domain: spi phys

text:
D Lasluisa

D Salas

N Mazet

D Aussel

Multidimensional analysis for the techno-economic study of the CSP plant

Keywords: Concentrated solar power plant, Thermochemical storage, SPOT market, Economical indicators, Black-box model, Optimal design

The profitability of a concentrated solar plant depends on the type of storage system and its daily operation. In addition, this profitability can be evaluated using different economical criteria such as the minimum time to recover the initial investment, the total profit at the end of the project and others. Given a solar field and a power block, the objective in this work is to determine an optimal design of the storage system for the concentrated solar plant. Two types of storage technology are considered, namely Thermochemical reactor or Two-tank molten salt system, two different production strategies, four different economical criteria and three price scenarios for the electricity market. And for each such combination (called profile), the model determines the optimal design of the concentrated solar plant. But, contrary to classical economical evaluation in literature, in this approach an optimal design consists of the optimal values of the physical design of the storage system but also the optimal operations of the plant (Production, Storage, Discharge). Thanks to this model and the results obtained, conditions can be exhibited under which thermochemical storage is economically more pertinent than Two-tank storage and also situations ensuring an economical equilibrium of the concentrated solar plant without subsidy.

Introduction

During the last decade, the use of renewable energy sources has become very popular due to the environmental problems caused mainly by the use of fossil energy sources. Therefore, the solar energy market specifically has grown rapidly and has great potential to expand in the coming years [START_REF] Khamlich | Technical and economic assessment of thermal energy storage in concentrated solar power plants within a spot electricity market[END_REF].

Unlike other renewable energy systems, CSP plants allow the incorporation of storage systems for electricity production, which store heat in materials that change temperature (sensible), phase (latent) or chemical composition (Thermochemical) [START_REF] Jacob | Technoeconomic impacts of storage system design on the viability of concentrated solar power plants[END_REF][START_REF] Zhang | Thermal energy storage: Recent developments and practical aspects[END_REF] and numerous studies have been made on the implementation of these storage technologies (see [START_REF] Khamlich | Technical and economic assessment of thermal energy storage in concentrated solar power plants within a spot electricity market[END_REF][START_REF] Jacob | Technoeconomic impacts of storage system design on the viability of concentrated solar power plants[END_REF][START_REF] Liu | Techno-economic feasibility of solar power plants considering PV/CSP with electrical/thermal energy storage system[END_REF] and references therein). Incorporating thermal en-ergy storage (TES) into a CSP plant allows to manage the mismatch between energy supply and demand [START_REF] Zaversky | Transient molten salt two-tank thermal storage modeling for csp performance simulations[END_REF]; it allows to shift electricity production to periods of higher prices, providing greater profit opportunities to the system; and finally, the reliability and flexibility of a plant increases by improving the quality of the provided services [START_REF] Khamlich | Technical and economic assessment of thermal energy storage in concentrated solar power plants within a spot electricity market[END_REF]. One of the most used systems today is the Sensible Heat Storage (SHS) that uses a storage of molten salt in two tanks. Its high implementation is due to the fairly economical and efficient storage medium. On the other hand, a storage system that has not yet been implemented industrially is Thermochemical storage, which is based on reversible reaction heat storage. The advantages of this type of system are the ability to store energy for longer periods of time, its high volumetric energy density compared to the thermal power used by the first thermal integration in Discharge phase [MW] q 2 effective thermal power dissipated in condensation process of the Thermochemical storage [MW] q 3 thermal power dissipated during the cooling process of the power block [MW] q D average thermal power released by the reactor during Discharge phases [MW] q R thermal power consumed by the power block [MW] q S average thermal power consumed by the reactor during Storage phases [MW] q dis thermal power to be dissipated in condensation process of the Thermochemical storage [MW] q u dis fraction of q dis recovered by the second thermal integration in Storage phase [ other technologies, its weak thermal losses and its operation within wider temperature ranges.

The profitability of a solar power plant depends largely on the storage system. According to the characteristics and functionalities of the storage sys-tems that will be considered in this work, our main objective is to optimize the dimensions and operations for each of these systems. To evaluate the profitability and viability of a CSP plant together with a storage system, many studies consider the economic indicator LCOE, see e.g. [START_REF] Prieto | Techno-economic analysis of a concrete storage concept for parabolic trough solar power plants[END_REF][START_REF] Jacob | Technoeconomic Impacts of Storage System Design on the Viability of Concentrated Solar Power Plants[END_REF][START_REF] Prieto | Thermal energy storage with phase change materials in solar power plants. Economic analysis[END_REF] with different thermal energy storage systems including Molten salt, concrete and phase change material. However, in a Spot market context, it is not enough to consider this type of simple indicator, since it does not take into account incomes, which can also be different depending on the type of storage system and production strategy. Hence the need to consider other economic indicators to evaluate a CSP plant project that may vary according to physical and economic conditions. In this work, the prices of electricity (costs and incomes) which will be considered are the ones of a Spot market and they will vary between hours, days of the week, seasons. Therefore the operation of the plant can change its economic benefit. Consequently, in addition to the LCOE indicator, in this work we consider indicators such as: NPV (Net Present Value), IRR (Internal Rate of Return), and CPB (Conventional Payback), since they capture both the production capacity of the plant and the impact of production strategies [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF][START_REF] Goyal | Financial feasibility of concentrated solar power with and without sensible heat storage in hot and dry Indian climate[END_REF]. Another consideration to be analyzed is the variation of electricity market prices, which will influence the income and costs generated by the plant, i.e., particularly according to the economic indicator under consideration, the economic benefit will be slightly or strongly affected. The previously described paradigm gives way to different optimization problems, each one involving two distinct sets of variables that are optimized simultaneously: the physical variables corresponding to the type of storage, and the operational/strategic variables, which define Storage/Production strategies according to the desired objective. Because some of the variables correspond to the operational use of the storage (and are therefore functions), the maximization (or minimization) problem is actually an optimal control problem. Several optimal control modelling has been already proposed in the literature. For example in [START_REF] Cirocco | Maximising revenue via optimal control of a concentrating solar thermal power plant with limited storage capacity[END_REF], the authors propose an optimal control problem corresponding to a simplified modelling of a CSP plant, with the objective of maximizing revenues, considering the variation of the solar irradiation and market prices. Also in [START_REF] Casella | Optimal operation of solar tower plants with thermal storage for system design[END_REF], some advanced optimal control techniques have been studies for the sizing evaluation but only with Two-tanks storage.

However, in some cases, solving an optimal control problem could be quite difficult. Moreover, due to the discontinuous nature of some functions, computational difficulties may arise. Therefore, in this work the notion of pre-scenarios, formulated in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF], will be used and will allow to transform the optimal control problem into a "classical" mathematical programming problem.

Let us observe that it is not the first time that optimal sizing or optimal dispatch are analysed for CSP plants with storage. Nevertheless, in most of these studies the model is simplified to be linear, see e.g. [START_REF] Sioshansi | The value of concentrating solar power and thermal energy storage[END_REF] and references therein, while our aim is to fully take into account the nonlinear (and more complex) behaviour of the thermal phenomena involved in the thermal storage systems.

The resolution of the optimization problems that will be developed in this work follows the alignments proposed in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF], these are: a prototype solar thermal plant is considered, which will be optimized over a lifetime of N years. During this time, each year is assumed to behave exactly the same with respect to the others, so only one year is optimized, and then repeated N times. This repetition takes into account an economic devaluation, which is captured by a discount rate. The model year is divided into periods (e.g. when divided into four, each would represent a season) and each period is represented by a cyclical stage (a week in general), which is repeated. The optimization of the model year consists of deciding the physical variables of the storage and the operation of the solar thermal power plant in each cyclic stage in order to maximize (or minimize) the economic criterion. So, in a first instance we will proceed in the creation and implementation of an algorithm for the maximization (or minimization) of the economical indicators for the prototype solar thermal power plant.

Thus doing so the main aim of this work is to carry out a comparative study, considering different integrated storage systems, economic criteria and operation strategies, each configuration being optimized. Following the terminology of "cooptimization" which is sometimes used in the community of energy design, the results will correspond to both the optimal sizing of the storage systems together with the optimal operation of the plant (Storage, Production and Discharge), depending on the hypotheses and objectives sought. To our knowledge, it is the first time in literature that such a comparative study between Two-tanks storage and Thermochemical storage is driven in a so general context considering so many different configurations (economic criteria, price scenarios, operations strategies...).

The paper is organized as follows. Section 2 aims to expose the material and methods used in this work.

It is composed of two subsections, the first one dealing with the black-box model representing a CSP plant and the second one describing the methodology used to structure the comparative analysis. In Section 3, the formulation of the optimization problem to be solved is detailed, giving a description of the dynamics of each of the storage systems under consideration, the composition of the objective function and the functionality for each of the variables that make up the problem. Results and discussions will be presented in Section 4 and finally a conclusion is given in Section 5.

Material and Methods

Following a black-box approach, in the first subsection, we show how the CSP plant can be decomposed into three main components. Then, in Subsection 2.2 we present the methodology to carry out a comparative study of the profitability and feasibility of a CSP plant, considering different economic criteria, production strategies and storage systems.

Note that an important feature in our approach is the use of prescenarios which allows to transform the control problem into a classic optimization problem.

Black-box model

The model of the daily operation of a CSP plant has been divided into three black boxes, which represent respectively the solar field and Receiver, the Storage system and the power block or Rankine cycle (see Figure 2.1). Their interaction are modeled via a quasi-static model. The operation of the plant is divided into four phases, which are the Inactive phase, the Storage phase, the Storage-Production phase and the Discharge phase, and it is assumed that the transition in between them is instantaneous (the transient phases are not considered except for the start-up of the power block). The Storage-and-Production phase occurs within the time interval of the Storage phase, that is the Storage-and-Production phase start after the beginning of the Storage phase and ends before the end of the Storage phase. Then, according to the available stored heat energy, discharge phases can be carried out. The modeling of these phases will be done in the forthcoming sections, whose representations will be different depending mainly on the type of storage system. For the comparative study, we will consider different technologies for the storage system, while the other components will remain fixed and interact with each technology. In subsections 2.1.1-2.1.2, the equations representing the operation of the solar field and power block components are presented. In addition, we provide several appendices containing the full description of our model. The main contributions and analysis of our work can be read without consulting this material. However, the content presented there might be useful for the reader interested in replicating our results, or in going further into modeling aspects.

The Solar Resource and the solar field

The solar field is composed of a set of heliostats, whose function is to project the sun's rays to a specific point or sector called a receiver at the top of a tower, heating it up to approximately 565 °C [START_REF] Gould | Solarreserve's 565 mwt molten salt power towers[END_REF]. The intensity of the sun's rays depends on the place where the plant is installed For example, Figure 2.2 shows the irradiation of an average day in the summer season in California/USA. This curve is expressed by the Direct Normal Irradiance function, DN I(•), which depends on the time and the mirror field. Thanks to this curve, we model the useful thermal power function q sol (•), whose expression represents the recovered thermal power from the solar field and considers the thermal losses at receiver. This is,

q sol (t) =K Hel • DN I(t), ∀t ∈ [0, H], (2.1)
where K Hel is an efficiency constant depending on the thermal efficiency of the receiver, the optical efficiency of the mirrors field and total surface of the heliostats (see [START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF]). Here, [0, H] is a time interval (measured in hours). Now, with the aim of establishing a curve that represents the average variations in climate over a specific period, we define a new function q u of the hourly average values of useful thermal power, which are simulated using SAM software. Thus, an approximation scheme is done with the polyfit function in MATLAB. The function of useful thermal power is

∀t ∈ [0, 24], q u (t) = p(t) if t ∈ [t sol ini , t sol f in ], 0 if not. (2.
2) with the interpolation polynomial p(•) of the average values and the times of sunlight, t sol ini , t sol f in .

The steam Rankine cycle or power block

The power block model has been reduced to a set of polynomial functions of degree 2 via quadratic regressions [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF]. These regressions were chosen to simplify the plant modeling of the different connections and functionalities [START_REF]System Advisor Model (SAM)[END_REF]. For the Rankine modeling, we mainly use three regressions, given by:

Ẇelec (t) = f b (q R (t), T 4 (t), T out (t)) = b 1 + b 2 T 4 (t) + b 3 T 2 4 (t) + b 4 T out (t) + b 5 T 2 out (t) + b 6 q R (t) + b 7 q 2 R (t) + b 8 T 4 (t)T out (t) + b 9 T 4 (t)q R (t) + b 10 q R (t)T out (t) (2.3) T 5 (t) = f c ( ṁ4 (t), T 4 (t), T out (t)) = c 1 + c 2 T 4 (t) + c 3 T 2 4 (t) + c 4 T out (t) + c 5 T 2 out (t) + c 6 ṁ4 (t) + c 7 ṁ2 4 (t) + c 8 T 4 (t)T out (t) + c 9 T 4 (t) ṁ4 (t) + c 10 ṁ4 (t)T out (t) (2.4) ṁ4 (t) = f f ( Ẇelec (t), T 4 (t), T out (t)) = f 1 + f 2 T 4 (t) + f 3 T 2 4 (t) + f 4 T out (t) + f 5 T 2 out (t) + f 6 Ẇelec (t) + f 7 Ẇ 2 elec (t) + f 8 T 4 (t)T out (t) + f 9 T 4 (t) Ẇelec (t) + f 10 Ẇelec (t)T out (t) (2.5)
For the full list of regressions, see Appendix D. For the case of thermochemical storage, these extra regressions model the integrations of the reactor and the Rankine cycle.

Methodology

For the development of the comparative study of the profitability and feasibility of a CSP plant, different economic criteria, operating strategies and storage systems will be considered. The choice of each of the above aspects leads to a different control problem. In Subsection 2.2.4, the transition from these control problems to classical optimisation formulations will be briefly explained. But before, subsections 2.2.1,2.2.2 and 2.2.3 introduce respectively the different economic criteria, the different production strategies and the different storage systems that will be considered throughout this comparative analysis.

Economic Indicator

When planning the construction of a plant, there are several points of view that can be used to "measure" the profitability of the project. For example, the minimum time in which the invested money can be recovered, the economic utility that is generated at the end of the lifetime, the unit value of the product, etc. For an overview of economic criteria and their interactions in the context of energy projects one can refer to [START_REF] Aussel | On the equivalence and comparison of economic criteria for energy projects: Application on pv/diesel hybrid system optimal design[END_REF][START_REF] Tsuanyo | Modeling and optimization of batteryless hybrid pv (photovoltaic)/diesel systems for off-grid applications[END_REF] Precise definition of these indicators can be found in [START_REF] Aussel | On the equivalence and comparison of economic criteria for energy projects: Application on pv/diesel hybrid system optimal design[END_REF], and also are detailed in Appendix C.

Production Strategies

In [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF], it was corroborated how the use of a production strategy for a CSP plant different from the classic one, can represent a higher profitability. This study was carried out considering the participation of the plant in a SPOT market. In this work three kind of production strategies will be considered , one of Classic strategies and two of Price Chasing. These are the following:

Classic production strategy: this is the typical strategy used in many plants around the world. It consists of, for each day k, doing a Storage phase and immediately after discharging all the stored energy until the system is empty.

Price Chasing operation strategy : whenever storage is implemented in a CSP, the classic production strategy is perfectly adapted to a situation where the produced energy is sold at a fixed price. But it leads to a low economical efficiency of the CSP, actually to the need of subsidies (see e.g. [START_REF] Khamlich | Technical and economic assessment of thermal energy storage in concentrated solar power plants within a spot electricity market[END_REF]). When the energy price varies, like in deregulated market context, an alternative approach which really takes advantage of the storage capacity is to prefer a production targeted to periods during which the produced energy can be sold at advantageous prices, namely the peaks of spot/deregulated electricity markets. Note also that weekly storage strategy can also take advantage of low demand days, Sunday typically, to to increase the discharge capacity during the week. Following [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF], this production strategy will be called «Price Chasing strategy». In order to explore more deeply the innovative Price Chasing strategy, we will here consider two different forms of it, one with one discharge per day and one with two discharges per day.

Storage Systems

In this work, two storage systems are considered, which are designed for the storage of sensible heat and chemical reaction enthalpy. The operation and composition of these systems will be briefly described below. In Section 3, the equations that describe the dynamics of these systems in the plant will be detailed.

Storage -indirect Two-tank molten salt

Sensible heat storage is a technology currently implemented in most CSP power plants around the world. They can store thermal energy for up to 15 hours using a heat transfer medium such as molten salt [START_REF] Liu | Review on concentrating solar power plants and new developments in high temperature thermal energy storage technologies[END_REF]. Molten salts have a high storage efficiency that allows sensible heat storage to produce electricity during the peak power demand after sunset [START_REF] Sioshansi | The value of concentrating solar power and thermal energy storage[END_REF]. In recent years, studies have been carried out to improve these types of systems [START_REF] Zaversky | Transient response simulation of a passive sensible heat storage system and the comparison to a conventional active indirect two-tank unit[END_REF][START_REF] Vannerem | Thermal cycle performance of thermocline storage: numerical and experimental exergy analysis[END_REF]. We consider here an indirect sensible heat stor-age system, composed of two heat transfer loop, a heat exchanger and two tanks for the storage material (low temperature and high temperature). The plants uses molten salt as storage material [START_REF] Guney | Solar power and application methods[END_REF]. The integration of such Two-tank molten salt storage is illustrated in Fig. 2.3 (see also [START_REF] Alva | An overview of thermal energy storage systems[END_REF]).

During daytime operation of the plant, the solar field concentrates the solar irradiation on the receiver. Then, the heat transfer fluid (HTF) passes through it and conducts the collected heat to the heat exchanger (nodes 3 and 6, in Figure 2.3). The heat is thus transferred from HTF to the molten salt (Storage phase), which flows from the cold tank, with temperatures not lower than 290 • C, to the hot tank, with temperatures not higher than 560 • C, where the heat will be stored. During the Storage-Production phase, one part of the HTF passes directly from solar field to the power block. For the Discharge phase, the molten salt flows from the hot tank to the cold tank and, by means of the heat exchanger, the stored heat is transferred to the HTF and finally is directed to the power block.

Storage -Thermochemical Reactor

Another type of heat energy storage is Thermochemical storage. It is not yet implemented in commercial CSP but its advantages in terms of heat storage and high temperature management promise its wide implementation in the near future. In [START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF], an economic evaluation was performed for CSP plant projects using an innovative Thermochemical system, and this work includes this storage system in the comparative study.

The Thermochemical storage is based on a process that involves a fixed-bed Thermochemical reactor implementing the calcium oxide/calcium hydroxide (CaO/Ca(OH) 2 ) and water (H 2 O) reactive pair. This storage is based on the association of two monovariant reversible transformations: a chemical reaction

CaO + H 2 O (g) Synthesis ⇌ Decomposition Ca(OH) 2 + ∆h • r (2.6)
and a liquid/gas phase change

H 2 O (g) Synthesis ⇌ Decomposition H 2 O (l) + ∆h • vap . (2.7)
where ∆h • r and ∆h • vap are respectively the standard molar enthalpy of reaction and vaporization. These transformations are defined by the thermodynamic operating conditions T c (constraint temperature) and P c (constraint pressure). The main advantages of CaO/Ca(OH) 2 Thermochemical storage are the intrinsic high energy density of this low cost and environmentally friendly reactive pair, and a non time-dependent energy storage capacity (as long as reactants are stored separately). Moreover, the power and temperature of the Storage/Discharge phases can be controlled not only by the mass flow rate and temperature of the heat transfer fluid, but also by the operating pressure of the reactor. In addition, these storage properties (power, energy density, ...) also depend on the physical characteristics of the reactor (dimension) and the way that the porous reactive solid is implemented in the reactor (apparent density, DEC, part of binder added to enhance heat transfer, ρ eng ). For more details, we refer to [START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF][START_REF] Thaker | Technoeconomic evaluation of solar-based thermal energy storage systems[END_REF][START_REF] Wu | Techno-economic analysis of an integrated liquid air and thermochemical energy storage system[END_REF].

From Control problem to the Optimization

Problem The aim of this study can be formally written as follows: given a location, a price curve, a solar field and a power block, the aim is to design the storage system (Two-tanks and Thermochemical) and its operational use in order to maximize (minimize) the economic indicator for the lifetime (N years) of the project. Due to the fact that some of the variables correspond to the operational use of the storage (and are therefore functions), this maximization (minimization) problem is actually an optimal control problem.

The variables of the control problem should be thus optimized in an time interval [0, H], with H being the number of hours of the plant project duration and defined as follows:

• ν -represent the vector of the physical variables that describe the storage system dimensions;

• µ -will represent the vector of operational variables composed of real-valued functions describing the operation of the storage system;

• τ -will stand the state of operation of the plant,

τ : [0, H] → {S, {S, P }, D}
describing the order and number of phases (Storage/Storage-Production/Discharge) for

each t ∈ [0, H].
Depending on the storage system, the physical and operation vectors will differ. For example, in the case of a Thermochemical storage system, the components of the vector of physical variables will correspond to the dimensions and amounts of chemical compounds for a Thermochemical reactor and the components of the vector of operations will correspond to the operating pressure and temperatures.

For the case of a storage with two molten salt tanks, the components of the vector of physical variables will correspond to the volume of the tank and dimensions of heat exchanger while the components of the vector of operation variables will correspond to the mass flows of the heat transfer fluid and molten salt, which are controlled by pumps.

The idea of Pre-scenarios, which has been introduced in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF], consists of restricting the "profile" of the operational and strategy functional variables in such a way that these restricted profiles can be described by vectors of real numbers.

In a pre-scenario, the functions fixing a certain operation (Storage/Production/Discharge) are defined on time intervals,

I S = [t S ini , t S f in ], I P = [t P ini , t P f in ] and I D = [t D ini , t D f in ].
Thus, along with some simplifying assumptions, such as that the Storage and Discharge phases occur at a constant power rate, the reformulated optimisation problem consists, for what concerns the operational variables, on deciding the size of the intervals of each of the phases, and the constant values of the operational values within these intervals.

In concrete: it will be assumed that all years have the same behavior in terms of market and climate, so it will only be necessary to optimize on one year (model year ). In turn, this year will be divided into stages (seasons). The ith season is represented by a stage S i of N days (i) days, that is imposed to be cyclic (the intial conditions coincide with the final ones). Also, the following considerations are taken into account: setting a stage S i and a day k ∈ {1, ..., N days (i)}, (a) The indices that will be representing the temporary variables, (t ini (i, j, k)) and t f in (i, j, k))), as well as other variables, for the different phases of Storage, Storage-Production, and Discharge, are:

• i : it represents the stage S, with i ∈ {1, ..., p}.

• k : it represents the day k of stage S i .

• j : it represents the order of the phase on day k.

(b) The number of Storage phases J S (i, k). For each j ∈ {1, ..., J S (i, k)}, the variables t S ini (i, k, j) and t S f in (i, k, j) are introduced as the initial time and final time of the jth Storage phase of the kth day of the stage S i .

(c) The number of Storage-Production phases J P (i, k). For each j ∈ {1, ..., J P (i, k)}, the variables t P ini (i, k, j) and t P f in (i, k, j) are introduced as the initial time and final time of the jth Storage-Production phase of the kth day of the stage S i .

(d) The number of Discharge phases J D (i, k). For each j ∈ {1, ..., J D (i, k)}, the variables t D ini (i, k, j) and t D f in (i, k, j) are introduced as the initial time and final time of the jth Storage phase of the kth day of the stage S i .

When a pre-scenario is fixed (that is when one decides about the number and order of storage/production/discharge phases), the admissible operations are restricted, considering only those which respect the pre-scenario strategy. Thus, these «pre-scenario type» operation functions can be fully described by real vectors/variables μ1 , ..., μp , τ1 , ..., τp , which are elements of a subset of the set of the continuous control variables µ and τ described above. Thanks to this representation, the natural control problem is then transformed into a "classical" optimization problem, with, among the variables to optimize, are the ones defining the initial and final time for each phase, see [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF].

Problem Formulation

This section is dedicated to the detailed structure of the optimization problem. In Subsection 3.1 we detail the equations governing each of the plant components. Then, the general methodoly to build the objective function, considering any economic indicator, is described in 3.2.

Finally, the constraints will be indicated in terms of the variables of the problem. Note that the complete optimization problems to solve is described in F.1 and F.2.

Modeling -Storage System

In order to reduce the notation of each variable and expression in the different phases of the plant, the sub-indices i, k, j will be omitted. Therefore, continuous equations are considered to represent the variation and evolution of temperatures, mass flows, powers, etc., for all t ∈ I = [t ini , t f in ]. Note that, for the Storage and Discharge phases occurring at a constant rate (see, [START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF]), it is sufficient that these expressions are verified for t = t ini , t f in .

3.1.1. Two-tank molten salt storage Here, the vector of physical variables ν is composed of two components, the volume of the tanks V [m 3 ], and the surface of the exchanger A exch [m 2 ] (see Figure 2.3). In our case we assume that the two tanks have the same volume, so only one variable is needed. Then, the vector of physical variables is simply ν = (V, A exch ).

The operation of the Two-tank system with molten salt is determined by the heat transfer fluid inside the heat exchanger, which is controlled from pumps that regulate mass flow rates. This exchange operates in a counter current mode. For the Storage phase, the fluid in the red circuit (see Figure 2.3) passes from node 3 to node 6, and the flow of molten salt passes through the exchanger from the cold tank (node 7) to the hot tank (node 8). In the case of the Discharge phase, the fluids pass through the same nodes in opposite directions. For this reason, we will differentiate the operational variables as follows:

• µ S (t) = ( ṁS 3 (t), ṁS salt (t), T 7 (t)) represents the operational variables of the mass flows during the Storage phase, where the molten salt ṁS salt moves from the cold tank (at temperature T 7 ) to the hot tank;

• µ D (t) = ( ṁD 3 (t), ṁD salt (t), T 8 (t)) represents the operational variables of the mass flows during the Discharge phase, where the molten salt ṁD salt moves from the hot tank (at temperature T 8 ) to the cold tank.

Therefore the vector of operational variables is defined as

µ(t) = ( ṁS 3 (t), ṁS salt (t), T 7 (t), ṁD 3 (t), ṁD salt (t), T 8 (t)).
Now, the physical and operational variables will allow to determine the thermal power involved in each of the different phases, and the amount of energy which is stored or discharged during a day of operation :

• q S (t) = q S (ν, ṁS 3 (t), ṁS salt (t), T 7 (t)) represents the storage power corresponding to heat transfer from the heat transfer fluid to the molten salt flow;

• q D (t) = q D (ν, ṁD 3 (t), ṁD salt (t), T 8 (t)) represents the discharge power corresponding to heat transfer from the molten salt to the heat transfer fluid. These two variables for the thermal powers, q S and q D , allow to calculate each operating mode. To simplify the modeling of the heat exchange, we assume that there is no heat loss. Therefore, the heat sent by the heat transfer fluid between nodes 3 and 6, and the heat received by the molten salt between nodes 7 and 8, is the same. This is represented by the following equations:

q S (t) = C htf ṁS 3 (t)(T 3 -T 6 (t)) (3.1a) q S (t) = C salt ṁS salt (t)(T 8 (t) -T 7 (t)) (3.1b)
where C htf , C salt are the heat capacity of the heat transfer fluid and molten salt, and ṁS 3 (•), ṁS salt (•) are the mass flow rates for each fluid. During the Storage phase, the CSP plant operates with a nominal temperature of T 2 = 560°C in the node 2. The thermal losses for the heat-transfer fluid are neglected and so, T 2 = T 3 , T 1 (t) = T 6 (t) and ṁ1 (t) = ṁ2 (t) = ṁ3 (t) = ṁ6 (t), for any t ∈ I S , as shown in Table 3.1. For the heat exchanger, each side of the exchanger has an average temperature, that is

Mass Flow

Temperature ṁ1 = ṁ2 = ṁ3 = ṁ6 T 3 = T 2 / T 1 = T 6
T htf (t) = T 3 + T 6 (t) 2 , and
T salt (t) = T 7 (t) + T 8 (t) 2 .
With the previous assumption, the thermal power inside the exchanger (which coincides with the exchanged heat q S ) is expressed as follows:

q S (t) = 1 2 hconv • A exch • ((T 3 + T 6 (t)) -(T 7 (t) + T 8 (t))), (3.2)
with h conv representing the convective heat transfer coefficient. Finally, the thermal power cannot exceed the available thermal power, leading to

C htf ṁS 3 (t)(T 3 -T 6 (t)) ≤ q sol (t) ∀t ∈ [t S ini , t S f in ]. (3.3)
Recall that q sol (•) is the thermal power coming from the solar field (see Eq. 2.2).

Storage-Production phase

During the Storage phase, when the thermal power obtained from the solar field generates a nonnegligible excess with respect to the q S consumption of the storage system, the operation of the plant switches to a Storage-Production phase. In this phase, we have one part of the recovered energy coming from the solar field which goes to the exchanger (nodes 3 and 6). The other part goes to the power block (nodes 4 and 5). All components in this phase are active. See Fig 3 .2. Let's assume that this phase happens in a time interval Note that, both in the Storage and Storage-Production phases, the same vector µ S will be used to describe the operation of the exchanger and the two tanks. Thus, the storage power during the Storage-Production phase coincides with q S (•), that is, q P (t) = q S ( ṁS 3 (t), ṁS salt (t), T 7 (t)), for all t ∈ I P . In addition, as in the Storage phase, the assumptions we are considering are summarized in Table 3 The mass flow through node 2 will be divided into two parts, the first one directed to the storage system and the second one to the power block. Therefore, according to the conservation of flow of mass and energy, we have the following relationships:

I P = [t P ini , t P f in ].
.2. Mass Flow Temperature ṁ1 = ṁ2 / ṁ3 = ṁ6 / ṁ4 = ṁ5 T 2 = T 3 = T 4
ṁ1 (t) = ṁS 3 (t) + ṁ4 (t) (3.4) ṁ1 (t)T 1 (t) = ṁS 3 (t)T 6 (t) + ṁ5 (t)T 5 (t) (3.5)
Now, since the thermal power cannot exceed the available thermal power, the inequality (3.3) is replaced by

C htf ṁS 3 (t)(T3 -T6(t)) + C htf ṁ4(t)(T4 -T5(t)) ≤ q sol (t), (3.6)
Finally, the following equations end the description of the operation of the plant in this phase,

q R (t) = min{q sol (t) -q S (t), q nom_max } (3.7) q R (t) = C htf ṁS 4 (t)(T 4 -T 5 (t)) (3.8)
q R (t P ini ), q R (t P f in ) ≥ q nom_min (3.9)

Ẇelec (t) = f b (q R (t), T 2 , T d env ) (3.10) ṁ4 (t) = f f ( Ẇelec (t), T 2 , T d env ) ∀t ∈ I P . (3.11)
with q R (•) + q S (•) the thermal consumption of the plant and q nom_min , q nom_max being respectively the minimum and maximum power value at which the power block operates. These values correspond to 20% and 105% of the nominal value of the power block. The last quadratic regressions are determined given that the CSP plant continues to operate with a nominal temperature at node 2 of T 2 = 560°C and the average environment daylight temperature T d env (which is exogenous), is known.

Discharge phase

In this phase, only the storage system and the Rankine cycle are active, as can be seen in the Fig. During this Discharge phase, the molten salt stored in the hot tank is conducted to the heat exchanger to transfer the stored heat energy to the heat transfer fluid. Subsequently this energy is transferred to the power block to finally produce electricity. Assuming that this phase happens in a time interval

I D = [t D ini , t D f in ],
the system of equations representing this phase is as follows:

q D (t) = C htf ṁD 3 (t)(T 3 (t) -T 6 (t)) (3.12a) q D (t) = C salt ṁD salt (t)(T 8 (t) -T 7 (t)) (3.12b) q D (t) = - 1 2 hconvA exch ((T 3 (t) + T 6 (t)) -(T 7 (t) + T 8 (t))) (3.12c) T 5 (t) = fc( ṁD 3 (t), T 4 (t), T i env ) (3.12d)
where q D (•) represent the thermal power transferred to the heat transfer fluid from the molten salt. For that reason, the relations (3.12a -3.12c) are similar to those expressed in (3.1a-3.1b and 3.2).

Note that, as shown in the Table 3.3, in this phase we have T 6 (t) = T 5 (t), T 3 (t) = T 4 (t) and q D (t) = q P (t), for all t ∈ I D .

Mass Flow

Temperature ṁ3 = ṁ4 = ṁ5 = ṁ6 T 3 = T 4 / T 5 = T 6 Table 3.3: Discharge phase assumptions.

Finally, using the quadratic regression f b , we calculate the electrical power produced in this phase by

Ẇelec (t) = f b (q D (t), T 4 (t), T i env ) ∀t ∈ I D .
where T i env is the temperature of the environment, which we assume constant while its value depends on the season i = 1 -4.

Volumes and Temperatures

During the Storage and Discharge phases, the volume and temperature of the two tanks vary. In each of the phases, the temperature will only vary in one tank while remaining constant in the other. The variation is produced mainly by the new volume entering one of the tanks and its temperature. The volume's balance is given by the following expressions:

V cold (t) = V cold (t S ini ) -V S new (t), if t ∈ I S V cold (t D ini ) + V D new (t), if t ∈ I D V hot (t) = V hot (t S ini ) + V S new (t), if t ∈ I S V hot (t D ini ) -V D new (t), if t ∈ I D
Here, the new incoming or outgoing volume of each of the tanks is given by

V S new (t) = t t S ini ṁS salt (t) ρ salt dt, V D new (t) = t t D ini ṁD salt (t) ρ salt dt,
where ρ salt is the density of molten salt.

Regarding the temperatures, since the variation in the cold tank is not really significant during the Storage phase, we assume that it remains constant during this phase, and similarly for the hot tank temperature in Discharge phase. Then, the temperature variation in the cold tank is,

T cold (t) = T cold (t S ini ) if t ∈ I S , and
T cold (t) = T cold (t D ini ) • V cold (t D ini ) • ρ salt + T7(t) • t t D ini ṁD salt (t) dt V cold (t) • ρ salt if t ∈ I D . (3.13)
For the hot tank,

T hot (t) = T hot (t D ini ) if t ∈ I D , and
T hot (t) = T hot (t S ini ) • V hot (t S ini ) • ρ salt + T8(t) • t t S ini ṁS salt (t) dt V hot (t) • ρ salt if t ∈ I S . (3.14)
Finally, these volumes cannot exceed the maximum or minimum capacity allowed in the two tanks, that is, setting the physical characteristics of the tank (V ), it is possible to determine these capacities,

V min (V ) ≤ V cold (t), V hot (t) ≤ Vmax(V ) ∀t ∈ I S,D .
According to [START_REF] Turchi | Csp systems analysisfinal project report[END_REF], a temperature range has been established for the stored molten salt in the tanks,

290°C ≤ T cold (t), T hot (t) ≤ 560°C ∀t ∈ I S,D

Thermochemical Storage

The model we use here has been mainly elaborated in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF]. In order to keep a self-contained exposition, we present the main elements of the model. When this type of storage is considered, the vector of physical variables ν is composed of four components: energy density DEC, apparent volumetric mass of conductive binder ρ eng , tubes' radius r sw and the equivalent length of the tubes L. The equivalent length of the tubes corresponds to the product of the number of tubes in a module by the number of modules and by the length of each tube. Then, the vector of physical variables is represented as ν = (DEC, ρ eng , r sw , L).

The operation of the Thermochemical reactor is defined by the pressure of the reactor P c (t) and the gap between operating and thermodynamic equilibrium temperatures ∆T eq (t). This gap of temperature ∆T eq (t) is expressed as the difference between the average temperature of the heat-transfer fluid (when passing through nodes 3 and 6), and the thermodynamic equilibrium temperature of the reversible reaction (2.6), T eq (P c (t)), which is given by

Teq(Pc(t)) = 94573 121.186 -8.314 • ln(Pc(t)) -273.15 • C (3.15)
Therefore it is assumed, as a first approximation, that

T 6 (t) + T 3 (t) 2 = Tc(Pc(t), ∆Teq(t)) = Teq(Pc(t)) + ∆Teq(t) (3.16)
where T c (P c (t), ∆T eq (t)) stands for the operating temperature of the reactor. However, the operational variables (P c (t), ∆T eq (t)) are different depending on the direction of the reaction that the reactor (2.6) performs. Thus, the following definitions are made:

• µ S (t) = (P S c (t), ∆T S eq (t)) stands for the operational variables during the Storage process (decomposition reaction).

• µ D (t) = (P D c (t), ∆T D eq (t)) stands for the operational variables during the Discharge process (synthesis reaction).

Finally, the vector µ of operational variables is defined as µ(t) = (P S c (t), ∆T S eq (t), P D c (t), ∆T D eq (t)).

The details of the dynamics for each of the phases with this storage system can be found in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF].

Objective function

To be coherent with our approach, the objective function must be an economic criterion taking into account, at the same time, the storage design (investment cost) and the dynamic operation of the plant (operational costs and possibly incomes).

For the different indicators presented in Subsection 2.2.1 (and detailed in Appendix C), we must determine the following key elements:

• Investment cost of the storage system;

• Operation/maintenance costs of the Solar field, the heat transfer systems and the Rankine cycle;

• Operation/maintenance costs of the storage system;

• Total production of electrical power;

• Revenues of the electrical power sold on the SPOT market.

First, the investment cost, denoted by C invest , is the initial cost of the project and it depends only on the physical variables of the system. Secondly, for indicators considering revenues, for each year k within the lifetime of the plant, the operation is valorized using the annual cash flow: the total revenues Rev k minus the total operation/maintenance costs Cost k . Finally, for efficiency indicators such as the LCOE, the operation of the plant is valorized by comparing the total cost of the project (investment and operation/maintenance) with the total energy produced by the plant during the lifetime.

The detailed computations of all the components of investment costs, revenues, operation/maintenance costs, and produced energy are described in Appendix E.

Bounds

Table 3.4 shows the bounds for the variables of the optimization problem, which differ depending on the storage system. The bounds for the variables corresponding to the Two-tank system have been deduced from [START_REF] Khamlich | Technical and economic assessment of thermal energy storage in concentrated solar power plants within a spot electricity market[END_REF][START_REF] Li | Dynamically coupled operation of two-tank indirect tes and steam generation system[END_REF] and from the energy balance in the system. On the other hand, for the Thermochemical system, these bounds are based on the know-how of PROMES-CNRS, which are presented in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF]. At the end of Table 3.4, the strategic variables (time variables) are presented, which correspond to the initial and final time of each phase of the plant.

t S ini t S f in [0, +∞[ t P ini t P f in [0, +∞[ t D ini t D f in [0, +∞[

Results and Discussion

The different simulations carried out for the comparative study with different «profiles», that is different economic indicators, different production strategies, different storage systems and even possibly different price scenarios or different correction coefficients β, are presented in this section. The model proposed in the previous sections has been implemented in the case of California, more precisely with the data corresponding to the network node of Daggett. Thus the solar resource (used with SM = 2.5) and market prices are the ones corresponding to this location. Nevertheless, as in many deregulated market around the world, the prices on the Californian electricity market has been subject to important perturbations in the recent years and it is quite difficult to determine a «mean profile»of prices. Therefore we decided to focus on a scenarios analysis, emphasizing the potential influence of such perturbations on the results. The Price-Pessimistic scenario (PP) corresponds to the price profiles of year 2016 (with low prices) which was also used in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF], thus allowing to compare both studies. An optimistic scenario has been elaborated on the base of the mean price in California in 2021. But in order to avoid that this sensitivity analysis would be perturbed by a difference of profiles instead of the difference of mean prices, the Price-Optimistic scenario (PO) corresponds to the prices profile of 2016 shifted by the difference of mean prices between 2021 and 2016, that is by 16.3$/M W h. The mean price of 2021 was preferred to the ones of 2022 which are quite extreme (reaching often 124$/M W h). Finally a Price-Medium scenario (PM) is considered by shifting the price profile of 2016 by the half of the difference between the mean price of 2021 and 2016, that is by 9$/M W h.

The real discount rate is set as ι r = 0.03 and the number of years as N = 30. In some simulations a sensitivity analysis on these parameters will be done. We consider p = 4 seasons, with a stage of N days = 7 each. For i = 1-4 (each index representing a season), the curves of useful power q sol (•, i) (see Subsection 2.1.1) and electricity price λ(•, i) (see Appendix B), are taken from [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF], whose data comes from the SAM software and the average electricity prices for each day during each season of year 2016 (Price-Pessimistic scenario).

Computational Implementation

The solution of the different optimisation problems were obtained with MATLAB ® version 9.9.0.1570001 (R2020b) using the f mincon function with the interior point algorithm [START_REF] Mathworks | MATLAB R2020b -Online Documentation[END_REF].

This implementation required a large computational processing capacity, mainly due to the number of variables of these problems (see F.1 and F.2), which varies depending on the type of storage system and production strategy. Moreover, for certain configurations, convergence problems occur. Hence, it has been needed to look for an alternative approach to obtain optimal results in a reasonable time.

When considering the Two-tank system, it was proposed to split the search for a best local minimum by two loops, as shown in Algorithm 1. In this case, the physical variable V , which represents the maximum volume of molten salt used during the model year, is left fixed and the problem is solved for the rest of the variables. In this way, we get a while loop with a subroutine of iterations corresponding to the command f mincon (the number of iterations for this subroutine can also be fixed).

For each iteration of the while loop of the variable V , the convergence is verified, based on different tolerance criteria, towards an optimal value of V * .

Algorithm 1: Double loop -Interior point

Input: x 0 = (V 0 , A 0 exch , ..., •) ∈ R n , k = 0, exitf lag = 2, ϵ > 0, V 0 = 10000, V A = 10000 + 2ϵ, x aux 0 = (A 0 exch , ..., •) ∈ R n-1 ; 1 while | V 0 -V A |> ϵ and k < maxiter and exitflag = 2, do 2 V 0 = V A ; 3 [x,fval,exitflag,output] = fmincon(f un, x aux 0 , .., •) V A = max{V i }, // i= 1,...,4. 4 k=k+1 5 V = V A ; 6 return x = (V, A exch , ..., •);
Now when considering Thermochemical storage, before executing the optimization process, the physical variables have been rescaled to avoid numerical instabilities.

In what follows, we provide a comparison of the results of the optimization process for different economic criteria. We compare the Two-tank system with the Thermochemical system under the different values of beta for the cost estimation of the reactor. Note that in the forthcoming section we will mainly illustrate our analysis by figures based on the values of economic criteria. The corresponding production profiles can be found in Appendix A.

LCOE sensitivity to discharge duration

For this first analysis, the economic indicator LCOE is used and the discharge duration for the two storage systems in question were varied. Different price scenarios (PP, PM, PO) were considered but their influence is minor since the LCOE criterion doesn't take into account revenues/incomes of the plant. Thus the different of price scenario only acts on the costs associated to the use of electricity (pumping, heliostats, dissipation). More important influence of the price scenarios will be observed in the forthcoming subsections where NPV criterion will be used. Here since we focus only on LCOE criterion, the price profile is not taken into account and the Discharge strategy is only the Classical one.

As can be observed in figures 4.1 and 4.2, in all cases the Two-tank system is the most economically efficient to implement. Furthermore, it can be observed that the compensation of the income generated from considering more time in the Discharge phase is maintained for all cases for the Two-tank system, hence its slow decrease. For the Thermochemical system, in the optimistic case (β = 1, continuous red curve), the results found indicate that if we consider a Discharge phase greater than 10 hours, the costs of producing electricity increase. This is reflected mainly when comparing the results of 10h and 12h, where we can observe a change in the behavior of the curve (continuous red, Figure 4.1 -4.2), and this is due to the cost of considering a larger Thermochemical reactor to discharge energy for a longer time is not compensated by the production of electrical energy. In the other cases (β = 2, 3), we can notice a more sensitive behavior of the curves that represent them, since when considering higher costs for the storage system, they present a peak for the 6 hours of duration and then a valley approximately in the 10 hours of duration, which is due to insufficient production of electric energy and to an excessively large design of the storage system. Finally, if we compare the most viable cases of both systems (12h duration for Two-tanks and 10h duration for Thermochemical reactor with β = 1), we have that the cost of energy production is reduced by approximately 27%. The results in Figure 4

Sensitivity analysis based on NPV criterion

In this subsection, the optimization (actually maximization) will be done using the NPV economical criterion, thus taking into account the revenues generated by electricity sales. Thus the different price scenarios will clearly have an influence on the results.

NPV sensitivity to discharge duration

The sensitivity analysis being here focusing on the Discharge duration, only the Classical strategy will be considered. Tables 4.1 -4.3 and figures 4.3-4.4

show the NPV values with respect to the variation of the discharge duration. For the PP scenario, we can notice the strong evolution of the curves corresponding to the Thermochemical system, as they decay rapidly after 6 hours of discharge duration (see Figure 4.3). The best configuration corresponds to "Reac-Class-β 1 " with a discharge duration equal to 6 hours. For the Two-tank system, the behavior of the blue curve is opposite to the others, since it has a slow and constant growth from a certain point. For this system, the best configuration corresponds to "Two-tanks-Class" with 12 hours discharge duration. It is clear that for both storage systems, the plant project does not recover the investment (negative NPV). A similar analysis happens for the PM scenario. Additional results have show that the variations of NPV when one considers either one Discharge phase or two Discharge phases is minor. Thus in the forthcoming subsection, and for simplicity, we will essentially consider configurations with only one Discharge phase.

For the PO scenario, the Thermochemical system continues to show greater sensitivity to the change in discharge duration. Here, the best configuration is "Reac-Class-β 1 ", with a duration of 8 hours. That is, the NPV value improved by 3 times compared to the PP scenario. For the case of "Reac-Class-β 2 ", we have a variation in the behavior of the NPV curve coinciding with the first case in that the best configuration corresponds to a duration of 8h. Finally, for the "Reac-Class-β 3 ", the results are analogous to those deduced with the PP scenario.

When considering the PO scenario the NPV values increase of more than 100 Me in most cases, which allowed that for a period of 30 years and under plant configurations with a duration in the Discharge phase of more than 6 hours, it was possible to obtain a positive NPV, i.e., the plant recovers the investment.

Optimal production profiles

In parallel to the optimization of the sizing of the storage systems, the optimization problems are also aimed at optimizing the plant production operations. According to the production strategies described in Subsection 2.2.2, the optimized strategies found for both storage systems will be analyzed.

Finally, the consideration of different price scenarios does not represent important variations in the operating profiles (as it is shown in Appendix A). Therefore, the price scenarios in question will not be specified in this subsection.

Thermochemical system

In this subsection, our aim is to combine the most innovating approaches, that are Thermochemical system and Price Chasing strategy. It can be observed that in the case of a single discharge (Figure 4.5), the discharge takes place immediately after the end of the Storage phase, mainly for two reasons: the first one is due to the highest peaks of the price curve, approximately after the beginning of the night; and the second one, due to the cost that would entail to restart the power block. Then, Figure 4.6 shows the operational profile for two Discharge phases per day, where Discharge phases are carried out in those time intervals where the value of the electricity price is higher. On can indeed observe that the optimisation aligns both Discharge phases with price peaks.

Similarly, in Figures 4. Here, the production strategy used is Price Chasing, with which a positive NPV is obtained for the price (PO) and cost coefficient β = 1. We can also see the large effect of considering a specific production strategy for the plant Indeed, when comparing Classical and Price Chasing strategies combined with Thermochemical system, then with Classical strategy positive NPV is never reached (see Table 4.1) while with Price Chasing strategy, in the PO scenario, a positive NPV is obtained for the configuration "Reac-PC-1D-β 1 " (see Table 4.4). Now, if we compare the configurations "Reac-Class-β 1 -6h" (see figures 4.3 and 4.4) and "Reac-PC-1D-β 1 " (see tables 4.4 and 4.5), we note that in this case, the implementation of the Price Chasing strategy increases the NPV by approximately 40 Me, for the PP and PO price scenarios.

Two-tank system

In Figures 4.9 and 4.10, the operational profiles of the plant with a Two-tank system are shown. These profiles correspond to the Classical 12-hour discharge duration strategy, which according to the sensitivity analysis performed in the Subsection 4.3.1, is the most cost-efficient. A negative effect of this strategy with fixed discharge duration is that the plant must discharge regardless of the market value price: first, in Figure 4.9, it can be observed that the Discharge phase for each day ends in the price valleys that occurs in the evening hours; second, the cost related to the large sizing of the storage system for discharges of such a length is quite high. If we compare this storage system with the best configurations for each production strategy, Two-Tanks-Class-12h and Two-tanks-PC-1D, this is opposite to what happens with the Thermochemical system, since the Classical strategy is more costeffective, increasing the NPV value by approximately 16 Me for the PO price scenarios and a period of 30 years (see Table 4.6). As can be checked in Table 4.6, in this case, the storage system has been sized larger for the case of the Price Chasing strategy, and therefore it is more costly. Table 4.6: Economic results for the Two-tank system considering different strategies of production and the PO price scenario. SM = 2.5, N = 30 years and ιr = 3%.

Conventional Payback

In the literature it is usually considered that CSP equipped with Thermochemical storage cannot reach a valuable CPB, i.e. a CPB value lower than the project duration, without subsidies. Nevertheless, based on the above presented results (see tables 4.4 and 4.5 with β = 1), NPV can reach a positive value within the lifetime of the project (see F). Our aim in this subsection is thus, focusing on Thermochemical storage, to make a sensitivity analysis on the CPB value.

In order to identify the time needed to obtain or recover the investment in a CSP plant, the optimization problem to be solved will be to maximize the NPV indicator, varying the parameter corresponding to the number of years, N , as well as considering different production strategies for each of the storage systems. For this indicator, the case studies corresponding to the Price Chasing strategies are also considered. Therefore, we will detail the most relevant configurations for the CSP plant according to the type of storage system. In the figures 4.11-4.12, we analyzed configurations with the Thermochemical system and the Price Chasing production strategy (one and two Discharge phases). For the PP scenario, we can notice that the different configurations require more than 50 years to recover the investment. Under such conditions, this type of systems cannot become economically competitive. A similar case happens with the Thermochemical system worked out in [START_REF] Bravo | Hybrid solar power plant with thermochemical energy storage: A multiobjective operational optimisation[END_REF]. For the PO scenario, the first configurations to recover the investment correspond to the optimistic cost cases (β = 1) with the Price Chasing strategy of one and two Discharge phases. A time horizon of 26 years, respectively 28 years is necessary to obtain a positive NPV. But for β = 2 (resp. 3), our results show that 30 years (resp. 40 years) are necessary. Finally, we can note that in the case of the PP scenario, we can observe a similar behavior of the curves for all the β cases, with a small difference between both curves of the same color, with the strategy of a single Discharge phase (1D) better than that of two Discharge phases (2D). In the case of the PO scenario, we can analyse that the difference between the one and two discharge strategies is negligible.

It is important to emphasize that, to our knowledge, it is the first time that a optimal configuration of a Thermochemical storage in a CSP plant attains a positive NPV with a "reasonable" time horizon (less that 30 years).

Internal Rate of Return

Another approach consists in analysing to what extend the possible fluctuations of the currency price can affect the economical efficiency of the CSP power plant with storage. Hence the importance of the parameter ι r , which is used to depreciate future money transactions and determine their current values. The following results correspond to the maximization of the NPV indicator by varying the real discount rate until the initial investment is recovered. In this subsection, we decided to present the sensitivity analysis of the Internal Rate of Return only for the PM scenario, that is the mean price scenario. Analyzing the evolution of the different curves for each configuration in Figure 4.13, we can observe that, as in the case study with the CPB indicator, the configuration of the Thermochemical system with the classic production strategy is the one that presents the most deficient results that is the plant can recover the investment only with a very small rates. On the other hand, the other configurations recover the initial investment, within a period of 30 years, considering a rate up to approximately ι r = 2%. Such a rate is particularly reasonable with regards to real life data. This shows that, for each of these configurations, the CSP plant will be economically efficient for a large set of values of rate ι r .

Finally the most favorable configurations, independently of the variations of the real discount rates, correspond to the Two-tank system with the Classical strategy (Two-tanks-Class-12h) and the Thermochemical system with the Price Chasing (Reac-PC-1D-β 1 ).

Comparison of some typical profiles

Comparing only the Price Chasing operation strategy with one Discharge phase for the two storage systems (see Table 4.7), the revenues produced with the Two-tank system are higher, but this implies a larger investment in the storage system. In contrast, the Thermochemical storage is designed to be smaller, attacking only the first peak of prices.

In other words, the Thermochemical system maintains a more efficient balance between investment costs and revenues, for all price scenarios. When considering the price scenario PO, we can see in Figure 4.15 several changes in the results. First, we have that all the evaluated configurations, except the Reac-Class-6h, obtain a positive value for the NPV after 26 years. Excluding the Reac-Class-6h case and analyzing the evolution of the curves of the other configurations, we can notice that the best configuration corresponds to Reac-PC-1D, with a small advantage on the second configuration, that is Two-Tanks-Class-12h. The Thermochemical system presents a higher cost-efficiency in the operation of the plant. Let us emphasize that the optimization problems are solved using the all purpose optimization tool fmincon of Matlab (with interior point methods). The problem being highly non-convex, the obtained solutions (physical/operational variables) are only stationary point or at the best local minimums and they depend on the initial point. As a consequence the optimisation in the case of Price Chasing strategy (thus with an higher degree of liberty) could possibly not reach the same optimal value as the more restricted optimisation with Classical strategy. This situation can be observed in Figure 4.14 between the curves «Two tank-PC-1D»and «Two tanks-Class-12h». The difference, in terms of NPV values, is nevertheless not significant for a -somehow classical-lifespan of the project of 30 years.

Conclusions

In conclusion, the main objective of this work was to develop a comparative study of the feasibility and economic efficiency of a CSP plant with different storage systems and production strategies, along with the impact of considering different pricing scenarios. In these cases studied, we have a parallel between technologies and strategies commercially implemented and those that are still in development phase.

First, when the sensitivity study was carried out using only the classical LCOE criterion, with Classical strategy and varying the discharge duration, in both the PP and PO scenarios, the Two-tank system with a duration of 12 hours for the Discharge phase was the most cost efficient to implement. On the other hand, it should be noted that for the Thermochemical system (optimistic case β = 1, 10 hours of Discharge phase) a very low value was also obtained. But of course, since the LCOE indicator does not take into account incomes, different price scenarios do not generate important variations of the LCOE (see Figure 4.1 and 4.2).

For the second sensitivity study, again with Classic strategy but using now the NPV criterion, in all prices scenarios the results also indicated that the Two-tank storage system was the most costeffective. Moreover, in the case of the PO scenario, we obtained a positive NPV for Discharge phases longer than 7 hours. For the Thermochemical system the most cost-efficient corresponds to a Discharge phase of 8 hours (the best configuration)while the optimal duration is of 6 hours if the PP scenario is considered.

For the third case study, using the CPB criterion and for the PO scenario, we obtained that, after a period of 26 years, the investment is recovered with the configurations Reac-PC-1D and Two-Tanks-Class-12h. When the IRR criterion is used, the best configuration corresponds to Two-Tanks-Class-12h, obtaining a NPV equal to zero for a real discount rate ι r = 2.1% over a time horizon equal to 30 years. Note that the configuration Reac-PC-1D reaches extremely similar performance.

These case studies also enlighten how the consideration of different prices scenarios impacted the sizing, operation and economic benefit of the CSP. Notice that considering the NPV, CPB and IRR indicators using the PO scenario, we found that it was possible to recover the investment mainly in a period of 30 years. It is important to emphasize that no subsidies are considered in this work.

One of the main contribution of this work is not only to provide a large spectrum tool (with possible variations of the economical criterion, of the electricity price, of the real discount rate) but more importantly to emphasize the real pertinence of an optimisation process in which the optimal design of the storage and the optimal operation are jointly (and not sequentially) decided/computed. And this allows to prove that an economical equilibrium for CSP is possible both with Two-tanks and Thermochemical technologies.

Let us also observe that our methodology can be easily adapted to other comparative studied. Indeed we are here proposing a general framework to asses the pertinence of storage systems in CSP, with a Multi-dimensional analysis and under the framework of co-optimization. Thus in this vein, some possible extensions could be first to integrate to the configuration of a hybrid storage system (Twotanks/Thermochemical), second, to investigate the feasibility of considering operating strategies for long periods of time like inter-seasonal strategies and third to reinforce the present analysis by using specific optimization methods to better embrace the high non-convexity of the problem.

C.2. Net Present Value

In [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF], the Net Present Value (NPV) was used as indicator to evaluate a CSP plant project. This metric considers the revenues of the power plant during its lifetime, according to the variable prices of the SPOT market, as well as the investment, operation and maintenance costs. More precisely,

N P V = -C invest + N k=1 (Rev k -Cost k ) (1 + ι r ) k (C.2)
where N is the number of years of lifetime of the plant, Rev k and Cost k are respectively the revenues and the costs (operational and maintenance costs) of year k. As before, C invest is the investment cost and ι r is the real discount rate.

C.3. Internal Rate of Return

The Internal Rate of Return (IRR) of a project is the real discount rate ι * r at which the Net Present Value of the project is zero. That is,

IRR = ι * r if and only if N P V (ι * r ) = 0,
where N P V (ι r ) is the expression of Eq.C.2 varying with respect to ι r This rate is used for depreciation of future flows and to determine their current values [START_REF] Aussel | On the equivalence and comparison of economic criteria for energy projects: Application on pv/diesel hybrid system optimal design[END_REF].

C.4. Conventional Payback

The conventional payback (CP B) evaluates the payback time based on the nominal discount rate ι. It does not take into account the inflation. The conventional payback is defined by:

CP B = inf{k ≤ N |N P V (k) ≥ 0},
where N P V (k) is the Net Present Value corresponding to the case of the project stopped after k years,

Appendix D. Rankine cycle quadratic regressions

To model the Rankine cycle, a sample run was made using the software SAM, followed by a polynomial approximation, as described in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF]. The values of the respective constants are described in Table D For the quadratic regressions used in the Two-tank model, we consider the temperature T out as the ambient temperature T env , which is fixed depending on the operating time interval of the plant and also takes different values depending on the day or night. On the other hand, in the Thermochemical storage system, T out corresponds to the outlet temperature in the Rankine steam loop (see, [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF]).

(t). • Ẇelec (t) = f b (q R (t), T 4 (t), T out (t)) = b 1 + b 2 T 4 (t) + b 3 T 2 4 (t) + b 4 T out (t) + b 5 T 2 out (t) + b 6 q R (t) + b 7 q 2 R (t) + b 8 T 4 (t)T out (t) + b 9 T 4 (t)q R (t) + b 10 q R (t)T out (t). • T 5 (t) = f c ( ṁ4 (t), T 4 (t), T out (t)) = c 1 +c 2 T 4 (t)
λ(i, t S ini (i, k)) • Cstart+ t S f in (i,k) t S ini (i,k) λ(i, t) • Cmov dt.
• Pumping cost of ṁS 3 (•) -Storage and Storage-Production phases: During the Storage and Production phases, the red circuit is only pumped by the pump P 1 , and therefore, associated only with ṁP 1 (•). This marginal cost is linear with respect to ṁS 3 (•) so

Cost P 1 (i, k) = C P 1 t P ini (i,k) t S ini (i,k) λ(i, t) ṁS 3 (i, k) dt + t P f in (i,k) t P ini (i,k) λ(i, t) ṁP 1 (i, k, t) dt + t S f in (i,k) t P f in (i,k) λ(i, t) • ṁS 3 (i, k) dt .
• Pumping cost of ṁR , Storage-Production phase: The power block must pump a vapor as heat-transfer fluid. Denoting by ṁR (t) the mass flow rate of this fluid, it is known that it is proportional to ṁ4 (t), t ∈ I P . Nevertheless, the power consumption of pumping ṁR is quadratic. Thus, during the Storage-Production phases one has

Cost P P R (i, k) = t P f in (i,k) t P ini (i,k) λ(i, t) A P R ṁ4(i, k, t) 2 +B P R ṁ4(i, k, t) + C P R dt. (E.1)
where A P R , B P R and C P R are three constant real numbers.

• Pumping cost of ṁR , Discharge phase: similar to the previous cost, we have to

Cost D P R (i, k, j) = J(i,k,j) j=1 t D f in (i,k,j) t D ini (i,k,j) λ(i, t) A P R ṁD 4 (i, k, j) 2 +B P R ṁD 4 (i, k, j) + C P R dt. (E.2)
with the same constants A P R , B P R and C P R as above.

• Pumping cost of ṁD 3 (•) -Discharge phase: In the Discharge phases, the active pump P 2 is at the node 4. As in the Storage phase, this cost is as follows, Ẇelec (t) dt .

Cost D P 2 (i, k) =C P 2 • J(i,k) j=1 t D f in (i,k,j) t D ini (i,k,j) λ(i, t) ṁD 3 (i, k, j)dt. (E.3)
(E.4)

• Start-up cost of the power block: Before starting the power block and produce any power energy, the power block must be heated. The pre-heating requirement is modeled as a delay of 20 min in the effective production. Nevertheless, if the power block is already partially heated, this delay is attenuated. To compute this start-up cost, first the non-attenuated one is computed, that is Then, the effective start-up cost the power block is obtained by considering an attenuation function φ(t) = 1 -exp(3t/2), which will take values depending on the difference in hours between the final and initial times of electric energy production (Production and Discharge phases), thus, the smaller the time difference (∆t) the more attenuation will tend to be negligible. These expressions have been established with reference to those set forth in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF].

E.3. Costs of Thermochemical System

For the case of Thermochemical Storage System, it was taken advantage of the various exo-and endothermal components of the power block and the Thermochemical system by recovering thermal energy between them (see blue and black circuits in Figure 3.4). That will result in reducing the amount of wasted heat released to the ambient. The costs related with these integrated systems are:

• Pumping cost of ṁ11 (•) -Storage-Production phase and Discharge Phase: The black cycle of Fig. 3.4 is active during the Storage-Production phases and the Discharge phases and the coolant is pumped by the pump P 3 between nodes n 12 and n 13 . As above, λ(i, t) ṁD 11 (i, k, j)dt.

Cost
• Dissipation cost of q 2 -Storage Phase: During the Storage phase, the ventilator is used to dissipates q 2 (•), which depends also ν and µ S (t),

Cost S Diss2 (i, k) = t S f in (i,k) t S ini (i,k)
λ(i, t)P Cq 2 (ν, P S c (i), ∆T S eq (i), q 2 (t)) dt,

where P C q2 (•) is a nonlinear function that computes the electrical power consumption of the ventilator.

• Dissipation cost of q 3 : Storage-Production and Discharge phase: The power consumption is modeled as the 2% of the heat power q 3 that must be dissipated. Thus The way to calculate q 3 (•) can be found in [START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF].

• Investment Cost: the total investment cost is calculated with the following formulation derived from SAM software:

Cost0 =C land + (C Str (ν) + C Rank + C SF )(1 + τc)[1 + τs + 0.8τt]
where τ c is the contingency rate (fixed at 7%); τ s is the rate which considers the EPC (engineer-procureconstruct) and owner costs (fixed at 13%); τ t is the sales tax (5%). The sub-costs of the power plant are calculated as follows:

(a) C land and C SF : to estimate the cost of the land and solar field, the SAM software was used to establish a linear approximation based on the solar multiple (SM ): (b) C Rank : the cost of the power block is proportional to the design gross capacity. In this study C Rank is 66Me.

(c) C Str : the cost of the storage system also considers the cost C acc2 of the ventilator that dissipates q 2 and the cost C hx of the heater that uses q 1 to evaporate the water in node n 10 . The final investment cost of the storage system is given by

C Str = 1.1(C reac (ν) + C acc2 + C hx ),
where the C reac of the reactor is estimated in terms of the physical variables via consultation with an industrial partner, associated with the laboratory PROMES-CNRS, namely COLDWAY. This company manufactures Thermochemical reactors for different industrial processes. Given a vector ν = (DEC, ρ eng , r sw , L), a polynomial regression was obtained based on the estimations done by this company. The cost of the reactor can be expressed as

Creac = βf h (DEC, ρeng,rsw)DEC • L • π(r 2 sw -r 2 dif f ),
where f h is a polynomial function, r dif f is a constant parameter and β is the ratio between the expected and estimated cost for the Thermochemical reactor. The correction coefficient β will take values between 1 and 3, with 1 being the optimistic case and 3 the pessimistic case [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF].

E.4. Costs of Two-tank System

• Pumping cost of ṁS salt (•) -Storage phase: In the Storage phases, the active pump is P 4 , which allows the flow of molten salt from cold tank to hot tank. This cost is as follows,

Cost S P 4 (i, k) =C P 4 • t S f in (i,k) t S ini (i,k)
λ(i, t) ṁS salt (i, k)dt.

• Pumping cost of ṁD salt (•) -Discharge phase: In the Discharge phases, the active pump is P 4 , allowing the flow of molten salt from hot tank to cold tank and similarly as in the Storage phase, this cost is as follows,

Cost D P 4 (i, k) =C P 4 • t D f in (i,k,j) t D ini (i,k,j)
λ(t) ṁD salt (i, k, j)dt.

• Investment Cost: The investment cost is according the three components, the solar field (which also considers the cost of the operation land) depending on the Solar Multiple Indicator (SM), the cost on the power block, which is constant, and the investment on the Two-tank storage system (a) C land , C SF and C Rank : these costs are the same as those proposed for the Thermochemical system in the previous subsection.

(b) C Str : this cost is based on the estimated results of thermal energy storage in tanks, made in [START_REF] Wu | Techno-economic analysis of an integrated liquid air and thermochemical energy storage system[END_REF] and the estimate made in SAM [START_REF] Turchi | Csp systems analysisfinal project report[END_REF]. The computation of the cost of the heat exchanger is driven by the following expression

C HExch = C 0 • ( Q Q 0 ) M • CEP CI CEP CI ref
where C is the cost of the exchanger with capacity Q (in this case, A exch ) and C 0 the known cost of an exchanger with capacity Q 0 . M is a constant that depends on the type of material in the exchanger. The evaluation of the investment cost for the Two-tank storage system is done on the basis of the estimated cost per unit of thermal energy, C N REL = 22$/kW h th , estimated in the NREL reports [START_REF] Turchi | Csp systems analysisfinal project report[END_REF]. Therefore, according to the above, the total cost of the Two-tank storage system is:

C Str (V, A exch ) = C HExch (A exch ) +E max (V ) • C N REL
with E max representing the maximum energy capacity storage for the volume V , produced per hour, during the time horizon.

• µ i = ( ṁS 3 (i, k), ṁS salt (i, k), T 7 (i, k), ṁD 3 (i, k, j), ṁD salt (i, k, j), T 8 (i, k, j)).

• τ i = (t S ini (i, k), t S f in (i, k), t D ini (i, k, 1), t D f in (i, k, 1), ..., t D ini (i, k, j), t D f in (i, k, j)), with j ∈ J(i, k),

where the time intervals for each of the phases are denoted as, I S = [t S ini (i, k), t S f in (i, k)], I P = [t P ini (i, k), t P f in (i, k)] and I D = [t D ini (i, k, j), t D f in (i, k, j)]. The optimization problem to solve takes the general form:

  Tout outlet temperature [ • C] T htf average temperature of the heat-transfer fluid inside the exchanger [ • C] T salt average temperature of the molten salt inside the exchanger [ • C] U SF Uniform Series Factor Vnew new volume of molten salt entering the tank [m 3 ] W elec electrical power injected to the market [MW] Greek Symbols β price factor for the estimation of the storage system's cost ιr real discount rate λ price curve [e/MWh] µ operational variables of the reactor ν physical variables of the reactor ρeng apparent volumetric mass of binder in the reactive bed [kg/m 3 ]

Figure 2 . 1 :

 21 Figure 2.1: Black-box model for a Concentrated Solar Power plant.

Figure 2 . 2 :

 22 Figure 2.2: Data and Interpolation model -Summer day. California, SM = 2.5.

Figure 2 . 3 :

 23 Figure 2.3: Schema in Storage-Production phase.

  3.1.1.1 Storage phase In this phase, the solar field and the storage system are active. The storage schema of the plant is shown in Fig. 3.1. Let's assume that this phase happens in a time interval I S = [t S ini , t S f in ].

Figure 3 . 1 :

 31 Figure 3.1: Scheme in Storage phase.

Figure 3 . 2 :

 32 Figure 3.2: Scheme in Storage-Production phase.

  3.3.

Figure 3 . 3 :

 33 Figure 3.3: Schema in Discharge phase.

Figure 3 . 4 :

 34 Figure 3.4: Concentrated Solar power plant with Thermochemical storage, including the thermal integration (original Figure from [9]).

Figure 4 . 1 :

 41 Figure 4.1: LCOE values, changing the discharge duration with the classic strategy, SM = 2.5, N = 30 years and ιr = 3%. PP scenario.

3 Figure 4 . 2 :

 342 Figure 4.2: LCOE values, changing the discharge duration with the classic strategy, SM = 2.5, N = 30 years and ιr = 3%. PO scenario.

3 Figure 4 . 3 :

 343 Figure 4.3: NPV values, changing the discharge duration with the classic strategy, SM = 2.5, N = 30 years and ιr = 3%. Price-Pessimistic scenario.

3 Figure 4 . 4 :

 344 Figure 4.4: NPV values, changing the discharge duration with the classic strategy, SM = 2.5, N = 30 years and ιr = 3%. Price-Optimistic scenario.

Figures 4 .Figure 4 . 5 :

 445 Figures 4.5 and 4.6 show the operational profiles of the plant with the Thermochemical storage system using Price Chasing as production strategy, for 7 days (upper graph) together with the electricity

Figure 4 . 6 :

 46 Figure 4.6: Winter season, Thermochemical system with Price Chasing strategy -2 Discharge phases.

7

 7

Figure 4 . 7 :

 47 Figure 4.7: Summer season, Thermochemical system with Price Chasing strategy -1 Discharge phase.

Figure 4 . 8 :

 48 Figure 4.8: Summer season, Thermochemical system with Price Chasing strategy -2 Discharge phases.

Figure 4 . 9 :

 49 Figure 4.9: Winter season, Two-tank storage system with Classical production strategy.

Figure 4 . 10 :

 410 Figure 4.10: Summer season, Two-tank storage system with Classical production strategy.

Figure 4 . 11 :

 411 Figure 4.11: Thermochemical system, Price Chasing strategy with one and two Discharge phases, different values for cost parameter β, SM = 2.5, ιr = 3% and PP scenario.

Figure 4 . 12 :

 412 Figure 4.12: Thermochemical system, Price Chasing strategy with one and two Discharge phases, different values for cost parameter β, SM = 2.5, ιr = 3% and PO scenario.

Figure 4 . 13 :

 413 Figure 4.13: These simulations correspond to the Classical and Price Chasing strategies with both storage systems. SM = 2.5, N = 30 years and PM scenario.

Figure 4 . 14 :

 414 Figure 4.14: These curves correspond to the different storage systems with the Price Chasing and Classical operation strategies. SM = 2.5, ιr = 3% and PP scenario.

Figure 4 . 15 :

 415 Figure 4.15: These curves correspond to the different storage systems with the Price Chasing and Classical operation strategies. SM = 2.5, ιr = 3% and PO scenario.

C

  land =9000(414 • SM + 51.6), C SolarF ield =10 6 (57.765 • SM + 14.373).

Cost0=

  C land + (C Str (V, A exch ) + C Rank + C SF )(1 + τc)[1 + τs + 0.8τt]

Table 3 .

 3 1: Storage phase -Conditions for temperatures and mass flows.

Table 3 .

 3 2: Storage-Production phase assumptions.

Table 3 .

 3

4: Optimization problem variables bounds.

Table 4 .

 4 1: Thermochemical reactor system with Classical strategy, β = 1.

	Time	4h	8h	12h
	Incomes PP.	135.14	170.15	177.08
	NPV PP[Me]	-151.95	-157.81	-246.66
	Incomes PM	180.74	230.92	235.61
	NPV PM[Me]	-110.78	-98.01	-392.02
	Incomes PO	218.0	277.90	287.51
	NPV PO[Me]	-77.28	-54.21	-146.66

Table 4 .

 4 2: Thermochemical reactor system with Classical strategy, β = 2.

	Time	4h	8h	12h
	Incomes PP	152.61	184.65	207.89
	NPV PP[Me]	-104.50	-91.96	-89.05
	Incomes PM	203.62	246.79	260.96
	NPV PM[Me]	-58.46	-39.81	-45.69
	Incomes PO	245.13	297.25	339.60
	NPV PO[Me]	-22.62	8.40	28.87

Table 4 .

 4

3: Two-tank system with Classical strategy.

Table 4 .

 4 4: Economic results considering different β for Thermochemical storage system and Price Chasing strategy with one Discharge phase. SM = 2.5, N = 30 years and ιr = 3%.

	Parameter	β = 1	β = 2	β = 3
	Incomes PP	195.98	171.03	163.59
	NPV PP [Me]	-94.12	-117.15	-134.63
	Incomes PM	274.03	246.60	224.26
	NPV PM [Me]	-29.40	-62.39	-84.16
	Incomes PO	334.94	321.91	287.15
	NPV PO [Me]	24.20	-14.44	-40.74
	Parameter	β = 1	β = 2	β = 3
	Incomes PP	190.65	165.79	162.28
	NPV PP [Me]	-99.61	-123.99	-138.06
	Incomes PM	271.48	243.46	224.63
	NPV PM [Me]	-30.73	-61.79	-83.33
	Incomes PO	332.71	295.76	286.84
	NPV PO [Me]	23.61	-14.65	-39.06

Table 4 .

 4

5: Economic results considering different β for Thermochemical storage system and Price Chasing strategy with two Discharge phases. SM = 2.5, N = 30 years and ιr = 3%.

  • ṁ4 (t) = f f ( Ẇelec (t), T 4 (t), T out (t)) = f 1 + f 2 T 4 (t) + f 3 T 2 4 (t) + f 4 T out (t) + f 5 T 2 out (t) + f 6 Ẇelec (t) + f 7 Ẇ 2 elec (t) + f 8 T 4 (t)T out (t) + f 9 T 4 (t) Ẇelec (t) + f 10 Ẇelec (t)T out (t).Table D.1: Coefficients of the quadratic regressions.

		a	b	c	d		e	f
	1	-2.99	-5.53	-1.58e2	1.94e1	-5.75	9.92e1
	2 -3.70e-3 1.44e-2	8.08e-1	1.10e-2	2.70e-1 -3.80e-1
	3	1.17e-6 -2.01e-5 3.73e-5	2.25e-6 -2.80e-3 5.27e-4
	4	6.75e-2	1.58e-2	3.47e-1	1.04		2.82	7.64e-1
	5	6.18e-5	2.82e-5	9.48e-4 -1.23e-4 -5.69e-6 4.87e-3
	6	4.88e-2	2.34e-1 -8.30e-2 -2.06e-2 -3.03e-3	1.31e1
	7	8.76e-3	3.18e-4	2.67e-4	1.82e-5			-3.83e-3
	8 -2.13e-4 -8.91e-5 1.32e-4				-1.72e-3
	9	2.05e-4	2.29e-4 -8.14e-5				-1.37e-2
	10 -5.24e-4 -9.73e-4 -4.69e-4				3.43e-2
		+c 3 T 2 4 (t)+c 4 T out (t)+c 5 T 2 out (t)+c 6 ṁ4 (t)+c 7	ṁ2 4 (t)+c 8 T 4 (t)T out (t)+c 9 T 4 (t) ṁ4 (t)+
	c 10 ṁ4 (t)T out (t).
	• T 12 (t) = f d ( ṁ4 (t), T 4 (t), T out (t))
	= d 1 + d 2 T 4 (t) + d 3 T 2 4 (t) + d 4 T out (t) + d 5 T 2 out (t) + d 6 ṁ4 (t) + d 7	ṁ2 4 (t).

• ṁ11 (t) = f e ( ṁ4 (t), T out (t)) = e 1 + e 2 T out (t) + e 3 T 2 out (t) + e 4 ṁ4 (t) + e 5 ṁ2 4 (t) + e 6 ṁ4 T out (t).

•

  Maintenance: The operation and maintenance costs are divided into a fixed cost by capacity, set at 59.4 e/kWe/year and a variable cost depending on the generated energy, fixed at 3.15 e/MWhe. Moreover, energy consumption of the heliostat field (tracking) and pumps are taken into account (using SAM default values). The variable costs are:

	Cost M (i, k) =
	3.15	t P f in (i,k)	Ẇelec (t) dt
		t P ini	(i,k)
		+	J(i,k)	t D f in (i,k,j)
			j=1	t D ini	(i,k,j)

  Cost Diss 3 (i, k) =

	t P f in (i,k) ini (i,k) t P	λ(i, t) • 0.02 • q 3 (t)dt
	+	J(i,k) j=1	t D f in (i,k,j) ini (i,k,j) t D	λ(t) • 0.02 • q 3 (t)dt.

Acknowledgments

The first and second author have been partially supported by the Center of Mathematical Modeling CMM, grant FB210005 BASAL funds for centers of excellence (ANID-Chile). The first author was partially supported by Beca Doctorado Nacional ANID-2020/21201478 and SOLSTICE project, ANR-10-LABX-0022. The second author was partially supported by the project FONDE-CYT 11220586 (ANID-Chile).

Appendix A. Supplementary Simulations

In this section, operational profiles for a CSP plant will be shown, under different considerations, such as: storage system, production strategy, price scenario, etc.

A.1. Thermochemical System -Price Chasing Strategy

Appendix B. Spot Market

There is a part of the electric energy that is offered in what is known as SPOT Market. These markets are also called "pay-as-clear", since the energy is paid according to the market price, instead of what is declared and/or ordered. Usually, the dispatch problem related to this market is solved for each hour of the next day, and so, the market price is in fact a piecewise constant function λ

where

In this work, it is assumed that the production of the CSP plant is not sufficient to influence the market price. Moreover, it is assumed that the production offer is always fully accepted/bought. Therefore, an exogenous price function λ(•) is used, according to which the production is paid. In our case of study, this price function λ(•) is calculated from the price per hour of the year 2016, at node price ALTA30ST_7_B1 (nearest to Daggett) of the Californian Market (see [33]). The price curves can be seen in

Appendix C. Economic Indicators

This section will describe the economic indicators that will be used for the comparative study of each of the CSP plant configurations, which depend on the storage system and production strategy. These indicators and the relation between them can be found in [START_REF] Aussel | On the equivalence and comparison of economic criteria for energy projects: Application on pv/diesel hybrid system optimal design[END_REF].

C.1. Levelized Cost of Energy

One of the most commonly used economic criteria in CSP plant projects is the Levelized Cost of Energy (LCOE), which represents the annualized cost of a unit of energy (kWh) produced by the system (see, e.g., [START_REF] Aussel | On the equivalence and comparison of economic criteria for energy projects: Application on pv/diesel hybrid system optimal design[END_REF]). More precisely, it relates the total amount of costs involved in the project and the total amount of energy produced, over the lifetime of the project. It is expressed as follows:

where N represents the duration of the project (years), ι r the nominal discount rate, C k is the annual cost of year k, C invest the initial investment of the project, and W k is the annual electrical energy produced.

Appendix E. Detailed Objective Function

In the present section, we detail each of the terms that make up the economic indicator equations (see Appendix C.1 and Appendix C.2), based on the reduced model of the plant and the pre-scenarios strategy.

Many of the following cost and revenue functions are common for both CSP with two-tank storage and CSP with thermochemical storage. Thus, many of them are already described in [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF]. However, for the sake of completeness, we present them again in this section.

E.1. Revenues

The revenues of the plant come from two sources: the electrical power produced during the Production-Storage phases,

and the electrical power produced during the Discharge phases

Thus,

E.2. Common Storage System Costs

The costs associated with plant operation during the different time intervals corresponding to each phase, I S , I P , I D , are listed below. The representation of these costs is the same, but the nature of the variables is not. The Inactive phase has no costs, since the plant is shut down. Therefore, there are no operating costs.

The following are the costs for each stage S i and day k ∈ 1, ..., N days (i):

• Heliostat operational cost: It consists in a constant energy requirement C start of start-up and a marginal energy requirement C mov of moving the Heliostats following the sun to supply the useful heat

Appendix F. Detailed optimization problems

In this section, we present the general form of the optimization problems for both storage systems. We use, only as illustration, the Net Present Value as economical indicator, and the price-chasing strategy as general pre-scenario. The other optimization problems are variations of these ones. All the optimization problems are derived from the Two-tank model described in this work, and the Thermochemical storage model from [START_REF] Salas | Economical optimization of thermochemical storage in concentrated solar power plants via pre-scenarios[END_REF][START_REF] Tapaches | The value of thermochemical storage for concentrated solar power plants: economic and technical conditions of power plants profitability on spot markets[END_REF].

F.1. Optimization Problem for Thermochemical Storage

In this case, according to Section 2.2.4 and the subsection 3.1.2, the variables to be optimized would be the following:

• ν = (DEC, ρ eng , ρ eng , L).

• µ i = (P S c (i, k), ∆T S eq (i, k), P D c (i, k, j), ∆T D eq (i, k, j)).

where the time intervals for each of the phases are denoted as,

The optimization problem to solve would be the following: if τ i = S, P, D.

F.2. Optimization Problem for Two-tank Storage

In this case, according to describe in the Section 2.2.4 and the subsection 3.1.1, the variables to be optimized would be the following: + T 7 (i, k) ≤ 600 C htf ṁS 3 (i, k)(T 3 -T 6 (i, k)) ≤ q sol (i, t)

if τ i = S q 4_min ≤ q R (i, k, t) C htf ṁS 3 (i, k)(T 3 -T 6 (i, k)) + C htf ṁ4 (i, k, t)(T 4 -T 5 (i, k, t)) ≤ q sol (i, t)

50 ≤ q D (i, k, j) 260 ≤ T 3 (i, k, j) 260 ≤ T 5 (i, k, j) = fc( ṁD 3 (i, k, j), T 4 (i, k, j), T n env ) ≤ 590 220 ≤ T 7 (i, k, j) = -q D (i,k,j) C salt ṁD salt (i,k,j)

+ T 8 (i, k, j) ≤ 600 t S ini (i, k), t S f in (i, k) ∈ [0, ∝] t D ini (i, k, j), t D f in (i, k, j) ∈ [0, ∝] ∀i ∈ {1, ..., p}, ∀k ∈ {1, ..., N Days (i)}, ∀j ∈ {1, ..., J(i, k)}