6.3 Methodology for multi-period multi-time slice problems
6.3.3 Integration of storage tanks
In order to be able to integrate storage units, each time slice needs a link with its previous and next time slice. This section describes the necessary equations to include storage into the MILP formulation of the heat cascade. The following assumptions are considered to establish the model:
Only liquid (sensible heat) storage will be considered. The heat load is therefore calculated considering a volume, a temperature difference and a specific heat load. The approach can however be extended to consider latent heat storage.
The storage system is discretized into a finite number of temperature levels that are modeled as interconnected storage tanks. Each virtual tank corresponds to the stored amount of
liquid at a specific temperature level. Figure 6.2 shows an example. The storage heat loads are modeled by considering the heat exchange needed (cold stream cl(t)) to transfer a given amount of liquid from one temperature level to the next temperature level. Similarly, a hot stream (hl(t)) that corresponds to the transfer of the liquid from temperature level to the next lower temperature level is defined.
A bound for the maximum volume in the storage system (Mtot) and the temperature dis-cretization of the tanks are defined by the user.
The approach can include an estimation of heat losses. They are proportional to the tem-perature and to the volume stored: The heat exchange area of the storage tank is calculated based on the predefined maximum volume capacity.
The investment costs of tanks are evaluated as a function of their maximum needed storage capacity heat demand heat demand heat demand heat demand
heat excess
Figure 6.2: Definition of the storage model
6.3.3.1 Mass balances of the storage model
The overall mass balance is given under the cyclic constraint in Equation (6.15), which states that the total liquid volume in the storage system has to be returned to its initial state at the end of the period. The heat amount received from the process is equal to the heat amount given back to the process (no heat accumulation at the end of the period).
Xnt
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cfhs is a unit conversion factor, dp,t the duration of time slice t in period p and fu,p,t the multi-plication factor of storage unit u. M˙h,l,u,p,t and ˙Mc,l,u,p,t correspond to the nominal flow-rates of the hot and cold stream respectively in time slice t. The cyclic constraint assumes that the total initial water content is equal to the total final water content in all virtual tanks. The water content (Ml,t) after each time slice t is calculated for each tank l in Equation (6.16). M0,l is the initial water content of tank l.
Ml,p,t= M0,l+
Furthermore, Equation (6.17) imposes a positive level for each tank, and Equation (6.18) limits the total volume in the storage system. This limit is fixed for solving the MILP problem and could be optimized in an outer problem with a sensitivity analysis or a multi-objective optimization.
Ml,p,t> 0 M0,l> 0 ∀l = 1..., nl ∀t = 1..., nt (6.17)
In order to guarantee that the storage network acts not as transfer network, Equation (6.19) becomes necessary. This is particularly important when heat exchange restrictions are defined in the problem.
When the storage network can be a transfer network for indirect heat recovery, Equation (6.19) can be omitted.
6.3.3.2 Definition of thermal streams for the heat integration
The heat exchange with other process or utility units is shown on Figure 6.2. The cold storage stream cl(t) is heated up by process excess heat and is going from a lower temperature tank l at temperature Tst,l to tank l + 1 at temperature Tst,l+1. Whereas the hot storage stream hl(t) corresponds to water coming from a higher temperature level l + 1 and delivering heat to the process units to reach a temperature of Tst,l. Regarding the previously presented heat cascade the hot storage stream is either accounted in the term Pnsh,s,k
hs,k=1fu,p,tQ˙h,s,k,u,p,t of Equation (6.8) when the storage units is defined in a sub-system. Or for storage unit of the heat transfer system it is accounted in the term Pnsh,hts,k
hhts,k=1fu,p,tQ˙h,hts,k,u,p,t of Equation (6.11). By analogy the cold storage
streams are included in Equation (6.8) or Equation (6.11).
As an example, a cold stream from tank l to tank l + 1 is defined with the inlet temperature Tin = Tst,l and the outlet temperature Tout = Tst,l+1. Using the specific heat capacity cp of the storage fluid, the corresponding heat amount is calculated with Equation (6.20). By analogy, the heat load is given for the hot stream in Equation (6.21).
Q˙c,l= fu,p,t· ˙Mc,l,u,p,t· cp· (Tst,l+1− Tst,l) (6.20)
Q˙h,l= fu,p,t· ˙Mh,l,u,p,t· cp· (Tst,l− Tst,l+1) (6.21)
6.3.3.3 Heat losses in storage tanks
Heat losses can be included in the approach. The heat losses of a storage tank at temperature Tst,l are proportional to the amount of liquid in the storage tank and depend on the temperature level.
The heat losses will be modeled by adding a cold stream to the heat cascade which corresponds to a given heat load to compensate the losses and to maintain the temperature of the tank.
The heat losses are a function of the surface area and the temperature of the tank. The heat loss model is a priori a non-linear dynamic model that has to be approximated to define a linear model in the set of equations. Therefore, Equation (6.24) has been developed to estimate the heat losses in the tank.
The heat losses depend on the isolation of the tank and would require a detailed calculation. The data from (EDF, 2011) is used, where a heat loss coefficient khl is estimated to 1/1000 kW/m2K.
The area and the volume of the tank can be calculated with Equations (6.22) and (6.23) as a function of the diameter and the height of the tank.
A = π· d · h (6.22)
V = π·d2
4 · h (6.23)
Knowing the current stored mass from Equation (6.16), the corresponding volume and the surface of the tank can be calculated. Then, by considering the heat loss coefficient khl, the heat losses can be associated in Equation (6.24). ρ is the density of the considered storage fluid in the tanks. In the following examples water is used. fhl is a factor to account for the heat losses on the top and bottom of the storage tank (here 1.1).
Q˙hl,l,p,t= khl·fhl· 4 · Ml,p,t
ρ· d · (Tst,l− Ta) ∀l = 1..., nl ∀t = 1..., nt (6.24)
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To take into account the heat losses, new cold streams corresponding to the heat losses and the associated temperature are added to the heat cascade. The multiplication factors of these cold streams have been fixed to consider the calculated heat losses.
It is assumed that each tank has a fixed diameter of 3m. This value is correct for storage tanks higher than 50m3 but under estimates the heat losses of smaller tanks. However, this value can be accepted for the purpose of the targeting method presented here.
6.3.3.4 Summary of the model
The above presented multi-period multi-time slice approach is generic and can be adapted through the definition of the minimum and maximum temperature levels of the storage tanks, temperature discretization (virtual number of tanks) and the maximum volume accepted for in the storage system.
As a first approximation, the temperature ranges can be determined, by analyzing the process grand composite curve. A fine temperature discretization allows to better identify the optimal temperature levels, but increases on the other side the problem size. With a compromise of well chosen temperature discretization levels, the optimization model allows to calculate real needed temperature levels in the storage system.
The advantage of adding the heat losses are on the one hand to make the problem more realistic and on the other hand, to ensure that the storage fluid will be used as soon as possible.
In industry, often stratified tanks are used, since only one tank with the total water volume is necessary. The main advantages are that less space is needed and also the investment costs are decreased. But on the other side, the heat losses are higher due to the thermocline where hot and cold water are in direct contact (Walmsley et al., 2010). By fixing the number of tanks to 2 (2 temperature levels), the proposed approach can be applied. The bottom of the tank corresponds to the lower temperature, while the top of the tank corresponds to the higher temperature. The estimation of the heat losses and the investment costs have to be modified.