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.

Multilevel inverters can provide an effective alternative of high power applications, providing a high quality voltage, increasing efficiency and robustness, and reducing interference electromagnetics [

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 [

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 [

Recently, the authors [

This paper proposes an adaptive observer in order to estimate the voltages capacitors of 3-levels inverter NPC (Figure

A 3-level NPC converter topologies.

This paper is organized as follows. In Section

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 [

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 [

The SVPWM control of the 3-level NPC inverter consists to control it in the (

Voltage space vectors in NPC inverter.

The objective of SVPWM technique is to approximate a reference space vector

For example in Sector 1 (Figure

Sector 1.

Let

Based on the averaged representation of the PWM switch [

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 [

Consider that

Command signals of the controlled devices in the case of

A simplified representation of averaged model.

If the current

The comparison between the current waveforms (Figure

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

The current error.

Figure

Evolution of the capacitors voltages

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

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

Type model | Running time (s) |
---|---|

Nonideal averaged model | 10 |

Circuit model (Simpower Matlab) | 90 |

The DC side of the three-level NPC inverter (Figure

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 (

The function

From the preceding composition (

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

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

Assume that the input

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

The system will be represented by a set of interconnected subsystems (Figure

Interconnected observer diagram.

If

Each observer subsystem

The proposed observer functions in the case if the inputs controls are regularly persistent (see Appendix

Now, we give the sufficient conditions which ensure the convergence of the interconnected observer

Assume that the input

The term

Then the following result can be established.

Consider the interconnected system

The proof of the lemma is given in Appendix

In this section, a detailed simulation was carried out to validate the proposed interconnected observer. The general block diagram is shown in Figure

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

The different behavior of the interconnected observer was performed. The capacitors values are

Figures

Real and estimated capacitors voltages

Real and estimated capacitors voltages

Figures

Estimation errors for

Estimation errors for

In order to study the observer behavior, we changed the load of the first leg (

Real and estimated capacitors voltages

Real and estimated capacitors voltages

Real and estimated capacitors voltages

Real and estimated capacitors voltages

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

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.

We introduce some definitions related with the inputs applied to the system. Consider a state-affine controlled system of the form

let

We then define the following:

The observability gramian

The universality index

Now, we give definitions of regular persistence of input

A measurable bounded input

This proof is based on the demonstration in [

The dynamics of the estimation error

Now, let

From the time derivative of

Now, adding and subtracting the term

Now, from Assumption

Then, we find

And replacing this expression in

Then, we get

For