followed by a maximum air flow control [START_REF] Kim | Comparative analysis on the part load performance of combined cycle plants considering design performance and power control strategy[END_REF]. The operating controls are then optimized considering their design values as upper bounds (ma > 0.7×ma.des). The off-design objective odj OD aims at ensuring the shocked conditions, maintaining TET at the highest possible value while minimizing the error with the part load CC output power The two optimization problems described above are solved using the Sequential Quadratic Programming embedded in the Matlab Optimization Toolbox [START_REF] Powell | A Fast Algorithm for Nonlinearly Constrained Optimization Calculations[END_REF]. For every design and every part load point of each unit the problems are run with a hundred starting points randomly taken within the bounds for the variables. Note that the off-design operation points are considered between 30 % and 100 % off the nominal load with a step of 5 %.

OD CC W .       2 2 2 2 2 2 D GT D GT GT D D D GTnet D GTnet GTnet D TET TET TET W W W obj                  2 2 2 2 2 2 D sh D sh sh D D OD CC OD CC CC OD G G G TET TET TET W W W obj       

UNITS COMPARED PERFORMANCES

Design Results

Table 4 summarizes the result obtained while running the model calibration in order to meet the performances advised by the manufacturer for all the five units. The variables in bold refer to the performances points used to calibrate the model while running the design procedures. The observed values are very close to the targeted data on Table I. On the whole the observed errors for the GT/CC works and TET are below 0.1 % while the maximum registered deviation for the efficiencies is only 0.6 %. The results also display the specific CO2 emission estimated for the different units. The observed values are close as it is expected for units with similar performances in terms of efficiency. Those SCE values are consistent with the ones found in the literature and field data in Singapore. [START_REF] Kannan | Gas fired combined cycle plant in Singapore: energy use, GWP and cost-a life cycle approach[END_REF] refers to a combined power plant with a SCE of 437 kg/MWh while the official energy statistics estimates the total emissions at 431 kg/MWh [START_REF] Dupuis | Comparing Costs and Generation Mix Alternatives for the Application of the Clean Power Plan in the PJM Electricity Market: Application of Unit Commitment[END_REF]. They also comply the references concerning power systems studies [START_REF] Deng | What is the Cost of Negative Bidding by Wind? A Unit Commitment Analysis of Cost and Emissions[END_REF] but remain higher than the best performances advised by the manufacturersdown to 325 kg/MWh [START_REF] Pickard | The future role of fossile power generation[END_REF].

Unit A Unit B Unit C Unit D Unit E

Off-Design Performance

Fig. 3 shows the results obtained while running the offdesign model for Unit A. Especially the operating variables variations are consistent with the considered control strategy [START_REF] Kim | Comparative analysis on the part load performance of combined cycle plants considering design performance and power control strategy[END_REF]. TET remain constant while the air mass flow and the compressions ratio decrease to reach load points below 100 % of the nominal load (Fig. 3a). Once the lower bound for ma is met (0.7), TET starts decreasing allowing the unit to work a lower output powers. Consequently, the CC efficiency drops around 20% for the unit works at 30 % of the nominal load. That strong decrement then justifies the consideration of part load performances when estimating the CO2 emissions. The same observation can be made concerning the SCE which increase significantly when the output power decreases (Fig. 4a). Plotting the SCE allows ranking the units from the dirtiest (Unit B) to the cleanest (Unit A). Multiplying the specific emissions by the corresponding output power for the units allows estimating the net emissions in tons. Note that the emissions for Unit B are significantly lower as the rated power is less important than for the other technologies (250 MW Vs 400 MW). Fig. 4b highlights the fact that the emissions Ei for for the i th unit can be expressed with a linear formulation (17) that will be used as an input parameter of the UC problem in the next section. Table V gives the values for the different ai, bi obtained with regression coefficients greater than 0.99.

i des CC CC i CC i b W W a W E    , ) (  

CCGT units Operation

Classical unit commitment aims at minimizing the cost of power generation while meeting the load and fulfilling the operating constraints [START_REF] Sheble | Unit commitment literature synopsis[END_REF]. The outputs are the schedules for a set of units U all along a time horizon T. Typical constraints for thermal units refer to the ramping limitation and minimum up/down times TUi, TDi defining the minimum duration for operating or non-operating phases [START_REF] Van Den Bergh | A Mixed-Integer Linear Formulation of the Unit Commitment Problem[END_REF]. More accurate approaches for thermal units consist in introducing the different phase for the operations as well as discriminating the start-up types [START_REF] Simoglou | Optimal Self-Scheduling of a Thermal Producer in Short-Term Electricity Markets by MILP[END_REF]. When a CCGT unit is committed it first enters in a synchronization phase where the turbine shaft reaches the synchronization speed with the grid frequency (Fig. 5). Then a soak phase occurs with the output power that increases until the minimum operating point. The dispatchable phase then starts with the i th unit power flowing within the range Pmin,i and Pmax,i. When the unit is decommitted the power then returns to zero during the desynchronization phase. In this work both synchronization and soak phase are combined as the only available duration is the total start time (dotted line model in Fig. 5). The start-up time tends to increase with the prior reservation time during which the unit is turned off. Three start-up types l = hot, warm, cold are then commonly considered with associated duration Ts l i. Table 6 gives the operating variables for the considered units. Note that with a daily UC cold start (after more than 120 h off) is not considered. The load is considered with a 30 min discretization (Δt = 30 min) based on the data for the whole year 2016 [START_REF]Singapore Energy Statistics[END_REF]. With such a time step the ramp constraints do not need to be considered. It appears that the studied units can fully operates on their whole range during only one increment of time. Also the duration of the desynchronization phase is neglected as they are usually lower than the start-up phase time [START_REF] Macak | Evaluation of Gas Turbine Startup and Shutdown Emissions for New Source Permitting[END_REF].

Problem Formulation

A great number of methodologies are commonly considered to solve UC problems from priority listing to heuristics such as interior points or genetic algorithms [START_REF] Padhy | Unit Commitment-A Bibliographical Survey[END_REF]. In the present study the MILP is considered. That method gained great important in the past decades with the solvers improvement. The problem is formulated in MATLAB using YALMIP [START_REF] Lofberg | YALMIP: a toolbox for modeling and optimization in MATLAB[END_REF] and is solved using CPLEX 12.7.1. As already mentioned the objective of the UC here is to meet the load with the minimum amount of CO2. The objective function is expressed as in [START_REF] Powell | A Fast Algorithm for Nonlinearly Constrained Optimization Calculations[END_REF] with the output power of the i th unit at time t Pi,t and the on/off status ui,t. An exhaustive nomenclature of the symbols used in the UC is given in Table 7.

                T t U i t i i i max t i i t u b P P a obj , , , Min :  
Note that the start-up costs (in tons of CO2) don't appear in the function as it is commonly done in such studies [START_REF] Yamashita | Trade-off Analysis of CO2 versus Cost by Multi-objective Unit Commitment[END_REF]. The main reason is that the information is not available for the considered units. The models previously developed are used to estimate the start-up emissions while extending the off-design operations a lower values of the delivered power (Fig. 6). Especially the SCE drastically increase to reach values more than five time greater than the ones observed at nominal operating points (Fig. 6a). With a non-null duration for the start-up phase the corresponding CO2 cost is computed when considering the output power trajectories from 0 to Pmin,i (Fig. 5). Consequently, and as expected, warm start-ups will be costlier with a longer duration. Then the first set of constraints attached to the UC problem describes the start up type selection based on the reservation times bounds T d l and T u l prior the start up. T u l = 8 h for hot start and T u l = 24 h for warm start. T d l = 0 h for hot start and T d l = 8 h for warm start. The constraints in [START_REF] Kannan | Gas fired combined cycle plant in Singapore: energy use, GWP and cost-a life cycle approach[END_REF] allows selecting the start-up type depending on the prior reservation time and ( 20) ensure that only one type is selected among S [START_REF] Sheble | Unit commitment literature synopsis[END_REF]. The unit immediately enters in start-up phase when vi,t is set to 1 and ( 21) and [START_REF] Simoglou | Optimal Self-Scheduling of a Thermal Producer in Short-Term Electricity Markets by MILP[END_REF] ensure that only the appropriate start phase is considered. The power delivered by the unit i in start-up phase l at time t is computed by multiplying the step power P l step,i by the duration of the phase before t (( 23) and ( 24)). 25) and ( 26) introduce the bounds for the units power considering both dispatchable and start-up phases while ( 27), ( 28) and (29) represent the typical tight formulation for the minimum up/down time and power balance [START_REF] Van Den Bergh | A Mixed-Integer Linear Formulation of the Unit Commitment Problem[END_REF]. The last constraints refer to logical operations of the units that cannot start-up or shut down at the same time or have to operates either in patchable or start-up phase. ((30), ( 31) and ( 32)) [START_REF] Van Den Bergh | A Mixed-Integer Linear Formulation of the Unit Commitment Problem[END_REF].

     d l u l T t T t t i l t i w v 1 , ,      S l l t i t i v v , ,       t Ts t l t i l t i l v us
t i t i i min t i P Ps P ud , , , ,      t i i max t i t i Ps P ud P , , , ,          t TU t t i t i i v u 1 , ,        t TD t t i t i i w u 1 , , 1   t U i t i D P    ,   0 , , , 1 ,      t i t i t i t i w v u u   t i t i t i ud us u , , ,     1 , ,   t i t i w v  

Obtained Results

At first the simulation is performed without considering the constraints for the start-up phases. Before commenting the results, it is necessary to recall the units ranking from the cleanest to the dirtiest. That ranking is establishes by plotting the specific carbon emissions for the different technologies in Fig. 4 -1) units A, 2) units C, 3) units E, 4) units D and 5) units E. The obtained results confirm that the cleanest units (A and C) are run in priority all along the day while the dirtiest (D and B) are much less solicited (Fig. 7). Without considering the start-up phases some units displays multiple starts and stops during the simulated day which is unrealistic and would dramatically increases the costs of generation (Fig. 7a and Fig. 7c). Introducing the start-up constrains solves the problem by avoiding multiple commitments within the same day for the dirtiest units as shown on Fig. 8. It can be noticed that the base load is still fed by the cleanest units while the peak load is meet with the commitment of units D in PLP and E1 in SRY (Fig. 8b and Fig. 8d). Note that the dirtiest units B in KPL are not used at all. The simulation allows computing the grid emission factor (GEF) which represent the amount of CO2 per kWh generated. The obtained values is 0.406 kg/kWh. That is consistent with the recorded GEF of 0.431 kg/kWh for the year 2016 in Singapore which also considers grid losses as well as 5 % of the electricity mixed coming from oil steam plants [START_REF]Singapore Energy Statistics[END_REF].

A 1 & A 2 B 1 & B 2 C 1 C 1 A 1 D 1 D 2 C 1 & C 2 E 1 E 2
Computational Performance This subsection brings some comments about the computational times for the simulations. As already mentioned the problem model is formulated in MATLAB using YALMIP [START_REF] Padhy | Unit Commitment-A Bibliographical Survey[END_REF] and is solved using CPLEX 12.7.1 with 16 threads in parallel. The computer has a RAM of 16 GB with a 3.2 GHz processor. Fig. 9 plots the computational performances (time, numbers of variables and equations) with an increasing number of units considered in the power dispatch while adapting the level of load demand to meet. Without considering any start-up phase the computational time remains very low (Fig. 9a). Introducing start-up constraints significantly increases the number of variables leading to more complex problems to solve. An arbitrary limit of solving time is set to 5 min and is reached for 9 units or more. The attention is then paid to the gap value to estimate the optimality of the return solutions as the error between the solution of the relaxed linear problem and the MILP solution (9 units, 1.3 % for 10 units, 0.7 % for 12 units, 0.4 % for 12 units, 0.6 % for 13 units). The observed times are much greater than the one advised in for the same type of studies and up to 180 units [START_REF] Sheble | Unit commitment literature synopsis[END_REF]. Here the complexity comes from the multiple equivalent optimal solutions. The non-unicity of the solution is obvious with identical units in game. Also, as observed in [START_REF] Delarue | Enhanced priority list unit commitment method for power systems with a high share of renewables[END_REF], the load level here does not require the commitment of all the units at any moment which hardens the solving by increasing the number of equivalent nodes in the branch and bounds MILP algorithm.

THIGHTENED FORMLATION

Motivations

The environmental power dispatch described in the previous sections aims at being integrated in wider power systems studies. Grid constraints might be considered as well as expansion planning problems involving energy storage systems (ESS), renewables or controllable loads [START_REF] Alnaser | Optimal sizing and control of energy storage in wind power-rich distribution networks[END_REF]. Thus the number of equations and variables would tend to increase and might lead to prohibitive computational times. With the relatively simple previous problem important solving times are observed with gap values around 1 % after 5 min of resolution (arbitrary limit). Thus simplifications should be made in order allow the computation of more complex studies. Especially the attention is paid to the reduction of variables and constraints to described the afore mentioned power dispatch.

From Units Models to Plants Representation

The main simplification lies on the aggregation of the different units within the same power plant. Then the power dispatch will consider four power plants instead of thirteen independent units. The cost function (i.e. the CO2 emissions) of each plant i takes account of the ak, bk coefficients for the corresponding individual units. A traditional priority list (PL) algorithm is used to determine the commitment of every unit depending on the total power delivered by the plant -with the units ranked from the cleanest to the dirtiest (Fig. 4). The main limitation of such PL approaches is that they do not consider any runtime and start up phases constraints [START_REF] Quan | An improved priority list and neighbourhood search method for unit commitment[END_REF]. Here those constraints are considered only for the first unit switched on when the plant is committed. Eventually the emissions cost function for the plant displays the shape on Fig. 10. The steps occurring when additional units are committed implicitly correspond to the attached start-up costs.

Formulation

The MILP formulation has to be modified in order to take the afore mentioned simplifications into account. The considered emissions function for each plant is similar to the traditional piecewise linear cost curve for generator in UC [START_REF] Carrión | A computationally efficient mixed-integer linear formulation for the thermal unit commitment problem[END_REF]. However, the discontinuities occurring when additional units are committed require the introduction of additional binary variables and specific constraints. A list of the added symbols (variables and parameters) is given in Table 8. The first added set of constraints ensure that the plant only works in one specific interval. Eq (33) and (34) allows applying the appropriate bounds while (35) allows identifying the operating interval if the corresponding plant i is committed (ui,t = 1). The power delivered in the specific operating interval equals to the power generated by the power plant Pi,t (36). Note that in the present modified formulation i refers to the different plants and not the units as before.

 t i k P , ,  Coefficients α 2,i , β 2,i Coefficients α 1,i , β 1,i Coefficients α k,i , β k,i Start-up Unit 1    i K k t i t i k u u 1 , , ,       i K k t i t i k P P 1 , , ,   
The objective function in ( 18) is modified and consists in summing the emissions for all the intervals of every plants (U = 4 plants here) (37). The objective function in ( 18) is modified and consists in summing the emissions for all the intervals of every plants (U = 4 plants here) (37). The slope coefficients αk,i are equivalent to the ones associated to the operating units in interval k (ak). The coefficients βk,i need to be computed by iteratively including the bk coefficient of the committed units with Pmax k,i the maximum power of unit k in plant i (38). Note that the problem simplification is enhanced by considering only one type of start-up when plants are switched on.

                     T t U i K k t i k i k d i k t i k maxk,i i k i t u P P P obj , , , , , , , :                1 1 , k j j j k i k b a b   

Simulations

Once more the tightened problem is implemented using YALMIP. The formulation displays a reduced number of variables and constraints compared to the previous problem with multiple start up types and thirteen independent units (Table 9). Especially the number of constraints is divided by four. The problem is run for a hundred of random days taken from the Singapore load profile for the year 2016 whose  characteristics are presented on Fig. 11. The daily profiles are consistent all along the year with very low standard deviations observed. Also note that the base load is important representing more than 70 % of the peak values. To enhance the diversity of test case the load demand Dt is adapted for every run as in (39). A uniform distribution U randomly adjusts the load level between 30 % and 100% of the maximum value for the considered set of units (maximum of 4.9 GW here).

    1 , 3 . 0 max U       t t U i K k max k,i t D D P D i  
The tight problem performance is compared to the one for the simple model where no start-up phase is considered with thirteen independent units. The multiple-start up types problem is used as a reference case to compute the average relative error over the runs. The average computational time to reach the convergence (gap below 10 -6 ) with the tight model is significantly reduced to 30 s even if it remains longer than the simple model computation (4s previously observed). The relative error (average after 100 runs) is very low below at 0.69 %. That value can be explained by the relatively low contribution of start-up emissions compared to the CO2 generated in dispatchable phase. For instance, for Unit A a hot start-up would generate 16 tons of carbon dioxide which represent 20 % of the hourly emissions at nominal point (154 tons/h). As expected the deviations with the reference problem tend to increase with higher numbers of start-ups in the considered commitment profiles but remains below 1 % (Table 10). Multi start model Tight model Fig. 12 shows the results obtained while running the complete multi-starts and the tight models. For the tested day the computed error is only 0.3 % even if the plant generation profiles can display strong deviations. For both Keppel and Pacific Light Power plants the two solutions are equivalent with the cleanest units working at full load all along the day (units A and C at 2×400 MW). For the tight formulation Sembcorp generation is limited to the power produced by the cleanest units on site (units A at 2×400 MW). The surplus of energy is provided by the units E in YLT power where it is provided by units D in Sembcorp plant while running the multiple start-up types model. At the end the result in term of CO2 emissions are close as the performance of units D and E are very close (Fig. 4).

CONCLUSIONS

The CCGT model described in this paper aims at estimating the CO2 emission coefficients that are used in an environmental UC. Compared to conventional UC studies with CO2 costs taken from the literature, the methodology developed here allows the representation of specific units. Especially the start-up emission doesn't have to be guessed and is estimated by introducing the start-up phase with a specific duration. Despite the sparsity of the information available from unit manufacturers, the calibrated models display very good agreement with real data in terms of GEF. Finally, a tightened formulation of the power dispatch allows reduced computational times while maintaining the same performance compared to a multiple start-up types model. With the individual power plants considered as aggregations of different units that approach is appropriate if grid constraints would have to be considered as well as loss computation. Indeed, the global objective of the study is to model the power system management on Jurong Island in Singapore. Further work should then focus on the grid modeling that would lead to additional operating constraints to be fulfilled in UC.

Fig. 1 .

 1 Fig. 1. Simplified diagram of a typical CCGT unit

Fig. 3 .Fig. 4 .

 34 Fig. 3. Performances of Unit Aa) Operating veraiblesb) efficiencisies

Fig. 5 .

 5 Fig. 5. CCGT unit operation Approximate Operation

Fig. 6 .

 6 Fig. 6. Extended Performancesa) SCEb) CO2 emissions 2500

Fig. 7 . 1 Fig. 8 .

 718 Fig. 7. UC without start-up phasesa) KPLb) PLPc) SMBd) SRY

Fig. 9 .

 9 Fig. 9. Computation infoa) no start-up phaseb) with start-up phases

Fig. 11 .

 11 Fig. 11. Singapore Electrical Demanda) Week daysb) Week ends

Fig. 12 .

 12 Fig. 12. Tight model performancea) KPLb) PLPc) SMBd) SRY

Table 1

 1 Nomenclature for the CCGT model

	Ta	ambient temperature	O C
	rc	compression ratio	-
	nc	compressor isentropic efficiency	-
	nm	mechanical efficiency	-
	ma	air mass flow	kg/s
	γc	air specific heat ratio in compression	-
	Wc	compressor consumed power	MW
	Cpa	specific heat of air	kJ/kg/K
	ncc	combustion chamber efficiency	-
	λ	air/fuel ratio in combustion chamber	-
	LHV	fuel lower heat value	kJ/kg
	TIT	turbine inlet temperature	O C
	mf	fuel mass flow	kg/s
	mg	flue gas mass flow	kg/s
	Cpg	specific heat of flue gas	kJ/kg/K
	γe	air specific heat ratio gas turbine expansion	-
	ngt	gas turbine isentropic efficiency	-
	TET	turbine exhaust temperature	O C
	WGT	shaft work of the gas turbine	MW
	WGTnet	gas turbine net output power	MW
	QGT	heat supplied to the gas turbine	kJ/s
	ngen	electrical generator efficiency	-
	ηGT	gas turbine efficiency	-
	QAV	available heat in the flue gas	kJ/s
	WSTnet	steam net output power	MW
	WCC	combined cycle net output power	MW
	ηCC	combined cycle efficiency	-
	SCE	specific carbon emission	kg/MWh
	.,des	subscript for design values	-
	for HSRG display up to three different pressure systems with
	corresponding low, intermediate and high pressure steam
	turbines. Thus the combine cycle (CC) units allow exploiting
	the available heat in the exhausts gas of heavy duty and
	enhancing the power generation efficiency from 40 % (ηGT) to
	60 % (ηCC)

Table 2 Identified units on site Unit A Unit B Unit C Unit D Unit E

 2

	WGTnet (MW)	240.0	165.0	260.0	256.0	292.0
	WSTnet (MW)	160.0	85.0	140.0	152.0	115.0
	WCC (MW)	400.0	250.0	400.0	408.0	407.0
	TET ( O C)	630.0	505.0	580.0	596.0	580.0
	ηGT (%)	37.0	36.4,	38.5	37.8	40.7
	ηCC (%)	32.0	53.1	58.8	56.7	59.0

  Inputs/Outputs of the CCGT model

	Fixed parameters		T a LHV W STnet
	Operating	variables	r c m a TIT		TET SCE
	Design Variables	n c n cc γ e n gt n m	Simplified CCGT model	W GTnet W CC η GT η CC
			n gen

Table 3

 3 Model validation

		Test Unit	Model	Error
	WGTnet (MW)	258.7	261.2	2.8 %
	WSTnet (MW)	404.2	406.2	0.9 %
	TET ( O C)	617.5	408.0	0.4 %
	ηGT (%)	37.0	38.0	2.7 %
	ηCC (%)	57.8	56.8	1.7 %

Table 4 Design results Unit A Unit B Unit C Unit D Unit E

 4

	rc	30.85	15.07	15.18	16.05	18.54
	ma (ks/s)	639.02	595.29 712.02 735.24 700.47
	TIT ( O C)	1390.00	1083	1261.2 1274.2	1296
	nc	0.95	0.91	0.8	0.83	0.9
	ncc	0.98	0.99	0.98	0.98	0.98
	γe	1.29	1.31	1.3	1.32	1.29
	ngt	0.83	0.89	0.94	0.9	0.91
	nm	0.99	0.97	0.99	0.96	0.99
	ngen	0.99	0.99	0.99	0.97	0.99
	WGTnet (MW)	239.99	165.00 259.99 256.00 291.98
	WSTnet (MW)	399.99	250.00 399.99 408.00 406.98
	TET ( O C)	630.00	504.99 580.00 596.00 579.99
	ηGT (%)	37.47	36.4	38.74	37.8	41.87
	ηCC (%)	61.20	53.10	58.42	56.70	57.20
	SEC (kg/MWh) 392.21	457.33 411.78 424.53 421.19

Table 5

 5 Linear coefficients for CO2 emissions

		Unit A	Unit B	Unit C	Unit D	Unit E
	ai (in tons/h)	126.56	94.00	135.16	141.51	143.34
	bi (in tons/h)	27.67	19.13	28.00	29.59	26.94

Table 6

 6 Operating parameters

		Unit A Unit B Unit C Unit D Unit E
	Ramp (MW/min)	50	30	10	22	10
	TUi , TDi (min)	60	120	120	90	120
	Ts hot i (min)	30	60	60	30	60
	Ts warm i (min)	90	120	120	90	120
	Pmin,i (MW)	80	75	120	142	122
	Pmax,i (MW)	400	250	400	408	407

Table 7

 7 Nomenclature for the UC problem

	ui,t	unit i committed at time t
	usi,t	unit i in start phase at time t
	us l i,t	unit i in start phase type l at time t
	udi,t	unit i in dispatchable phase at time t
	vi,t	unit i started at time t
	v l i,t	unit i started type l at time t
	wi,t	unit i shut down at time t
	Pi,t	power generated by unit i at time t
	Psi,t	power generated by unit i in start phase at time t
	ai , bi	coefficients for the CO2 emissions of unit i
	Pmin i	minimum power of unit i in dispatchable phase
	Pmax i	maximum power in dispatchable phase of unit i
	TUi, TDi minimum up/down time of unit i
	Tsl	start up type l duration
	Dt	demand load at time t
	P l step i	power increment of unit of during start up type l
	T d	l, T u l	bounds for start up type l intervals
	T, U, S	set of time steps, units and start-up types

Table 8

 8 Added symbols in the tightened problem

	Pδk,i,t	power in interval k of plant i at time t
	uδk,i,t	plant i operating in interval k at time t
	αk,i, βk,i	Emissions coefficients of interval k of plant i
	Pδ d k,i,, Pδ u k,i	lower/upper limits of interval k of plant i
	us l i,t	unit i in start phase type l at time t
	Ki	number of units in plant i

Table 9

 9 Tightened model performance

		Multiple Start-up Model	Tight Problem
	Continuous variables		1248		384
	Binary variables		5616		960
	Constraints		8867		2060
	CPU Time		> 300s		30s
	Error (%)		-		0.69
	Number of starts	1	2	3	4	≥ 5
	Error (%)	0.28	0.62	0.73	0.81	0.84

AKNOWLEDGEMENT

The authors acknowledge the support by the Singapore National Research Foundation under its Campus for Research Excellence And Technological Enterprise (CREATE) program and the Cambridge Centre for Advanced Research in Energy Efficiency in Singapore (CARES).