Energy potential of long-period oscillations (on the example of Kakhovka plain reservoir, Ukraine)

: The energy potential of long-period oscillations is estimated by comparing it with watercourse power. The relaxation time of long-period waves is chosen for the estimation time interval, during which their amplitude decreases e (Euler's number) times from the initial one. According to calculations, the amount of energy produced during this time by the watercourse is 9.35–18.71 million kW  h, while the amount of energy of long-period oscillations is 3–6 times less – 1.60–5.48 million kW  h. The components of the economic factor of using long-period waves and currents for electricity production are the predictability of their magnitudes and location of maxima, long-term availability, concentration.


Introduction
Concerned about the expected shortage of fossil fuels, we continue to look for alternatives for electricity production. Not much time has passed since the opening of the first power plant, a converter of moving water energy into electricity (1878, a small hydroelectric power plant on the Coquet River in England), compared to the history of creative humanity. However, the list of available surface water energy resources has expanded significantly due to the increasing technological capacity to use them (see Fig. 1). A strong periodicity and a certain height of tidal waves determined the use of tidal energy. The first tidal hydroelectric power plant was built in 1966 at the mouth of the Rance river, France [2].
Wind waves and capillary waves, unlike tides, are good because they occur in any water area. The first wave hydroelectric power plant was opened in 2008 near Aguçadoura, Portugal [2].
Analysis of the wave spectrum of Michigan and Ontario lakes (USA) revealed powerful long-period storm surges and standing waves (seiches) in their composition, which can also be used for electricity production [3,4].
The purpose of the paper is to evaluate the energy potential of long-period waves and currents of the Kakhovka plain reservoir (Ukraine), the largest of the Dnieper river cascade.

Method
Estimation of the potential of long-period waves is performed by comparing it with watercourse power [5]: where water density (10 3 kg/m 3 ); ggravitational acceleration (g9.8 m/s 2) ; Qwater consumption; H fall of water in the selected area.
Specific energy flux per unit width of wavefront and unit wavelength along its propagation direction over one oscillation period can be calculated by the formula [3,4]: where A=h/2, h, Tamplitude, height and period of the wave, respectively.
Seiche period is a function of the morphometric characteristics of the reservoir. The period of longitudinal seiches in a closed rectangular reservoir with a horizontal bottom (see Fig. 2) is calculated by the Merian formula [6]: where Mwave mode; L, Dlength and depth of the reservoir, respectively. Fig. 2. Profiles of the first two modes of longitudinal seiches in a closed reservoir [6].
The dynamics of the damping seiche is represented by the formula [7]: where Atamplitude at time t; A0initial oscillation amplitude at time t=0; angular frequency (=2/T); damping coefficient. The damping coefficient is inversely proportional to the relaxation time τ, during which the amplitude decreases e times from the initial one.
The wavefront power is calculated by the formula [8]: where the specific energy flux is taken under the sign of the module, which takes into account the change in the direction of wave motion relative to the non-excited surface. The wavefront power takes into account its lengths lw, which for a rectangular basin is equal to the width of the basin W, and the number of which is determined by the wave mode M (see Fig. 2).
Oscillations of seiche waves are accompanied by a reversible current (forward and backward movements of water), as shown in Fig. 3. The power density of the water flow moving at velocity V can be calculated by the formula [10]: Let us estimate the velocity of the waves, caused by the seiche current, by the formula [6]: Current power is calculated by the formula [8]: where the flow power density is taken under the module sign. The flow power takes into account the flow cross-sections sc, which for a rectangular basin are numerically equal to the product of the width and depth of the basin, and the number of which is determined by the wave mode M.
Waves and currents availability for a specified period of time t is calculated by the formula [11]: where twctotal duration of oscillation sessions over a specified period of time.  The excitement of seiches by the storm surge is shown in Fig. 4, the standing wave profiles, as suggested by M. Longuet-Higgins [13], are represented by the corresponding vibrational positions of the mathematical pendulum. Fig. 4. Standing wave oscillations: on the leftmathematical pendulum whose positions correspond to the standing wave profiles in Fig. 3; on the rightwater levels in the anti-node (1) and the flow velocity in the node (2), at fixed times t=0.25n, n=0,1,2,…  The dynamics of the seiche power of the Kakhovka reservoir during the oscillation relaxation time is shown in  The wave and current phases correspond to the oscillation profiles of Fig. 4. The authors of the papers [3,4], considering the surge energy on the Great Lakes (USA, Canada), examine extreme cases. Using their idea, let us consider the energy of long-period oscillations of the Kakhovka reservoir for a 106 cm surge (see Table 3).

Conclusions
1. Calculations showed the significant energy potential of the "storm surge -seiche wave and current" sequence. The amount of energy produced by the watercourse for 1.5-3 days is 9.35-18.71 million kWh, compared with the amount of energy produced by long-period waves and currents (1.60-5.48 million kWh).
2. Predictability of wave values, wave availability, location of maxima (see Table 4) are components of the economic factor of using them for electricity production.
3. The energies of waves and currents are distributed over the reservoir area, while the energy of falling water is concentrated in a relatively narrow line of the dam. To concentrate the energy of long-period waves, the authors of the papers [3,4] suggest using tidal basins.