8.3.1 Application of Geomembranes in Water/Energy Storage of PSH Technology

A very important polymeric material, geomembranes, which belong to the group of geosynthetics (e.g., geotextiles, geogrids, geonets, geosynthetic clay liners, geopipe, geofoam, geocomposites), can significantly contribute to the further development of PSH technology.

In this subsection, the most commonly used geomembranes are described, and their characteristics for technological applications of water storage are analyzed. For these purposes, geomembranes are used in combination with other geosynthetics to ensure the safety and durability of water reservoirs, and thus ensure sustainable energy supply.

Table 8.1 presents some types of polymeric materials used for the technological preparation and production of geomembranes; such a final product can be later used for various purposes (Koerner, 2005).

Geomembranes are made from relatively thin continuous polymeric sheets, and their primary function is liquid and/or gas containment.

Due to good physical-chemical-mechanical and environmental characteristics of the geomembranes, their production and sales are rapidly increasing worldwide, as shown in Figure 8.4 (Polyethylene, 2014).

Geomembranes have been used in some of the largest hydroelectric and dam reservoir projects in the world. Hydroelectric power plants can require very large reservoirs. Water reservoirs are often lined with geomembranes to avoid leakage and erosion (Figure 8.5). High-density polyethylene (HDPE) and linear low-density polyethylene (LLDPE) geomembranes are effective and affordable options in reservoirs for water storage.

Geomembranes can provide an impermeable barrier for water retention that is superior to concrete or asphaltic lining in both cost and lower permeability. They are relatively impermeable in the range 1 × 10^− 12 to 1 × 10^− 15 m/s, which is three to six orders of magnitude lower than the typical clay liner.

Critically important for the proper design of geomembrane-lined side slopes of reservoir is the soil-to-geomembrane shear strength. Geomembranes can also be made from the impregnation of geotextiles with asphalt, elastomer, or polymer sprays, or as multilayered bitumen geocomposites.

The basic use of geomembranes should be their retention of liquids/solids over a long time, and they need to have high endurance properties. One of the best additives for plastics to resist ultraviolet (UV) degradation is carbon black. Black geomembranes will get very hot when exposed to sunlight because they absorb sunlight radiation. They will have more expansion/contraction due to temperature variations. New, innovative technologies enable the implementation of the different geomembranes (white geomembranes). In addition to having esthetic advantages, the reflective liner will also reflect solar radiation and reduce the temperature of the geomembrane.

Another innovative solution is an electrically conductive geomembrane (in which electrically reactive carbon is incorporated) that, once installed, can be rigorously tested for spark ruptures and leaking through the geomembrane.

For any technological application, geomembranes must have certain properties in order to satisfy demanding requirements for their safe use (Figure 8.6).

In order to maintain their function, geomembranes’ test methods and standards (e.g., ASTM and ISO) are available or are being developed by standards-settings organizations around the world.

There are important factors to consider when deciding which type of geomembrane to use in a water storage application. Table 8.2 presents the physical, chemical, mechanical, and other properties for the selected geomembranes.

The geomembrane properties are most involved with resistance or susceptibility to tear, puncture, and impact damage in thickness. For this reason, many agencies require a minimum under any circumstance (e.g., in the United States, minimum thickness is 0.5 mm; in Germany, for similar applications, minimum thickness is 0.75 mm). The lifetime of geomembranes is very long—about 25-30 years. HDPE geomembranes can far outlast other engineering materials in comparable situations.

8.3.2 Reservoir Volume

Before selecting the geomembrane type, the desired liquid volume to be contained versus the available land area must be considered. Such calculations are geometric by nature and result in a required depth on the basis of assumed side slope angles. For a square or rectangular section with uniform side slope, the general equation for volume is

$V=H\cdot L\cdot W-S\cdot H^2\cdot L-S\cdot H^2\cdot W-2\cdot S^2\cdot H^3\left({\mathrm{m}}^3\right)$

where

V = reservoir volume (m³)

H = average height (i.e. depth) of the reservoir (m)

W = width at ground surface (m)

L = length at the ground surface (m)

S = slope ratio (horizontal to vertical).

8.3.3 Hydro Energy Storage

Hydro energy that is stored in the reservoir can be calculated according to

$E_{\mathrm{H}}=\rho \cdot g\cdot H_{\mathrm{T}\mathrm{E}}\cdot V\left(\mathrm{k}\mathrm{W}\mathrm{h}\right)$

where V (m³) is water volume in the upper reservoir, H_TE (m) is total elevation difference between the lower and upper water levels, g (m/s²) is gravity acceleration, and ρ (kg/m³) is water density.

8.5.1 Water Balance

The upper reservoir basically represents daily and seasonal energy storage that can balance summer surpluses and winter shortages of SE. However, its charge and discharge dynamics (water volume change from one time period i − 1 into the other i) are dependent on the inputs and outputs of water in the reservoir. Therefore, the dynamics for time stage i can be expressed by the water balance equation

$V_{\left(i\right)}=V_{\left(i-1\right)}+V_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}+V_{\mathrm{R}\mathrm{H}\left(i\right)}+V_{\mathrm{N}\mathrm{A}\mathrm{T}\left(i\right)}+R_{\left(i\right)}-V_{\mathrm{E}\mathrm{V}\left(i\right)}-V_{\mathrm{HE}\left(i\right)}\text{,}$

providing that

$V_{\mathrm{M}\mathrm{I}\mathrm{N}}\le V_{\left(i\right)}\le V_{\mathrm{MAX}}\text{,}$

where V_(i) and V_(i − 1) are water volumes in the upper reservoir in time steps i and i − 1, respectively; V_NAT(i) is the volume of natural watercourses flowing into the upper reservoir (rivers, precipitation, etc.); V_ART(SE)(i) is the volume of artificial watercourses created by the pumping system (PS), which is generated by the energy from the solar generator; R_(i) is the volume of rainfall falling into the upper reservoir; V_RH(i) is the volume of rain harvesting technology that comes into the upper reservoir; V_EV(i) is the volume of evaporation from the surface of the upper reservoir; V_HE(i) is the volume of water discharged to the turbines to produce electric energy; and V_MIN and V_MAX are the minimum and maximum water volume in the upper reservoir, respectively.

8.5.2 Water Storage of PSH as Energy Storage of SE Generator

According to Glasnovic et al. (2013), interrelations between the value of the optimal nominal electric power of the SE generator and the working volume of the PSH upper reservoir, V₀, can be calculated from the following expression:

$P_{\mathrm{e}\mathrm{l}\left(\mathrm{N}\mathrm{O}\mathrm{M}\right)}=-\psi \cdot \mathrm{L}\mathrm{n}\left(V_0\right)+{\psi }^{\ast }$

where Ψ and Ψ* are parameters on the basis of location characteristics and technological features, and V₀ is the operating volume of the upper reservoir.

Therefore, after determining the optimal values of the nominal electric power of the ST generator, V₀ can be determined from the expression

$V_0={\mathrm{e}}^{\frac{{\psi }^{\ast }-P_{\mathrm{e}\mathrm{l}\left(\mathrm{N}\mathrm{O}\mathrm{M}\right)}}{\psi }}\text{,}$

From the operating volume V₀ and reserve volume V_r of the upper reservoir

$V_0=V_0+V_{\mathrm{r}}\text{,}$

the required total size of the upper reservoir of PSH can be easily determined:

$V_0={\mathrm{e}}^{\frac{{\psi }^{\ast }-P_{\mathrm{e}\mathrm{l}\left(\mathrm{N}\mathrm{O}\mathrm{M}\right)}}{\psi }}+V_{\mathrm{r}}$

8.5.3 Model Formulation

Considering the fact that this is a multistage process in terms of time and that the objective and constraint functions are nonlinear in relation to decision variables, the most appropriate model is based on dynamic programming that uses the simulation-optimization model. Optimization is performed by dynamic programming, which is well adapted to the characteristics of the problem, while the necessary constraints are solved by simulation.

The elements of the mathematical simulation-optimization model are shown in Figure 8.8. The system inputs are controlled and uncontrolled (natural), while outputs are also uncontrolled. They are variable in time according to the state of the system and its surroundings. This means that the processes within the system can be represented in relation to time, introducing time stage i.

According to Glasnovic et al. (2013), the general formula of nominal electric power of the solar generator (PV or ST) can be calculated:

$P_{\mathrm{e}\mathrm{l}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}=\frac{\rho g{H}_{\mathrm{T}\mathrm{E}\left(i\right)}}{{\eta }_{\mathrm{S}\mathrm{E}}{\eta }_{\mathrm{P}\mathrm{S}\mathrm{I}}{R}_{\mathrm{coll}\left(i\right)}{E}_{\mathrm{S}\left(i\right)}}{V}_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}\left(\mathrm{kW}\right)$

where ρ is the water density (ρ = 1000 kg/m³) and g is the gravitational constant (g = 9.81 m/s²); H_TE(i) is the average total head (water level difference in upper and lower storage plus hydrodynamic losses in the pumping system (m)); η_SE is the efficiency of the collector field of solar power plants; η_PSI is the efficiency of the pumping system and inverter; R_coll is the ratio of mean daily irradiation of tilted plane of tracking parabolic collectors and horizontal plane (for PV generator R_coll = 1); E_S(i) is the mean daily radiation on the horizontal plane (terrestrial radiation); V_ART(SE)(i) is the artificial water inflow, the daily water quantity pumped from the lower into the upper reservoir by the solar generator; and i is the time stage (increment).

Thus, power consumption of the integrated solar-hydro system, for a given manometer height and efficiency, depends on the ratio of the amount of water that solar power can pump and mean daily irradiation on the horizontal plane. This means that higher consumer energy consumption will require greater amounts of water to be pumped into the upper reservoir V_ART(SE)(i), and that the specific size of the solar radiation E_S(i) (with certain specific local conditions) would be proportional to the needs and higher rated power of the solar generator P_el(SE)(i).

The system state has already been described by the corresponding balance Equation (8.1), while oscillations of the upper reservoir level are shown in Equation (8.2). This type of constraint also entered the initial conditions and requirement that the reservoir be full at the beginning of the observed period:

$V_1=V_{\mathrm{MAX}}$

In that sense, the period with maximum available of solar energy should be taken as the starting point of the calculation. In view of the condition of sustainability of the whole system, the reservoir should also be full at the end of the observed period:

$V_N=V_{\mathrm{MAX}}$

where V_N is reservoir volume in the last time stage.

The decision variable constraint is linked to the motor/pump unit capacity (i.e., the appertaining pipeline), and can be expressed as follows:

$0\le V_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}\le V_{\mathrm{ART}\left(\mathrm{MAX}\right)}$

where V_ART(MAX) is the maximum amount of water that can be pumped by the motor/pump unit (m³) in a certain period of time.

As Eck and Zarza (2006) showed that solar collectors produce thermal power for direct nominal irradiance values above 250 W/m², it is necessary to introduce the constraint of minimum average daily solar irradiation of about 2 kWh/m² day, which can be expressed as (Glasnovic et al., 2013)

$\mathrm{If}{E}_{\mathrm{S}\left(i\right)}\le 2\mathrm{k}\mathrm{W}\mathrm{h}/{\mathrm{m}}^2/\mathrm{day}\Rightarrow P_{\mathrm{e}\mathrm{l}\left(\mathrm{S}\mathrm{T}\right)\left(i\right)}=0$

In addition to these constraints, it is necessary even to introduce a limit to the amount of water that can lead to the upper reservoir of PSH technology rain harvesting V_RH(i), which is governed by the surface that provides ground/land around the PSH technology as follows:

$0\le V_{\mathrm{R}\mathrm{H}\left(i\right)}\le V_{\mathrm{ART}\left(\mathrm{MAX}\right)}$

where V_ART(R)(MAX) is the maximum amount of water that may accumulate geosynthetics at a location along the upper reservoir of PSH.

The general model for determining the optimal power of the solar PV or ST generator (i.e., PV or ST power plant), when integrated with the PSH, can be written with recursive formulas of the optimization process by dynamic programming: (Glasnovic et al., 2013)

$f_i\left(V_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}\right)=\mathrm{M}\mathrm{I}\mathrm{N}\left\{\underset{V_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}}{\mathrm{MAX}}\left[P_{\mathrm{e}\mathrm{l}\left(\mathrm{S}\mathrm{E}\right)i}\left(V_{\mathrm{ART}\left(i\right)}\right),f_{i-1}\left(V_{i-1}\right)\right]\right\}$

with constraints and time step i:

$\begin{array}{l}\begin{array}{ll}\left(\mathrm{a}\right)\hfill & V_{\left(i\right)}=V_{\left(i-1\right)}+V_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}+V_{\mathrm{R}\mathrm{H}\left(i\right)}{V}_{\mathrm{N}\mathrm{A}\mathrm{T}\left(i\right)}+R_{\left(i\right)}-V_{\mathrm{E}\mathrm{V}\left(i\right)}-V_{\mathrm{HE}\left(i\right)}\hfill \\ \left(\mathrm{b}\right)\hfill & P_{\mathrm{e}\mathrm{l}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}=\frac{\rho g{H}_{\mathrm{T}\mathrm{E}\left(i\right)}}{2{\eta }_{\mathrm{S}\mathrm{E}}{\eta }_{\mathrm{P}\mathrm{S}\mathrm{I}}{R}_{\mathrm{coll}\left(i\right)}{E}_{S\left(i\right)}}{V}_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}\hfill \\ \left(\mathrm{c}\right)\hfill & \left[\mathrm{If}{E}_{\mathrm{S}\left(i\right)}\le 2\mathrm{K}\mathrm{W}\mathrm{h}/{\mathrm{m}}^2/\mathrm{day}\Rightarrow P_{\mathrm{e}\mathrm{l}\left(\mathrm{S}\mathrm{T}\right)\left(i\right)}=0\right]\hfill \\ \left(\mathrm{d}\right)\hfill & 0\le V_{\mathrm{R}\mathrm{H}\left(i\right)}\le V_{\mathrm{R}\mathrm{H}\left(\mathrm{MAX}\right)}\hfill \\ \left(\mathrm{e}\right)\hfill & V_{\mathrm{M}\mathrm{I}\mathrm{N}}\le V_{\left(i\right)}\le V_{\mathrm{MAX}}\hfill \\ \left(\mathrm{f}\right)\hfill & V_1=V_N=V_{\mathrm{MAX}}\hfill \\ \left(\mathrm{g}\right)\hfill & 0\le V_{\mathrm{ART}\left(\mathrm{S}\mathrm{E}\right)\left(i\right)}\le V_{\mathrm{ART}\left(\mathrm{MAX}\right)}\hfill \end{array}\hfill \\ i=1,2,\dots \dots ,N\hfill \end{array}$

where f_i (V_(i)) is the optimal return function per state variable V_(i) in time stage i, and f_(i − 1)(V_(i − 1)) is the optimal return function per state variable V_(i − 1) in time stage i − 1. In Equation (8.16), (a) is actually Equation (8.1); (b) is Equation (8.7); (c) basically represents the constraint that applies only to the ST power plant (not to the PV power plant), where it operates on irradiated SE of 2 kW/m²/day (Glasnovic et al., 2011); (d) limits the amount of water that can give rain harvesting technology V_RH(i) between the minimum (zero) and maximum V_RH(MAX) value; (e) is the state variable constraint (water volume in the upper reservoir) between the maximum V_MAX and minimum V_MIN value; (f) is the condition that the first time stage V₁ equals the last V_N; that is, at the beginning and the end of the period, the reservoir volume is maximum V_MAX (keeping in mind that the calculation should not be done as usual in calculating the storage hydro power plant (i.e., from the beginning until the end of the year, when there is a rainy period when the hydro reservoirs are full), but calculation should start from July 1 and finish with the same date, because at that time there is maximum SE and hydro reservoirs will be full); and (g) is the control variable constraint, i is time stage (increment), and N is the number of time steps.

The relationship between the PV and ST generators and working volume of the upper reservoir is given in Equations (8.3) and (8.4). However, if geosynthetics are involved in system “rain harvesting” technology, then this relationship changes, as shown in Figure 8.9 for ST and PV generators.

This can be explained by the fact that the capture of solar energy by ST power generator is weaker in winter (for several months, ST generator practically does not work because there is a lower threshold of solar radiation, i.e., 250 W/m², Eck and Zarza, 2006), and the energy produced by the ST generator is more stochastic than the energy produced by the PV generator.

8.6.2 The Dynamics of Charging and Discharging the PSH System

It is particularly interesting to observe the dynamics of charging and discharging the upper reservoir in the case of different working volumes V₀, which are shown in Figure 8.10. For this purpose, the observed changes in the working volume are from 10% to 90% in steps of 10% for both cases, respectively, and for PV-PSH and for ST-PSH systems; see Figure 8.10.

As shown, PV-PSH systems have a smoother transition from one time step to another compared to ST-PSH systems, which can also be explained by greater stochastic energy production from the ST generator. On the right side of Figure 8.10, the trends of increasing optimal nominal power generators are visible. As expected, the optimal nominal power ST generators are larger than the necessary PV power generators for the same reason that ST generators are inferior to PV generators in winter.