# Using Averaged Modeling for Capacitors Voltages Observer in NPC Inverter

Authors: Bassem Omri; Kaiçar Ammous; Anis Ammous
Journal: Advances in Power Electronics
 (2012)

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## 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.


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## Body

## 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 (Figure1). 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 Section2, 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, Sk1 and Sk2, 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 Sk3 and Sk4 on, the state is called “N.” Moreover, the switching state is defined as “O” when Sk2, and Sk3 both power devices, are on.Figure 2
Voltage space vectors in NPC inverter.The objective of SVPWM technique is to approximate a reference space vectorVs 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 (Figure3), the Vs vector can be divided on V1a,V12 or V1b,V12. We choose the second solution, since V1b is greater in amplitude than V1a. At each commutation period of the inverter, the vector Vs, projected on two adjacent vectors, ensures the calculation of the dwelling time.Figure 3
Sector 1.LetTmod be a modulation period, T1b dwelling time of the first vector V1b,T12 dwelling time of the second vector V12, and Tnul=Tmod-T1b-T12 dwelling time of zero vectors. We define the duty cycles ρ1 and ρ2 related to times T1b and T12, respectively, of the vectors V1b and V1a by:
(1)ρ1=T1Tmod,ρ2=T12Tmod.
For one modulation period Tmod, the output voltage of the inverter is given by:
(2)TmodVS=(T1bV1b+T12V12+T3VNull).
And the dwelling time of each vector is given by
(3)T1b=23TmodE(122Vsα-322Vsβ),T12=22TmodEVsβ.Tnull=Tmod-T12-T1b.
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. Isa is the average value of the load current over the switching period. This model applies a controlled voltage source (Va1 and Va2) and a controlled current source (Ia1 and Ia2). 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 thatTmod 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 ((Si1,Si2), (Si2, Si3), or (Si3,Si4)) by the switching period Tmod, respectively (Figure 5).Figure 5
Command signals of the controlled devices in the case ofIsa>0.V
d and Vds 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):
(4)ρi=ToniTpwm;ρMi=ToffiTpwm,ρM=ρM1+ρM2+ρM3.
When only output current behavior is considered, a simplified representation of the averaged model (Figure 6) can be used,  where
(5)〈Ina〉=(Ina1+Ia2-Ia1),〈Va〉=(Va1+Va2),〈Ia〉=〈Ia2〉.
The averaged values of the voltage source Va and the current source Ia of the bridge (A) are given as follows.Figure 6
A simplified representation of averaged model.If the currentIsa is positive,
(6)〈Va〉=2Vdsρ1+(E+2Vd)ρ3+(E2+Vd+Vds)ρ2+(E+2Vd)ρM,〈Ia〉=-Isaρ3,〈Ina〉=Isaρ2.
If the current Isa is negative,
(7)〈Va〉=-2Vdρ1+(E-2Vds)ρ3+(E2-Vd-Vds)ρ2+(2Vd)ρM,(8)〈Va〉=-2Vdρ1+(E-2Vds)ρ3+(E2-Vd-Vds)ρ2+(2Vd)ρM,(9)〈Ia〉=Isaρ3.
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 (Figure7) 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.Figure9 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 voltagesC1 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 (Table1). 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).
Type model
Running time (s)
Nonideal averaged model
10
Circuit model (Simpower Matlab)
90


## 4. Observer Design of Capacitors Voltages

The DC side of the three-level NPC inverter (Figure1) has a DC source E and two connections capacitors C1 and C2. The switching variable γk represents the state of the multilevel inverter active switches, Skj, k∈{1,2,3}, j∈{1,2,3,4}. The three states of the binary switching γk can be defined as
(10)γk={1(Sk1=1∧Sk2=1)∧(Sk3=0∧Sk4=0)0(Sk1=0∧Sk2=1)∧(Sk3=1∧Sk4=0)-1(Sk1=0∧Sk2=0)∧(Sk3=1∧Sk4=1).
Applying the Kirchhoff laws to the three-level inverter circuit (Figure 1) and doing some mathematical manipulations, the dynamic equations of the AC currents, Isa, Isb, and Isc, and the capacitor voltages, VC1 and VC2, are defined as functions of the circuit parameters and switching variables γk:
(11)[dIsadtdIsbdtdIscdtdVC1dtdVC2dt]=[-RL00ψ11Lψ12L0-RL0ψ21Lψ22L00-RLψ31Lψ31L-Γ11C1-Γ12C1-Γ13C100-Γ21C2-Γ22C2-Γ23C200]×[IsaIsbIscVC1VC2]+[0001C11C2]×idc,(12)Γ1k=γk(γk+1)2,Γ2k=γk(-γk+1)2,ψki=13(2Γik-∑j=1,j≠k3Γik).
In order to obtain a simplified model which contains less equations, we change the switching sequences in the (α,β) coordinates:
(13)X123=[T]Xαβ,
where T is the Concordia transformation:
(14)T=23(10-1232-12-32).
Applying (11) to (8), the new model is represented in (α,β) coordinates by
(15)[dIαdtdIβdtdVC1dtdVC2dt]=[-RL0Γ1αLΓ2αL0-RLΓ1βLΓ2βL-Γ1αC1-Γ1βC100-Γ2αC2-Γ2βC200]×[IαIβVC1VC2]+[001C11C2]×idc,
where
(16)Γiα=23(Γi1-Γi22-Γi32),Γiβ=12(Γi2-Γi3).
This model can be written as
(17)X˙=A(S)X+H(S,X),y=CX,
where X=(IαIβVC1VC2)T, C=(1100),H(S,X)=[001/C11/C2]. 						(18)A(S)=[-RL0Γ1αLΓ2αL0-RLΓ1βLΓ2βL-Γ1αC1-Γ1βC100-Γ2αC2-Γ2βC200].
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:
(19)Θ=rang([CCACA2CA3])=3≺4.
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:
(20)Σ1:[dIαdtdIβdtdVC1dt]=[-RL0Γ1αL0-RLΓ1βL-Γ1αC1-Γ1βC10][IαIβVC1]+[Γ2αLΓ2βL0]VC2+[001C1]×idc,(21)Σ2:[dIαdtdIβdtdVC2dt]=[-RL0Γ2αL0-RLΓ2βL-Γ2αC2-Γ2βC20][IαIβVC2]+[Γ1αLΓ1βL0]VC1+[001C2]×idc.
Each subsystem can be written as
(22)Σj{χ˙j=Aj(Sk)χj+Γi(Sk,χj),y=Cjχj,j=1,2k=1,2,3i=1,2≠j,
where χj=(IαIβVCj)T represents the state of the jth subsystem, Sk are the instantaneous inputs applied to the system, and y is the measurable output, and
(23)Aj(Sk)=[-RL0ΓjαL0-RLΓjβL-ΓjαCj-ΓjβCj0],Γi(Sk,χj)=[ΓiαLΓiβL0]VCi+[001Cj]×idc,(24)j=1,2,k=1,2,3,i=1,2≠j,
and C=(110).The functionΓi(Sk,χj) is the  interconnection function. It depends on the control Sk and the state vector of each subsystem Σj.From the preceding composition (22), the following system:
(25)φj{Zj=Aj(Sk)Zj+Γi(Sk)-Pj-1Cj(y-y^),Pj=-θjPj-AT(Sk)Pj-PjA(Sk)+CjCjT,
is an observer for the system (16) for j=1,2, where  Pj-1Cj is the observer gain which depends on the solution of the Riccati equation (25) for each subsystem Σj. The parameter θj must be a positive constant and sufficiently greater such that for any positive symmetric matrix Pj(0), the following condition is verified [23]:
(26)∀θ≥θ0∃γ≻0,δ≻0,t0≻0:∀t≥t0,γI≤P(0)≤δI.
The parameter θj 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 inputsSj 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 Pj is nonsingular.Now, a further result based on regular persistence is introduced.Lemma 1.
Assume that the inputSj is regularly persistent for system (22) and consider the following Lyapunov differential equation:
(27)P˙j=-θjPj-AjT(Sk)Pj-PjAj(Sk)+CjCjT.
With Pj≻0, then, ∃θj0≻0 such that for any symmetric positive definite matrix Pj(0), ∃θj≻θj0, ∃αj, βj, t0≻0: ∀t≻t0,
(28)∀t≻t0,αjI≺Pj(t)≺βjI,
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 (Figure10):
(29)Σ{χ˙1=A1(Sk)χ1+Γ2(Sk,χj)χ˙2=A2(Sk)χ2+Γ1(Sk,χj).
The idea of this paper is to construct an observer for all the system Σ from the separate observers of the subsystem Σj.Figure 10
Interconnected observer diagram.Ifφj is an exponential observer for Σj, then the following interconnected system:
(30)φ{χ˙1=A1(Sk)χ1+Γ1(Sk,χj)+P1-1C1T(y^-y)χ˙2=A2(Sk)χ2+Γ2(Sk,χj)+P2-1C2T(y^-y)P1=-θ1P1-A1T(Sk)P1-P1A1(Sk)+C1C1TP2=-θ2P2-A2T(Sk)P2-P2A2(Sk)+C2C2T.
is an observer for the interconnected system Σ.Each observer subsystemφj is defined by (25).Remark 2.
The proposed observer functions in the case if the inputs controls are regularly persistent (see AppendixA), which is equivalent to that all subsystem φj is observable.Now, we give the sufficient conditions which ensure the convergence of the interconnected observerφ. For that, we introduce the following assumptions.Assumption 1.
Assume that the inputSj is a regularly persistent input for subsystem Σj and admits an exponential observer φj 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 termΓj(Sk,χi) does not destroy the observability property of subsystem φj under the action of the regularly persistent input Sj. Moreover, Γj(Sk,χi) is Lipchitz with respect to χi and uniformly with respect to Sj 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 systemΣ 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 AppendixB.


## 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 Figure11. 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 areC1=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 θj=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.Figures12 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 voltagesVC1 (V) using nonideal average model and circuit model (Simpower Matlab).Figure 13
Real and estimated capacitors voltagesVC2 (V) using nonideal average model and circuit model (Simpower Matlab).Figures14 and 15 show the estimated errors for the estimated capacitors voltages VC1 and VC2. The error for each model is calculated by (VCi-real)-(VCi-estimated).Figure 14
Estimation errors forVC1 (V) using nonideal average model and circuit model (Simpower Matlab).Figure 15
Estimation errors forVC2 (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 voltagesVC1 (V) using nonideal average model with disturbance load.Figure 17
Real and estimated capacitors voltagesVC2 (V) using nonideal average model with disturbance load.Figure 18
Real and estimated capacitors voltagesVC1 (V) using nonideal average model with disturbance load and its elimination.Figure 19
Real and estimated capacitors voltagesVC2 (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 (Table1) 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.



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