Using Averaged Modeling for Capacitors Voltages Observer in NPC Inverter

Abstract

This paper developed an adaptive observer to estimate capacitors voltages of a three-level neutral-point-clamped (NPC) inverter. A robust estimated method using one parameter is proposed, which eliminates the voltages sensors. An averaged modeling of the inverter was used to develop the observer. This kind of modeling allows a good trade-off between simulation cost and precision. Circuit model of the inverter (implemented in Simpower Matlab simulator) associated to the observer algorithm was used to validate the proposed algorithm.

1. Introduction

Multilevel inverters can provide an effective alternative of high power applications, providing a high quality voltage, increasing efficiency and robustness, and reducing interference electromagnetics [1–4]. There are three main topologies of multilevel inverters: diode-clamped inverter [2, 5, 6], flying capacitor inverter [7, 8], and cascade multilevel inverter [6, 9]. Among the multilevel inverters, the most popular topology is the diode-clamped inverter which is called as a neutral-point-clamped (NPC) inverter in three levels. It is proposed by Nabae et al. [10]. This type of inverter avoids the complexity associated with the series connection of semiconductor switches or the bulky coupling transformer, produces low distortion harmonics, and has an average cost.

However, in NPC inverter, the DC-link voltage is divided by capacitors, and each capacitor is composed of series connection construction. Therefore, if the voltage unbalancing occurs between each capacitor, the line-to-line output voltage waveform has many harmonic components and the power devices in NPC inverter cannot guarantee the safe operation. So, one NPC inverter requirement is to produce a good control maintaining the optimal balance of capacitor voltages [11–13]. Thus, regulation of these capacitors voltages requires information about such voltages. For that there are two possibilities either measuring or estimating such voltages.

Regarding the first possibility, it presents several disadvantages: besides the known difficulties when measuring a high-voltage level, if the number of levels increases the number of capacitors increases in NPC inverter. Then it will be necessary to use multiple-voltage sensors to measure the capacitor voltages, which reduces the reliability. Furthermore, the use of the voltages sensors presents many difficulties of establishment in front of the use of the current sensors. On the other hand, the second alternative uses software sensors, called observers, which replace the physical sensors (the voltage sensors). It is suitable to estimate the capacitors voltages through only the phase currents measurement. So, the observer is independent of the number of the level of NPC inverter. Unlike the traditional direct voltage sensing depends on the levels number.

An observer is a data-processing algorithm to reconstruct the variable state system from the input, mathematical model, and output measurement of the real system. It is an attractive solution because it benefits from a mathematical model of the system. It is less expensive and more reliable because it is applied by a digital computer. Actually, the calculation of the digital signal processors (DSP) is the execution of such sophisticated and complex algorithms with the high-degree precision. Many works related to the observer design for the nonlinear systems were published [14–16].

Recently, the authors [17] designed a robust observer based on adaptive backstepping approach for three-level NPC inverters. It is used for capacitor voltages estimation. This approach gave good results. However, the authors used an instantaneous reference model of NPC inverter. Therefore, the simulation cost is significant. Moreover, the adaptive backstepping algorithm used many parameters (adaptation gains, design constant) to design the observer. The algorithm becomes so complex when the number of levels increases.

This paper proposes an adaptive observer in order to estimate the voltages capacitors of 3-levels inverter NPC (Figure 1). The adapive algorithm is robust, simple to implement, and uses one parameter called forgetting factor. An averaged reference model of three-level NPC inverter is used for simulation. This kind of modeling allows a good tradeoff between simulation cost and precision.

Figure 1 

A 3-level NPC converter topologies.

This paper is organized as follows. In Section 2, the PWM control strategies are presented. An averaged reference model of three-level inverter NPC is given in Section 3. Section 4 shows the observer design for the NPC inverter. Finally, simulation results are given in order to illustrate the performance of the proposed observer to estimate NPC capacitors voltages.

2. The PWM Control Strategies of Multilevel Inverters

The proliferation of power electronic devices has led to a demand for more effective pulse width modulation strategies. The current waveform can be improved by increasing the frequency of the carrier wave; this approach reduces copper losses at the expense of increased switching losses. To overcome the limitations of existing switching strategies the technique known as space vector pulse width modulation (SVPWM) is becoming widely used in industry [18, 19].

SVPWM is a highly efficient method of generating the six-pulsed signals for the inverter stage of the motor drive. Conventional switching techniques treat each phase as a separately generated sinusoid that is displaced by 120 degrees. However, a change in the voltage of one-half bridge due to switching invariably influences the other two-phase voltages [20]. SVPWM evaluates the switching schema as a whole, which results in better use of the DC bus and generates significantly less harmonic distortion than the sine triangle method [20].

The SVPWM control of the 3-level NPC inverter consists to control it in the (𝛼,𝛽) coordinates which can generate 27 possible voltages vectors (Figure 2). They are divided in 6 large vectors (PNN,PPN,NPN,NPP,NNP,PNP), 6 medium vectors (PON,OPN,NPO,NOP,ONP,PNO), 12 small vectors (with the redundancy: (PPO,OON), (OPO,NON), (OPP,NOO), (OOP,NNO), (ONO,POP), (POO,ONN)) and 3 zero vectors (OOO,NNN,PPP). When the power devices, 𝑆_𝑘1 and 𝑆_𝑘2, are connected to the positive DC-link rail the state is noted by “P.” In contrast, when the output is connected to the negative DC-rail by turning both S_𝑘3 and S_𝑘4 on, the state is called “N.” Moreover, the switching state is defined as “O” when S_𝑘2, and S_𝑘3 both power devices, are on.

Figure 2 

Voltage space vectors in NPC inverter.

The objective of SVPWM technique is to approximate a reference space vector 𝑉_𝑠 somewhere within the hexagon of Figure 2 using a combination of the 27 switching vectors. Indeed, when the reference voltage is in one sector, several couples of vector are possible.

For example in Sector 1 (Figure 3), the 𝑉_𝑠 vector can be divided on 𝑉_1𝑎,𝑉₁₂ or 𝑉_1𝑏,𝑉₁₂. We choose the second solution, since 𝑉_1𝑏 is greater in amplitude than 𝑉_1𝑎. At each commutation period of the inverter, the vector 𝑉_𝑠, projected on two adjacent vectors, ensures the calculation of the dwelling time.

Figure 3 

Sector 1.

Let 𝑇_mod be a modulation period, 𝑇_1𝑏 dwelling time of the first vector 𝑉_1𝑏,𝑇₁₂ dwelling time of the second vector 𝑉₁₂, and 𝑇_nul=𝑇_mod−𝑇_1𝑏−𝑇₁₂ dwelling time of zero vectors. We define the duty cycles 𝜌₁ and 𝜌₂ related to times 𝑇_1𝑏 and 𝑇₁₂, respectively, of the vectors 𝑉_1𝑏 and 𝑉_1𝑎 by: 𝜌₁=𝑇₁𝑇_mod,𝜌₂=𝑇₁₂𝑇_mod. (1) For one modulation period 𝑇_mod, the output voltage of the inverter is given by: 𝑇_mod𝑉_𝑆=𝑇_1𝑏𝑉_1𝑏+𝑇₁₂𝑉₁₂+𝑇₃𝑉_Null. (2) And the dwelling time of each vector is given by 𝑇_1𝑏=2√3𝑇_mod𝐸12√2𝑉_𝑠𝛼−√32√2𝑉_𝑠𝛽,𝑇₁₂=2√2𝑇_mod𝐸𝑉_𝑠𝛽.𝑇_null=𝑇_mod−𝑇₁₂−𝑇_1𝑏. (3) For more details see [21].

3. The Nonideal Average Model

Based on the averaged representation of the PWM switch [22], Figure 4 shows the proposed averaged model of the one-leg three-level multilevel converters. 𝐼_𝑠𝑎 is the average value of the load current over the switching period. This model applies a controlled voltage source (𝑉_𝑎1 and 𝑉_𝑎2) and a controlled current source (𝐼_𝑎1 and 𝐼_𝑎2). Each three-level converter leg has four controlled switches and six uncontrolled switches. The four controlled switches are directly governed by external control signals. The uncontrolled switches are indirectly governed by the state of the controlled switches and the circuit conditions.

Figure 4 

The proposed average model of the 3-level PWM converter.

The average values of the different voltage and current sources of the proposed averaged model of the bridge (A) are given by [22].

Consider that 𝑇_mod is the switching period of the active switch and 𝜌 (𝜌₁,𝜌₂,𝜌₃) the duty cycle, which is the ratio of the on-time of the active couple switches ((𝑆_𝑖1,𝑆_𝑖2), (𝑆_𝑖2, 𝑆_𝑖3), or (𝑆_𝑖3,𝑆_𝑖4)) by the switching period 𝑇_mod, respectively (Figure 5).

Figure 5 

Command signals of the controlled devices in the case of 𝐼𝑠𝑎>0.

𝑉_𝑑 and 𝑉_𝑑𝑠 represent respectively, the voltage across the diode and the MOSFET in on-state. These values can be deduced from the devices static characteristics 𝐼(𝑉): 𝜌_𝑖=𝑇_on𝑖𝑇_𝑝𝑤𝑚;𝜌_𝑀𝑖=𝑇_off𝑖𝑇_𝑝𝑤𝑚,𝜌_𝑀=𝜌_𝑀1+𝜌_𝑀2+𝜌_𝑀3. (4) When only output current behavior is considered, a simplified representation of the averaged model (Figure 6) can be used, where 𝐼_𝑛𝑎=𝐼_𝑛𝑎1+𝐼_𝑎2−𝐼_𝑎1,𝑉_𝑎=𝑉_𝑎1+𝑉_𝑎2,𝐼_𝑎=𝐼_𝑎2. (5) The averaged values of the voltage source 𝑉_𝑎 and the current source 𝐼_𝑎 of the bridge (A) are given as follows.

Figure 6 

A simplified representation of averaged model.

If the current 𝐼_𝑠𝑎 is positive, 𝑉_𝑎=2𝑉𝑑𝑠𝜌₁+(𝐸+2𝑉𝑑)𝜌₃+𝐸2+𝑉𝑑+𝑉𝑑𝑠𝜌₂+(𝐸+2𝑉𝑑)𝜌_𝑀,𝐼_𝑎=−𝐼_𝑠𝑎𝜌₃,𝐼_𝑛𝑎=𝐼_𝑠𝑎𝜌₂. (6) If the current 𝐼_𝑠𝑎 is negative, 𝑉_𝑎=−2𝑉𝑑𝜌₁+(𝐸−2𝑉𝑑𝑠)𝜌₃+𝐸2−𝑉𝑑−𝑉𝑑𝑠𝜌₂+(2𝑉𝑑)𝜌_𝑀,𝑉_𝑎=−2𝑉𝑑𝜌₁+(𝐸−2𝑉𝑑𝑠)𝜌₃+𝐸2−𝑉𝑑−𝑉𝑑𝑠𝜌₂+(2𝑉𝑑)𝜌_𝑀,𝐼_𝑎=𝐼_𝑠𝑎𝜌₃. (7)(8)(9) In order to demonstrate the accuracy of the proposed averaged model, a circuit model of the inverter implemented in Simpower Matlab simulator, using the circuit represented in Figure 1 is used. For the simulation in this section and Section 5, we use 2 GHz Core2 Duo processor and 3 GB RAM.

The comparison between the current waveforms (Figure 7) obtained by the Simpower simulator and the current given by the proposed model shows the good accuracy of the averaged model behavior for converter electric analysis. The current error between the two types’ models is given in Figure 8.

Figure 7 

The output current obtained with the non-ideal averaged model and the circuit model after filtering (3-levels converter).

Figure 8 

The current error.

Figure 9 shows the capacitors voltages (𝐶₁ and 𝐶₂) waveforms obtained by the model Simpower simulator and the averaged model. The simulation results given by the proposed averaged model are in good agreement with Simpower simulations.

Figure 9 

Evolution of the capacitors voltages 𝐶1 and 𝐶2 (using Simpower Matlab and the averaged model).

It is evident that multilevel converter analysis using fine models gives accurate results, but these simulations cost is unaffordable. When long time range simulation is needed, the proposed averaged model can be used, and the obtained results show the accuracy of the proposed model (Table 1). So, we used this model like a reference model in the observer design.

Table 1 

CPU running time (s) necessary to simulate a period (0.02 s).

4. Observer Design of Capacitors Voltages

The DC side of the three-level NPC inverter (Figure 1) has a DC source 𝐸 and two connections capacitors 𝐶₁ and 𝐶₂. The switching variable 𝛾_𝑘 represents the state of the multilevel inverter active switches, 𝑆_𝑘𝑗, 𝑘∈{1,2,3}, 𝑗∈{1,2,3,4}. The three states of the binary switching 𝛾_𝑘 can be defined as 𝛾_𝑘=1𝑆_𝑘1=1∧𝑆_𝑘2=1∧𝑆_𝑘3=0∧𝑆_𝑘40𝑆=0_𝑘1=0∧𝑆_𝑘2=1∧𝑆_𝑘3=1∧𝑆_𝑘4𝑆=0−1_𝑘1=0∧𝑆_𝑘2=0∧𝑆_𝑘3=1∧𝑆_𝑘4=1. (10) Applying the Kirchhoff laws to the three-level inverter circuit (Figure 1) and doing some mathematical manipulations, the dynamic equations of the AC currents, 𝐼_𝑠𝑎, 𝐼_𝑠𝑏, and 𝐼_𝑠𝑐, and the capacitor voltages, 𝑉_𝐶1 and 𝑉_𝐶2, are defined as functions of the circuit parameters and switching variables 𝛾_𝑘: 𝑑𝐼_𝑠𝑎𝑑𝑡𝑑𝐼_𝑠𝑏𝑑𝑡𝑑𝐼_𝑠𝑐𝑑𝑡𝑑𝑉_𝐶1𝑑𝑡𝑑𝑉_𝐶2=−𝑅𝑑𝑡𝐿𝜓00¹¹𝐿𝜓₁₂𝐿𝑅0−𝐿0𝜓₂₁𝐿𝜓₂₂𝐿𝑅00−𝐿𝜓₃₁𝐿𝜓₃₁𝐿−Γ₁₁𝐶₁−Γ₁₂𝐶₁−Γ₁₃𝐶₁−Γ00₂₁𝐶₂−Γ₂₂𝐶₂−Γ₂₃𝐶₂×𝐼00_𝑠𝑎𝐼_𝑠𝑏𝐼_𝑠𝑐𝑉_𝐶1𝑉_𝐶2+0001𝐶₁1Γ𝐶2×𝑖𝑑𝑐,_1𝑘=𝛾_𝑘𝛾_𝑘+12,Γ_2𝑘=𝛾_𝑘−𝛾_𝑘+12,𝜓_𝑘𝑖=132Γ_𝑖𝑘−³_{𝑗=1,𝑗≠𝑘}Γ_𝑖𝑘. (11)(12) In order to obtain a simplified model which contains less equations, we change the switching sequences in the (𝛼,𝛽) coordinates: 𝑋₁₂₃=[𝑇]𝑋_𝛼𝛽, (13) where 𝑇 is the Concordia transformation: √𝑇=2√3−1102√32−12−√32. (14) Applying (11) to (8), the new model is represented in (𝛼,𝛽) coordinates by 𝑑𝐼_𝛼𝑑𝑡𝑑𝐼_𝛽𝑑𝑡𝑑𝑉_𝐶1𝑑𝑡𝑑𝑉_𝐶2=−𝑅𝑑𝑡𝐿0Γ_1𝛼𝐿Γ_2𝛼𝐿𝑅0−𝐿Γ_1𝛽𝐿Γ_2𝛽𝐿−Γ_1𝛼𝐶₁−Γ_1𝛽𝐶₁−Γ00_2𝛼𝐶₂−Γ_2𝛽𝐶₂×𝐼00_𝛼𝐼_𝛽𝑉_𝐶1𝑉_𝐶2+001𝐶₁1𝐶₂×𝑖𝑑𝑐, (15) where Γ_𝑖𝛼=√2√3Γ_𝑖1−Γ_𝑖22−Γ_𝑖32,Γ_𝑖𝛽=1√2Γ_𝑖2−Γ_𝑖3. (16) This model can be written as 𝑋̇=𝐴(𝑆)𝑋+𝐻(𝑆,𝑋),𝑦=𝐶𝑋, (17) where 𝑋=(𝐼_𝛼𝐼_𝛽𝑉_𝐶1𝑉_𝐶2)^𝑇, 𝐶=(1100),𝐻(𝑆,𝑋)=⁰⁰_{1/𝐶11/𝐶2}. 𝐴(𝑆)=−𝑅𝐿0Γ_1𝛼𝐿Γ_2𝛼𝐿𝑅0−𝐿Γ_1𝛽𝐿Γ_2𝛽𝐿−Γ_1𝛼𝐶₁−Γ_1𝛽𝐶₁−Γ00_2𝛼𝐶₂−Γ_2𝛽𝐶₂00. (18) For different configurations, the capacitors are not always traversed by the load current. The current can pass by one or two capacitors as it cannot traverse any capacitor if the top switchers of each leg are all closed or all open. Thus, some configurations give observability problem of the studied system. This can be proven by the test of rank, whenever, at least, a capacitor is traversed by the load current: 𝐶Θ=rang𝐶𝐴𝐶𝐴²𝐶𝐴³=3≺4. (19) It is clear that Θ is not of full rank; thus the system is not observable. So, to design the observer, a convenient representation of the model must be given.

To overcome this difficulty, the idea is to consider two interconnected affine models and to construct the new observer based on those models.

Now, consider that the system (15) can be represented as a set of the interconnected subsystems as follows: Σ₁∶𝑑𝐼_𝛼𝑑𝑡𝑑𝐼_𝛽𝑑𝑡𝑑𝑉_𝐶1=−𝑅𝑑𝑡𝐿0Γ_1𝛼𝐿𝑅0−𝐿Γ_1𝛽𝐿−Γ_1𝛼𝐶₁−Γ_1𝛽𝐶₁0𝐼_𝛼𝐼_𝛽𝑉_𝐶1+Γ_2𝛼𝐿Γ_2𝛽𝐿0𝑉_𝐶2+001𝐶₁Σ×𝑖𝑑𝑐,₂∶𝑑𝐼_𝛼𝑑𝑡𝑑𝐼_𝛽𝑑𝑡𝑑𝑉_𝐶2=−𝑅𝑑𝑡𝐿0Γ_2𝛼𝐿𝑅0−𝐿Γ_2𝛽𝐿−Γ_2𝛼𝐶₂−Γ_2𝛽𝐶₂0𝐼_𝛼𝐼_𝛽𝑉_𝐶2+Γ_1𝛼𝐿Γ_1𝛽𝐿0𝑉_𝐶1+001𝐶₂×𝑖𝑑𝑐. (20)(21) Each subsystem can be written as Σ_𝑗𝜒̇_𝑗=𝐴_𝑗𝑆_𝑘𝜒_𝑗+Γ_𝑖𝑆_𝑘,𝜒_𝑗,𝑦=𝐶_𝑗𝜒_𝑗,𝑗=1,2𝑘=1,2,3𝑖=1,2≠𝑗, (22) where 𝜒_𝑗=(𝐼_𝛼𝐼_𝛽𝑉_𝐶𝑗)^𝑇 represents the state of the jth subsystem, 𝑆_𝑘 are the instantaneous inputs applied to the system, and 𝑦 is the measurable output, and 𝐴_𝑗𝑆_𝑘=−𝑅𝐿0Γ_𝑗𝛼𝐿𝑅0−𝐿Γ_𝑗𝛽𝐿−Γ_𝑗𝛼𝐶_𝑗−Γ_𝑗𝛽𝐶_𝑗0,Γ_𝑖𝑆_𝑘,𝜒_𝑗=Γ_𝑖𝛼𝐿Γ_𝑖𝛽𝐿0𝑉_𝐶𝑖+001𝐶_𝑗×𝑖𝑑𝑐,𝑗=1,2,𝑘=1,2,3,𝑖=1,2≠𝑗, (23)(24) and 𝐶=(110).

The function Γ_𝑖(𝑆_𝑘,𝜒_𝑗) is the interconnection function. It depends on the control 𝑆_𝑘 and the state vector of each subsystem Σ_𝑗.

From the preceding composition (22), the following system: 𝜑_𝑗𝑍_𝑗=𝐴_𝑗𝑆_𝑘𝑍_𝑗+Γ_𝑖𝑆_𝑘−𝑃_𝑗⁻¹𝐶_𝑗𝑦,𝑃𝑦−_𝑗=−𝜃_𝑗𝑃_𝑗−𝐴^𝑇𝑆_𝑘𝑃_𝑗−𝑃_𝑗𝐴𝑆_𝑘+𝐶_𝑗𝐶^𝑇_𝑗, (25) is an observer for the system (16) for 𝑗=1,2, where 𝑃_𝑗⁻¹𝐶_𝑗 is the observer gain which depends on the solution of the Riccati equation (25) for each subsystem Σ_𝑗. The parameter 𝜃_𝑗 must be a positive constant and sufficiently greater such that for any positive symmetric matrix 𝑃_𝑗(0), the following condition is verified [23]: ∀𝜃≥𝜃₀∃𝛾≻0,𝛿≻0,𝑡₀≻0∶∀𝑡≥𝑡₀,𝛾𝐼≤𝑃(0)≤𝛿𝐼. (26) The parameter 𝜃_𝑗 determines the convergence rate of the observer. The authors [23] proposed guidelines for the tuning of the parameters of the adaptive observer.

Note that this observer is the deterministic version of the Kalman filter for state affine systems. It is clear that the system observability depends on the applied inputs. Then the convergence of this observer can be proved assuming that the inputs 𝑆_𝑗 are regularly persistent (see Appendix A); it is a class of admissible inputs that allows to observe the system (for more details, see [19]). This guarantees the works of observer, and the observer gain is well defined, that is, the matrix 𝑃_𝑗 is nonsingular.

Now, a further result based on regular persistence is introduced.

Lemma 1. Assume that the input 𝑆_𝑗 is regularly persistent for system (22) and consider the following Lyapunov differential equation: 𝑃̇_𝑗=−𝜃_𝑗𝑃_𝑗−𝐴^𝑇_𝑗𝑆_𝑘𝑃_𝑗−𝑃_𝑗𝐴_𝑗𝑆_𝑘+𝐶_𝑗𝐶^𝑇_𝑗. (27) With 𝑃_𝑗≻0, then, ∃𝜃_𝑗0≻0 such that for any symmetric positive definite matrix 𝑃_𝑗(0), ∃𝜃_𝑗≻𝜃_𝑗0, ∃𝛼_𝑗, 𝛽_𝑗, 𝑡₀≻0: ∀𝑡≻𝑡₀, ∀𝑡≻𝑡₀,𝛼_𝑗𝐼≺𝑃_𝑗(𝑡)≺𝛽_𝑗𝐼, (28) where 𝐼 is the identity matrix.

The proof of this lemma follows the same steps of Theorem  3.2 in [24].

The system will be represented by a set of interconnected subsystems (Figure 10): Σ𝜒̇₁=𝐴₁𝑆_𝑘𝜒₁+Γ₂𝑆_𝑘,𝜒_𝑗𝜒̇₂=𝐴₂𝑆_𝑘𝜒₂+Γ₁𝑆_𝑘,𝜒_𝑗. (29) The idea of this paper is to construct an observer for all the system Σ from the separate observers of the subsystem Σ_𝑗.

Figure 10 

Interconnected observer diagram.

If 𝜑_𝑗 is an exponential observer for Σ_𝑗, then the following interconnected system: 𝜑𝜒̇₁=𝐴₁𝑆_𝑘𝜒₁+Γ₁𝑆_𝑘,𝜒_𝑗+𝑃₁⁻¹𝐶^𝑇₁𝜒̇𝑦−𝑦₂=𝐴₂𝑆_𝑘𝜒₂+Γ₂𝑆_𝑘,𝜒_𝑗+𝑃₂⁻¹𝐶^𝑇₂𝑃𝑦−𝑦₁=−𝜃₁𝑃₁−𝐴^𝑇₁𝑆_𝑘𝑃₁−𝑃₁𝐴₁𝑆_𝑘+𝐶₁𝐶^𝑇₁𝑃₂=−𝜃₂𝑃₂−𝐴^𝑇₂𝑆_𝑘𝑃₂−𝑃₂𝐴₂𝑆_𝑘+𝐶₂𝐶^𝑇₂. (30) is an observer for the interconnected system Σ.

Each observer subsystem 𝜑_𝑗 is defined by (25).

Remark 2. The proposed observer functions in the case if the inputs controls are regularly persistent (see Appendix A), which is equivalent to that all subsystem 𝜑_𝑗 is observable.

Now, we give the sufficient conditions which ensure the convergence of the interconnected observer 𝜑. For that, we introduce the following assumptions.

Assumption 1. Assume that the input 𝑆_𝑗 is a regularly persistent input for subsystem Σ_𝑗 and admits an exponential observer 𝜑_𝑗 for 𝑗=1,2. If an input is regularly persistent, it excites sufficiently the system to obtain information necessary to constructing the not measured variables using the designed observer (see Appendix A). In this case, an observer of the form 𝜑 can be designed and the estimation error will be bounded.

Assumption 2. The term Γ_𝑗(𝑆_𝑘,𝜒_𝑖) does not destroy the observability property of subsystem 𝜑_𝑗 under the action of the regularly persistent input 𝑆_𝑗. Moreover, Γ_𝑗(𝑆_𝑘,𝜒_𝑖) is Lipchitz with respect to 𝜒_𝑖 and uniformly with respect to 𝑆_𝑗 for 𝑗=1,2. This condition is verified because these functions are linear.

Then the following result can be established.

Lemma 3. Consider the interconnected system Σ and the two Assumptions 1 and 2 are verified. Then the system 𝜑 (30) is an observer for the system Σ (29).

The proof of the lemma is given in Appendix B.

5. Simulation Results

In this section, a detailed simulation was carried out to validate the proposed interconnected observer. The general block diagram is shown in Figure 11. In this study the 3-level NPC inverter block was replaced by the averaged model of the inverter to develop the observer. The block of the interconnected observer is given by (30). In order to show the accuracy of the observer a fine model (circuit model) of the inverter (implemented in Matlab Simpower) was used to estimate the capacitors voltages. The estimated capacitors voltages of the two models are compared to the real capacitors voltages. We define the real capacitor voltage by the capacitor voltage given by the voltages sensors of the circuit model implemented in Simpower Matlab.

Figure 11 

Block diagram of three-level NPC inverter and interconnected observer.

The different behavior of the interconnected observer was performed. The capacitors values are 𝐶₁=𝐶₂=1000 μF. The NPC Inverter is connected at three equilibrium phases where each phase composed by an inductor 𝐿=10mHand a resistor 𝑅=20Ω. To estimate the NPC capacitor voltages according to (25) the observer parameter gain 𝜃_𝑗=15000 for 𝑗=1,2 is used. SVPWM was used to control the multilevel inverter described in Section 2. The switching frequency was fixed at 10 kHz. Simulations were carried out using matlab/simulink. Matlab simulator is an important tool where different system models (electrical, mechanical, thermal, etc.) can be developed. In fact, Matlab simulator enables averaged model representation.

Figures 12 and 13 show the comparison between the estimated capacitors voltages 𝐶₁ and 𝐶₂ (using nonideal average model and circuit model) and the real capacitor voltage.

Figure 12 

Real and estimated capacitors voltages 𝑉𝐶1 (V) using nonideal average model and circuit model (Simpower Matlab).

Figure 13 

Real and estimated capacitors voltages 𝑉𝐶2 (V) using nonideal average model and circuit model (Simpower Matlab).

Figures 14 and 15 show the estimated errors for the estimated capacitors voltages 𝑉_𝐶1 and 𝑉_𝐶2. The error for each model is calculated by (𝑉_𝐶𝑖-real)−(𝑉_𝐶𝑖-estimated).

Figure 14 

Estimation errors for 𝑉𝐶1 (V) using nonideal average model and circuit model (Simpower Matlab).

Figure 15 

Estimation errors for 𝑉𝐶2 (V) using nonideal average model and circuit model (Simpower Matlab).

In order to study the observer behavior, we changed the load of the first leg (𝐿=10 mH, =20 Ω) to (𝐿=10 mH, 𝑅=40 Ω) at 𝑡=0.1 s (Figures 16 and 17). In other terms, we have not got equilibrium three phases. Then, Figures 18 and 19 show at 𝑡=0.1 s the change of the load of the first leg (𝐿=10 mH, =20 Ω) to (𝐿=10 mH, 𝑅=40 Ω) and at 𝑡=0.15 s we return to the initial load.

Figure 16 

Real and estimated capacitors voltages 𝑉𝐶1 (V) using nonideal average model with disturbance load.

Figure 17 

Real and estimated capacitors voltages 𝑉𝐶2 (V) using nonideal average model with disturbance load.

Figure 18 

Real and estimated capacitors voltages 𝑉𝐶1 (V) using nonideal average model with disturbance load and its elimination.

Figure 19 

Real and estimated capacitors voltages 𝑉𝐶2 (V) using nonideal average model with disturbance load and its elimination.

Simulations results show the usefulness of this design. In any condition, the observer is able to follow the real capacitor voltages accurately. Moreover, the good agreement between estimated capacitors voltages obtained by the proposed averaged model and the circuit model from Simpower Matlab is illustrated. We notice that simulation time with fine model (circuit model implemented in Simpower Matlab) is eight times (Table 1) greater than the simulation with the averaged model. This is an important reason to use averaged modeling especially in complex converters architecture. Converter states observer and control becomes, more and more easy.

6. Conclusion

The control and monitoring of the capacitors voltages in multilevel NPC converter are essential. Measuring voltages becomes expensive and impractical because of the high voltages and power levels handled in such applications. Thus the advantage of using an observation technique becomes evident.

This paper proposed an adaptive observer to estimate the capacitor voltages in NPC inverter. This observer can reconstruct the capacitor voltages accurately from the nonideal average model inputs and replace the needed capacitor voltage sensors. Simulation results show the good precision of the proposed average model. It can be used for the NPC converters analysis instead of fastidious simulations in circuit type simulators where simulations cost is unaffordable. The obtained results are acceptable, and let us conclude that the observer is well suited for purposes control of the capacitor voltages in multilevel NPC inverter.

Appendices

A. Observability Definitions

We introduce some definitions related with the inputs applied to the system. Consider a state-affine controlled system of the form Σ𝑥̇=𝐴(𝑢)𝑥+𝐵(𝑢),𝑦=𝐶𝑥, (A.1) where 𝑥∈ℝ^𝑛, 𝑢∈ℝ^𝑚, 𝑦∈ℝ^𝑝 with 𝐴∶ℝ^𝑚→𝑀(𝑛,𝑚), 𝐵∶ℝ^𝑚→𝑀(𝑛,1) continuous, and 𝐶∈𝑀(𝑝,𝑛), where 𝑀(𝑘,𝑙) denotes the space of 𝑘×𝑙 matrices with coefficients in ℝ𝑘 (resp., 𝑙) are the number of rows (resp., columns). From now on, we will assume that 𝐵(𝑢) = 0 without loss of generality.

Notation. let 𝜑_𝑢(𝜏,𝑡) denotes the transition matrix of 𝑑𝜑𝑑𝜏^𝑢(𝜏,𝑡)=𝐴(𝑢(𝜏))𝑥+𝜑_𝑢(),𝜑𝜏,𝑡_𝑢(𝜏,𝑡)=𝐼, (A.2) with the classical relation 𝜑_𝑢(𝑡₁,𝑡₂)𝜑_𝑢(𝑡₂,𝑡₃)=𝜑_𝑢(𝑡₁,𝑡₃).

We then define the following:(i)The observability gramian Γ(𝑡,𝑇,𝑢)=_𝑡^𝑡+𝑇𝜑^𝑇_𝑢(𝜏,𝑡)𝐶^𝑇𝐶𝜑_𝑢(𝜏,𝑡)𝑑𝜏. (A.3)(ii)The universality index 𝛾(𝑡,𝑇,𝑢)=min_𝑖𝜆_𝑖(Γ(𝑡,𝑇,𝑢)), (A.4)where 𝜆_𝑖(𝑀) stands for the eigenvalues of a given matrix 𝑀. The input functions are assumed to be measurable and such that 𝐴(𝑢) is bounded on the set of admissible inputs of ℝ⁺.

Now, we give definitions of regular persistence of input 𝑢.

Definition 4 (regular Persistence). A measurable bounded input 𝑢 is said to be regularly persistent for the state-affine system (Σ) if there exists 𝑇≻0, 𝛼≻0, 𝑡₀≻0 such that 𝛾(𝑡,𝑇,𝑢)≻0 for every 𝑡≥𝑡₀.

B. Regular Persistence

This proof is based on the demonstration in [25].

The dynamics of the estimation error 𝜀_𝑗=𝑍_𝑗−𝜒_𝑗 is given by 𝜀_𝑗̇=(𝐴(𝑆_𝑗)−𝑃_𝑗⁻¹𝐶^𝑇_𝑗𝐶_𝑗)𝜀_𝑗+ΔΓ_𝑗(𝑆_𝑗,𝜒_𝑗,𝑍_𝑗), where ΔΓ_𝑗(𝑆_𝑗,𝜒_𝑗,𝑍_𝑗)=Γ_𝑗(𝑆_𝑗,𝑍_𝑗)−Γ_𝑗(𝑆_𝑗,𝜒_𝑗) for 𝑗=1,2.

Now, let 𝑉=𝑉₁+𝑉₂ be a Lyapunov function for the interconnected system Σ, where 𝑉_𝑗(𝜀_𝑗)=𝜀^𝑇_𝑗𝑃_𝑗𝜀_𝑗 is a Lyapunov function for each subsystem Σ_𝑗.

From the time derivative of 𝑉_𝑗(𝜀_𝑗), it follows that 𝑉̇_𝑗(𝜀_𝑗)≤−𝜃_𝑗𝑉_𝑗(𝜀_𝑗)+𝜀^𝑇_𝑗𝑃_𝑗ΔΓ_𝑗(𝑆_𝑗,𝜒_𝑗,𝑍_𝑗) for 𝑗=1,2.

Now, adding and subtracting the term ΔΓ_𝑗(𝑆_𝑗,𝜒_𝑗,𝑍_𝑗)^𝑇𝑃_𝑗ΔΓ_𝑗(𝑆_𝑗,𝜒_𝑗,𝑍_𝑗) we have 𝑉̇_𝑗𝜀_𝑗≤−𝜃_𝑗𝑉_𝑗𝜀_𝑗+𝜀^𝑇_𝑗𝑃_𝑗ΔΓ_𝑗𝑆_𝑗,𝜒_𝑗,𝑍_𝑗±ΔΓ_𝑗𝑆_𝑗,𝜒_𝑗,𝑍_𝑗^𝑇𝑃_𝑗ΔΓ_𝑗𝑆_𝑗,𝜒_𝑗,𝑍_𝑗. (B.1) Next, regrouping the appropriate terms, 𝑉̇_𝑗𝜀_𝑗≤−𝜃_𝑗−1𝜀_𝑗²_𝑃𝑗−𝜀_𝑗²_𝑃𝑗+2𝜀^𝑇_𝑗𝑃_𝑗ΔΓ_𝑗𝑆_𝑗,𝜒_𝑗,𝑍_𝑗−ΔΓ_𝑗𝑆_𝑗,𝜒_𝑗,𝑍_𝑗²_𝑃𝑗+ΔΓ_𝑗𝑆_𝑗,𝜒_𝑗,𝑍_𝑗²_𝑃𝑗. (B.2) From this inequality, we get 𝑉̇_𝑗(𝜀_𝑗)≤−(𝜃_𝑗−1)‖𝜀_𝑗‖²_𝑃𝑗+‖ΔΓ_𝑗(𝑆_𝑗,𝜒_𝑗,𝑍_𝑗)‖²_𝑃𝑗.

Now, from Assumption 2, we have ‖ΔΓ_𝑗(𝑆_𝑘,𝜒_𝑗,𝑍_𝑗)‖²_𝑃𝑘≤(𝜆_𝑙‖𝜀_𝑙‖²_𝑃𝑗)_{𝑙∈{1,2}≠𝑗}.

Then, we find 𝑉̇_𝑗(𝜀_𝑗)≤−(𝜃_𝑗−1)‖𝜀_𝑗‖²_𝑃𝑗+(𝜆_𝑙‖𝜀_𝑙‖²_𝑃𝑗)_{𝑙∈{1,2}≠𝑗}.

And replacing this expression in 𝑉=𝑉₁+𝑉₂ it follows that: 𝑉̇(𝜀)≤²_𝑗=1−𝜃_𝑗−1𝜀_𝑗²_𝑃𝑗+𝜆_𝑙𝜀_𝑙²_{𝑃𝑗𝑙∈{1,2}≠𝑗}. (B.3) Take into account that the inputs are regularly persistent, and then from Lemma 1 the matrix 𝑃_𝑗 is bounded. Using the lemma on equivalence of norms, that is, there exists a positive constant 𝜇_𝑙 such that 𝜀_𝑙²_𝑃𝑗≤𝜇_𝑙𝜀_𝑙²_𝑃𝑙. (B.4)

Then, we get 𝑉̇(𝜀)≤²_𝑗=1−𝜃_𝑗−1𝜀_𝑗²_𝑃𝑗+𝜆_𝑙𝜀_𝑙²_{𝑃𝑙𝑙∈{1,2}≠𝑗}. (B.5) or 𝑉̇_𝑗𝜀_𝑗≤−²_𝑗=1𝜃_𝑗−1−𝜇_𝑙𝜆_𝑙𝜀_𝑗²_𝑃𝑗. (B.6) Finally, we have 𝑉̇𝜀_𝑗≤𝑉(𝜀(𝑡))𝑒^{−𝛾(𝑡−𝑡0)}. (B.7)

For 𝛾=min(𝛾₁,𝛾₂), where 𝛾_𝑗=(𝜃_𝑗−1)−𝜇_𝑗𝜆_𝑗, taking 𝜀=col(𝜀₁,𝜀₂), it is easy to see that ‖𝜀(𝑡)‖≤𝐾𝜀𝑡₀𝑒^{−𝛾(𝑡−𝑡0)}. (B.8)