Comparative Performance of a Hybrid Renewable Energy Generation System with Dynamic Load Demand

This article presents the modeling and simulation of a hybrid generation system, which uses solar energy generation, wind energy, and the regulation of a proton exchange membrane (PEM) cell to raise the demanded load, empowering the use of these hydride systems worldwide. This generation system was simulated for different locations in Puerto Bolivar (Colombia), Bremen (Germany), Beijing (China), and Texas (USA), for two demand profiles. The data used for the simulation was calculated using the mathematical solar model proposed by Beistow and Campbell for solar radiation. In contrast, for the wind resource evaluation, the Weibull probability distribution was used to calculate the most probable wind speed for each day, according to the historical data for each of the studied locations. Considering these data, the process transfer functions were used for tuning the control parameters for the hydrogen and oxygen production system. For the evaluation of the performance of these controllers, the indices of the absolute value of the error (IAE), the integral of the square of the error (ISE), the integral of the absolute value of the error for time (ITAE), and the integral of the square of the error for time (ITSE) were used. It was found that in the second load profile studied, better performance of the ITSE performance parameter was obtained, with stabilization times lower than those of the first profile.


Introduction
Global energy consumption has increased significantly due to population growth [1]. The increase in energy demand has prompted the search for alternatives for energy generation, the development of thermal processes according to the rational use of energy, preservation of the environment [2], and the improvement of thermal efficiency by mean of thermoeconomics modeling [3] and thermoeconomic optimization [4]. The development of new-generation systems using renewable energies has been one of the most widely studied options [5,6] because many countries have developed an energy policy that encourages the generation of environmentally friendly options based on clean-development mechanisms [7]. Thus, the contribution of these energy sources globally has been growing. Between 2016 and 2017, as recorded by Bloomberg New Energy Finance (BNEF), there was an increase in renewable energy of 2%, which equates to a rise of almost 157 GW in global power generation [8].
Electricity production from solar energy increased from 1050 GW in 1992 to 112,150 GW by 2013. Of this increase, 78% was produced by Japan, the United States, Germany, Spain, and Italy [9]. This was achieved thanks to various projects that were developed in different countries. In South Korea, a neural network model was built to calculate horizontal solar irradiation, taking images from the Korean

Methodology
The following are the geographic locations used in this work. These were selected based on the availability of data and variable environmental conditions. Similarly, the data obtained in these areas and the description of the generation process are presented together with the control systems. In addition, the mathematical models applied to the solar panel, the wind turbine, and the hydrogen cell are presented; these models were used for the simulation of the hybrid generation system.

Description and Information About the Region
For the evaluation and comparison of the hybrid trigeneration system, four countries were selected: USA, Germany, China, and Colombia. The first three countries were chosen for their great influence on their continents; on the other hand, Colombia was chosen because it is the country in which the study was carried out. According to the report of the National Renewable Energy Laboratory (NREL) of the United States, the central region of the Central American country has the greatest wind potential [24], i.e., the state of Texas, USA, which is why Texas was chosen for the current study. A study by Germany's federal environment agency also showed that the northern part of Germany, which includes the cities of Berlin, Bremen, and Hamburg, among others, is the area with the greatest wind power generation capacity. This is the reason why the city of Bremen was chosen for the current study [25]. Similarly, for the selection of the Chinese city in this study, an investigation on the wind speed pattern was taken into account, and it was found that the eastern part of China, in which Beijing is located, has great wind potential, so Beijing was chosen for the current study [26]. Finally, for Colombia, the department chosen was Guajira, taking into account the wind atlas of this country [27]. Figure 1 shows the coordinates where the data on solar irradiation and wind speed were collected. The wind speed and solar radiation data were obtained using the methodology set out in Section 2.4, with an hourly frequency, and correspond to a one-year interval, starting on 1 January 2019, and ending on 31 December of the same year. The wind speed data were calculated for a height of 50 m for the simulation.

System Description
The hybrid power generation system, presented in Figure 2, consists of a wind turbine and a solar panel. These two primary sources of generation feed the load demanded by the system. When the climatic conditions of the area are not sufficient to cover the demand, the hydrogen cell comes into operation to supplement the missing energy. Due to climatic conditions, excess energy may be used for the electrolysis of the water in the hydrogen cell. Figure 2 shows the schematic layout of the generation system studied. The input and output in the hydrogen cell system are managed by a proportional integral derivative (PID) control system, in which a constant load is used as a set point to facilitate the calculations.  The wind speed and solar radiation data were obtained using the methodology set out in Section 2.4, with an hourly frequency, and correspond to a one-year interval, starting on 1 January 2019, and ending on 31 December of the same year. The wind speed data were calculated for a height of 50 m for the simulation.

System Description
The hybrid power generation system, presented in Figure 2, consists of a wind turbine and a solar panel. These two primary sources of generation feed the load demanded by the system. When the climatic conditions of the area are not sufficient to cover the demand, the hydrogen cell comes into operation to supplement the missing energy. Due to climatic conditions, excess energy may be used for the electrolysis of the water in the hydrogen cell. Figure 2 shows the schematic layout of the generation system studied. The input and output in the hydrogen cell system are managed by a proportional integral derivative (PID) control system, in which a constant load is used as a set point to facilitate the calculations. The wind speed and solar radiation data were obtained using the methodology set out in Section 2.4, with an hourly frequency, and correspond to a one-year interval, starting on 1 January 2019, and ending on 31 December of the same year. The wind speed data were calculated for a height of 50 m for the simulation.

System Description
The hybrid power generation system, presented in Figure 2, consists of a wind turbine and a solar panel. These two primary sources of generation feed the load demanded by the system. When the climatic conditions of the area are not sufficient to cover the demand, the hydrogen cell comes into operation to supplement the missing energy. Due to climatic conditions, excess energy may be used for the electrolysis of the water in the hydrogen cell. Figure 2 shows the schematic layout of the generation system studied. The input and output in the hydrogen cell system are managed by a proportional integral derivative (PID) control system, in which a constant load is used as a set point to facilitate the calculations.

Model Description
To carry out the simulation of the hybrid trigeneration system, a mathematical model was made for each of the three energy sources used in the work. Solar energy is obtained through the photovoltaic method, which transforms direct solar radiation into electrical energy through panels. The model adopted to simulate the photovoltaic solar panel was based on the studies conducted by Pandiarajan and Muthu [28]. All of the relevant phenomena in the modeling of a photovoltaic panel are shown in Table 1.
Saturation current, (4) In this model, the light generated current in a PV module is calculated from the Equation (1), and the inverse saturated current is defined from the Equation (2). Furthermore, it is taken into account that the saturation current varies with the temperature of the panel, for which it is modeled and expressed, as seen in Equation (3). In Equation (4), the cell output current is described. The photovoltaic panel model allows us to describe mathematically the operation of the panel array included in the system studied and to observe its curves of current-voltage and power-voltage operating under standard conditions. The simulation allows for the observation and analysis of one or more variables as a function of time, where the input variable to the system is the annual radiation profile in the selected places, and the main output is power generated, which is regulated in the hybrid generation system according to the energy load profile. The parameters required for the simulation were obtained from the literature and technical information provided by the manufacturer of the equipment [28].
Similarly, wind energy is transformed into electrical energy by a turbine, which transforms the kinetic energy of the wind into rotational energy by the propellers. In turn, this rotational energy is transformed into electrical energy by a generator. The dynamic model used for the analysis in the simulation was the one presented by Khan and Iqbal [29], which is expressed in Equation (5).
This model was developed for a Southwest Wind Power Inc. AIR 403 wind turbine. The power generated varies with wind speed and behaves as described in Figure 3.
The power generation capacity of this turbine has a maximum speed limit of 40 miles per hour. At or above this wind speed, power generation becomes impossible. For the hydrogen cell, the model uses the expanded form of the Nemst Equation [30], which models the thermodynamic potential E and is presented in Table 2 as Equation (6). As for the concentration of dissolved oxygen at the gas-liquid interface, the Henry Law model [31] is used, which is shown in the table as Equation (7).
In addition, this model takes into account the excess voltage caused by the activation and the internal resistance, whose models were developed experimentally and are presented in Equations (8) and (9), respectively. Similarly, the output voltage of the fuel cell is modeled according to Equation (10), and the hydrogen production rate, according to Faraday's law, is presented in Equation (11). Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 25   (11) In addition, this model takes into account the excess voltage caused by the activation and the internal resistance, whose models were developed experimentally and are presented in Equations (8) and (9), respectively. Similarly, the output voltage of the fuel cell is modeled according to Equation (10), and the hydrogen production rate, according to Faraday's law, is presented in Equation (11).
A more detailed description of the models adopted for the development of this study can be found in the literature. The dynamic model of the fuel cell [32] is a function of the external fuel and the oxidant supply; in other words, as long as both flows are fed continuously and in equivalent proportions, the fuel cell will operate correctly in the hybrid energy generation system, which is regulated by means of the PID controller. This cell is composed of two electrodes: an anodic side in which hydrogen is fed, and a cathodic side in which oxygen is fed. The two electrodes act as platinum catalysts and are separated by a thin membrane of an electrolytic ionic conductor. The cell considered in this dynamic model in this research is a PEM (proton exchange membrane or electrolytic polymer membrane), which allows protons to flow through the cell, but stops the movement of electrons, reactants, and products through the cell [29].
Overvoltage activation, n act [V] n act = −0.96514 + 0.00312T − 0.000187· ln(i)    (11) In addition, this model takes into account the excess voltage caused by the activation and the internal resistance, whose models were developed experimentally and are presented in Equations (8) and (9), respectively. Similarly, the output voltage of the fuel cell is modeled according to Equation (10), and the hydrogen production rate, according to Faraday's law, is presented in Equation (11).
A more detailed description of the models adopted for the development of this study can be found in the literature. The dynamic model of the fuel cell [32] is a function of the external fuel and the oxidant supply; in other words, as long as both flows are fed continuously and in equivalent proportions, the fuel cell will operate correctly in the hybrid energy generation system, which is regulated by means of the PID controller. This cell is composed of two electrodes: an anodic side in which hydrogen is fed, and a cathodic side in which oxygen is fed. The two electrodes act as platinum catalysts and are separated by a thin membrane of an electrolytic ionic conductor. The cell considered in this dynamic model in this research is a PEM (proton exchange membrane or electrolytic polymer membrane), which allows protons to flow through the cell, but stops the movement of electrons, reactants, and products through the cell [29]. ohmic (10) Hydrogen production rate, n H 2 [mol/s] n H 2 = n F ·n c ·i e 2F (11) A more detailed description of the models adopted for the development of this study can be found in the literature. The dynamic model of the fuel cell [32] is a function of the external fuel and the oxidant supply; in other words, as long as both flows are fed continuously and in equivalent proportions, the fuel cell will operate correctly in the hybrid energy generation system, which is regulated by means of the PID controller. This cell is composed of two electrodes: an anodic side in which hydrogen is fed, and a cathodic side in which oxygen is fed. The two electrodes act as platinum catalysts and are separated by a thin membrane of an electrolytic ionic conductor. The cell considered in this dynamic model in this research is a PEM (proton exchange membrane or electrolytic polymer membrane), which allows protons to flow through the cell, but stops the movement of electrons, reactants, and products through the cell [29].

Control System Description
For the control system of the hybrid generation system, steady-state operational values are required. These values were defined for both the fuel cell and the photovoltaic cell based on research results presented in the literature [32]. The values for the fuel cell are presented in Table 3. For the photovoltaic cell, the assigned constant values are presented in Table 4. A PID control is used to control the hybrid system, and the tuning of these controllers was done using the Ziegler-Nichols open-loop method [33]. The approach of the plant to a first-order process plus dead time was made by a set of open-loop tests.
First, the gain, tao, and dead time were determined for the first-order equation, following the procedures outlined in Smith and Corripio [34]. The procedure was performed for a variation in the hydrogen flow rate from 0.005 to 0.008 mol/s while keeping the oxygen flow constant. Then, the oxygen flow rate was changed while keeping the hydrogen flow constant. The transfer functions obtained for hydrogen and oxygen are presented in Equations (12) and (13), respectively.

G(S) oxygen =
125 · e −1.21s 18.20 · s + 1 (12) Having obtained the values, tuning was carried out using the Ziegler-Nichols rules [35], resulting in the tuning parameters for the oxygen and hydrogen PID controllers shown in Table 5. To evaluate the performance of the PIDs for the different load profiles, four indices were used to calculate the time taken to reach system stabilization [36]. The first index is defined by the integral of the square of the error, as seen in Equation (14).
Likewise, the integral of the time-weighted absolute error is described according to Equation (16).
Finally, the integral of the time-weighted squared error is defined by Equation (17).
where t sim is the simulation elapsed time, n represents the number of iterations, and w is the output values in each iteration executed.

Data Resources
One of the most important inputs for this research is the wind speed and solar radiation data. These data were the operational input variables for the hybrid trigeneration system, and they were used to evaluate the system performance and to conduct the analysis. The USA, Germany, and China were chosen because they are economies with a high impact on their regions and have made significant progress in generating clean energy. To these three nations was added Colombia, because it is the country in which the research as carried out, and it presents promising wind [37] and solar generation potential [38].
Having selected the countries, we proceeded to choose the places in these countries where we would collect data; the frequency distribution of wind speed was considered, as shown in Figure 4.
For this, the solar potential of each country was taken into account, and data were collected in the places with the highest potential. There are several models to predict solar radiation. One of the main models was proposed both by Anstrom [27] and Prescot [28] and uses daily sun hours. This method is also known as AP. Similarly, Bristow and Campbell [29] proposed a method called BC correlation, in addition to the method proposed by Hargreaves and Samani [30] known as HS correlation. These methods use the minimum and maximum daily temperature in both cases, as well as theoretical values of incident extraterrestrial radiation.
The model used to obtain the data in this investigation was Bristow and Campbell, which uses variables typical of a physical phenomenon, as seen in Equation (18).
where a is the transmissivity of the atmosphere, b and c are constants assigned to each region, H O is the daily extraterrestrial energy and depends on the day of the year and the latitude, and ∆T is the difference between the maximum and minimum daily temperature. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 25 As for wind speed, the prediction was made by taking into account the Weibull probability distribution, which is affected by shape parameters (k or ) ranging from 1 to 3.6 and is dimensionless. This probability is also dependent on a scale parameter (c, or ) [39]. This probability is described by Equation (19).
where , , and > 0 are in m/s. This function describes the relative probability of a random variable such as wind occurring at a given time; that is, it characterizes the probable behavior of wind data remeasured by the meteorological station. Thus, the probability of a random variable to fall within a particular region is given by the integral of the density of this variable in that region.
The wind potential is determined to calculate the Weibull distribution parameters by mean of the least-squares method. Weibull is a continuous and triparametric distribution. In other words, it is completely defined by three parameters, and it is the most-used model in the field of wind energy [40]. For this research, the distribution of two parameters was selected, which corresponds to the scale parameter c in m/s that determines the average wind speed in the place of study and the shape parameter k that indicates the degree of dispersion of the records. With these parameters, the frequency with which certain wind speeds are manifested in the hybrid generation system can be established.
These calculated wind speed data are presented in Figure 5. This figure shows the wind speeds for Texas (USA), Bremen (Germany), Beijing (China), and Puerto Bolivar (Colombia) throughout the year 2019. In Figure 5, in the case of China, you can see the maximum wind speed of 12.9 m/s was reached on Monday 13 May. Another important value is the minimum wind speed, which happened on the first day of January, registering 2.6 m/s. As for the average wind speed, its value was 6.7 m/s. As for wind speed, the prediction was made by taking into account the Weibull probability distribution, which is affected by shape parameters (k or α) ranging from 1 to 3.6 and is dimensionless. This probability is also dependent on a scale parameter (c, θ or β) [39]. This probability is described by Equation (19).
where x, α, and θ > 0 are in m/s. This function describes the relative probability of a random variable such as wind occurring at a given time; that is, it characterizes the probable behavior of wind data remeasured by the meteorological station. Thus, the probability of a random variable to fall within a particular region is given by the integral of the density of this variable in that region.
The wind potential is determined to calculate the Weibull distribution parameters by mean of the least-squares method. Weibull is a continuous and triparametric distribution. In other words, it is completely defined by three parameters, and it is the most-used model in the field of wind energy [40]. For this research, the distribution of two parameters was selected, which corresponds to the scale parameter c in m/s that determines the average wind speed in the place of study and the shape parameter k that indicates the degree of dispersion of the records. With these parameters, the frequency with which certain wind speeds are manifested in the hybrid generation system can be established.
These calculated wind speed data are presented in Figure 5. This figure shows the wind speeds for Texas (USA), Bremen (Germany), Beijing (China), and Puerto Bolivar (Colombia) throughout the year 2019. In Figure 5, in the case of China, you can see the maximum wind speed of 12.9 m/s was reached on Monday 13 May. Another important value is the minimum wind speed, which happened on the first day of January, registering 2.6 m/s. As for the average wind speed, its value was 6.7 m/s.  Similarly, for solar radiation, Figure 6 shows the calculated radiation levels for the four locations. Figure 6 shows that the region with the highest radiation is Puerto Bolivar (Colombia), this trend was maintained for most of the year; it was only surpassed by Texas (USA) in July.
Once the wind speed and radiation data were obtained in the places chosen for the study, two scenarios were proposed in which the simulations for the generation of the hybrid system would be carried out. These two scenarios are load profiles that made demands on the system during the simulation year. Figure 7 shows the daily behavior of the load profiles selected for the simulation. The selected load profiles correspond to the average energy consumed for the years 2019 (load profile 1) and 2018 (load profile 2) of the lighting system of the renewable energy laboratory of the Universidad del Atlántico. Similarly, for solar radiation, Figure 6 shows the calculated radiation levels for the four locations. Figure 6 shows that the region with the highest radiation is Puerto Bolivar (Colombia), this trend was maintained for most of the year; it was only surpassed by Texas (USA) in July.
Once the wind speed and radiation data were obtained in the places chosen for the study, two scenarios were proposed in which the simulations for the generation of the hybrid system would be carried out. These two scenarios are load profiles that made demands on the system during the simulation year. Figure 7 shows the daily behavior of the load profiles selected for the simulation. The selected load profiles correspond to the average energy consumed for the years 2019 (load profile 1) and 2018 (load profile 2) of the lighting system of the renewable energy laboratory of the Universidad del Atlántico. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 25

Results and Discussions
Based on the dynamic conditions established for the different places of interest by the control system, an evaluation of the behavior of the molar flows of and was carried out to identify the energy contribution of the PEM cell. The PEM cell complemented the energy generation provided by the wind turbine and the solar cell, achieving with these three, the dynamic load required by the system. The PID control system regulated this molar flow. This regulation resulted in the behavior presented in Figure 8. For the case of flow observed in Figure 8a, the molar flow was maintained at its minimum value to keep the PEM cell in operation without making energy contributions to the hybrid generation system in the four locations studied. On 19th October (day 292 of the year), the flow in Texas increased from 7.50 × 10 mol/s to 5.50 × 10 mol/s; this same increase was presented on 21 October (day 294 of the year) and 12 October (day 285 of the year) for Bremen and

Results and Discussions
Based on the dynamic conditions established for the different places of interest by the control system, an evaluation of the behavior of the molar flows of and was carried out to identify the energy contribution of the PEM cell. The PEM cell complemented the energy generation provided by the wind turbine and the solar cell, achieving with these three, the dynamic load required by the system. The PID control system regulated this molar flow. This regulation resulted in the behavior presented in Figure 8. For the case of flow observed in Figure 8a, the molar flow was maintained at its minimum value to keep the PEM cell in operation without making energy contributions to the hybrid generation system in the four locations studied. On 19th October (day 292 of the year), the flow in Texas increased from 7.50 × 10 mol/s to 5.50 × 10 mol/s; this same increase was presented on 21 October (day 294 of the year) and 12 October (day 285 of the year) for Bremen and

Results and Discussions
Based on the dynamic conditions established for the different places of interest by the control system, an evaluation of the behavior of the molar flows of H 2 and O 2 was carried out to identify the energy contribution of the PEM cell. The PEM cell complemented the energy generation provided by the wind turbine and the solar cell, achieving with these three, the dynamic load required by the system. The PID control system regulated this molar flow. This regulation resulted in the behavior presented in Figure 8. Similarly, for the flow of presented in Figure 8b, the molar flow was kept at its minimum value to keep the PEM cell in operation without providing energy to the hybrid generation system in the four locations studied. The increment of the molar flow of for 22 October went from 7.50 × 10 mol/s to 2.91 × 10 for Texas. Similarly, for Bremen, it went from 7.50 × 10 mol/s to 8.32 × 10 mol/s on 23 October, and from 7.50 × 10 mol/s to 1.94 × 10 mol/s on 14 October for Beijing. In the case of Puerto Bolivar, on 29 November there was an increase from 7.50 × 10 mol/s to 2.14 × 10 mol/s. The days indicated with an increment for and are the first days on which the PEM cell contributed to the generation to achieve the required load.
Two of the limitations to the proposed control system for the hybrid system based on nonconventional energy sources were to model the components of the system and to know the meteorological variables that allow for prediction of the energy generation of the system. Thus, Song-Yul Choe [41] performed dynamic modeling of a PEM-type fuel cell, where the load demand, fuel flow, and temperature play an important role when applying a control system. The dynamics and performance of the designed controllers were evaluated and analyzed by simulations using dynamic fuel cell system models, similar to the one proposed in the current study, using a multistep current and an actual load profile.
Likewise, Granda-Gutiérrez et al. [42] presented a model of a fuel cell integrated to a solar panel array and studied the effect of solar irradiation and solar cells on the voltage, current, and power of the fuel cell. One of the main limitations of that work, compared to the results presented in the current article, is that the system considers the wind resource in the hybrid system, and it also allows for the study of the time evolution of the complementarity of the energies and the performance of the control system to satisfy the energy demand.
For the case of load profile 2, Figure 9 illustrates the molar flow behavior of and . Figure 9a shows the variation of flow required for the PEM cell. Initially, for the four locations, the flow presented is the minimum to keep the PEM cell working, and from 10th July, the flow for Texas showed an increase from 7.50 × 10 mol/s to 9.65 × 10 mol/s. Similarly, on 31 August and 4 September there was a change compared to the minimum flow value of in Bremen and Beijing, respectively; it went from 7.50 × 10 mol/s to 1.29 × 10 mol/s for Bremen and from 7.50 × 10 mol/s to 1.70 × 10 mol/s for Beijing. In the case of Puerto Bolivar, the increase of the minimum molar flow value of occurred on 7 October and went from 7.50 × 10 mol/s to 1.70 × 10 mol/s. Similarly, for the flow of H 2 presented in Figure 8b, the molar flow was kept at its minimum value to keep the PEM cell in operation without providing energy to the hybrid generation system in the four locations studied. The increment of the molar flow of H 2 for 22 October went from 7.50 × 10 −5 mol/s to 2.91 × 10 −3 for Texas. Similarly, for Bremen, it went from 7.50 × 10 −5 mol/s to 8.32 × 10 −4 mol/s on 23 October, and from 7.50 × 10 −5 mol/s to 1.94 × 10 −3 mol/s on 14 October for Beijing. In the case of Puerto Bolivar, on 29 November there was an increase from 7.50 × 10 −5 mol/s to 2.14 × 10 −3 mol/s. The days indicated with an increment for H 2 and O 2 are the first days on which the PEM cell contributed to the generation to achieve the required load.
Two of the limitations to the proposed control system for the hybrid system based on non-conventional energy sources were to model the components of the system and to know the meteorological variables that allow for prediction of the energy generation of the system. Thus, Song-Yul Choe [41] performed dynamic modeling of a PEM-type fuel cell, where the load demand, fuel flow, and temperature play an important role when applying a control system. The dynamics and performance of the designed controllers were evaluated and analyzed by simulations using dynamic fuel cell system models, similar to the one proposed in the current study, using a multistep current and an actual load profile.
Likewise, Granda-Gutiérrez et al. [42] presented a model of a fuel cell integrated to a solar panel array and studied the effect of solar irradiation and solar cells on the voltage, current, and power of the fuel cell. One of the main limitations of that work, compared to the results presented in the current article, is that the system considers the wind resource in the hybrid system, and it also allows for the study of the time evolution of the complementarity of the energies and the performance of the control system to satisfy the energy demand.
For the case of load profile 2, Figure 9 illustrates the molar flow behavior of H 2 and O 2 . As for the flow of , in Figure 9b it can be seen that for the case of Texas, on day 220 the first increase in molar flow was presented, going from 7.50 × 10 mol/s a 1.26 × 10 mol/s. In the case of Bremen and Beijing, the first increases occurred on 3 September and 6 September, respectively; going from 7.50 × 10 mol/s to 7.77 × 10 mol/s for the Bremen and from 7.50 × 10 mol/s to 3.00 × 10 mol/s for Beijing. On the other hand, in Puerto Bolivar, this increase in molar flow did not occur until 8 October, from 7.50 × 10 mol/s a 1.10 × 10 mol/s. The variations in the intensities of solar radiation and wind speed that motivate these variations in the molar flows of and change the amount of power generated by each of the sources. These combinations are variable for each of the geographical areas analyzed, as well as for each load profile. These powers were analyzed for each of the areas studied. Figure 10 shows the power generated by the system for Puerto Bolivar (Colombia) for the demanded load profile 1. Figure 10 highlights what happened between 19 July and 7 September, days on which the generated power exceeded the power demand stipulated in the load profile. This was due to above-average winds and a slight increase in average solar radiation added to the low demand at this time of year, which led to the observed overshoot.  As for the flow of H 2 , in Figure 9b it can be seen that for the case of Texas, on day 220 the first increase in molar flow was presented, going from 7.50 × 10 −5 mol/s a 1.26 × 10 −3 mol/s. In the case of Bremen and Beijing, the first increases occurred on 3 September and 6 September, respectively; going from 7.50 × 10 −5 mol/s to 7.77 × 10 −4 mol/s for the Bremen and from 7.50 × 10 −5 mol/s to 3.00 × 10 −3 mol/s for Beijing. On the other hand, in Puerto Bolivar, this increase in molar flow did not occur until 8 October, from 7.50 × 10 −5 mol/s a 1.10 × 10 −4 mol/s. The variations in the intensities of solar radiation and wind speed that motivate these variations in the molar flows of H 2 and O 2 change the amount of power generated by each of the sources. These combinations are variable for each of the geographical areas analyzed, as well as for each load profile. These powers were analyzed for each of the areas studied. Figure 10 shows the power generated by the system for Puerto Bolivar (Colombia) for the demanded load profile 1. Figure 10 highlights what happened between 19 July and 7 September, days on which the generated power exceeded the power demand stipulated in the load profile. This was due to above-average winds and a slight increase in average solar radiation added to the low demand at this time of year, which led to the observed overshoot. As for the flow of , in Figure 9b it can be seen that for the case of Texas, on day 220 the first increase in molar flow was presented, going from 7.50 × 10 mol/s a 1.26 × 10 mol/s. In the case of Bremen and Beijing, the first increases occurred on 3 September and 6 September, respectively; going from 7.50 × 10 mol/s to 7.77 × 10 mol/s for the Bremen and from 7.50 × 10 mol/s to 3.00 × 10 mol/s for Beijing. On the other hand, in Puerto Bolivar, this increase in molar flow did not occur until 8 October, from 7.50 × 10 mol/s a 1.10 × 10 mol/s. The variations in the intensities of solar radiation and wind speed that motivate these variations in the molar flows of and change the amount of power generated by each of the sources. These combinations are variable for each of the geographical areas analyzed, as well as for each load profile. These powers were analyzed for each of the areas studied. Figure 10 shows the power generated by the system for Puerto Bolivar (Colombia) for the demanded load profile 1. Figure 10 highlights what happened between 19 July and 7 September, days on which the generated power exceeded the power demand stipulated in the load profile. This was due to above-average winds and a slight increase in average solar radiation added to the low demand at this time of year, which led to the observed overshoot.  Similarly, Figure 11 shows the power generated by the system in Beijing (China) for load profile 1. As in the previous case, increases in solar radiation and wind speed and a decrease in load demand coincide. This point is relevant due to the fact that the excess in the demanded charge was maintained from 19 July to 7 September, unlike the cases that occured on the previous days, which were only daily peaks in wind speed. This figure also shows the influence of the solar radiation peak highlighted in Figure 6; this increase in radiation occurred at the time of year when the load demand for the simulation was higher, so the system needed the PEM fuel cell operation to achieve the load demand. However, the tunning parameters of the control system are responsible for ensuring that the hybrid system follows the energy demand, regardless of the wind or solar resource peaks.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 25 Similarly, Figure 11 shows the power generated by the system in Beijing (China) for load profile 1. As in the previous case, increases in solar radiation and wind speed and a decrease in load demand coincide. This point is relevant due to the fact that the excess in the demanded charge was maintained from 19 July to 7 September, unlike the cases that occured on the previous days, which were only daily peaks in wind speed. This figure also shows the influence of the solar radiation peak highlighted in Figure 6; this increase in radiation occurred at the time of year when the load demand for the simulation was higher, so the system needed the PEM fuel cell operation to achieve the load demand. However, the tunning parameters of the control system are responsible for ensuring that the hybrid system follows the energy demand, regardless of the wind or solar resource peaks. Likewise, Figure 12 shows the power generated by the system in Texas (USA) for load profile 1. Figure 11 highlights the last days of January, together with the month of February, in which there was an increase in wind speed, causing a generated power higher than the demanded load profile. The rest of the time, the generation contributed by the solar radiation and the wind was complemented with the contribution of the PEM cell; this contribution is evident in the increase of the flows of and presented in Figure 8a,b.  Likewise, Figure 12 shows the power generated by the system in Texas (USA) for load profile 1. Figure 11 highlights the last days of January, together with the month of February, in which there was an increase in wind speed, causing a generated power higher than the demanded load profile. The rest of the time, the generation contributed by the solar radiation and the wind was complemented with the contribution of the PEM cell; this contribution is evident in the increase of the flows of H 2 and O 2 presented in Figure 8a Similarly, Figure 11 shows the power generated by the system in Beijing (China) for load profile 1. As in the previous case, increases in solar radiation and wind speed and a decrease in load demand coincide. This point is relevant due to the fact that the excess in the demanded charge was maintained from 19 July to 7 September, unlike the cases that occured on the previous days, which were only daily peaks in wind speed. This figure also shows the influence of the solar radiation peak highlighted in Figure 6; this increase in radiation occurred at the time of year when the load demand for the simulation was higher, so the system needed the PEM fuel cell operation to achieve the load demand. However, the tunning parameters of the control system are responsible for ensuring that the hybrid system follows the energy demand, regardless of the wind or solar resource peaks. Likewise, Figure 12 shows the power generated by the system in Texas (USA) for load profile 1. Figure 11 highlights the last days of January, together with the month of February, in which there was an increase in wind speed, causing a generated power higher than the demanded load profile. The rest of the time, the generation contributed by the solar radiation and the wind was complemented with the contribution of the PEM cell; this contribution is evident in the increase of the flows of and presented in Figure 8a,b.  In the case of Bremen (Germany), the results are presented in Figure 13. In Figure 13 it can be seen that in the first 100 days of the year, there were peaks in wind speed, which triggered an increase in the energy generated that exceeded the load profile. This increase was the highest presented in the four locations analyzed and is due to the increase in wind speed presented on those days, which, as in the previous peaks, generated a high activity in the electrolyzer observed in the purple area. Similarly, Figure 13 shows that the contribution of solar energy made at the peak of solar radiation highlighted in Figure 6 is not as high as the contribution made by the wind speed when it reached more than 10 m/s in the following days.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 25 In the case of Bremen (Germany), the results are presented in Figure 13. In Figure 13 it can be seen that in the first 100 days of the year, there were peaks in wind speed, which triggered an increase in the energy generated that exceeded the load profile. This increase was the highest presented in the four locations analyzed and is due to the increase in wind speed presented on those days, which, as in the previous peaks, generated a high activity in the electrolyzer observed in the purple area. Similarly, Figure 13 shows that the contribution of solar energy made at the peak of solar radiation highlighted in Figure 6 is not as high as the contribution made by the wind speed when it reached more than 10 m/s in the following days. These generation peaks occur due to increased wind and solar radiation causing an overshoot of the load demanded by the profile. In Table 6, a sum of the total energy generated monthly by the solar panel, the wind turbine, and the PEM cell for load profile 1 is presented. In addition, a comparison is made with the load demanded for the same period in each of the areas studied. This comparison allows us to observe that Bremen is the location with the highest excess energy generated due to the peaks of wind speed and solar radiation in the whole simulated period, reaching 9801 W, well above the 1389 W in Texas, equivalent to 14% of the excess energy generated in Bremen. Likewise, 3549 W was generated in excess in Beijing to reach 36% of what was generated in Bremen. Finally, in Puerto Bolivar, 4828 W was generated, reaching 49% of what was generated in Bremen. In spite of this, Bremen is the location with the second-highest number of months with a generation equal to the demand due to the load profile reaching 5 months, only surpassed by Texas that reached 6 months without excess generation, which shows that the high wind and solar activity presented in the first 3 months was decisive to achieve the 9801 W of excess generation in Bremen.
On the other hand, Table 7 presents for load profile 2 the total sum of the energy generated monthly for each of the actors in the hybrid generation system. It shows Bremen as the location with the greatest excess of energy generated, reaching 7071 W, quite far from the excess generated in Texas, which only reached 204 W, meaning only 3.41% of the excess generated in Bremen. Likewise, in Beijing, 1600 W of excess energy was generated, representing 22% of the excess generated in Bremen. On the other hand, Puerto Bolivar was the closest location to Bremen, with an excess generation of 2646 W, being 37% of what Bremen generated. Besides this, Puerto Bolivar was the location that presented an excess of generation in more months, with 10 months exceeding the demand presented by the load profile 2. Only in September and December was this demand not exceeded. The opposite was the case with Texas, where the demand for the profile was exceeded in only three months and the PEM cell had to cover the demand imposed by the load profile. These generation peaks occur due to increased wind and solar radiation causing an overshoot of the load demanded by the profile. In Table 6, a sum of the total energy generated monthly by the solar panel, the wind turbine, and the PEM cell for load profile 1 is presented. In addition, a comparison is made with the load demanded for the same period in each of the areas studied. This comparison allows us to observe that Bremen is the location with the highest excess energy generated due to the peaks of wind speed and solar radiation in the whole simulated period, reaching 9801 W, well above the 1389 W in Texas, equivalent to 14% of the excess energy generated in Bremen. Likewise, 3549 W was generated in excess in Beijing to reach 36% of what was generated in Bremen. Finally, in Puerto Bolivar, 4828 W was generated, reaching 49% of what was generated in Bremen. In spite of this, Bremen is the location with the second-highest number of months with a generation equal to the demand due to the load profile reaching 5 months, only surpassed by Texas that reached 6 months without excess generation, which shows that the high wind and solar activity presented in the first 3 months was decisive to achieve the 9801 W of excess generation in Bremen.
On the other hand, Table 7 presents for load profile 2 the total sum of the energy generated monthly for each of the actors in the hybrid generation system. It shows Bremen as the location with the greatest excess of energy generated, reaching 7071 W, quite far from the excess generated in Texas, which only reached 204 W, meaning only 3.41% of the excess generated in Bremen. Likewise, in Beijing, 1600 W of excess energy was generated, representing 22% of the excess generated in Bremen. On the other hand, Puerto Bolivar was the closest location to Bremen, with an excess generation of 2646 W, being 37% of what Bremen generated. Besides this, Puerto Bolivar was the location that presented an excess of generation in more months, with 10 months exceeding the demand presented by the load profile 2. Only in September and December was this demand not exceeded. The opposite was the case with Texas, where the demand for the profile was exceeded in only three months and the PEM cell had to cover the demand imposed by the load profile.  For the evaluation of the performance of the controllers, the four indices selected for this evaluation are presented in Figure 14. For each of these indexes, the stabilization times were calculated, and the sum of all these times during the simulation year was plotted, obtaining two indicator values for each location studied. Figure 14a shows the IAE values for load profile 1 in black and for load profile 2 in gray; it can be seen that the stabilization time for load profile 2 is approximately 21% less on average, with the smallest time difference in Puerto Bolivar, presenting 17% less time for the second load profile. Figure 14b shows that the trend presented in the IAE is maintained in the ISE. In this indication, the stabilization time is more significant for load profile 1, and on average, the time for load profile 2 is approximately 31% less, reaching the most considerable difference in Texas where the stabilization time for the second load profile is 45% less. This significant difference in stabilization time occurs for two main reasons. The first is the difference in the load profiles; the changes in the load are smoother than in the second load profile than they are in the first. The second reason for this shorter stabilization time is that the power generated by the system in Texas for the second load profile almost never exceeds the demanded load, which allows reaching the setpoint more easily.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 25 For the evaluation of the performance of the controllers, the four indices selected for this evaluation are presented in Figure 14. For each of these indexes, the stabilization times were calculated, and the sum of all these times during the simulation year was plotted, obtaining two indicator values for each location studied. Figure 14a shows the IAE values for load profile 1 in black and for load profile 2 in gray; it can be seen that the stabilization time for load profile 2 is approximately 21% less on average, with the smallest time difference in Puerto Bolivar, presenting 17% less time for the second load profile. Figure 14b shows that the trend presented in the IAE is maintained in the ISE. In this indication, the stabilization time is more significant for load profile 1, and on average, the time for load profile 2 is approximately 31% less, reaching the most considerable difference in Texas where the stabilization time for the second load profile is 45% less. This significant difference in stabilization time occurs for two main reasons. The first is the difference in the load profiles; the changes in the load are smoother than in the second load profile than they are in the first. The second reason for this shorter stabilization time is that the power generated by the system in Texas for the second load profile almost never exceeds the demanded load, which allows reaching the setpoint more easily. On the other hand, in Figure 14c the ITAE indicator is shown, in which it can be observed that, as in the two previous cases, the second load profile generates a time of stabilization less than the first, due to the smooth variation in the load demanded from the system. In this indicator, the load profile is 27% less on average for the four locations, pointing to Puerto Bolivar as the location that presented a smaller difference than the others, reaching a 19% difference between the profiles. This is due to the constant surpassing of the demanded load profile, caused by the constant increases in wind speed in this area. On the other hand, in Figure 14c the ITAE indicator is shown, in which it can be observed that, as in the two previous cases, the second load profile generates a time of stabilization less than the first, due to the smooth variation in the load demanded from the system. In this indicator, the load profile is 27% less on average for the four locations, pointing to Puerto Bolivar as the location that presented a smaller difference than the others, reaching a 19% difference between the profiles. This is due to the constant surpassing of the demanded load profile, caused by the constant increases in wind speed in this area.
Finally, Figure 14d shows, for the ITSE indicator, the most significant difference in stabilization time between the two load profiles. On average, the time for the second load profile is 44% less than the time required by the control system to achieve stabilization for the first profile. In this indicator, the difference in stabilization time between the two profiles presented in Texas is 57%. This difference is due to the fact that solar and wind power generation was always below the load profile.
The results presented in this article were developed from a dynamic hybrid system model similar to the one proposed by M.T. Iqba et al. [29], who presented the simulation of a photovoltaic component system, considering only a Southwest Wind Power Inc. AIR 403 wind turbine, a PEM fuel cell, and an electrolyzer. The dynamic model was evaluated in Simulink, and the transient responses of the system under a step change of the power load and wind speed were studied, regulating the system with a classical PID control system. In addition, G. E. Valencia et al. [43] tested the hybrid system model without a solar photovoltaic source and improved the energy performance using the generalized predictive control system, also simulated in MATLAB/Simulink software, which is a widely software used to simulate model of energy generation systems [44]. However, the application of this hybrid system guarantees the maximum use of the renewable energy resources available in the places selected in this study, since they are places with high solar radiation.

Conclusions
When reviewing the wind and solar behavior of the four selected locations, it was concluded that the level of solar radiation presented by Puerto Bolivar was higher than that presented by Texas, Beijing, and Bremen. Only Texas presented a similar behavior due to the fact that both regions are desert zones. As for the wind speed, the behavior presented was much more variable, with each region presenting its peak at different times of the year. The highest peaks occurred in the first three months of the year in all four locations.
Similarly, the simulation allowed us to conclude that the implementation of a PEM cell as a support to wind and solar generation system was very effective in meeting variable energy demand. The PEM cell, together with a PID system, was able to deliver the necessary energy in spite of the fact that in some seasons of the year, the climatic conditions to cover the demand were not available, as was observed in Texas for the load profile 2. This control system was strongly affected by the high peaks in the input variables over time. This can be appreciated in the first three months of the year in Bremen. The high wind supply made it difficult for the hybrid trigeneration system to maintain a stable power output by increasing the settling time, which could cause problems by exceeding the demanded load.
In addition, the evaluation of the performance of the controller through the indicators IAE, ISE, ITAE, and ITSE, showed that the PID used presented a shorter error stabilization time when the changes in the load were smoother, and the error was given by radiation levels, and wind speeds lower than necessary, forcing the PID to vary the molar flow of H 2 and O 2 .
The results allow us to conclude that the difficulty of controlling a multivariable process such as a hybrid generation system is not limited only in the number of variables, but also by the interaction that exists between the subsystems (wind turbine, fuel cell, and solar photovoltaic system). The degree of interaction will determine whether the control strategy to control the process should be decentralized or centralized. Therefore, it is necessary to have interaction measures to help make such a decision, as is the case of the relative gain matrix, which is a heuristic technique without a strong theoretical basis, but rigorous connections have been established between the relative gain matrices and the stability for systems from their transfer functions. Thus, a relative gain matrix is proposed to advance in the study of the stability of multivariable control systems of hybrid energy generation processes.
In the ITSE, it was observed that the second load profile in the study generated a lower stabilization time than the first one, due to the smooth variation in the load demanded from the system. In this indicator, the load profile was 27% lower on average for the four locations, with Puerto Bolivar standing out as the location that presented the smallest difference from the others, reaching a 19% difference between the profiles due to the variability in the study period for wind and solar resources.   (11) excess voltage caused by the activation and the xperimentally and are presented in Equations (8) f the fuel cell is modeled according to Equation o Faraday's law, is presented in Equation (11). opted for the development of this study can be uel cell [32] is a function of the external fuel and h flows are fed continuously and in equivalent the hybrid energy generation system, which is is composed of two electrodes: an anodic side in oxygen is fed. The two electrodes act as platinum electrolytic ionic conductor. The cell considered ton exchange membrane or electrolytic polymer the cell, but stops the movement of electrons, ohmic

Appendix A
The power generated by the hybrid system under the load profile 2 is presented below. Figure A1 shows the power generated for Puerto Bolivar (Colombia).

Appendix A
The power generated by the hybrid system under the load profile 2 is presented below. Figure  A1 shows the power generated for Puerto Bolivar (Colombia). The results for Beijing (China), are shown in Figure A2.  Figure A3 shows the power results for Texas (USA).   Figure A3 shows the power results for Texas (USA). Finally, Figure A4 shows the power generated for Bremen (Germany). Finally, Figure A4 shows the power generated for Bremen (Germany).