Optimal Start-Up Policies for a Solar Thermal Power Plant

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Optimal Start-Up Policies for a Solar Thermal Power Plant Maricarmen Lopez-Alvarez, Antonio Flores-Tlacuahuac, Luis Ricardez Sandoval, and Carlos Rivera-Solorio Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04357 • Publication Date (Web): 29 Dec 2017 Downloaded from http://pubs.acs.org on December 31, 2017

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Industrial & Engineering Chemistry Research

Optimal Start-Up Policies for a Solar Thermal Power Plant Maricarmen López-Alvarez,† Antonio Flores-Tlacuahuac,∗,‡ Luis Ricardez-Sandoval,¶ and Carlos Rivera-Solorio‡ †Departamento de Ingenieria y Ciencias Quimicas, Universidad Iberoamericana, Ciudad de Mexico, Mexico ‡Escuela de Ingenieria y Ciencias, Tecnologico de Monterrey, Campus Monterrey, N.L., 64849, Mexico ¶Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario Canada N2L 3G1 E-mail: [email protected] Phone: +52(1) 55 4347 2804

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Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Sustainability and depletion of fossil fuels have propelled the use of renewable energy sources to meet energy demands. Solar radiation is perhaps the most economical and widely available alternative energy source. Energy in the form of solar radiation can be recovered using either photovoltaic or thermal processes. Nowadays, both approaches can only capture a small fraction of the available solar radiation. In this work, we have addressed the dynamic optimal operation of thermal solar plants during start-up. During a normal operating day when solar radiation becomes available, power should be available as soon as possible to meet consumer demands. One of the major problems related to thermal solar plants is the lack of power when solar radiation is off. To overcome this problem, energy storage tanks are considered in the design of the thermal plant. We assume that a conventional Rankine cycle can be used for power generation from low-temperature energy sources. In this work, a dynamic optimization framework is deployed to identify optimal dynamic start-up policies in thermal solar plants. Since the dynamic plant model is composed of a set of partial and differential equations, we have deployed the method of lines and the direct transcription approach for spatial and time discretization, respectively. The results indicate that fast optimal control policies allow to realize power production in a more efficient fashion than simple heuristic-based start-up policies.

Keywords: Solar Energy, Dynamic Optimization, Energy Storage, Renewable Energy


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Introduction Depletion of conventional fossil fuels, pollution and climatic change have motivated the need to develop new types of renewable and sustainable technologies for energy production, 1 . 2 Another major reason for development of new ways of energy production is related to the continuous growth of human population. In the next decades, we will need to deploy all known ways of energy production to meet energy demands, mainly the use of available renewable energies to reduce the green-house effect . Due to the increase in pollution, a few countries have set as an official state policy to reduce the use of fossil fuels for power production by 50%. 3 It has been reported that the amount of solar radiation from sun reaching the earth surface is 173,000 TW, 4 which should be enough to meet the human population energy demands. However, the main challenge in solar radiation is to find practical and economic ways for capturing and storing such large amounts of energy. Moreover, only 30% of total solar radiation can be recovered, by direct reflection, using either photovoltaic cells or thermal solar plants, 4 . 5 Photovoltaic cells, 6 7 absorb photons from sun-light to release electrons that can be used in an electronic device to produce power. Similarly, thermal solar plants, 8 9 reflects the sun rays on a surface that absorbs heat and transfers it to a fluid for energy recovery purposes. These technologies have been mainly used to meet small or medium size energy demands. Photovoltaic cells have low efficiency (25-30%) 10 when transforming solar radiation into power. However, their capital cost is not overly expensive and they are easy to operate. They are mostly used in homes and offices. However, solar thermal plants are more often used in industry due to higher solar radiation capture (75-80 %). Many different process configurations have been proposed for improving solar thermal power production, 11 , 12 . 13 Previous contributions have emphasized improving the performance through the design of thermal plants 14 15 . 13 In previous studies 8 , 16 a summary of different control techniques that have been applied to control the outlet temperature of solar plants during the last years is presented.



Limitations like weather dependence and radiation have motivated the use of energy storage schemes for the operation of solar thermal plants when solar radiation is not available. 17 Thus, there is a motivation to develop highly efficient solar radiation systems for large-scale industrial applications. An approach that can be used to improve the efficiency of solar thermal plants consists in using working fluids featuring heat transfer characteristics that are more suitable than water, 18 , 19 e.g. pure organic fluids or even mixtures of organic fluids. Another novel approach consists in using nano-fluids 20 21 22 23 24 . 25 While most of the solar thermal designs produce low-temperature streams, which can be used for steam production, such streams can be also be used in a Rankine cycle framework for power production, 26 . 27 Although there are works in the literature related to the closed-loop control of thermal solar plants, the optimal start-up of such plants is yet to be addressed. A suitable start-up policy is key for an optimal and reliable operation, not only in the first days or weeks of operation, but also throughout its entire lifetime. 28 An optimal start-up policy can improve the efficiency of solar thermal plants since they can achieve full operation from shut-down in short periods of time and with minimum energy requirements. Heuristic-based start-up policies may be inadequate or non-optimal since they do not explicitly account for operational constraints. Optimal start-up policies can be calculated off-line and later implemented using feedback control systems, 29 , 30 . 31 This approach assumes that process disturbances and/or model uncertainty will not occur during the start-up procedure. On the other hand, when the start-up policy is calculated on-line such policy is evaluated and implemented in realtime, 32 . 33 Using this approach process disturbances and model uncertainty can be taken into account when computing such start-up control policies. The first approach is easier to handle leading to a dynamic optimization problem that does not require control actions to be computed within a limited CPU time. The on-line approach represents the way realistic start-up procedures should be implemented, but it is more difficult to implement given that control actions must be estimated from optimization within a relatively short period of CPU time.That is, one of the major challenges faced by this approach is that the optimal control 4 Environment ACS Paragon Plus

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problem needs to be solved between two successive sampling intervals. In the present work, we will only address the computation of off-line optimal start-up policies. In this work we use the deterministic dynamic mathematical model of a previously proposed solar thermal power facility to compute optimal start-up policies, 34 , 9 , 34 . 32 Moreover, we have added energy storage facilities and assumed that warm water will be used in a conventional Rankine cycle for power production. We propose both nonlinear programming and mixed-integer nonlinear programming (MINLP) optimization formulations to address the challenge of computing minimum time start-up policies. The optimization problems were solved by well known optimization algorithms embedded in the GAMS optimization environment. Our results indicate that shorter start-up times can be achieved using optimal start-up policies instead of simple step-like start-up policies. This result is important since power demands can be met as soon as warm water reaches target temperature values trough the Rankine cycle operation. The paper is organized as follows. Section addresses the operation mode of a typical thermal plant with energy storage tank facilities. The mathematical models of the thermal solar and the storage tank are presented in section . The formulation for optimum start-up is presented in section and the discussions of obtained results is shown in the section . The concluding remarks are summarized in section .

Thermal power plant description The thermal power plant flowsheet considered in this study is shown in Figure 1. During start-up, water is fed to the parabolic array through solar collectors from either an external water source or recycled from the energy storage tanks. Water fed to the system is heated in the solar collector through direct solar radiation. The solar collector consists of a long metal pipe with water flowing inside the tube. In this unit, the temperature of the metallic wall is raised by heat transfer by conduction, and then heat is transferred by convection within the fluid. The warm water coming from the solar collector is sent to the energy storage 5 Environment ACS Paragon Plus


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tank which is used to store energy for power production (i..e through a Rankine cycle) when sunshine radiation is off, and to provide enough water at high temperature for quick startup of the plant when enough solar radiation becomes available. The aim is to reduce start-up time taking advantage that the temperature of the recycled water stream is higher than fresh water stream temperature. In this work, we assume that both tanks are perfectly isolated such that heat losses are negligible. Although we only use two energy storage tanks, it is clear that more tanks with different types of arrangements can also be considered. We also assume that power production takes place in a Rankine cycle where warm water from the thermal plant provides thermal energy which will be transformed into power. In this work, the Rankine cycle was not explicitly modeled to simplify the calculations of the optimal startup policies; instead, based on experience with Rankine cycle operation, a constant water temperature drop was considered. Future work in this research will explicitly consider the integration of this process within the solar thermal power plant. 1

Energy Storage Tank

Solar collector

Water

2

Energy Storage Tank

Rankine Cycle

Mixer

Figure 1: Solar thermal plant with two energy storage tanks. The problem to be addressed in this work can be cast as follows. Given: • A deterministic first-principles dynamic mathematical model of a solar thermal plant including energy storage facilities. • Water availability (working fluid) 6 Environment ACS Paragon Plus

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• Operating and geometric parameter values The problem consists in the calculation of the discrete time-variant input water flowrate (i.e. the manipulated variable), such that the thermal solar plant is taken from temperature environment conditions to a target operation temperature (i.e. the control variable) in the shortest time, holding a set of constraints related to process operation and bounds on decision variables.

Dynamic Model Solar thermal collector Figure 2 shows a schematic diagram of a typical solar collector, which is actually a tube or arrangement of tubes where radiation is reflected allowing thermal energy capture trough the metallic walls by heat conduction. In the present work, we do not address the design of the parabolic device and just assume constant radiation profile. Next, the dynamic mathematical model of the thermal solar collector is presented. This model is based on work done by, 34 , 9 , 34 32 for an Acurex type collector located at the Solar Plataform of Almeria Spain (PSA). 32 The next modeling assumptions were considered: 35 • Sunlight radiation is uniform • Water is the thermal fluid • The flow is hydrodynamic and thermal fully developed • Plug flow (i.e. radial temperature variations across the solar collector are neglected) • Perfect insulation (i.e. axial heat lost in both sides of the wall and the fluid are negligible) • The fluid behavior is time and axially dependent 7 Environment ACS Paragon Plus


• The metal (i.e. collector) temperature is only time dependent • The energy storage tanks are perfectly isolated (i.e. no heat losses) • The contents in the tanks are perfectly mixed (i.e. no spatial variations) The dynamic model is described by the following system of partial and ordinary system of differential equations. The subscripts f and m stand for fluid and metal, respectively. Note that the metal temperature is only time, but not spatially, dependent. On the other hand, the temperature of the fluid (i.e. water) is both time and spatially dependent. For simplicity, only fluid temperature changes along the longitudinal axis are considered (i.e. radial fluid temperature changes were neglected). Table 1 lists the model parameters of the system. Hence, the dynamic thermal solar collector model reads:

Solar Irradiation

h

gt Len

Solar Irradiation Fluid Temperature at the end of the Solar Collector

Ambient Temperature

re rtu pe

al A

tic Op

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Reflecting Surface

Solar Collector Tube

Warm Fluid Tf(0,t)

Fluid Velocity v(t)

Hot Fluid Tf (L,t)

Pipe wall temperature Tm(t)

0

Fluid Temperature at the start of the Solar Collector

x

L

Figure 2: Schematic diagram of a typical solar collector.

∂Tf ∂Tf (t, x) + ρf Cf u(t) (t, x) = Di πht (Tm (t, x) − Tf (t, x)) ∂t ∂x

(0.1)

dTm (t, x) = η0 GI − D0 πhl (Tm (t, x) − Ta (t)) − Di πht (Tm (t, x) − Tf (t, x)) dt

(0.2)

ρf Cf Af ρ m C m Am

Initial conditions

Tm (0, x) = 25 C

(0.3)

Tf (0, x) = 25 C

(0.4)


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We will assume that at the beginning of the startup procedure the collector is filled with water at 25 C and that the wall temperature is also 25 C. Boundary condition

Tf (t, 0) = Tin (t)

(0.5)

where Tin (t) represents the temperature of the water feed stream which can be either constant or time dependent. Table 1: Solar plant model variables and parameters 34 Symbol Value L 200 ρm 1100 ρf 1000 Cm 440 Cf 1100 Ain 0.0038 Af 0.0013 η0 0.67 G 0.9143 hl 20.773 ht 1283.2 Di 0.04 D0 0.07 Ta 25 I 800 t x Tm (t) Tf (t, x)

Units m kg/m3 kg/m3 J/K Kg J/K Kg m2 m2

Description Solar collector length Metal density Fluid density Metal specific heat capacity Fluid specific heat capacity Metal cross-sectional area Fluid cross-sectional area Mirror optical efficiency m Mirror optical aperture W /m K Combined coefficient of convection and radiation W/m2 K Heat convection coefficient fluid and metal surface-water m Inner diameter of the pipe line m External diameter of the pipe line C Environmental temperature W/m2 Solar irradiance sec Time m Axial coordinate C Metal temperature C Fluid temperature

Storage Tank Power production when sunshine is not available is clearly one of the main drawbacks related to solar thermal power plants. To overcome this problem, a system that stores energy has been proposed, 17 . 9 A simple approach consists of the use of isolated storage tanks. The idea is that flow of the working warm fluid does not stop when solar radiation is not avail9 Environment ACS Paragon Plus


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able making feasible overnight operation for power and/or steam production. As shown in Figure 1, we have considered only two energy storage tanks. Additional energy storage tanks in several different arrangements can be included in the process layout. One of the storage tanks is located at the end of the solar thermal collector. The aim of this tank is to accumulate warm water during the day when solar radiation is available. The proposed storage tank model relates the amount of energy stored to the warm water accumulated in the tank. The content of the first energy storage tank is used in a conventional Rankine cycle for power production. After leaving the Rankine cycle, water is stored in a second energy storage tank located after the Rankine cycle. The aim of the second tank is to deploy the still warm water to speed up the start-up procedure. In this work, we assume a temperature decrease of 10o C of the working fluid after leaving the Rankine cycle. The model used for handling energy storage in a tank is made up of energy and mass balance equations, where tank content is assumed to be perfectly mixed (i.e. no spatial temperature variations). Based on the above, the dynamic mass and energy balance for the storage tank are as follows: dVi = Fi,in − Fi,out , i = T a1 , T a2 dt d(Vi Ti ) ρf Cpf = Cpf (Fi,in Ti,in − Fi,out Ti ) − Ut At (Ti − Tenv ), i = T a1 , T a2 dt ρf

(0.6) (0.7)

where V is the tank volume, T is the tank temperature, t is time, F is the mass flow, Cp is the heat capacity, ρ is density, A is the heat transfer area, U is the global heat transfer coefficient. The subscripts in, out, f , env and t stand for inlet, outlet, fluid, environment and tube, respectively. Moreover, the subscript i represents set membership, and T a1 , T a2 refer to the first and second energy storage tanks, respectively.


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Dynamic Optimization Formulation In this section we describe the optimization formulation employed for calculating the optimal start-up policy of the solar thermal plant described in the previous section. The conceptual mixed-integer dynamic optimization (MIDO) problem is formulated as follows:

min Ω =

x,u,y

R tf 0

F(x, u)dt

(0.8)

Subject to : g(x, u) = 0

(0.9)

h(x, u, y) ≤ 0

(0.10)

xL ≤ x ≤ xU

(0.11)

uL ≤ u ≤ uU

(0.12)

where Ω is the objective function, x ∈ Rn is the vector of system states, u ∈ Rm is the vector of manipulated variables, y is the vector of binary variables, tf is the final time over which the dynamic optimization takes place, F : Rn+m → R1 represents the particular form of the objective function, g ∈ Rng represents the equality constraints, i.e. the mass and energy balance equations; h ∈ Rnh is the inequality constraints and typically specifies the feasible operating region. The superscripts L and U stand for lower and upper values. We should stress that without the inclusion of binary variables (y) the above optimization problem reduces to a nonlinear programming problem (NLP). The reason why binary variables were included was to take into account logic decisions, concerning switching decisions in tanks operation, which can be formulated in term of disjunctions. 36 Therefore, the above optimization formulation corresponds to an artificial MIDO problem, since disjunctions can also be expressed in terms of purely nonlinear constraints using complementary constraints. 37 In order to solve the above dynamic optimization problem, a discretization approach was deployed such that a mixed-integer nonlinear programming (MINLP) was obtained. The discretization was performed at several levels. The integral operators were approximated 11 Environment ACS Paragon Plus


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by Radau quadrature, 38 . 39 The method of lines, 40 41 was used for the spatial discretization of the partial differential equation, and the resulting set of discretized time-dependent ordinary differential equations was again discretized using the transcription approach. 42 After proper discretization (which involves the correct selection of the number of spatial discretization points and the number of finite elements and internal collocation points for temporal discretization to be discussed in the results section) the following discretized MINLP was obtained:

Maximize

Ωx,u =

PNc PNe PNs i=1

j=1

k=1

F(xijk , uijk )

(0.13)

Subject to : i = 1 . . . Nc ; j = 1 . . . Ne ; k = 1 . . . Ns

(0.14)

hijk (xijk , uijk , y) ≤ 0, i = 1 . . . Nc ; j = 1 . . . Ne ; k = 1 . . . Ns

(0.15)

xL ≤ xijk ≤ xU ,

i = 1 . . . Nc ; j = 1 . . . Ne ; k = 1 . . . Ns

(0.16)

uL ≤ uijk ≤ uU ,

i = 1 . . . Nc ; j = 1 . . . Ne ; k = 1 . . . Ns

(0.17)

gijk (xijk , uijk ) = 0,

where Nc , Ne and Ns are the number of internal collocation points, the number of finite elements and the number of spatial discretization points, respectively.

Objective functions In this work, we have addressed the dynamic performance of the thermal solar plant during start-up by considering the advantages/disadvantages of using the following objective functions. The aim was to perform the start-up procedure and achieve full operation as fast as possible. Moreover, smooth control actions should also be deployed to avoid saturation control issues. In what follows, the superscript “ ∗ ” on the vector of states and manipulated variables (x and u, respectively) denote target values.


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Omega A h

ΩA = αx (xNˆp ,Nˆ

,ˆ f Nb

∗ 2

−x )

i

(0.18)

The aim of this objective function is to measure the deviation from the target value of the two states (i.e. fluid and wall temperature) measured in the last point along the solar collector, ˆb variable. such a point is denoted by the N ˆp Moreover, this deviation is only considered at the final time of the start-up procedure; N ˆf is the number of internal represents for the maximum number of finite elements whereas N collocation points deployed for time discretization using the transcription approach. αx is a weighting function used to regulate, up to some extent, the shape of the dynamic system response. The aim is to store enough energy to be able to run the Rankine cycle when solar energy is not available; hence, the target temperature considered at the end of the operating time. This objective function stresses the fact that both the water leaving the solar collector and the metal wall at the end of the collector at the final integration time should satisfy their corresponding target values. However, since the control action is not included in the objective function, the resulting optimal start-up policy tend to render aggressive control actions.

Omega B Z ΩB = αx 0

tf

(xNˆb (t) − x∗ )2 dt

(0.19)

This objective function resembles the past ΩA objective function. However, the deviation from the target value at the exit of the solar collector is evaluated at every point in time during the start-up procedure as indicated by the integral symbol along all the finite elements (Nf ) and internal collocation points (Nf ).



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Omega C

ΩC

h = αx (xNˆp ,Nˆ

,ˆ f Nb

∗ 2

∗ 2

− x ) + αu (uNˆp ,Nˆ − u )

i

f

(0.20)

This objective function has a purpose similar to the ΩA objective function. In addition, we have also considered in the present objective function that at the end of the start-up period the manipulated variable (u) should be close to its target value. αu is a weighting function used to stress the relevance of this term in comparison to the other term as represented by the deviation in the system states with respect to their target values.

Omega D Z ΩD = 0

tf

h

αx xTNˆb xNˆb

T

i

+ αu u u dt

(0.21)

This objective function is similar to the ΩB objective function. Moreover, in the present objective function we have also considered the full trajectory of the manipulated variable with the aim of smoothing the control actions. It should be stressed that in the past equation xNˆb stands for a column vector representing a time point to point difference between the states ˆb and the target state value. Similarly, u stands for a time values at the point denoted by N point to point difference column vector of the manipulated variable and the manipulated variable target. As usual, the superscript T denotes a vector transpose operation.

Disjunctions in the mode of operation During the start-up of the thermal solar plant, the tanks could be either full or empty depending upon the operating conditions. If the tanks are not full up to its maximum liquid level, the dynamics mass balance is as follows (for shortening notation in all remaining equa-


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tions of this section the index i is defined as follows: i = T a1 , T a2 ): dVi = Fi,in dt this equation should be enforced until the tank is filled to its maximum volume. When this occurs the following form of the mass balance should be enforced: dVi = Fi,in − Fi,out dt Hence, to avoid operating beyond the physical constraints set by the design of the tanks, we have modeled the tanks as an disjunctive optimization system. The aim of disjunctions is to control valve opening at the outlet of the storage tank, thus avoiding spillover if the storage tank reaches its maximum capacity. The disjunctive programming model for implementing the logic switching decisions is as follows:









¬Yi     

dVi dt

Yi

= Fi,in Vi ≤ Vi∗

_        

dVi dt

= Fi,in − Fi,out Vi ≥ Vi∗

    

(0.22)

where the superscript “∗” stands for target or maximum value. The above set of disjunctions are implemented using the big-M approach: 42

−Mi (1 − yi ) ≤ Fi,out − Fi,in ≤ Mi (1 − yi )

(0.23)

−Mi yi ≤ Fi,out ≤ Mi yi

(0.24)

where Mi is an ”sufficiently enough large” number. The above set of constraints can be reformulated as follows: 15 Environment ACS Paragon Plus


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Fi,out − Fi,in ≥ Mi (1 − yi )

(0.25)

Fi,out − Fi,in ≤ Mi (1 − yi )

(0.26)

Fi,out ≤ −Mi yi

(0.27)

Fi,out ≤ −Mi yi

(0.28)

Constraints to switch the binary variable are as follows: Vi∗ − Vi ≤ Mi (1 − yi )

(0.29)

Vi − Vi∗ ≤ Mi yi

(0.30)

Remarks on Start-Up procedure • Warm water flowrate is sent to the Rankine cycle only until the level in the storage tank reaches its target value. Here it should remarked that in the present work no efforts were done to incorporate the modeling of the Rankine cycle. Moreover, we have assumed that there is a 10 degrees temperature drop when warm water is used to provide heating to the working fluid circulating trough the Rankine cycle. No modeling assumptions were made regarding warm water flowrate. • As explained in the past item, we did not use any modeling assumption regarding minimum or maximum values of flowrate and temperature of the warm water stream. We just have assumed that with the values featured by those variables some power generation will be obtained. Of course, we fully recognize the importance of modeling the Rankine cycle and to incorporate it as part of the start-up optimal control model. • It should be remarked that we have addressed two start-up cases. In the first case, we have assumed that no warm water from storage tanks is available, and hence during the start-up procedure only external low temperature fresh water is used (this case is 16 Environment ACS Paragon Plus

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addressed in scenarios 1 and 2). In the second case, we assume that, to start-up the thermal power plant, enough warm water is available from storage tanks. Hence, in this case no external fresh water was used (this case is addressed in scenarios 3-6). Therefore, no disjunctions were needed for the second case.

Results and Discussion Next, a set of scenarios is presented for each one of the objective actions described above. The aim is to identify the advantages/disadvantages of each scenario for the optimal startup of the thermal solar plant. Note that the outlet target temperatures for both the fluid and solar collector wall was set to 80 C. The maximum filling volume of each tank is 50 m3 and the upper bound on the maximum allowed water flowrate is 0.2 m3 /s. Moreover, the limitations and benefits of each scenario were compared on the following basis: (a) Start-up time, (b) Water and (c) Energy consumption for carrying out the start-up procedure. The start-up time refers to the processing time needed to take the system from shut-down conditions until approaching the target operating conditions as close as possible. In addition, it should be stressed that all weighting functions in Eqn (4.18)-(4.21) were set to unit values.

Initial conditions first , Storage Tank

TTa1 (0) = 25 C

(0.31) (0.32)

Initial conditions Second , Storage Tank

TTa2 (0) = 60 C

(0.33) (0.34)



Scenario 1: Open-loop dynamic simulation In this scenario, an open-loop dynamic simulation was carried out with a storage tank located after the solar collector, i.e. the second storage tank was not considered. This was done by fixing the feed stream water volumetric flowrate to 0.1 m3 /min. Initially, the storage tank was assumed to be empty; therefore, when the water feed stream valve is open, the liquid level in the tank will start to increase. When reaching 50 m3 (maximum allowed volume of the tank) the control valve of the outlet flowrate of the first tank opens. Since we assume that inlet/outlet volumetric flowrates remain the same, this keeps the liquid level inside the tank constant. The disjunction approach presented in the previous section was implemented to account for the change in operation in the storage tank. The resulting Mixed Integer Nonlinear Programming (MINLP) problem was solved using the standard branch and bound (SBB) solver embedded in the GAMS optimization environment. For the discretization of the dynamic model, 100 finite elements and 3 internal collocation points were used. 42 Adding more elements and/or collocation points in the discretization scheme did not improve the accuracy in the results but increased the computational costs. The open-loop responses for this scenario are shown in Figure 3. As shown in this Figure, using a constant flowrate of 0.1 m3 /min in the water feed stream does not allow the system to reach the target fluid since the outlet temperature is around 57 C.

Scenario 2: Dynamic optimization without water feedback In this case, we carried out a dynamic optimization run using the objective functions ΩA and ΩC and without water recycling from the first storage tank to the thermal solar collector ; the corresponding results are shown in Figures 4 and 5, respectively. As shown in both cases, the water feed stream policy identified from the proposed MINLP formulation allows to approximately reach the fluid, metal and storage tank target temperatures . However, the start-up time are relatively long and the control actions feature large and frequent oscillations leading to on/off control operation and valve saturation issues.


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These results indicate that the control actions are mostly driven by the ΩA objective function. That is, given that the control actions are not explicitly considered in the ΩA objective function, wide and frequent oscillations of such control are expected. Moreover, note that we did not include the water flowrate output response from the storage tank mainly because the output control valve remains closed until the storage tank reaches its maximum filling volume (50 m3 ), which is the same value as the target volume. Once it reaches this condition, the input and output flowrates are the same so that the liquid level remains constant in the first storage tank. Storage Tank

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Figure 3: Open-loop Dynamic Simulation (a) Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid Temperate inside the Storage Tanks (c) Storage Volume (d) Output Flow in the Tank



Storage Tank

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30 25 20

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Figure 4: Dynamic optimization without water feedback. Objective Function ΩA (a) Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid Temperate inside the Storage Tank (c) Storage Volumen(d) Input flow in the Tank .


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Storage Tank

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Figure 5: Dynamic optimization without water feedback. Objective Function ΩC (a)Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid temperate inside the storage tanks (c) Storage Volumen(d) Input flow in the Tank.



Scenario 3: Dynamic simulation deploying water feedback In this scenario, the warm water stream coming out from the water storage tank located after the solar collector is recycled and used as the main water feed stream to the thermal solar plant. Initially, we assumed that the storage tank is full of water at 30◦ C; the water flowrate was kept constant at 0.1 m3 /min. Therefore, the results shown in Figure 6 correspond to an open-loop dynamic simulation framework. The results feature a start-up time that is almost 70% shorter than the time obtained for scenario 1 (see Figure 3), for which a final maximum temperature of 76 C, only 4 C below the desired value, was obtained. Storage Tank

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Figure 6: Dynamic simulation deploying water feedback (a) Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid temperate inside the storage tanks. The flow is constant.

Scenario 4: Dynamic optimization with water feedback In this scenario, we carried out the computation of the dynamic start-up policy by using the ΩA objective function and only one water storage tank located after the solar collector. The responses of this scenario are shown in Figure 7. From this graph (red lines) it is clear that the control actions predicted by the dynamic optimization formulation allows the system to reach the desired fluid, metal and storage tank target temperatures. However, control valve saturation issues are evident but only for short time periods. In what follows, we aim to reduce the start-up time using different objective functions. 22 Environment ACS Paragon Plus

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Storage Tank

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0.12 0.1 0.08 0.06 0.04 0.02 0 0

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Figure 7: Dynamic Optimization with water feedback (a) Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid temperate inside the storage tanks (c) Input flow in the Tank.



Scenario 5: Dynamic simulation with water feedback using two storage tanks In this scenario, the thermal solar plant is equipped with two water storage tanks (see Figure 1). Initially, we assumed that both tanks contain water at their maximum filling volume. Both tanks are deployed for energy storage. Assuming that enough water is available, they could be used for power generation when sunshine is no longer available. Another potential use of the tank contents is to speed up the start-up procedure of the thermal solar plant. To this aim, we assumed that the initial temperature of the tank located at the outlet of the thermal solar collector is 80 C. Similarly, we assumed that the initial temperature of the recycle stream tank is 65 C. Figure 8 depicts the dynamic optimization results of the thermal solar plant system while keeping the feed water flowrate constant at 0.1 m3 /min. As shown in Figure 8(a), the temperature of the fluid and metal wall feature an off-set that remains during the operation of the system. This is due to the implementation of the second system storage tank, which helps the collector inlet temperature to be higher than the initial conditions. Solar Collector

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Figure 8: Dynamic Simulation with water feedback using two Storage Tanks. (a) Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid Temperate inside the Storage Tanks. The flow is constant.


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Scenario 6: Dynamic Optimization with two storage tanks In this scenario, we carried out a dynamic optimization run with feedback from the second storage tank. We aimed to reduce the start-up time deploying different objective functions. As shown in Figure 9, practically all the four objective functions are able to drive the thermal power plant from shut-down conditions to the target operating conditions. However, the optimal dynamic response using the ΩD objective function shows the largest deviations from the target fluid and metal temperatures, respectively. This is because, as shown in Figure 9(d), the control action turns out to be aggressive. On the other hand, the best optimal dynamic response was achieved when the ΩB objective function was implemented. In fact, as shown in Figure 9(a), the optimal plant dynamic response is smooth and fast for the case of ΩB (blue lines) . The explanation of this behavior has to do with the smooth and non-aggressive control actions shown in Figure 9(d). From Figure 9 we note that the optimal oscillatory dynamic responses using the ΩA and ΩC objective functions lie somewhere between the two previously discussed optimal responses using the ΩB and ΩD objective functions. Moreover, in an attempt to improve the start-up procedure of the thermal plant, we decided to reduce the flowrate of water circulating through the system and observe whether or not this water flowrate reduction could improve plant operation. Accordingly, a water flowrate upper bound of 0.14 m3 /min was enforced. As shown in Figure 10 , this time the ΩB and ΩD objective functions practically lead to the same optimal start-up policy. In fact, the temperature peak shown in Figure 9(a) when using the ΩD objective function was not observed for this case. Once again, as previously discussed, using the ΩB and ΩD objective functions lead to the best and smooth optimal start-up policies, which are directly related to the smooth control actions shown in Figure 10(d) for these objective functions. Moreover, as shown in Figure 10(a), the start-up time was also 30% reduced. As in the previous cases, using the ΩA and ΩC objective functions resulted in optimal oscillatory behavior.



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Figure 9: Dynamic Optimization with two storage tank. (a) Fluid and Metal Temperature (b) Temperature of the first storage tank (c) Temperature of the Second storage tank (d)Input flow in the First Tank (e)Input flow in the Second Tank with ΩA and ΩB . (f)Input flow in the Second tank with ΩB and ΩC .


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Figure 10: Dynamic Optimization with two storage tank. (a) Fluid and Metal Temperature at the end of the Solar Collector (b) Liquid temperate inside the first storage tank (c) Liquid temperate inside the second storage tank (d) Input flow in the First tank (e) Input flow in the Second tank with ΩA and ΩD . (f)Input flow in the Second tank with ΩB and ΩC .



Water/Energy consumption and start-up times When addressing the optimal start-up of a solar thermal power facility, it is convenient to compare the optimal and dynamically feasible start-up policies based not only on the startup time but also on the water and energy consumption demands required to perform such optimal start-up procedures. Although water is not a relatively expensive resource and solar radiation can be considered as free source of energy, it is important to implement start-up policies featuring minimum amount of water and energy resources. In Table 2 we have collected all this information to make a fair evaluation of the optimal start-up policies discussed above. From Table 2 we note that scenarios 1 and 2 require external or fresh water. However, scenarios 3 through 6 do not demand the use of external or fresh water. This is because water is provided from the energy storage tanks. Also note that scenarios 1 and 2 require large amounts of solar radiation and start-up times, even though the target conditions were not actually achieved. Therefore, these results indicate that simple heuristic start-up procedures should not be used since they tend to require large energy demands and start-up times. On the other hand, most optimal start-up policies correspond to those shown in Figures 7(ΩD ), 7(ΩB ) and 10(ΩD ), since they require minimum energy demands and short start-up times. The results shown in Figure 7(ΩD ) outperform significantly any of the results obtained from scenarios 1 and 2. Accordingly, the results shown in Table 2, suggest that scenarios 4 and 6 resulted in better optimal start-up procedures in terms of minimum water, energy consumption and start-up times.

Conclusions The wide availability of solar radiation makes feasible energy conversion using either photovoltaic or thermal solar facilities. Thermal solar is easy to implement and relatively simple to operate. Without expensive equipment, solar radiation can be used to heat water. Even when not so large water temperatures are reached, low temperature processing streams can 28 Environment ACS Paragon Plus

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Table 2: Comparison of solar energy consumption [MJ] for optimum start-up of a solar thermal power plant. The superscript “*" stands for those cases where steady-state target conditions were not achieved in 13.3 h, whereas the acronym NOF indicates the cases where a objective function was not used. The values in parenthesis are either the start-up time [h] (†) or maximum time [h] (‡) used for computing solar start-up energy. Furthermore, Hw represents the volume of water [m3 ] used during the implementation of the start-up policy. Scenario 1 2 2 3 4 5 6 6

Figure 3∗ 4∗ 5∗ 6∗ 7 8∗ 9 10

NOF 1689 (13.3‡ )

ΩA

ΩB

ΩC

ΩD

1689 (13.3‡ ) 1689 (13.3‡ )

Hw 80 78.9 77.4

1689 (13.3‡ ) 1542 (12.2† )

24 (0.2† )

1623 (12.8† )

16 (0.12† )

‡

1689 (13.3 ) 1623 (12.8† ) 81 (0.64† ) 1639 (13‡ ) 1656 (13.1† ) 65 (0.51† ) 1656 (13.1† )

1672 (13.2‡ ) 49 (0.38† )

be deployed in conventional Rankine cycles for energy recovery purposes. One of the main operating problems faced by thermal solar plants has to do with the start-up procedure employed to take the plant from shut-down conditions up to full or complete operation. Ideally, the start-up procedure should be done as quickly as possible such that power is available to meet energy demands. Therefore, it is important to design start-up procedures to achieve this aim in the best possible way. On the other hand, it is also important to produce power when the solar radiation is off. Both objectives can be achieved using a combination of thermal solar plant and an energy storage system. In this work, we have addressed the efficient and optimal calculation of the minimum time start-up control policy of a thermal solar plant from shut-down conditions to full operation. The evaluation of the complete energy system aids in the selection of optimal operating conditions of the water volumetric flow, where the operating range must be between +/- 25% of the value of 0.1 m3 /min, which is the nominal operating condition. These avoids issues that can damage the equipment, and ensures a safe operation. To appreciate the benefits of the results obtained in this work, the optimal startup policies were compared against simple empirical start-up procedures. In future work, we will address the impact of process uncertainty on the performance of optimal start-up procedures.



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TOC graphic 1

Energy Storage Tank

Solar collector

2

Water

Rankine Cycle

Energy Storage Tank

Mixer

Solar Collector

120

Storage Tank

120

Target Fluid ΩA

110

110

Fluid ΩB

80

Fluid ΩD Metal ΩD

70 60

ΩC

ΩD

80 70 60

90 80 70 60

50

50

40

40

40

30

30 2

4

6

8

10

30 0

12

2

4

6

8

10

12

0

2

4

Time [hours]

Time [hours] Flow in the Tank

0.14

ΩC

ΩD

90

50

0

ΩB

100

Temperature [C]

Metal ΩC

Temperature [C]

Temperature [C]

Fluid ΩC

ΩA

110

ΩB

100

Metal ΩB

90

Second Storage Tank

120 ΩA

Metal ΩA

100


0.2

6

8

10


0.2 ΩA

ΩB

0.18

ΩD

0.12

0.08

0.06

0.04

ΩA ΩB

0.02

ΩC ΩD

0

0.16

0.14

0.14

0.12 0.1 0.08

2

4

6

8

Time [hours]

10

12

0.12 0.1 0.08

0.06

0.06

0.04

0.04

0.02

0.02

0 0

ΩC

0.16

Flow [m3 /min]

Flow [m3 /min]

0.1

12

Time [hours]

0.18

Flow [m3 /min]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 0

2

4

6

8

10

12

Time [hours]


0

2

4

6

8

Time [hours]

10

12

Optimal Start-Up Policies for a Solar Thermal Power Plant

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