1. Introduction

The emerging advances in photovoltaic (PV) cell panel manufacturing processes have pushed forward the technology. It is considered a feasible option for energy production instead of fossil fuels due to its safe, sustainable, and clean supply capabilities. The feasibility is reflected in the advancement of installed solar capacity by at least ten times from 2010 to 2018 [1]. However, the current PV panel technology forfeits up to 25% of its generated energy due to a deficient performance and dependence on climatic conditions [2]. Thus, the maximum power point tracking (MPPT) problem should be addressed in order to enhance the PV cell arrays’ energy generation. By having an adequate MPPT strategy, the power transfer efficiency from the PV cell arrays to consumption loads is improved. The MPPT focuses on matching the PV panel’s output load by controlling a DC voltage converter stage, ensuring the maximum power transfer [2–4].

Different approaches for the MPPT techniques based on application requirements and system constraints define the scope and objective of the control technique [5]. A first approach is focused on the maximum power point (MPP) of a single stage of PV cells, where the objective of the control is the extraction of the maximum power of a specific block of PV cells (or a single PV cell) of the whole array, under any environmental condition. The stage is generally governed by a control on a DC-DC converter from a microcontroller and works specifically over the block independently from the rest of the PV cell array [2, 6–16]. Since the objective of this strategy relies on the local control of the MPPT on a low area of PV cells, the robustness of the solution relies on the control capability to reach the MPP under disturbances over the whole controlled stage, i.e., disturbances on radiation or output load. Consequently, the shading issue is neglected from our scope, and later improvements on the PV arrays could be required to cover shading issues for more extensive areas. Traditional methods, such as constant voltage tracking, open-circuit voltage tracking, short-circuit current tracking, lookup table, current scanning, and curve fitting, are recommended when addressing the shading issues. Although the effectiveness of intelligent control strategies such as genetic algorithms, fuzzy logic algorithms, artificial neural networks, and upgraded P&O has been verified by experiments in many cases, such algorithms still have the disadvantages of high complexity and slow convergence speed [13]. Furthermore, the local MPPT approach is helpful in applications such as space satellites, solar vehicles, and solar water-pump systems [4].

A second approach is focused on the global MPP of the whole PV cell array or a big block of the array, where partial shading (PS) has an appreciable effect due to the area of the array. The PS distorts the global MPPT of the PV cell array due to the reflection of the power-voltage characteristics of any single shaded stage of PV cells over the characteristics of the whole PV cell array. It results in multiple peaks that could distract the MPPT control unit, while it may remain into a local MPP instead of a global one; see for instance [3, 17–26]. These techniques attain a balance of the power distribution between shaded, partially shaded, and fully illuminated stages of PV cells over the area of the whole array. The strategy is based on the flexibility to modify the architecture of the array [3]. Intelligent techniques, such as fuzzy logic control, artificial neural networks, and particle swarm optimization, have the advantage of working with imprecise inputs, becoming more suitable methods for applications where PS is a critical issue [5].

Robust controllers have been previously proposed to improve the performance of the PV cell arrays on the local MPP. For example, the sliding mode controller (SMC) represents a robust algorithm that efficiently responds to environmental changes and load variations, [7, 8, 19, 27]. The stability and robustness of the SMC have been verified through simulations and experiments, reported in [7, 8, 19, 28]. The SMC nonlinear control strategy proposes differential mapping algorithms that require constant and simultaneous sensing of voltage and current at the output of the PV cell array to perform the MPPT, [9, 20]. Such a nonlinear strategy includes extra steps to determine future states from previous ones based on historical data [7, 19].

On the other hand, in [10], an interconnection and damping assignment passivity-based control (IDA-PBC) is proposed to address the local MPPT problem. Their IDA-PBC method provides a strategy that brings the system to the desired energy equilibrium by sensing the instantaneous current/voltage responses from the PV cell, avoiding the estimation of future and past states.

The strategies mentioned above, based on “electrical current monitoring”, presents several inconveniences thoroughly discussed in [29, 30]. Examples of such inconveniences are the loss of connections, replacement complexity, inconsistent sensibility and resolution at low current, parasitic magnetic energy remanence in the cores, periodical calibration requirements, and erratic frequency responses. Moreover, external current sensors are usually expensive, which increases the inherent cost of the PV power system [11, 29, 31]. Finally, implementing algorithms of future-step prediction required a higher operating frequency and memory from the controller.

Consequently, the elimination of current sensors to track the MPP has been proposed as a counterpart to increase the cost benefits of PV systems [31]. Previous surveys have classified the so-called current-sensorless techniques according to their features and type of MPPT control strategies. See, for instance, [4, 5, 31] and the references therein. Specifically, the temperature/radiation monitoring (TRM) technique demonstrates a lower dependency on the topology and direct sensing connections. Controllers based on sensing radiation are seen as a more effective solution due to their economic advantage, fast response, and noninvasive characteristics [4]. Hence, the radiation parameter might be familiar to any PV cell connected to the same array, allowing a single sensor for control purposes.

In this paper, we propose an energy-based modelling approach for a PV cell array system connected to a DC-DC boost converter, based on the port-Hamiltonian formalism of [32–34]. The formalism is adopted here since the system has energy storage and dissipating elements (inter)connected with input-output power port-pairs. The port-Hamiltonian framework is the natural selection modelling approach because it allows a clear physical interpretation of the system’s energy flow towards a robust and scalable control design. Thus, inspired by [35, 36], a novel current-sensorless control algorithm in the framework of an IDA-PBC strategy is able to track the local MPP under variations of solar irradiance and output loads. The inputs for the control are the output voltage of the PV cell array and the sensed solar irradiance. The control objective is reached by setting the impedance matching between two stages: the PV cell array and a DC-DC boost converter system.

Since the control is based on sensing irradiation as a control input, the strategy allows connecting several PV cell stages to the same irradiation sensor with a separated control per stage. Our proposed strategy is effective since it ensures the extraction of the maximum power from any PV cell or stage. Nonetheless, the strategy requires that environmental conditions and disturbances remain homogeneously over the PV cell stages interconnected to the same radiation sensor. Furthermore, the proposed strategy demands lower frequency and power consumption than other robust control laws such as the SMC strategy, requiring noncomplex hardware to implement the control algorithm. The controller could be extended to several independent MPPT stages interconnected to a single solar irradiance sensor, decreasing the cost of sensing devices. Such an extension implementation is left out of the current work. The outline of this paper is as follows. Section 2 recapitulates the port-Hamiltonian modelling framework. In the same section, we summarize the IDA-PBC technique of [35, 36] and the SMC strategy of [8, 12]. In Section 3, we have proposed a novel port-Hamiltonian modelling approach for the PV cell array connected to a solar irradiance sensor and a DC-DC boost converter system. From our port-Hamiltonian modelling approach, we have developed a sensorless control loop in Section 4, ruled by a control algorithm designed under the IDA-PBC strategy, that avoids the current monitoring as in [10]. Furthermore, in Section 5, simulation results are given to demonstrate the performance of the proposed IDA-PBC compared to the SMC. The system is characterized, implemented, and controlled with a noncomplex hardware platform, and the experimental results are presented in Section 6. Finally, in Section 7 key concluding remarks and future work are provided.

2. Mathematical Modelling and Control Strategies

Firstly, it is necessary to cover the bases of the port-Hamiltonian modelling framework, as a preamble to develop a useful mathematical model to design a IDA-PBC strategy, for a further comparison with the well-known SMC strategy.

3. A Port-Hamiltonian Modelling Approach to a PV Cell Connected to a DC-DC Boost Converter System

We first defined a pH approach to the PV cell modelling problem. Such an energy-based modelling strategy is previously introduced in [38] where we have proposed a model of a pumped hydro storage system powered by solar radiation. Also, in [38] we have simulated the system’s open-loop model in order to evaluate its performance where a control strategy is not yet designed and implemented to attain MPPT. Instead, in [38] we have applied a conveniently fixed frequency square signal to activate the switch of the DC-DC buck converter system.

3.1. PV Cell Modelling Approach

The equivalent circuit of the PV stage shown in Figure 1 is based on the model of [39–41]. The PV cell electrical performance follows the five-parameter model approach as in [42–44]. Into the PV cell structure represented by the dashed rectangle, the current i_ph is generated by the solar irradiance, while the diode D₁ represents the equivalent model of the p-n join semiconductor layers. Furthermore, an equivalent resistor R_p is connected in a shunt configuration, followed by a serial resistor R_S at the system’s output power. Meanwhile, the output of the the PV cell is linked to the next stage across the DC-link capacitor C_pv.

Based on the modelling approach of [42–44], the inner current i_s of the PV cell is defined as

$$ (24)

In (23), the photogenerated current, i.e., i_ph, is given by

$$ (25)

where G_Sun represents the solar irradiance, G_nom the reference irradiance given by the manufacturer, I_SC,nom the PV cell output short circuit measured at G_nom, K₀ the thermal correction factor, T the environment temperature, and T_v the PV cell temperature. The current across the diode i_D in (23) is described by

$$ (26)

where n is the diode quality factor and V_t the thermal voltage of the silicon layer junction. The saturation current i₀ in (25) is described by

$$ (27)

where V_g is the silicium band gap, k the Boltzmann constant, and the constant i_α in (26) given by

$$ (28)

where V_OC,nom is finally the PV cell open circuit voltage. The parameters and variables in (24), (25), (26), and (27) are fully explained in [43]. To determine the system’s dynamics, we first consider Kirchhoff’s current law on node a. The resulting equation is

$$ (29)

Moreover, Kirchhoff’s voltage law on the close path I yields

$$ (30)

We substitute now (29) in (28) such that

$$ (31)

Furthermore, from the circuit’s node b, we know that

$$ (32)

$$ (33)

If we define the electrical charge q_pv as a state variable in terms of the PV cell output voltage v_pv and the capacitor C_pv such that q_pv = C_pvv_pv, then (32) could be rewritten in terms of (30). Finally, the resulting dynamics of the electrical charge is

$$ (34)

We realize now the dynamics of the system’s second state variable that represents the electrical flux, i.e., ϕ_pv, which is given by

$$ (35)

Since the charge ϕ_pv in the capacitor C_pv is the only energy-storing element of the system in Figure 1, then the Hamiltonian function H_pv(q_pv, ϕ_pv) of the PV cell stage is

$$ (36)

Thus, the partial derivatives of the Hamiltonian function in terms of the state variables q_pv and ϕ_pv are

$$ (37)

$$ (38)

which means that the dynamics in (33) can be rewritten in terms of $$ and $$ as

$$ (39)

with k being conveniently chosen as k = −1 to keep the skew-symmetric properties of a new $$ matrix to be defined later on. In (38), we have left out the arguments of H_pv for notation simplicity.

The dynamics in (38) and (34), together with the energy function (35), are used to formulate the pH system Σ_PV of the PV cell, given by

$$ (40)

where $$, and $$ is a skew-symmetric matrix with the form

$$ (41)

Based on the dynamics of (39) with an input-output port pair (u_pv, y_pv) = (i_s, v_p) and the Hamiltonian (35), and since R_p + R_s ≥ 0, we see how the power balance (9) clearly holds.

For control design purposes, it is now necessary to find an expression to track the amplitude of v_pv according to the solar irradiance in order to obtain the so-called current-sensorless control strategy. Such expression is developed in the following subsection.

3.3. DC-DC Boost Converter Modelling Approach

A boost configuration is selected here to achieve a DC-DC voltage step-up conversion which is regulated by a switching device. The converter is able to draw the MPP from the PV cell by adjusting its duty cycle for given solar radiance levels. For further details regarding the working principles of the model, we refer to [45, 46] and the references therein.

Consider now the DC-DC boost converter configuration circuit shown in Figure 2. v_pv at the left side of the circuit represents the PV cell output voltage, and R_L at the right side represents a resistive output load connected to the PV cell via the DC-DC boost converter. Based on [8, 12], the system can be described by the dynamics of i_bc and v_bc. In addition to this, we define a two state switching device represented by S_bc = {0, 1}. Following the Biot-Savart Law, the dynamics of i_bc and v_bc are given in terms of the state variables ϕ_bc = L_bci_bc (magnetic flux in the inductor) and q_bc = C_bcv_bc (charge in the capacitor), respectively. By Kirchhoff’s Laws, it follows that

$$ (45)

$$ (46)

We now define the Hamiltonian function H_bc(q_bc, ϕ_bc) of the DC-DC boost converter in Figure 2 as

$$ (47)

where, together with the dynamics of (44) and (45), we formulate the pH system Σ_BC of the boost DC-DC converter with an output load as

$$ (48)

where $$, and $$ is a skew-symmetric matrix with the form

$$ (49)

Based on the dynamics of (47) with its input-output port pair (u_bc, y_bc) = (v_pv, i_bc) and the Hamiltonian (46), and since R_L > 0, the power balance (9) clearly holds.

The following subsection provides the resulting interconnected system containing the PV cell and the DC-DC boost converter.

4. Proposed Control Approach

We have formulated in Section 2.3 a specific type of SMC strategy for our PV cell plus the DC-DC boost converter system for the MPPT problem according to [8, 12]. Now, we introduce our novel IDA-PBC design to the interconnected system (49), and the Hamiltonian function (50). Our IDA-PBC strategy is inspired by the work of [10, 35, 36], whose main design steps are recapitulated in Section 2.2.

First, we develop the IDA-PBC method of Section 2.2 to the interconnected pH system Σ as in (49). The feedback function β(x) in (10) is obtained in order to control the system (49). We compute below the requirements on structure preservation (14), integrability (15), equilibrium assignment (16), and Lyapunov stability (17) [label = ( )].

5. Simulation Results

We simulate here the behaviour of the terms C_1A and C_1B from the control function β(v_pv), the transfer and power functions of the PV cell, and the response of the PV cell output power at different solar irradiance and output impedance levels. The simulation integrates the control action of our IDA-PBC and the SMC strategies over the pH system (50). Then, a behavioural multi-physics numerical model is developed, fed with the key parameters of Table 1 from Section 6. In order to get accurate simulation results, the sampling of the states for the control strategies ({v_pv, G_sun} in IDA-PBC and {v_pv, i_pv} in SMC) is restricted in frequency into the programming code.

From (56) and (59), it can be demonstrated that the function β(v_pv) has a discontinuous behaviour at the right and left sides of a given equilibrium state $$, respectively, in terms of C_1A and C_1B, as shown in Figure 3. Therefore, C_1A must be aplied to the control law (61) when the instant value of $$ and C_1B in the case $$ to ensure the continuity of the control law over the whole tracking trajectory.

In Figure 4, the responses of the PV cell output power P_pv for the strategies IDA-PBC and SMC in (b) are compared under level changes on the solar irradiance as shown in (a), from 960 W/m² to 670 W/m². Here, the PV cell’s output power P_pv of both the IDA-PBC as in (61), and the SMC as in (23), shows no perceptible differences. The sample rate of the simulation for both control strategies is set to 10 MHz, in order to ensure an SMC suitable performance due to the extra-computation requirements for the differential equation calculation.

In Figure 5, we show the transfer functions i_pv vs v_pv in (a) and P_pv vs v_pv in (b) of a PV cell, in order to determine the MPP of the array. Such characteristic curves of the PV cell are based on the dynamics of (26) to (28), with the key parameters given in Table 1. The figure also shows the values of v_pv, i_pv, and P_pv at the MPP for a solar irradiance G_sun = 1000. We test and evaluate here the capacity of the control laws (61) and (23) to follow the PV cell functions’ behaviour.

In order to further compare the performance of the IDA-PBC and the SMC strategies, the DC-DC boost converter load R_L is simulated as a time function, following a step perturbation from 10Ω to 5Ω (Figure 6(a)). As a result, the power signal P_pv remains unmovable for both the IDA-PBC and the SMC (Figure 6(b)). Once again, the sample rate of the simulation for both control strategies is set to 10 MHz, due to the extra-computation requirements for the differential equation calculation required by the SMC.

The radiation is represented by 100 W/m² steps from 100 to 1000 W/m² for both (a) and (b).

6. Experimental Results

The experimental setup used to measure the radiation and control of the MPP is shown in Figure 7. The setup consists of (a) commercial PV cell JD-7 W; (b) an open-source electronic prototyping platform Arduino UNO; (c) an Adafruit data logger shield; (d) irradiance sensor Spectrum 210; (d) hall current sensor module ACS712 5A; (e) multimeter BK Precision 393; (f) oscilloscope BK Precision 2540C; (g) laptop DELL Latitude 5491; (h) generic electronic components (resistors, capacitors, inductors, semiconductors); (i) protective boxes [48]. We have used real-time data from a sensor of the Costa Rican’s National Meteorological Institute (https://www.imn.ac.cr/web/imn/inicio), to calibrate our irradiance sensor Spectrum323 210. The experimental set up from Figure 7 allows to track the physical parameters v_pv and i_pv under constant solar irradiance of ≈1000 W/m², ≈850 W/m², and ≈700 W/m². Meanwhile, a variable resistance load connected at the output of the PV cell induced different impedance matching conditions. Hence, the experimental PV cell output power function was determined by fitting the PV cell output power P_pv at the MPP, as shown in Figure 8. An MPP is determined from the experimental data for P_pv ≈ 5.7 W when the voltage v_pv ≈ 4.0 V and i_pv ≈ 1.41 A. In addition to this, the values I_SC,nom ≈ 1.95 A and V_OC,nom ≈ 6.5 V are calculated at the top end and bottom end of the impedance matching. These aforementioned parameters allow the characterization of the PV cell as explained in the follow-up.

In order to evaluate the response of IDA-PBC strategy, the system setup in Figure 7 is firstly exposed to a constant solar irradiance. Then, the PV cell device is rotated, generating an irradiance step from G_Sun ≈ 960 W/m² to G_Sun ≈ 670 W/m², as shown in Figure 10(a). Consistently, the response of the output power P_pv to this irradiance step is measured. It is experimentally demonstrated that the IDA-PBC strategy implemented into the hardware platform tracks the MPP (Figure 10(b)), according to the P_pv experimental trend lines of Figure 8.

Finally, Figure 11(a) shows an output impedance step from 10 Ω to 5 Ω that is induced at the output of the boost DC-DC converter at a constant solar irradiance of G_Sun ≈ 960 W/m². Once again, the implemented IDA-PBC strategy controls the system and keeps the PV cell output power P_pv unmovable around 5.5 W (Figure 11(b)). Such result is consistent with the MPP reported on the P_pv experimental functions of Figure 8.

7. Concluding Remarks and Future Work

We have proposed a novel control law for the local MPPT problem in an energy-based setting, which takes advantage of the structure preservation properties of the system under study. The simplicity in the calculations allows an adequate response under uncertainties in radiation inputs. We have also simulation results on an SMC control law strategy. Our novel control law and the SMC are simulated, and we demonstrated that both strategies are able to reach the addressed MPPT problem. However, for the physical implementation, our IDA-PBC approach requires no direct electrical current measurements, being, in this case, less invasive and requiring less monitoring sensors than the SMC strategy.

The control strategy is tested for a single PV cell’s MPPT. However, the solution can be extended to applications where PS is not critical. For instance, an application with PV cell arrays governed with separated DC-DC converters, exposed to the same radiation level or localized far away from PS sources. It means that several PV cell arrays could be controlled with a single sensor. Since the sensor is externally connected, its installation does not depend on a specific place. Moreover, a single sensor requires a single connection line. Such installation simplicity reduces implementation costs, e.g., wiring over the array, in applications such as low power production PV farms and roof PV cell arrays.

The IDA-PBC strategy is physically implemented in a prototype governed by a hardware platform, with a maximum switching rate on the control signal of 122 Hz. The sense sampling time in our control strategy is faster than the SMC strategy. Such an advantage is due to the algorithm’s simplicity, resulting in a low-cost implementation and a low dynamic-power consumption. Moreover, the PV cell is characterized to obtain the MPP at several solar irradiance inputs. The prototype’s performance is tested by setting step changes in the dynamics of the system: firstly, in the solar irradiance input over the PV cell, and secondly, in the impedance output of the boost DC-DC converter. In both cases, the output power remains over the MPP of the PV cell, according to the characterization of the PV cell. Our control law requires the PV cell’s parameter identification. Such identification follows mainly from the manufacturer’s datasheet. The parameter identification procedure should be made once if the PV array is homogeneous in terms of PV cells.

Future work will consider the design and implementation of the proposed control law in a higher power physical system and under reactive output loads conditions.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.