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The complementarity of energy resources used in hybrid power generation can result in optimization of power capacity and reservation capabilities. This article is dedicated to the study of hybrid hydro PV systems. The goal is to establish the relationship between system performance and complementarity of energy resources. The study was carried out with computer simulations based on a method that uses ideal functions developed to describe the energy resources and determines a limit of performance. The results confirm expectations that performance, as measured by the total time of failure to meet demand, will be better as energy resources are complementary. Charts relating energy complementarity with failures are presented. The subsequent research work shall proceed to at least two different phases. In the first one, the method exposed in the present work shall be applied to real data and compared to the operation of existing hybrid plants. In the second phase, results shall be confronted with design parameters of hydro PV plants based on complementary resources. A next stage would be the enlargement of the method applied in this work for systems based on other energy resources, such as wind energy and ocean wave energy.

Hybrid systems are an alternative technically and economically feasible for power generation in remote sites and in places where the power supply quality is compromised. Research on the performance of hybrid systems has enabled the use of energy resources that were not feasible in non-hybrids. The combination of dynamic characteristics of different energy resources has also allowed the overcoming of difficulties in meeting customer demand profiles.

The definition and the adjustment of strategies for the operation of hybrid systems are forcibly more complex than those for non-hybrid systems due to the complexity of the problems faced and the non-linear characteristics of some system components. The energy available from the sources involved may present some type of time or space complementarity or both and this complementarity may influence the sizing and the operation of the resulting power system.

The combination of hydroelectric and photovoltaic sources of energy in a generation system may reduce the cost of energy available in plants implemented in low hydroelectric potential sites for which main system interconnection costs prove prohibitive. Research on hybrid hydro PV plants can evaluate the use of water reservoirs and battery banks as alternatives for energy storage within a system and can also evaluate the advantages of generation from complementary energy sources.

A hybrid hydroelectric photovoltaic plant is a generation system based on a hydroelectric power plant and a set of photovoltaic modules operating together to satisfy the demand of an ensemble of consumer loads. The complementarity between the energy resources may then be beneficial for the sizing of components and for the operation of this type of system, possibly including various devices for energy storage. Reference [

If the hydro and solar energy availabilities present time-complementary, the storage devices may be sized to optimize the operation of the hybrid system. The water reservoir may be sized for a shorter period, implying lower costs and lesser environmental impacts if the hydro generator operates with a PV generator and a battery bank provided there is a complementarity between the two energy sources. Reference [

This article applies the method proposed by the author [

The performance is evaluated by comparing the total time of failure in meeting consumer demand, measured by the failure index, F, as defined by Equation (1),

where TF is the total time of failure measured in the time interval considered in the analysis and TA é the total time considered in the analysis, usually equal to one year.

The hybrid system under study, sketched in

The simulations adopt the pu system [

The operation of the hydro generator is obviously different from the usual operation of a hydroelectric power plant, as discussed for example by Reference [

The theoretical limit of performance [

certain source if it were insensitive to random events typical of the environment and insensitive to periods of extreme energy scarcity or extreme availability.

The maximum and minimum average values for the total instantaneous available power to the PV generator were obtained from monthly data [^{2}) and of 0.37108 (corresponding to an incident solar radiation of 371.08 W/m^{2}) are considered for the respective maximum and minimum instantaneous power available to the PV generator.

The simulated system is designed from the proportions between the total annual available and demanded energies (p_{ad}); the total annual available hydro and solar energies (p_{sh}); and the difference between the maximum and the minimum availabilities (p_{Mm}). For a constant demand profile the power required by the loads is considered equal to one. The available hydro and PV powers are defined as a function of the aforementioned proportions and present sinusoidal variations along the year.

_{H}), the black stands for the maximum daily photovoltaic power (p_{P}) and the green is the power consumed by the loads (p_{L}). It can be seen that there are no shutoffs of the hydroelectric generator and no supply failures.

The complementarity index, k, is defined by Equation (2), where k_{t} is the index for complementarity in time, k_{e} is the index for energy complementarity and k_{a} is the index for amplitude complementarity.

Equation (3) defines k_{t}, where D is the day of the year for maximum availability, d é the day for minimum availability and the subscripts h and s distinguish hydropower and solar energy. Equation (4) defines k_{e}, where E is the total annual energy. The definition of k_{a} is more complex and requires a query to the Reference [

In the simulated system the annual energy available for consumption is equal to the demanded annual energy (p_{ad} = 1.00) and also the complementarity index is equal to one (k_{C} = 1.00), which means perfect complementarity. Consequently, the time complementarity index k_{t}, the energy complementarity index, k_{e}, and the amplitude complementarity index, k_{a}, are all equal to one.

The total annual solar energy available is equal to the total annual hydro energy available for the simulated system (p_{sh} = 1.00) and the differences between the maximum and the minimum solar energy availability and between the maximum and the minimum hydro energy availabilities are also equal (p_{Mm} = 1.00).

The days of maximum availability of solar and hydro energy, as well as their days of maximum and minimum hydro and solar energy availability, have lagged 180 days, leading to a perfect time complementarity (k_{t} = 1.00). The ratio p_{sh} leads to a perfect energy complementarity (k_{e}_{ }= 1.00) and the ratio p_{Mm} leads to a perfect amplitude complementarity (k_{a} = 1.00).

This system was considered the “starting point” for the simulations, a pattern for easy comparisons with other results. The choice was made after a survey of different combinations of components and their importance for achieving the expected results.

The study of the effect of different degrees of complementarity on the performance of the energy system considered is based on the effects of the variations in the energy availability and of different combinations of installed power of hydro and of photovoltaic generator sets, in addition to the effects of the range of water discharges turning the hydroelectric generator.

The effect of variations in the time-complementarity index are illustrated in the results of reference [

for systems with 1, 2, 3, 5 and 10 days of batteries capacity for the results of references [3-9].

It is apparent that F is reduced with the increase in the time complementarity index and, consequently, with the improvement of the time complementarity characteristics of the energy availabilities. A lower capacity battery bank produces an upper shift of the curve due to a failure index increase while a bigger capacity bank produces a down shift towards smaller failure indexes.

It can be stated that the use of time-complementary energy resources contributes to the reduction of the failure index of a system, allowing the reduction of the size of the battery bank and the reduction of each generator installed power (as compared to equivalent one-generator systems of one type or the other).

For example, a system with k_{t} = 0.25 and somewhat less than 14% of failures, corresponding to approximately 50 days over a year, with a one season water reservoir may operate without supply failures. As another example, a system with k_{t} = 0.50 and 8% of failures would require a water reservoir with a capacity of six months. A feasibility study should determine the appropriate size for the storage in either of these two cases, but reservoirs for a few weeks should significantly reduce failures.

The effect of variations in the energy complementarity index is depicted in _{sh} and index k_{e} and the results for F appear listed in the legend of each result. Four results are shown, corresponding to the values of 1.43, 1.67, 0.70 and 0.60 for the ratio p_{sh} and the values of 0.8235 and 0.7500 for the energy complementtarity index. For reference, the system of _{sh} and k_{e}.

The values of p_{sh} imply different average energy availabilities. For the first two results the photovoltaic energy supplied tends to increase while the supplied hydro power tends to decrease. Since the solar availability presents great daily and yearly variations, when its relative importance increases, so does the failure index. These variations cause corresponding, however subtly, amplified variations in the battery stored energy.

In the other two results the available photovoltaic power tends to decrease and the hydroelectric power tends to increase. As a growing fraction of the system energy tends to be “firm”, in view of its hydro origin, the failure index tends to decrease. It can be seen that from (c) to (d), chiefly in (d) an increase in the battery stored energy in the left side of the graph, as a consequence of the higher hydro availability.

It should, however, be emphasized that the variations of F are quite small and no changes are seen in the battery stored energy. In general, the variations in the energy-complementarity index cause variations in the fractions of the total energy supplied by each of the two generators. As the total available energy along the year remains constant, no important modifications occur either in the battery behavior or in the system performance.

It can be seen, however. that a bigger contribution by the hydroelectric power component leads to a failure free system while a bigger contribution of the energy of photovoltaic modules tends to require a bigger battery bank to avoid the failures. To some extent this limits the

applicability of hybrid hydroelectric photovoltaic generating plants to situations in which the hydroelectric power availability is insufficient.

These results show how a bigger contribution of the energy of photovoltaic modules implies a bigger energy storage capacity. The utilization of a reservoir may become easier to justify in the cases of near perfect timecomplementarity as the suitable effect of the energycomplementarity may be artificially obtained by the accumulation of hydraulic energy. A small increase in storage capacity may be sufficient to accommodate daily and seasonal variations in solar availability.

The small influence of the variations of energy complementarity on the failure index is quite evident. Failures remain under 0.40%, as can be seen from the results of

The effect of variations in the amplitude complementarity index is shown in Figures 6-9, summarized in _{Mm} and index k_{a} and the results for F appear listed in the legend of each result. Eight results are shown for the amplitude complementarity index values of 0.99, 0.96, 0.91, 0.84 (_{Mm}, and 0.9878, 0.9412, 0.8448, 0.6923 (_{Mm}. Other eight results are shown to detail the values of index k_{a} close to unity, with 1.01, 1.02, 1.03, 1.04 (_{Mm}. For reference, the _{Mm} and the index k_{a}.

Importantly, the simulated systems with different ranges of energy availability, as they appear in different

values for the ratio p_{Mm}, do not involve differences in the gap between the maximum availability (since k_{t} = 1.00) or differences in energy availability (since k_{e} = 1.00).

Different values of p_{Mm} do not affect the average hydro availability but represent the difference between its maximum and minimum values. In the four initially shown results (

In the subsequent four results (

Failures occur in these cases always at the end of the period of sunlight, by depletion of energy stored in batteries. This feature can be proven by way of lines in the graphs of these figures. Similarly, the hydroelectric generator is disconnected at the end of the period of sunlight, since the batteries are fully charged.

It shall be emphasized that the failure levels are intermediate between the values caused by time complementtarity variations (which go as high as 15% in the worse situations) and those due to energy complementarity variations (that are under 0.12%).

Failures in

A reservoir with capacity of one week is sufficient to reduce failures to nearly zero for systems with more than

75% complementarity, which includes the results of

The use of real data, with daily variations of availability added to idealized values that were considered in the simulations, will change these results, requiring more storage capacity to compensate failures.

Obviously, it is possible to obtain storage capacity with other devices or other storage media. These water reservoirs above, with capacities of seven or fifteen days, could be substituted for example by banks of batteries. However, these banks would be extremely large compared to the reservoirs. Could also be used flywheels or air compressed or the system could be designed to include water or air heating.

There is a difference between the battery load variations over a year in