PV system and single-phase grid characteristics.
This chapter addresses the problem of controlling single-phase grid-connected photovoltaic system through a full bridge inverter with L-filter. The control objectives are threefold: (i) forcing the voltage in the output of photovoltaic panel to track a reference. This reference has been obtained from the maximum power point tracking strategy; (ii) guaranteeing a tight regulation of the DC-link voltage; and (iii) ensuring a satisfactory power factor correction (PFC) at the grid such as the currents injected must be sinusoidal with the same frequency and the same phase as the grid voltage. The considered control problem entails several difficulties including: (i) the high dimension and strong nonlinearity of the system; (ii) the changes in atmospheric conditions. The problem is dealt with by designing a synthesized nonlinear multi-loop controller using singular perturbation technique, in which a three-time-scale dynamics is artificially induced in the closed-loop system. A formal analysis based on the three-time-scale singular perturbation technique and the averaging theory is developed to proved that all control objectives are asymptotically achieved up to small harmonic errors (ripples). The performance of the proposed approach and its strong robustness with respect to climate changes are evaluated based on the various simulations results carried out under Matlab/Simulink software.
- single-phase grid-connected photovoltaic system
- nonlinear control
- three-time-scale singular perturbation technique
- power factor correction
- averaging theory
- stability analysis
Due to dramatic increase in energy consumption and thrust to reduce carbon and greenhouse gas emissions from the traditional electric power generation systems, photovoltaic (PV) power generators have gained a great popularity in recent years. Indeed, photovoltaic systems produce electric power without harming the environment, transforming a free inexhaustible source of energy, solar radiation, into electricity. Furthermore, the major advantage of the photovoltaic systems is to meet the basic power requirement of non-electrified remote areas, where grid power has not yet reached. Also, there are other advantages such as the declining cost and prices of solar modules. On the other hand, the importance of PV systems in the solar industry makes these systems more efficient and reliable, especially for utility power in distributed generation (DG) at medium and low voltages power systems . All these considerations assure a promising role for PV generation systems in the near future.
On the other hand, many technical problems, such as untimely failures, could be found on electronic systems related in particular to the transfer and conversion of this energy to the network. Today, most conversion systems often suffer from low yields in real production sites. To meet the requirements of the new international standards on expected performance on associated conversion systems, it is important to make a research effort to solve the many control problems associated with the static power converter and bring this area to a degree of sufficient maturity to make them industrial products in their own right. One of the difficulties caused by the use of a photovoltaic conversion chain is focused on the problem of non-perfect control of the chain between the photovoltaic generator itself and the continuous or alternative type of load.
The efficiency of a PV plant is affected mainly by three factors: the efficiency of the PV panel, the efficiency of the static power converter and its control, and the efficiency of the maximum power point tracking (MPPT) algorithm.
PV grid-connected systems represent the most important field applications of solar energy [2, 3, 4]. In general, the power converter interface, from PV module (the DC source) to the load or to the grid, consists of two-stage converters: The first-stage DC/DC converter is usually used to boost the PV voltage and to implement the maximum power point technique. While the second stage is used to convert this power into high-quality AC voltage, with power factor correction (PFC) respecting to the power supply grid (i.e. sinusoidal and in phase with the AC supply voltage).
Maximum power point tracking (MPPT) is required to match the PV array power to the environmental changes achieving to extract the maximum power output from a solar cell . To this end, different MPPT techniques have been proposed such as incremental conductance , perturbation and observation (P&O) [7, 8], the hill-climbing, and some other special methods, such as neural networks, fuzzy logic technique . Among all available techniques, a simple and effective MPPT of incremental conductance algorithm is applied to attain the maximum power of PV array in different solar irradiance and temperature condition parameters.
In order to provide a stable controller of DC/DC and DC/AC converters, many linear control methods have been proposed using many methods, such as a fuzzy-proportional integral controlled , a simple PR controller  where the performances have been illustrated by experimental result. However, in both proposed controllers, the problem of maximizing PV power transfer is not accounted for in the controller design. In contrast to linear control, nonlinear approaches can optimize the dynamic performance of system, such as sliding mode , fuzzy-sliding mode , feedback linearization , singular perturbation technique , and many others works [16, 17, 18]. In light of the previous descriptions, no theoretical analysis is made to formally prove that the closed-loop control performances are actually achieved.
In this chapter, a multi-loop nonlinear controller is designed and developed via singular perturbation technique (Chapter 11 in Refs. [19, 20]), as was shown in Refs. [21, 22], where three-time-scale dynamics is artificially induced in the closed-loop system. The control objectives are threefold: (i) achieving the MPPT for the PV array; (ii) ensuring a tight regulation of the DC-link voltage; and (iii) ensuring a grid connection with unity power factor (PF). These objectives must be met despite changes of the climatic variables (temperature and radiation). A theoretical stability analysis, for the closed-loop system, is provided using the three-time-scale singular perturbation technique [23, 24] and averaging technique (Chapter 10 in Refs. [19, 25]. The three-time-scale analysis allows to construct a suitable composite Lyapunov function candidate for the closed-loop photovoltaic system, and the stability properties of the resulting subsystems are analyzed providing mathematical expressions for the upper bounds of the singularly perturbed parameters.
Compared to previous works, the contribution of the new nonlinear controller enjoys several interesting features including the following:
A theoretical analysis will prove, using three-time-scale singular perturbation and averaging technique, that the desired multiple objectives are achieved. Such a formal analysis was missing in the previous works [12, 13, 14, 15].
The nonlinearity of the controlled system was preserved in the controller design in order to keep all the properties of the studied system, whereas it is partly or totally ignored in previous controllers .
By including of three-time-scale dynamics in the full-order closed-loop system can ensure to achieve desired properties, such as robust zero steady-state error of the reference input realization, desired output performance specifications (overshoot, settling time), and insensitivity of the output transient behavior with respect to parameter variations and external disturbances.
The content of this chapter is outlined as follows: in Section 2, the grid-connected PV system is described and modeled. Section 3 is devoted to the cascade nonlinear controller design and its performances are formally analyzed in Section 4. The global performance of the closed-loop photovoltaic system will be illustrated by numerical simulation using MATLAB/SIMULINK tool in section 5. A conclusion and a reference list end the chapter.
2. System description
This section describes the modeling of photovoltaic system connected to the grid. The power circuit topology used in the proposed single phase grid connected to the photovoltaic array is shown in Figure 1. It consists of the following components: (i) a photovoltaic array which consists of an arrangement of -series and -parallel strings; (ii) an input capacitor and a DC-DC boost converter used to increase the voltage level and achieve MPPT for photovoltaic array; (iii) a DC link capacitor ; and (iv) a single-phase full-bridge inverter including four power semiconductors with filter that is used to provide the energy to the grid and ensure power factor correction.
Typical (Ip-Vp) characteristics of solar cells arranged in -parallel and -series can be found in many places (see, e.g. ). The PV array module considered in this paper is of type KC200GT. In this chapter, a simple and effective MPPT of incremental conductance algorithm is applied to attain the maximum power of PV array in different solar irradiance and temperature condition parameters.
By analyzing the circuit and applying the well-known Kirchhoff laws, the system of Figure 1 can be described by the following set of differential equations:
where and are, respectively, the photovoltaic generator voltage and current. and are, respectively, DC link voltage and current. designates the input current chopper, is DC link capacitor, and are, respectively, the equivalent series resistances (ESR) of input inductance and the filter inductance . and are, respectively, the voltage and current of the grid. Here, the grid voltage is defined by , where and denote the constant amplitude and the constant angular frequency. The switching functions and are defined by:
The instantaneous model (1)–(4) cannot be used directly for the development of continuous control laws since it involves, as input variables, the binary signal and . To overcome this inconvenience, the average model is used . Therefore, the state variables , , , and are replaced by their average values , , , and over a cutting period. The control inputs and denote the average values of and , respectively.
3. Controller design
3.1 Input inductor current regulation and PFC objectives
3.1.1 Control law design
The first control objective is to enforce the photovoltaic voltage to track, as closely as possible, the optimal point (called Regulator 2). However, it is well-known that the boost converter has a non-minimum phase feature. Such an issue is generally dealt with by resorting a cascaded loop design strategy that starts with the input current loop (Regulator 1), as it is shown in Figure 2. More specifically, the controller makes the input inductor current to track a reference signal , the latter is determined from (Regulator 2).
In parallel with the input current controller, the network current controller (Regulator 3) will be designed for power factor correction requirement that amounts to forcing the network current to match the reference signal of the form . It means that the grid current should be sinusoidal and in phase with the AC grid voltage , with is a signal. In fact, the latter is allowed to (and actually will) be time-varying but it must converge to a positive constant value. For both objectives (i.e. the input current regulation and PFC objectives), let us consider Eqs. (7), (9) in the following form:
where , , , and .
Remark 1. Under condition that the DC-link voltage remains all the time positive, it can be showed that and are satisfied.
Let us introduce the following current tracking errors:
Then, the reference model can be constructed in the following form
where and denote the time constants and they are selected based on the desired settling time, respectively, for the currents and . Based on (13), the realization errors of the desired behaviors of and , namely and , are given by
Therefore, the control problem corresponds to the insensitivity condition defined by
Hence, there is an isolated root called the inverse dynamic solutions for given by
The control variable, namely , has emerged in Eq. (16). At this point, an appropriate control law with the first derivative in feedback has to be found, so that the (, )-systems are made asymptotically stable. As the objective is to drive the error to zero, it is natural to choose the Lyapunov functions candidate
are made negative-definite using the following control laws:
3.1.2 Singular perturbation system of the inner current loops
with , , , .
Now, we go to the fast time . Then for , the ultra-fast dynamic subsystem (UFDS) is given by
After the rapid decay of transients in expression (25), the steady state (more precisely, quasi-steady state) tends toward an equilibrium . The manifold defined by is called the slow manifold, which is given by
During the fast transient in expression (25), the variables are treated as the frozen parameters.
The equilibrium point given by expression (27) involves a division by the DC link voltage , from a practical point of view this division is not a problem because the DC link voltage remains all the time positive for the power converter to work correctly.
By substituting of this equilibrium into Eq. (24), the slow dynamic subsystem (SDS) of inner loops takes place on the slow manifold, according to the equation
Proposition 1. Consider the closed-loop system composed of Eqs. (23) and (24). For , the system takes the singular perturbation form where the UFDS is defined by equation (25), while the SDS of inner loop is defined by equation (28). Under the considerations given by Remark 2, one has the following properties
If the design parameters () are positives, the UFDS (25) will be exponentially stable, and converge exponentially fast to .
The behaviors of () are prescribed by the stable reference equations of the form . Then, the requirements are maintained.
3.2 MPPT and DC bus voltage regulation objective
The second step consists in completing the inner control loops by outer control loops for PV voltage (Regulator 2) and DC-link voltage (Regulator 4). The aim is now to enforce the photovoltaic voltage and the DC-link voltage to track, respectively, the optimal point and a given reference voltage , such that tuning laws for the ratio and must be designed. According to the three-time-scale design methodology that is employed in this work, the general formulation of the three-time-scale singular perturbed systems requires the system to possess three different time scales. To this end, the voltage loops will be slow compared to the transients of the current loops. Therefore, the design parameters for voltage loops, in particular (, , and ) must satisfy: , and . In addition, the steady states for the current and yield, respectively, and . Therefore, the SDS of inner loop given by equation (28) will be reduced to
Now, the first step is to establish the relation between the ratio (which acts as the control input of the outer loop) and the DC-link voltage (representing the output of the outer loop).
3.2.1 Relation between and , and the control law
The relation between the ratio and the DC-link voltage is the subject of the following proposition.
Proposition 2. We consider the second equation of expression (29) and the power factor correction requirement defined by .
The relation between and is described in the following resultsE30
Therefore, the squared-voltage varies, in response to the tuning ratio β, according to the following first-order time-varying nonlinear equation:
The second step is to establish control laws for the outer loops, in which represent the new control inputs, while represent the new output variables. To this end, introduce the following tracking errors
Then, let the desired behavior of be assigned by
The error of the desired dynamic realization it follows
Then, the insensitivity condition is given by
Similar to the previous subsection and bearing in mind the fact that and their first derivative must be available, and in order to meet the requirement (35), we should apply the control law given by the following structure
3.2.2 Singular perturbation system of the outer voltage loops
with . The fast dynamic subsystem (FDS) is obtained by transforming the slow time-scale to the fast time-scale , then, by setting
Notice that expression (39) has an isolated equilibrium at , which will be determined (in the mean) in Appendix. As the FDS (39) is nonlinear, the stability properties of its equilibrium can be checked through the analysis of the Jacobian matrix of the linearized version defined as follows
with , . Taking into account that is positive, we conclude that all eigenvalues of satisfy for the negative values of and . Therefore, is Hurwitz matrix. By substituting of the equilibrium into expression (38), the reduced SDS of outer loops takes place on the slow manifold, according to the equation
Proposition 3. Consider the system closed loop composed of expressions (37) and (38). For , this system takes the singular perturbation form, where the FDS is given by expression (39) and the reduced SDS of outer loop is given by expression (42). One has the following properties
If the design parameters and are negative, the FDS (39), will be exponentially stable and converge exponentially fast to .
The behaviors of () are prescribed by the stable reference equations of the form . Then, the requirements () are maintained.
4. Control system analysis
The objective of the global stability of closed-loop system can be analyzed in the following theorem. It is shown that the control objectives are achieved (in the mean) with an accuracy that depends on the network frequency and the small parameters .
Theorem. Consider the overall control system composed of the Pv panel, boost DC-DC converter and DC-AC inverter, described by the model (7)–(10), in closed loop with the multi-cascade multi-loop composed of:
The inner regulators (23), where (, , , , ) are the design parameters;
The outer regulators (37), where (, , , , ) are the design parameters.
Then, one has the following property
The augmented state vector , , and undergoes the following state equationsE43E44E45
, , .
Let the control design parameters be selected, such that the following inequalities hold , , , , , , and .
where , , and with , , , , , .
See Appendix for the proof.
5. Simulation and discussion of results
The experimental setup is described by Figure 2 and the nonlinear controller, developed in Section 3, including the control laws (23) and (37), will now be evaluated by simulation in MATLAB/SIMULINK platform using the electromechanical characteristics of Table 1.
|PWM switching frequency|
The numerical values used for the design parameters are chosen as follows: , , , , , , , , , . These values have proved to be suitable based on several trials respecting the singular perturbation technique. In this simulation, we consider the KC200GT type of PV array module with and .
The performances of the proposed controller are illustrated by Figures 3–5. Figure 3 shows that the DC-link voltage is well regulated and quickly settles down after each change in the signal reference (stepping from to at ). The wave frame of the output current is showed in Figure 4. The current is sinusoidal and in phase with the network voltage complying with the PFC requirement. This is further demonstrated by Figure 5, which shows that the ratio takes a constant value after transient periods following the changes in reference signals.
5.1 Radiation variation effect
Figure 6a–c illustrates the resulting closed-loop control performances in presence of radiation changes. Specifically, the radiation takes a low, medium, and high value (equal to 800, 1000 and step to 600 W/m2 at times 0, 0.4, and 0.7 s, respectively), meanwhile the temperature is kept constant, equal to (i.e. ). Figure 6a shows that the captured PV voltage varies between 182, 212, and 138 V. These values correspond to the maximum points. Figure 6b shows that the DC-link voltage regulation is recovered after a short transient period following each change of the irradiation. Figure 6c shows that the current amplitude changes whenever the radiation varies. It is seen that the output current and the grid voltage are actually sinusoidal and in phase. Hence, the converter connection to the supply network is done with a unitary power factor.
5.2 Temperature variation effect
The perfect MPPT is illustrated by Figure 7a. Here, the temperature steps from 298.15 to 318.15 K, then to 308.15 K while the radiation λ is kept constant equal to . Figure 7b shows that the DC-link voltage is tightly regulated: it quickly settles down after each change in the temperature. Figure 7c illustrates the current amplitude changes whenever the temperature varies. The current remains (almost) sinusoidal and in phase with the network voltage complying with the PFC requirement.
In this chapter, an advanced controller is developed for PV grid-connected system. The latter is described by fourth-order nonlinear averaged model. The multi-loops nonlinear controller has been designed and developed using three-time singular perturbation technique and averaging theory.
Using the theoretical analysis (via three-time-scale singular perturbation technique and averaging theory) and simulation, it is proved that the controller does meet the performances for which it was designed, namely: (i) Maximum power point tracking of PV array; (ii) tight regulation of the DC bus voltage; (iii) perfect power factor in the grid; and (iv) global asymptotic stability of the all system.
Several simulation results have been made that illustrate the high performances of the proposed controller in ideal operating conditions (in the presence of meteorological constant) and its robustness against radiation and temperature change.
Part 2: The stability of the time-varying system (43)–(45) will now be performed in two steps using the averaging theory (e.g. Chapter 10 in Refs. [19, 25]) and the singular perturbation theory (e.g., Chapter 11 in Refs. [19, 23, 24]). The next step consists in using the averaging theory. To this end, let us introduce the time-scale change . Using this time, one gets , , and with . In view of expressions (43)–(45), it is seem that the functions , , and as functions of are periodic with period–2π, let us introduce the averaged functions:
Since the systems here studied present equilibrium different from zero and in order to satisfy this requirement, a change of variables is introduced such that defines the new system in terms of its error dynamics. Therefore, the error dynamics are defined by introducing: , , and , where the constant , , , , , , , and represent the desired average values of the state variables. Then, since expressions (43)–(45) and according to expressions (47)–(49), the system can be rewritten into its error-dynamics formulation thus defining the closed-loop error dynamics as:
In order to get stability results regarding the system of interest (43)–(45), it is sufficient (thanks to averaging theory) to analyze the averaged system (50)–(52). Now, the asymptotic stability of the resulting three-time-scale photovoltaic single phase grid system (50)–(52) is discussed, which is based on the sequential (double) time-scale analysis similar to the one presented in Refs. [23, 24], it is an extension of the two-time-scale analysis presented .
The use of theory of the three-time-scale singular perturbations for the stability analysis is based on the idea that, for , the trajectories in , , and of the system (50)–(52) can be approximated by three models: the slow dynamic subsystem (SDS) of full system, the fast dynamic subsystem (FDS), and the ultra-fast dynamic subsystem (UFDS). We can thus find Lyapunov functions for each one of the singularly perturbed subsystems.
For the UFDS, it is necessary to ensure that the dynamic of expression (50) does not to shift from the quasi-steady-state equilibrium . Then, the associated Lyapunov function candidate is obtained by introducing a change of variables , so that its equilibrium is centered at zero. By letting , the UFSD in function of is defined as follows
in which is treated as a fixed parameter. Thus, the associated Lyapunov function can be defined by
In view of and , it is clear that the solutions and can be chosen as
with and are positive. Similar to the ultra-fast subsystem, it is necessary to ensure that the dynamic of expression (51) does not to shift from the equilibrium . Then, by introducing a change of variables , so that its equilibrium is centered at zero and by letting , the ultra-fast subsystem can be rewritten in the following form
Since expression (61), it is seem that the fast subsystem is nonlinear. According to the proposition 4 (part i), it is shown that this subsystem can be made asymptotically stable by letting (for ). Moreover, we can find a Lyapunov function, which takes the following quadratic form
with the solutions to the associated Lyapunov given as
in which is treated as a fixed parameter. Finally, the new slow dynamic subsystem is obtained by substituting of the ultra-fast subsystem equilibrium and the fast subsystem equilibrium into expression (52)
It is easy to define the associated Lyapunov function for SDS as follows
where , , , , and , , , are the positive constants.
Based on these Lyapunov function candidates, the double application of the standard two-time-scale stability analysis is divided in two stages: in the first stage, the stability analysis focusses on proving the stability properties of the degenerated -subsystem (slow-fast subsystem). The results obtained will be used in the second stage in order to prove the stability properties for the full -system (slow-fast-ultra-fast subsystem).
In the first stage, the standard method for two-time-scale systems is applied in which the previously derived Lyapunov functions for the slow and fast subsystems, that is, and , respectively, must satisfy certain inequalities.
Isolated equilibrium of the origin for the-subsystem
The origin is an isolated equilibrium of the -subsystem
moreover, is the root of which vanishes at , and where is a function.
Reduced system condition for the-subsystem
Using the Lyapunov function (67) and substituting yields
where and .
Boundary-layer system condition for the-subsystem
where and are defined as follows
For physical point of view and domain of working principle, it is supposed that all physical state variables are bounded in domain of interest, where , , and . Therefore, and can take positives minimum possible values and . This can be done by ensuring the appropriate selection of , , , and as follows
Therefore, the boundary-layer system condition for the -subsystem is defined as follows
where and .
Now, we consider the composite Lyapunov function candidate of the -subsystem given as follow
with . The derivative of presents new terms which represents the effect of the interconnection between the slow and fast dynamics. These interconnections are assumed to satisfy the following conditions
where the constants , , and are non-negative.
Therefore, from the singular perturbation theory (e.g. Theorem 11.3 in Ref. ), it follows that the derivative of VSF is negative-definite for
The stability of full system is analyzed now. In this step, the results obtained in above section will be used in order to prove the asymptotic stability properties of the full -system, which, for convenience, is first rewritten as
where , similarly to the general asymptotic stability analysis presented in the above section, the -system is treated like a two-time-scale singularly perturbed system, where the -subsystem is treated as the new slow augmented reduced order. The stability of the full system implicates that the previously derived Lyapunov functions for the new slow and new fast subsystems, that is, and , respectively, must satisfy certain inequalities.
Isolated equilibrium of the origin for the-subsystem
The origin is an isolated equilibrium of the -subsystem
Moreover, is the unique root of , which vanishes at , and where is a function.
Reduced system condition for the-subsystem
Using Lyapunov function candidate given by expression (76), it is easily shown that
, , .
Substituting the expressions of and by using absolute value version of Young’s inequality , it shows that
where , , , , , and
Now, it needs to ensure that . This can be done by ensuring the appropriate selection of , , , , , and , such that and have sufficient large positive values, and have sufficient small positive values, and and are limited as follows , (for ). Therefore, the reduced system condition for the system is defined as follows
where and .
Boundary-layer system condition for the -system
where and .
Now, we consider the composite Lyapunov function candidate of the full system given as follow
with . The derivative of presents new terms, which represents the effect of the interconnection between the slow and fast dynamics. These interconnections are assumed to satisfy the following conditions:
with and .
Therefore, it can be inferred that the equilibrium , , and of full system, is asymptotically stable for all . Therefore, Part 2 immediately follows from the averaging theory (e.g. Theorem 10.4 in Ref. ). Theorem is established.