**Solar Photovoltaic Cell Parameters Extraction Using Di**ff**erential Evolution Algorithm** †

**Rachid Herbazi 1,\*,**‡ **, Youssef Kharchouf 2,**‡**, Khalid Amechnoue <sup>1</sup> , Ahmed Khouya <sup>1</sup> and Adil Chahboun <sup>2</sup>**


Published: 22 December 2020

**Abstract:** This work presents a method for extracting parameters from photovoltaic (PV) solar cells, based on the three critical points of the current-voltage (I-V) characteristic, i.e., the short-circuit current, the open circuit voltage and the maximum power point (MPP). The method is developed in the Python programming language using differential evolution (DE) and a three-point curve fitting approach. It shows a good precision with root mean square error (RMSE), for different solar cells, lower than to those cited in the literature. In addition, the method is tested based on the measurements of a solar cell in the Faculty of Science and Technology of Tangier (FSTT) laboratory, thus giving a good agreement between the measured data and those calculated (i.e., RMSE = 7.26 <sup>×</sup> 10−4) with fewer iterations for convergence.

**Keywords:** differential evolution; photovoltaic; parameters extraction; single diode

## **1. Introduction**

In recent decades, access to renewable energy (RE) sources has attracted a lot of attention. In fact, growing concerns over the environmental situation and the energy crisis due to the limited quantities of fossil fuel reserves have made the development and adoption of RE an urgent priority. The PV option has proven to be a leader among many promising renewable technologies to replace fossil fuels due to lower PV technology prices and improved efficiency of solar cells thus leading to a significant growth in the PV industry [1–3]. PV are mainly divided into three technologies; poly-crystalline, mono-crystalline and thin films. Since these technologies rely mostly on the PV effect in silicon P-N junctions, their behavior can be modeled using electric diode circuits [4,5]. These circuits include different components where each is representative of a certain physical mechanism within the cell such as exciton recombination and cell bulk resistance. Knowing the exact values of these parameters is important for many applications. The simulation and emulation of PV cells is crucial for energy yield prediction, quality control during manufacturing [2] and the study of PV cell degradation. However, the values of these parameters are not available in the manufacturers' technical data sheets. Open-circuit voltage, short-circuit current, and maximum power are given only under standard test conditions (STC). Obviously, the actual PV modules operation is generally far from these ideal conditions, which makes real-time fast parameter extraction necessary.

#### **2. The Single Diode Model**

The simplest model used for PV solar cells is the single diode model (SDM) [6] as shown in Figure 1. The current generator represents the photocurrent (*Iph*) that is due to the PV effect, the diode represents the P-N junction and its electric field through the diode ideality factor (n) and the saturation current (*I*0), the series and shunt resistances (*Rs* and *Rsh*) represent cell bulk resistance and leakage current, respectively. The SDM is used for the three dominant technologies in the PV industry (namely, mono-Si, poly-Si and thin films), but for other emerging technologies which do not rely on the electric field of the P-N junction for exciton separation [7,8], other more specialized models are necessary. This model is described by the Shockley diode equation for a PV module with *Ns* cells in series and *Np* in parallel:

$$I = I\_{ph}N\_p - I\_0N\_p \left[ \exp\left(\frac{q\left(V + IR\_sN\_s/N\_p\right)}{nkTN\_s}\right) - 1\right] - \frac{V + IR\_sN\_s/N\_p}{R\_{sl}N\_s/N\_p} \tag{1}$$

**Figure 1.** Equivalent circuit SDM of PV module.

#### **3. Background**

For the parameters extraction of the model, one can distinguish two main strategies. Firstly, the analytical approach based on information from several key points on the I-V curve such as the short-circuit current (*Isc*), the open circuit voltage (*Voc*), the maximum power point (MPP) as well as the gradient of the curve at those points. This approach is simple and allows rapid calculations, however, several assumptions and simplifications can be made, thus leading to incorrect results [3,9,10]. This approach is also sensitive to measurement noise since it only relies on a few selected points [11].

The second strategy is the numerical approach, the problem of which is formulated as an optimization problem by trying to minimize the error between the calculated and measured values. This approach allows for the usage of the wide array of well-studied deterministic and stochastic optimization algorithms. The deterministic algorithms include methods such as the Newton Raphson method [12], Iterative curve-fitting [13], Conductivity method [14], Levenberg Marquardt algorithm [15], etc. These are gradient-based methods, which adds continuity, differentiability and convexity requirements on the error functions. In contrast, stochastic methods provide global search capability and do not require gradient information, meaning the error function can be non-differentiable, and even non-continuous, which include; the genetic algorithms (GA) [2,6,16], particle swarm optimization (PSO) [17,18], DE [1], artificial bee swarm optimization (ABSO) [19]. The issues encountered by these stochastic methods are mainly related to the vastness of the search spaces, which cost a lot of time to cover and require a few thousand iterations before converging. They can also be trapped in local minima but are much less susceptible to this problem when compared to their deterministic counterparts.

In this context, this work is based on the DE algorithm for the extraction of SDM parameters, namely *Iph*, *I*0, *a*, *Rs* and *Rsh*. The extraction method used is based on the three critical points (*Voc*, *Isc*, *Pmpp*) of the I-V curve whose model is forced to pass through these cited points. Furthermore, the method has been developed and implemented in the Python programming language, thus giving more precision and the need for few iterations for convergence.

#### **4. The Proposed Method**

The DE algorithm application to the SDM would result in a five-dimensional search space. In this work, three points of the I-V characteristic are used to analytically calculate three of the parameters whose search space is reduced to two dimensions. The ideality factor (a) and the series resistance (Rs) are the two parameters introduced into DE algorithm.

#### *4.1. Di*ff*erential Evolution*

The DE introduced by Storn and Price in 1995 [20] is a relatively recent stochastic and evolutionary algorithm. It is very similar to other evolutionary algorithms like GA in that it also has crossing and mutation operations as shown in Figure 2.

**Figure 2.** Main operations of DE algorithm.

## 4.1.1. Initialization

First, we generate the initial population randomly trying to cover the entire search space. We may achieve this using Formula (2) assuming the *rand* (0, 1) term provides a uniform distribution of outputs.

$$V\_j = V\_{\text{min},j} + rand(0, 1)(V\_{\text{max},j} - V\_{\text{min},j}) \tag{2}$$

The initial population is spread uniformly over the entire search space, which helps the algorithm avoid being stuck in local minima. DE is very efficient, which means the algorithm converges well within Genmax = 100 iterations, even with a small population NP = 50. The DE strategy is "DE/best/bin" which means that we choose the fittest vector in a population to generate mutants for the next generation, and that binomial crossover is performed on the trial vectors.

## 4.1.2. Mutation

Next, we create a mutant vector by taking the fittest vector in the population and adding the difference of two other random vectors from the population. The difference is scaled with a mutation factor *F* ∈ [0, 1]. This the key step that differentiates DE from other evolutionary methods according to the following formula:

$$M = V\_{fittest} + F(V\_{r1} - V\_{r2}) \tag{3}$$

#### 4.1.3. Crossover

In this step, we created a trial vector by crossing over the elements from the mutant vector into the original. The probability of crossing over each element is set by the crossover rate, and we usually take high values (*CR* = 0.8) in order to promote diversity in the population [21]. A trial vector is generated as follows:

$$T\_i = \begin{cases} M\_i \text{ if } (rand(0, 1) \le \mathbb{C}R) \\\ X\_i \text{ otherwise, } X \text{ being the original vector} \end{cases} \tag{4}$$

#### 4.1.4. Penalty

One issue with DE is that unphysical values of the parameters can be obtained (i.e., negative Rs). To circumvent this, we apply a penalty to any unphysical solution by assigning it a large fitness value.

## 4.1.5. Selection

The last step is to select the vector based on whether or not its fitness value is superior to its previous generation counterpart. A possible method to achieve this is:

$$X\_{\\$^{\text{cp}+1}} = \begin{cases} \text{ } \text{ } \text{if } f(\text{} \text{I}) \prec f(\text{X})\\ \text{ } \text{ } X \text{ } \text{otherwise} \end{cases} \tag{5}$$

Note that, when the fitness function *J* is calculated, it evaluates to 100 for unphysical values as per the penalty phase. Storn and Price (2006) [21] recommend high crossover (*CR* = 0.7) and mutation factor (*F* ≥ 0.4) values. The limits used to penalize solution vectors are a ∈ [1, 2], *Rs* ∈ [0, 0.5] as well as the previously stated negative values of currents and resistances. We should note that the method is very sensitive to the choice of the MPP; then, the proposed method includes different running the algorithm times each with a slightly different MPP.

#### *4.2. Three-Point Curve Fitting*

In order to minimize the complexity of the algorithm, it is possible to reduce the dimensionality of the problem using a few geometrically critical points in the I-V curve, namely the short-circuit, open-circuit and maximum power regions. These three points will allow us to determine three SDM parameters:

$$I\_i = I\_{\rm pli} N\_P - I\_0 N\_P \left( e^{\frac{V\_i + I\_i R\_0 (N\_{\rm s}/N\_p)}{dV\_I}} - 1 \right) - \frac{V\_i + I\_i R\_s \left( N\_s / N\_p \right)}{R\_{\rm sli} \left( N\_s / N\_p \right)}, \text{ where } i = 1, 2, 3 \tag{6}$$

Using algebraic manipulation, one can extract the three parameters *Iph*, *Rsh*, and *I*0:

$$I\_{\rm plt} = \left[ I\_0 \text{Np} \left( e^{\frac{V\_1 + I\_1 R\_s (N\_b/N\_p)}{aV\_1}} - 1 \right) + \frac{V\_1 + I\_1 R\_s \left( N\_s / N\_p \right)}{R\_{sh} \left( N\_s / N\_p \right)} + I\_1 \right] \frac{1}{N\_p} \tag{7}$$

$$R\_{\rm sl} = \frac{(V\_1 - V\_2)\left(N\_p / N\_s\right) + R\_s(I\_1 - I\_2)}{I\_2 - I\_1 - I\_0 N\_P \left(e^{\frac{V\_1 + I\_1 R\_0 (N\_{\rm s} / N\_p)}{xV\_1}} - e^{\frac{V\_2 + I\_2 R\_0 (N\_{\rm s} / N\_p)}{xV\_1}}\right)}\tag{8}$$

$$I\_0 = \frac{a(I\_2 - I\_1) - \beta(I\_3 - I\_1)}{N\_p \left[ a \left( e^{\frac{V\_1 + I\_a R\_0 (N\_b / N\_p)}{aV\_t}} - e^{\frac{V\_2 + I\_2 R\_0 (N\_b / N\_p)}{aV\_t}} \right) + \beta \left( e^{\frac{V\_1 + I\_a R\_0 (N\_b / N\_p)}{aV\_t}} - e^{\frac{V\_3 + I\_3 R\_0 (N\_b / N\_p)}{aV\_t}} \right) \right]} \tag{9}$$

where

$$\alpha = V\_3 - V\_1 + R\_s \Big(\frac{N\_s}{N\_p}\Big) (I\_3 - I\_1) \tag{10}$$

and

$$\beta = V\_2 - V\_1 + \left(\frac{N\_s}{N\_p}\right)(I\_2 - I\_1) \tag{11}$$

Using this technique, the differential algorithm is executed in a two-dimensional search space, which considerably reduces the algorithm complexity.

#### **5. Results and Discussion**

#### *5.1. The Schutten Solar STP6–120*/*36 Module*

First, our code developed in Python gives a set of 22 data points of the I-V characteristic of the Schutten Solar STP6–120 module, which contains 36 solar cells, connected in series at a temperature of 55 ◦C. A comparison between the experimental dataset and the computed I-V curve is illustrated in Figure 3. We clearly see that the algorithm fits the dataset very well, despite most of the points being close the maximum power points and few being in the short-circuit and open-circuit regions. Table 1 compares the computed parameters using our code with those reported in [1].

**Figure 3.** A comparison between the experimental data (green dots) and the calculated characteristic (blue line) for the Schutten Solar STP6–120/36 module.

**Table 1.** Parameter extraction results of the Schutten Solar STP6–120/36 module with those reported in [1].


The fitness of a population is calculated by averaging each solution vector's fitness in a specific generation. Its evolution is shown in Figure 4 where rapid convergence (within 30 iterations) is noticed towards the final value.

**Figure 4.** The evolution of average population fitness throughout the 100 iterations.

#### *5.2. The R.T.C. France Commercial Silicon Solar Cell*

For this case, we study the R.T.C France commercial silicon solar cell. The experimental dataset was taken at a temperature of 33 ◦C. Subsequently, the method used manages to extract very precise parameter values (Figure 5). A comparison between the computed values python DE algorithm and the proposed method in [1] is shown in Table 2. We can see the DE algorithm is slightly more accurate, and this could be explained by the absence of the constraints that are introduced in [1] by utilizing the three pivot points. It turns out that by giving up some precision on the pivot points; we can slightly gain in overall accuracy.

**Figure 5.** Comparison between experimental data (green dots) and calculated I-V curve (blue line) for the R.T.C France solar cell.

**Table 2.** Parameter extraction results of the R.T.C France commercial silicon solar cell with those reported in [1].


#### *5.3. FSTT Laboratory Silicon Cell*

In this section, we tested in our FSTT laboratory, using RaRe Solutions solar simulator, the proposed method of the DE algorithm to extract the parameters values of a solar cell at a temperature of 30 ◦C.

A comparison between experimental data and the computed I-V curve is shown in Figure 6. The parameter values corresponding to this solution are laid out in Table 3 and are associated with a RMSE of 7.26 <sup>×</sup> <sup>10</sup><sup>−</sup>4. The evolution of population fitness is shown in Figure 7.

**Figure 6.** Comparison between experimental data (green dots) and the algorithm calculated I–V curve (blue line) for the FSTT solar cell.

**Table 3.** Extracted single diode parameter values for the FSTT cell.

**Figure 7.** Evolution of average population fitness in the 100 iterations.

#### **6. Conclusions**

This work represents a developed method of extracting SDM parameters from PV solar cells using DE and the three-point curve fitting approach. The method is implemented in Python requires very few iterations for convergence and giving more precision compared to the results cited in the literature. Furthermore, the Python code developed gives very consistent results on several executions. However, the method seems very sensitive to choice of MPP in the experimental dataset, which makes it necessary to perform several executions, each with a different choice of MPP. It should be noted that the use of the three points seems to slightly reduce the method precision. Whereas a naive DE scheme implies a five-dimensional research space and requires a few hundred iterations, which is slightly higher in precision terms. This can probably be explained by the noise measurement in the pivot points constraining the proposed method.

**Author Contributions:** Conceptualization, R.H. and Y.K.; methodology, R.H. and Y.K.; software, R.H. and Y.K.; validation, A.C., K.A., A.K., Y.K. and R.H., formal analysis, R.H., Y.K., A.C., K.A. and A.K.; investigation, R.H. and Y.K.; writing—original draft preparation, R.H.; writing—review and editing; A.C., K.A. and A.K.; supervision, K.A., A.K. and A.C.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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## *Proceedings* **Energetic Sustainability of Systems** †

## **Attila Albini \*, Edina Albininé Budavári and Zoltán Rajnai**

Doctoral School on Safety and Security Sciences, Obuda University, 1034 Budapest, Hungary; budavari.edina@phd.uni-obuda.hu (E.A.B.); rajnai.zoltan@bgk.uni-obuda.hu (Z.R.)


Published: 25 December 2020

**Abstract:** An important problem in our world is that humanity's energy consumption is constantly rising. Therefore, nowadays there is an increasing emphasis on the problem of reusability and efficient energy management. The present paper studies the energy sustainability of systems by developing a unique test model. Using this test model, the theoretical problems of closed systems are investigated. With a theoretical experiment, the temporal motion of rigid systems is monitored and the behavior of flexible systems is analyzed. Finally, the study of the energy interaction of the general system and its environment shows the basic condition for the system's overall sustainability.

**Keywords:** closed system; dual system; ideal system; energetic model; long-term sustainability

#### **1. Introduction**

An important problem in our world is that humanity's energy consumption is constantly rising. Therefore, nowadays there is an increasing emphasis on the problem of reusability and efficient energy management [1–3]. Among other things, increasing efficiency is one of the requirements for the existence of long-term sustainable systems. Due to energy hunger, the need for sustainability is emerging in all areas of life. Demand is emerging everywhere from the industry's new paradigmatic concept to the smart city concept [4–7]. From other aspects, the need for sustainability must be emphasized in the underlying legal regulation [8] and in education [9–11].

The long-term sustainability of systems can be examined by the energy balance of the system. The aim of the present study is to examine the energy relationship between the ideal system and its environment in theoretical experiments, and to use this to demonstrate the energy conditions for the long-term sustainability of systems. The result can help to understand the relationship between the flexibility, efficiency and sustainability of systems. Furthermore, the result can help to show the dependence of the energy requirement of sustainability.

There are several arguments in favor of the energy aspect. The analysis of the movement of systems over time differs depending on the structural changes [12–14]. Structurally static motion can be modeled with the loops of cybernetics. This modeling leads to a system of differential equations with a static structure [15–21]. The result of organic change is a system of equations with a variable structure that cannot be handled easily. The study of such system changes is more based on textual modeling of human-based systems' change management. Long-term sustainability requires structural change. The energetic modeling of this is problematic because of the above. The series of theoretical experiments seeks a solution to this problem.

### **2. Methods**

The study is based on the results of energy balance modeling according to [3]. Said modeling showed the energy relationship of the system components and the whole system and grouped the energy issues. Theoretical experiments have analyzed the energy balance [1–3] of several types of system connections. As a first theoretical experiment, the problem of closed systems was investigated. Ideal dual relationships were modeled in the second series of theoretical experiments. In the third phase of the experiments, one of the actors in the dual relationship was replaced by the outside world. With this method, the operation of real systems can be deduced in several steps. It is also possible to formulate the energy conditions for long-term sustainability.

To achieve this:

	- a. the dual relationship of rigid systems must be examined;
	- b. the dual relationship between fully flexible systems needs to be examined;

Thus, the two systems in the study are the same as the system and the world outside it. Generalization is the aim of this theoretical experimental step, from which conclusions can be drawn regarding the general sustainability requirements of systems.

## **3. Results**

The structure of the model produced for the tests is the same as the general simplified model of the systems. In terms of energy balance, the behavior of the system can be characterized by the behavioral pattern of the general energy model. The mechanism of the interaction of the systems can be modeled with the model of cybernetic loops [15,16,19]. Theoretical experiments should be performed on the resulting system model.

Using the energy model of general systems, it can be shown that ideal closed systems are completely safe [22–25]. However, information on such systems cannot be obtained in reality, so their existence has not been proven. Thus, closed systems are only worth talking about at the level of a theoretical model. An examination of the pairwise interaction of the systems showed that rigid systems devour each other, while flexible systems are in balance with each other. Theoretical examination of the system and its environment has shown that the energetic condition for the long-term sustainability of the systems is that the stored energy of the system shows a continuous and unlimited increase. Summarizing the results:


## **4. Discussion**

The series of experiments is based on a modeling procedure not used so far. It uses the general energy and structural model of the systems simultaneously. The logical sequence of the series of theoretical experiments is also unique. The cases that resulted in anomalies are also examined during the experiments. For this reason, the sequence of experiments provides a way to answer the problem and get closer to real systems.

#### *4.1. Model*

Since the aim is to study the energy conditions, the application of the operating model used in the energy balance modeling of the systems may be expedient. According to this, the energy entering the system is equal to the sum of the change in stored energy and the energy leaving the system. The relevant equation is:

$$\text{E(ir)} = \Delta \text{E(store)} + \text{E(out)}.\tag{1}$$

In performing theoretical experiments, the logical separation of each experiment is based on the structure of the systems. For this reason, it is worth using the structure of the information system simplified system model [12–14] to model the structures of the systems. According to this, systems can be easily structurally modeled by defining the system, its properties, and the designation of its interface. In this case, the energy exchange of the system is only possible through its interface.

Finally, to model the interaction of systems, it is worth borrowing the mechanism of action of the cybernetic loops used in control theory [15–19]. According to this, negative feedback control loops were used to control the system according to the chain of action within the system. Based on the evaluation of the output, an intervention was made at the input of the chain to achieve the desired effect. This model assumes that the system is continuously exposed to external disturbances. The general model of the study is shown in Figure 1.

**Figure 1.** Model of the study. Based on [3].

#### *4.2. Closed Systems*

The interface of a perfectly closed system model is an empty set. This means that such systems are unsuitable for energy exchange. As a result, there is no exchange of information between the closed system and its environment. They know nothing about each other. Therefore, the discussion of closed systems is purely theoretical. In practice, such systems cannot be observed. A simplified structural model of closed systems is shown in Figure 2.

**Figure 2.** Simplified structural model of closed systems. Based on [3].

#### *4.3. Dual Systems*

At the beginning of the study of the interaction of systems, it is worth creating a general model. According to this model, there is no system in the imaginary world other than the two systems studied. The effect of one system is in interference with the other system and vice versa. Thus, the energy exchange also takes place only between the two systems. That is, systems only communicate energy to each other. The life of this dual system is determined by the variability of the energy transfer capacity

and the structure of each system. The relevant equations are (2) and (3). The system identifiers (E1 and E2) are in the lower index.

$$\mathbf{E}\_2\text{(out)} = \mathbf{E}\_1\text{(in)} = \Delta\mathbf{E}\_1\text{(store)} + \mathbf{E}\_1\text{(out)},\tag{2}$$

$$\mathbf{E}\_1(\text{out}) = \mathbf{E}\_2(\text{in}) = \Delta \mathbf{E}\_2(\text{store}) + \mathbf{E}\_2(\text{out}).\tag{3}$$

The model of the dual systems is shown in Figure 3.

**Figure 3.** Model of the dual systems. Based on [3].

#### *4.4. Rigid Dual Systems*

Rigid systems are systems in which it is not possible to flexibly change the energy storage capacity of the system. As a result, it is not possible to quickly change the structure of the system. Almost all of the energy entering the system serves the operation of the system, which energy leaves the system after work. In the case of a dual examination of these systems, systems that are the same in terms of energy balance operate in balance with each other. This is based on Equations (2) and (3). The consequence is Equation (4). Due to the rigidity of the structures, the equilibrium condition is Equation (5).

$$
\Delta E\_1 \text{(store)} + \Delta E\_2 \text{(store)} = 0,\tag{4}
$$

$$
\Delta E\_1 \text{(store)} = \Delta E\_2 \text{(store)} = 0 \tag{5}
$$

This balance is not ideal. Although a small deviation causes the excess energy to be absorbed by the other system without damage, the difference between the systems involved in the study gradually increases. As a result, the higher energy system swallows up the lower energy system. In the event of a high degree of difference in systems, this effect is immediate and severe. So, it is characteristic of rigid systems that they seek to achieve singularity.

## *4.5. Flexible Dual Systems*

Flexible structured systems are systems that are capable of rapid structural change. This is conditional on the possibility of storing a large part of the incoming energy. The energy demand for the operation of the system is much lower than the energy storage capacity. In the case of a dual examination of these systems, the identical systems are also in balance. Here, however, small differences are immediately offset because the ability of systems to change is rapid. The system can store the incoming excess energy without structural damage. The equilibrium condition is (6).

$$
\Delta \text{E}\_1 \text{(store)} + \Delta \text{E}\_2 \text{(store)} = \Delta \text{E}\_1 \text{(out)} = \Delta \text{E}\_2 \text{(out)}\tag{6}
$$

As a result, the energy balance of systems with different parameters and the amount of stored energy are balanced in the long run. So, flexible systems maintain a stable state of equilibrium with their environment.

## *4.6. Complementary Dual Systems*

During the complementary test, one of the dual systems must be replaced with the complement of the other system. This means that the interaction between the system and the outside world is studied as shown in Figure 4. This type of approach is already closer to real-world interactions than previous studies. The system under study has neither a perfectly rigid structure nor a perfectly flexible structure. Several conclusions can be drawn when conducting the theoretical experiment.

**Figure 4.** Model of the complementary dual systems. Based on [3].

The model of the complementary dual systems is shown in Figure 4.

The system is safe until a larger amount of energy arrives in its direction than it can store or pass. At higher energy doses, the system reacts as a rigid structure. Its structure is forced to change. The system then moves in the direction of the singularity. With a lower dose of energy, the system can store the excess. The structure of the system varies according to its own capabilities. In this case, the system moves towards balance with its environment. The general equation of the complementary dual system model is (7).

$$0 < \mathcal{E}\_{\text{system}}(\text{in}) = \mathcal{E}\_{\text{environment}}(\text{out}) < \infty. \tag{7}$$

The results of the study show the basic condition for the long-term energy sustainability of a general system. The system is most secure under the conditions if it is possible to ensure the storage and transmission of excess energy for all time intervals. This means that the change of the system must also be of this nature. All this suggests exponential behavior. This involves continuous flexibility. At the same time, the energy level of the environment can be characterized by infinity at the scale of the system. Therefore, the continuous flexibility of the system and thus its continuous sustainability can be increased by increasing the energy of the system continuously and unlimitedly.

#### **5. Summary**

Industry 4.0 represents a new paradigm in terms of production. The smart city concept represents the same paradigm in terms of consumption, among other things. One of the most important features of this paradigm is to ensure long-term sustainability in all aspects [1–7].

In the present study, a unique test model has been developed to study the energy sustainability of systems. Using the test model, the theoretical problems of closed systems were first investigated. Subsequently, the temporal motion of rigidly structured systems was analyzed, which showed that such systems strive for singularity—they swallow up each other. Then, the study of flexible systems revealed that these systems strive for balance with their environment. Finally, a closer examination of the system showed that the basic condition for the energy sustainability of systems is that the system's own energy should increase continuously and unlimitedly. This result is consistent with the experience that the energy of long-life systems increases exponentially.

**Funding:** This research received no external funding.

## **References**


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## *Article*
