Non-linear controller for storage systems with regulated output voltage and safe current slew-rate for the battery

1. Introduction

Nowadays, there are different applications that require an Energy Storage System (ESS) like microgrids, standalone renewable energy systems, electric transport systems, uninterruptible power supply, among others. Such a necessity of ESS have produce a continuous growth of the global installed storage capacity, which was estimated as 15.3 GWh in 2017, excluding pumped hydro [1]. From such a capacity it is important to highlight that lithium-ion batteries have been widely used in the last years to implement ESS, since in 2016 88 % of the deployed storage capacity correspond to this technology [1].

In many applications (e.g. microgrids, electric vehicles, stand-alone photovoltaic systems) batteries are used to regulate the voltage of a DC bus by compensating the differences between generation and load. Such a compensation is performed by charging/discharging the batteries when the load is greater/less than the generation. The power provided/extracted to the DC bus from the batteries is performed through a charging/discharging system that is formed by a power converter and a control system.

Usually the battery voltage is less than the DC bus voltage, therefore, bidirectional boost converters are used to implement the charging/discharging system [2]. Nevertheless, a battery or battery bank may have voltages equal or greater than the DC bus; therefore, step-up/down converters are a more flexible solution for charging discharging systems [3, 4]. Moreover, the current required from the battery to regulate the DC bus depends on the load profiles that disturb the DC bus voltage. Therefore, fast variations in the load may produce high current derivatives in the battery that surpass the slew-rate limitations defined by the manufacturers [5]. Those high current derivatives may significantly reduce the battery lifetime [5]; then, the current slew-rate limitation is an important part of a charging/discharging system.

One option to address this problem is to construct Hybrid Energy Storage Systems (HESSs) with batteries and supercapacitors (SCs), where the low frequency currents required by the load are provided by the battery and the high frequency currents (transients) are supplied by the SC [2]. In general, a HESS can be used in any application with load perturbations that demand high slew-rate currents from the batteries. However, in the literature the main applications of HESS are the DC bus voltage regulation in: microgrids [5, 6, 7, 8, 9, 10], electric vehicles [11, 12, 13, 14, 15, 16, 17, 18], systems with a generic load [19, 20], and systems with renewable energy sources like photovoltaic [21], wind turbines [22] and fuel cells [20].

In microgrids, the HESS compensates the differences between the unpredictable energy sources and loads to keep the microgrid stability; while in electric vehicles the HESS provide the variable power required by the motor and absorbs the energy from the regenerative brakes. Finally, in systems with unpredictable energy sources the HESS may compensate the excess/lack of power production by charging/discharging the HESS to supply energy required by the load.

In the literature there are different HESS architectures for the HESS, as shown in [2]; however, the parallel architecture is the most widely used. In this architecture, the battery and the SC are connected to the DC bus using two power converters with the outputs connected in parallel [5, 6, 7, 8, 10, 11, 12, 14, 18, 20, 21, 23]; therefore, the battery and the SC can be independently controlled to use the SC as a power source to compensate fast perturbations in the load and the battery to supply/absorb the long-term energy required/provided by the DC bus. HESS with series architecture are also reported in the literature, which use one converter to connect the SC to the DC bus and the battery is connected in parallel to the DC bus (i.e. without converter) [13, 19, 24] or vice-versa [15]. In those cases, the storage device without converter (SC or battery) cannot be controlled, hence, it is difficult to guarantee that the fast perturbations of the load are supplied by the SC to protect the battery. Moreover, in other papers the HESS topology is not clearly explained [22].

Regarding the power converters, most of the proposed HESS use boost [7, 8, 10, 11, 12, 15, 18, 21, 22, 23, 24] converters to connect the battery and the SC to the DC bus; while others use interleaved boost [13, 14], buck [19, 25], or a buck/boost operating as step-up converter [5, 6] to couple the SC and batteries to the DC bus. Therefore, most of the HESS the charging/discharging system work for battery and SC voltages less than the DC bus. Nonetheless, in some ESS technologies (like SC) the voltage significantly vary with the state-of-charge, as consequence its voltage may be less, greater or equal than the DC bus voltage. In those cases, some authors use step-up/down converters to increase the flexibility of the charging/discharging system [3, 4]. Additionally, it is worth noting that boost converters have the disadvantage of injecting discontinuous current to the DC bus; while buck converters produce discontinuous currents in the storage devices.

Moreover, there are different control systems for HESS reported in the literature, some of them use linear controllers, while the others propose nonlinear controllers. The authors in [5, 6] propose cascade control structures with linear controllers, where the outer loop regulates the DC bus voltage in [6, 10, 13] and tracks a power reference in [5]; while the inner loops in [5, 6, 10, 13] track the current of the battery and SC with PI controllers.

Nevertheless, linear controllers cannot guarantee the system stability for all the operating conditions (i.e. no global stability), and their dynamic performance may vary depending on the operating point, since linear controllers are designed by using linearized models.

Some authors use nonlinear control systems based on Sliding-Mode Control (SMC) theory. On the one hand, in [11, 12, 24] the control system of the HESS is a multiple input multiple output SMC to regulate the DC bus voltage and control the current provided/absorbed by the battery and the SC. However, the design procedure of the control systems is not clearly explained and the DC bus voltage show significant oscillations where there are disturbances in the DC bus.

Moreover, the SMCs proposed in [11, 12] do not generate the switching signals of the power converters, since the SMCs are used to generate the PWM signals for the switches; as consequence, the system doesn’t slide around a surface within a desired hysteresis band. On the other hand, in [21] the authors use two SMC controllers when the battery reaches its maximum and minimum currents (i.e. discharging and charging) and two passivity based multiple input multiple output controllers to regulate the DC bus voltage and the storage devices’ currents. However, the design procedure is complex and the control system is formed by four controllers, which complicates its implementation.

The control systems proposed in [8, 20, 23] combine linear controllers and SMC in a cascade structure. The inner loops are SMC controllers of the battery and SC currents and the outer loops are PI regulators of the DC bus voltage. Nonetheless, the SMC controllers proposed in [8, 23] are used to generate the duty cycle of the power converters and not their switching signals. Additionally, the SMC proposed in [20] doesn’t include the design procedure, nor the stability analysis.

Other authors propose HESS control systems based on adaptive SMCs for HESS in series architectures, where the SC [19] or the battery [15] are connected to the DC bus through power converters and the other storage device (i.e. battery in [19] and SC in [15]) is connected in parallel to the DC bus. The controller proposed in [19] modifies the sliding surface function depending if the SC current is between its maximum and minimum values, or if it reaches one of the limits. Although the authors provide the stability analysis and the design of the SMC parameters, the voltage of the SC is not controlled and the transition between one sliding surface to the other is not analyzed. The controller proposed in [15] uses a SMC for the battery current and a SMC-based observer for the battery voltage and the HESS output current to regulate the DC bus voltage. Nevertheless, the control only considers resistive loads and the SMC is used to caL.ulate a duty cycle and not to generate the switching signals to slide around a surface.

Some approaches combine SMCs with other nonlinear controllers to construct the HESS control strategy. In [22] the authors use the desired output power of the HESS and a SMC theory to generate a surface and its derivative, which are used as inputs of a fuzzy controller to generate the reference of the battery current. Then such a reference is tracked by a PI regulator. The authors in [18] use two SMCs for the SC and battery currents and a Lyapunov based controller to regulate the DC bus voltage. The paper includes the stability analysis to show the global stability of the system. However, the SMCs proposed in [18, 22] are used to caL.ulate a duty cycle and not to generate the switching signals of the converters; moreover, the design procedure is not provided [22] or not clearly explained [18].

It is important to highlight that any the HESS systems described before guarantee that the battery currents fulfill the slew-rate limitation provided by the manufacturer when there are fast perturbations in the DC bus. Therefore, the battery may be submitted to a high slew-rate current, which reduces its lifetime.

This paper proposes a charger/discharger system for a HESS using a series architecture. The battery is connected to an auxiliary capacitor (not necessarily a SC) through a buck/boost converter. In turn, the auxiliary capacitor is connected to the DC bus by another buck/boost converter. The control system if formed by two cascade controllers, where the inner loops are SMCs of the converters’ inductors currents and the outer loops are linear controller to limit the battery current slew-rate, with one converter, and regulate the DC bus voltage, with the other converter. The paper provides the controllers’ design procedure and their validation with simulation results. The proposed HESS is able to regulate the voltage of a DC bus when the battery is within its maximum and minimum state-of-charge, while keeping the battery current slew-rate below its maximum value. Moreover, buck/boost converters improves the flexibility of the system, since it facilitates the connection of batteries, auxiliary capacitors, and DC buses with different voltages.

2. Power circuit

Figure 1 presents the proposed double-stage battery power system aimed at improving both the battery operation conditions and the voltage quality of the load.

The first stage is a bidirectional buck/boost converter designed and regulated to limit the slew-rate of the battery current to a safe value. Such a first stage has a highfrequency capacitor C_HF connected in parallel with the battery, which is in charge of absorbing the switching ripple of the current generated by the buck/boost converter.

However, the mean current value requested to the batterycould exhibit high frequency transients, e.g. step current transients; therefore, this stage must be controlled to ensure that the mean battery current fulfills the slew-rate restriction required for a safe operation. The buck/boost converter is formed by two Mosfets, M_1b and M_2b, and by an inductor L_b . Moreover, this stage provides or extracts current to an auxiliary capacitor C_a , which enables to isolate the battery from the load current i_dc.

This stage is controlled with a cascade structure: an inner controller regulates the inductor current, while an outer controller limits the slew-rate of the battery current and ensures a power balance into the auxiliary capacitor. The current controller produces the control signals of the Mosfets, u_b for M_1b and 𝔲̅= 1− _ub for M_2b, while the outer controller produces the current reference i_Rb for the current controller.

The second stage is also a bidirectional buck/boost converter, which is designed and regulated to provide a stable voltage to the load. This buck/boost converter is formed by two Mosfets, M_1c and _2c , and by an inductor L_c . This stage provides or extracts current from the auxiliary capacitor C_a to be delivered to the output capacitor C_dc . Moreover, this stage must compensate the perturbations of the load current i_dc , hence behaving as an ideal voltage source.

The control of this stage is also based on a cascade structure: an inner controller regulates the inductor current, while an outer controller regulates the load voltage depending on some performance criterion, e.g. maximum voltage deviation.

The current controller produces the control signals of the Mosfets, uc for M_1c and 𝔲̅.= 1− u for M_2c , while the outer controller produces the current reference i_Rdc for the current controller. Finally, the load is modeled with the capacitor C_dc in parallel with the current source idc. Such a circuit is able to reproduce the behavior of any load by correctly defining the current profile of i_dc .

Figure 1 also shows the difference of the power flows imposed by the controllers: the second stage provides the complete load power, hence no frequency restriction is imposed to the load current; instead, the first stage only provides the averaged load power, hence only low frequency current components are allowed. However, the outer controller of the first stage will charge, respecting the current slew-rate (slowly), the auxiliary capacitor over time. Therefore, the current requested by the load is provided protecting the battery from high-frequency transients. Moreover, it must be pointed out that this solution adopts the same number and type of converters reported in other solutions such as [8, 10, 12, 18, 26, 27, 28, 29, 30, 31, 32]; hence no additional costs are introduced due to the power stages. Similarly, the controllers have the same nature that the solutions reported in [8, 15, 18]; hence no additional costs are introduced due to the control stages. The following section presents the mathematical model of the buck/boost stages and describes the design of the inner current controller.

3. Stage model and current controller

Taking into account that both stages of the power circuit in Fig. 1 are based on bidirectional buck/boost structures, this section presents a general model of a single buck/boost stage. Such a model is used to design the current controller for the single stage, which is the same for both stages of the complete power circuit.

Figure 2 presents the circuital model of a single buck/boost stage, where the input voltage source is v_i , the Mosfets are M₁ and M₂ , the inductor is L, the control signals are u and 𝔲̅ = 1 − u, the output capacitor is C and the output voltage is v_o . For this general circuit, the reference of the current controller is i_R , and the reference of the outer controller is v_R . Table 1 presents the correspondence of the parameters and variables of the single stage model with the first and second stages parameters and variables. The C_HF capacitor is not considered in the dynamic analysis since it is connected in parallel with the battery, hence its voltage is imposed by the battery and it does not introduce a new state variable.

Expressions (1) and (2) report the switched differential equations that describe the behavior of both the inductor current i. and output voltage v_o of the buck/boost stage.

To regulate the inductor current of the stage, a sliding mode controller (SMC) is proposed, which provides robustness to changes on both the parameters and the operation conditions [33]. Moreover, a well designed SMC ensures global stability, which is needed in this application due to the wide range of operation conditions: charging the source (negative inductor current), discharging the source (positive inductor current), stand-by mode (null inductor current), boosting voltage, reducing voltage or ensuring the same voltage at both the input and output ports. To fulfill those conditions, this paper proposed to design the SMC using the following switching function . and sliding-surface S_F :

To ensure the global stability of the SMC three conditions must be fulfilled [34]: transversality, reachability and equivalent control. Those conditions are analyzed in the following subsections.

3.1. Modelo de un Supercondensador utilizando la DFC

The transversality condition verifies the control signal u is present into the switching function derivative This is a necessary condition for controllability, otherwise the controller will not be able to modify the trajectory of the system. The mathematical formalization of this condition is given in (4).

The first step to evaluate the transversely condition is to caL.ulate, explicitly, the switching function derivative:

Replacing expression (6) into (4) leads to inequality (7), which is always positive since L, 𝑣_𝑖 and 𝑣_𝑜 are positive quantities.

Therefore, expression (7) verifies that the proposed SMC always fulfills the transversality condition.

3.2. Reachability conditions

The reachability conditions verifies that the system, under the action of the SMC, is able to reach the surface S _F = {F = 0}. In practical terms, those conditions verify that the switching function derivative is positive when the switching function is operating under the surface (F < 0); similarly, the switching function derivative must be negative when the switching function is operating above the surface (F > 0).

The formalization of the reachability conditions depends on the sign of the transversality value: a positive transversality value, as the one given in (7), implies that a positive change on u (from 0 to 1) produces a positive switching function derivative, and a negative change on u (from 1 to 0) produces a negative switching function derivative . In conclusion, the formalization of the reachability conditions is given in (8).

Replacing the switching function derivative (6) into (8) leads to restriction (9), which defines the minimum and maximum derivatives of the current reference ensuring the reachability conditions. It must be noted that those limits are the minimum and maximum derivatives of the inductor current, which are caL.ulated from (1) with u = 0 and u = 1, respectively.

Therefore, the previous equation ensures that the proposed SMC can reach the sliding surface, and also it demonstrates that the SMC provides the maximum speed achievable with the converter.

3.3. Equivalent control condition

The equivalent control evaluates the local stability by analyzing the duty cycle saturation. This is performed by caL.ulating the averaged value ueq of the control signal u, which is equal to the duty cycle d: where T_sw is the switching period. The equivalent control must be analyzed when the system is operating into the surface F = 0 with a parallel trajectory, i.e. [33].

Then, replacing (6) into leads to the following expression for u_eq :

The classical equivalent control condition is evaluated as 0 < ueq < 1, which evaluates that the duty cycle is not saturated, i.e. 0 < d < 1. This is a necessary condition for both controllability and local stability. Finally, replacing equation (10) into 0 < u_eq < 1 results in the same limits for given in (9). This is expected since Sira-Ramirez demonstrated in [34] a dc/dc converter under the action of a SMC that fulfills both transversality and reachability conditions also fulfills the equivalent control condition. In conclusion, the proposed SMC based in (3) is globally stable, hence it guarantees that i_L = i_R .

4. Outer controllers

There are two outer controllers to be designed: the battery slew-rate controller and the output voltage controller. Those outer controllers are designed considering the equivalent dynamics of the SMC, i.e. i_L = i_R . Moreover, since the control signals of the Mosfets are defined by the SMC, then the equivalent dynamics are analyzed by averaging the state-variables into the switching period, i.e. u_eq = d. Therefore, by replacing i_L by i_R and u by d into equation (2) leads to the equivalent dynamics of the capacitor voltage given in (11), where the duty cycle d is the steady-state value given in (12).

Such an equivalent voltage dynamics, represented in the Laplace domain, is given in (13), where V_o, I_R and I_o are the Laplace representations of v_o, i_R and i_o , respectively.

The outer controller, whose transfer function is named Goc, defines the current controller reference IR as follows: IR = Goc · (V_R - V_o ). However, taking into account that the voltage reference v_R of the circuital model in Fig. 2 is a constant value (hence also V_R is constant), the smallsignal closed-loop dynamics (including the outer controller) of both V_o and I_R depend only on I_o , i.e. the current perturbations imposed by the load, as follow:

The following subsections describes the design of the outer controllers for both stages.

4.1. First stage: slew-rate controller

For this stage, the adopted controller is a constant gain 𝛼 (i.e. G_oc = 𝛼), which leads to the closed-loop dynamics of V_o and I_R reported in (16) and (17), respectively.

Taking into account that the main objective of this controller is to limit the slew-rate of the battery current, the design of 𝛼 is performed using equation (17). The design considers the worst-case scenario for a load perturbation, i.e. the fastest transients possible, which corresponds to a step current with amplitude |Io| (i.e. Io = |Io| /.). Replacing the Io = |Io| / _s into equation (17), and applying the inverse Laplace transformation, the time-domain waveform of the current reference given in (18) is obtained.

The derivative of the current reference is reported in (19), which must be limited to constrain the slew-rate of the battery current. Since the exponential term of (19) decreases with increments in t, the maximum value of (19) occurs at t = 0 as it is reported in (20).

The battery current is obtained from Fig. 2 as i_i = d· i_L . Since under the action of the SMC i_L = i_R , hence dii To respect the safe slew-rate S _R of the battery current, the following expression must be fulfilled:

Replacing expression (20) into (21) leads to the design expression for a:

Finally, from the steady-state error in the output voltage of this first stage, which corresponds to the voltage of the auxiliary capacitor, is caL.ulated from (16) as:

Such a voltage deviation does not introduce problems in terms of the load voltage regulation since the second stage is also a buck/boost topology. Moreover, the auxiliary capacitor C_a could be designed to limit that voltage deviation into a desired range.

4.2. Second stage: voltage controller

For this stage, the adopted controller is a gain, a zero and a pole as given in (24), which leads to the closed-loop dynamics of Vo reported in (25).

Since the output (load) voltage dynamic has a second order transfer function, this solution adopts a damping ratio equal to one (p = 1), which leads to the following transfer function:

For the worst-case scenario for a load perturbation, defined as Io = |Io| / _s , the time-domain waveform of the load voltage (27) is defined as follows:

This controller is designed to ensure a safe maximum load voltage deviation ∆V_dc . The caculation of such a quantity requires to calulate the time derivative of v_o as follows:

Solving equation (28), and replacing such a time value into (27), leads to the maximum deviation ∆V_dc reported in (29), which is used to caculate b for a desired ∆V_dc . Then, equation (26) is used to caculate . for that particular b value.

Finally, since b in (26) depends on the duty cycle, the caL.ulations must be done for the worst-case scenario, i.e. lowest input voltage v_i, which for this second stage corresponds to the lowest auxiliary capacitor voltage.

4.3. Controller implementation

The classical hysteresis comparator is adopted [33] to implement the SMC. Such a technique introduces a hysteresis band H around the sliding-surface to limit the switching frequency, which transforms the sliding surface (3) to the practical expression given in (30). Basedon that practical sliding-surface, and on the reachability, conditions given in (8), the control laws for the SMC of both the first and second stages are reported in (31) and (32), respectively.

5. Result

The validation of the design procedure considers the following parameters, but any other values can be adopted: L_b = L_c = 100 µH, C_HF = C_a = C_dc = 100 µF, v_b = v_C = v_dc = 12 V, therefore v_Rb = v_{Rd c} = 12 V. The performance criteria are: ∆V_{d c} = 0,5 V for load current transients with amplitude |Io| = 1A, and a maximum slew-rate of the battery current S _R = 4 A/ms.

Applying the design procedure reported in the previous sections, the parameter . = 0,8 for the slew-rate controller of the first stage was caL.ulated. Similarly, the parameters for the load voltage controller of the second stage are . = 3,6788 × 10³ and . = 3,5490. Figure 3 presents the simulation of expressions (18) and (19) considering the designed . parameter and a step-current perturbation in the load with amplitude |Io| = 1A: this simulation confirms that the reference signals fulfill the slew-rate limitations. Such a condition is explicit in the waveforms at the middle of the figure, where the derivatives of both references are under the design limit . Finally, the waveforms at the bottom of the figure confirms that the battery current slew-rate is always under the safe limit S_R = 4 A/ms, hence fulfilling the first design criterion.

Figure 4 presents the simulation of expression (27) considering the designed a and b parameters for the same stepcurrent perturbation in the load: the simulation confirms that the proposed controller constraints the voltage deviation to the designed limit ∆V_dc = 0,5 V, hence fulfilling the second design criterion. Figure 5 present the circuits for both controllers, where the SMC control laws are implemented using S-R Flip-Flops, classical comparators and adders; while the slew-rate and voltage controllers are implemented using linear functions. Then, the complete power circuit of Fig. 1 and the controllers circuits of Fig. 5 were implemented in the power electronics simulator PSIM adopting H = 0,3 A to limit the switching frequency up to 300 kHz.

Figure 6 presents the circuital simulation of the complete battery power system performed in PSIM. The simulation considers the battery in the three possible operation conditions: discharge (positive current), charge (negative current) and stand-by (null current). The simulation verifies the correct regulation of the load voltage by fulfilling the maximum deviation limit ∆V _dc = 0,5 V, hence confirming the simulation of Fig. 4. The figure also presents the auxiliary capacitor voltage, where it is observed the support provided by this capacitor, which enables the power circuit to regulate the slew-rate of the battery current. Figure 6 also presents battery current, which fulfills the dynamic restriction of the maximum slew-rate SR; such a restriction is also evident in the maximum slew-rate of i_Lb, hence confirming the simulation of Fig. 3. The switching functions of both SMC are also reported in Fig. 6, which shows that both converters always operate inside the hysteresis bands: such a behavior verifies the global stability of the SMCs, hence the safe operation of both the battery and the load is guaranteed.

The global stability of the proposed solution, demonstrated by both the analytical and simulation results, significantly improves the reliability and safety of the hybrid power system in comparison with other solutions based on linear controllers, such as the ones reported in [10, 13, 17, 29, 35, 36]. Moreover, the limitation of the current slope, provided by proposed solution, also reduced the battery degradation in comparison with solutions where such a limitation is not considered, e.g. [8, 10, 13, 15, 17, 18, 30, 31].

In any case, with the aim of illustrating the improvement provided by the proposed solution, Fig. 7 presents a comparison of the performance of both the proposed and a classical solution. To provide a fair comparison, the classical solution also considers SMC for the current controllers, but such a classical solution does not consider limitation on the current derivative of the battery, which is the case found in literature. The simulation considers a fast current transient in the load current (steplike change), and both the proposed and classical solutions provide a satisfactory regulation of the bus voltage. However, since the classical solution does not include a limitation on the battery current derivative, the slew-rate of the battery current is much higher than the safe limit S_R . Instead, the proposed solution fulfills such a limitation, hence protecting the battery. In conclusion, despite both solutions provide an accurate bus voltage regulation, only the proposed solution protects the battery from high-frequency current transients since such a derivative limitation is considered into the SMC design.

6. Conclusions

A HESS with a series architecture formed by a battery, an auxiliary capacitor (not necessarily a SC), two buck/boost converters and a control system has been introduced. The control system is formed by two cascaded controllers, one for the converter connecting the battery with the auxiliary capacitor, and the other for the convert interfacing the auxiliary capacitor with the DC bus. The inner loops of both controllers are SMCs of the inductors’ current, while the outer loop of the battery converter is a PI controller that limits the current slew-rate, and the outer loop of the auxiliary capacitor converter is a PI controller to regulate the DC bus voltage. A detailed design procedure has been introduced, and the validation with simulation results has been performed. Those results show that the proposed system fulfills the control design requirements regarding slew-rate limitation, DC bus voltage regulation, and current tracking for both the battery and the auxiliary capacitor. Moreover, the buck/boost converters provide additional flexibility to connect batteries and auxiliary capacitors with voltages lower, equal or higher than the DC bus voltage.

References

Additional information

How to cite: C. Ramos, J. Bastidas, D. González, “Non-linear controller for storage systems with regulated output voltage and safe current slew-rate for thebattery,” Rev. UIS Ing., vol. 19, no. 3, pp. 117-130, 2020. doi: https://doi.org/10.18273/revuin.v19n3-2020012

Revista UIS Ingenierías
ISSN: 1657-4583
Vol. 19
Num. 3
Año. 2020

Non-linear controller for storage systems with regulated output voltage and safe current slew-rate for the battery

Carlos AndrésJuan DavidDaniel Ramos-PajaBastidas-RodríguezGonzález-Montoya
Facultad de Minas, Universidad Nacional de ColombiaUniversidad Nacional de ColombiaInstituto Tecnológico Metropolitano,ColombiaColombiaColombia