Submodule Voltage Sensor Faults Diagnosis in Modular Multilevel Converters

Submodule Voltage Sensor Faults Diagnosis in Modular Multilevel Converters Faizan Mehmood∗, Muhammad Umair Mutarraf†, Marco Liserre†,Lenos Hadjidemetriou∗ ∗KIOS Research and Innovation Center of Excellence, University of Cyprus, Nicosia, Cyprus †Chair of Power Electronics, Kiel University, Kiel, Germany Abstract—Modular Multilevel Converters (MMCs) have emerged as a key technology for grid-integrated renewable energy systems in high-power applications due to their superior performance and scalability. The effective operation of an MMC for stable and reliable renewable energy integration is crucial and depends on its controller. The MMC controller relies on feedback from submodule (SM) capacitor voltage sensors, as well as voltage and current sensors installed in each arm and on the direct current (DC) and alternating current (AC) sides of the converter. However, the failures or divergence of these sensory devices from their normal operation can affect the MMC operation, leading to capacitor voltage imbalance, increased circulating currents, and reliability issues. Therefore, this paper proposes a model-based fault diagnosis algorithm to timely and accurately detect multiple SM sensor faults in the MMC system. Firstly, an online estimator with time-varying estimator gain matrix is developed for SM capacitor voltages estimation, considering the effects of sensor noise and uncertainty. Secondly, the online estimator is utilized to design the SM voltage sensor fault diagnosis scheme based on analytical redundancy relationships, established through pairs of residual signals and adaptive thresholds. Finally, the proposed SM sensor fault diagnosis scheme has been validated through simulation case studies, showing its effectiveness in diagnosing sensor faults and in improving the reliability and safety of MMCs. Index Terms—Fault diagnosis, modular multi-level converter, sensor faults, submodule capacitor voltage. I. INTRODUCTION The modular multilevel converter (MMC) is considered one of the most promising solutions for high-power applications due to its ability to cascade submodules (SMs) to achieve different voltage levels and operate efficiently at low switching frequencies with reduced losses. Recently, MMC technology has gained attention and applicability in largescale RES integration and high-voltage direct current (HVDC) interconnections [1], [2]. Both applications utilize extensive cascaded SMs to withstand high DC-link voltage and enhance harmonic performance. However, the deployment of a large number of SM voltage sensors is necessary to measure the SM capacitor voltage, as it is needed by the MMC controller. The faults in SM voltage sensors or divergence from their normal operation can result in an improper operation of the MMC. This work is funded by the European Union’s Horizon Europe research and innovation programme under grant agreement No 101172757 (HYNET) and by the Horizon Europe Research and Innovation Programme under grant agreement No 101075747 (TRANSIT). This work was also performed within the OMRES project funded by the CETPartnership, under the Joint Call 2023. The CETPartnership’s research projects are co-funded by the European Commission (Grant Agreement No.101069750), the Research and Innovation Foundation of Cyprus (EP/CETP/0923/0059), the Innovation Fund Denmark, and the Agence Nationale de la Recherche. This improper operation not only causes an imbalance in SM capacitor voltages and injects an asymmetric output current but also significantly compromise converter’s reliability. In recent years, considerable efforts has been devoted to address the reliability concerns of MMC submodules through (i) special SM configurations to handle internal faults (e.g., multiple SM failures) [3], [4], (ii) redundancy-based strategies [5], and (iii) fault-tolerant designs incorporating SM bypass structures [6]. While these techniques effectively enhance MMC reliability, they increase costs, hardware complexity, and susceptibility to SM sensor faults. Conversely, data-driven methods for SM fault detection and localization [7] can minimize hardware requirements but face challenges like transmission delays, service disruptions, and extensive hyperparameter tuning. Although, these works are effective in addressing switching devices and capacitor failures in the SMs, they have limitations in ensuring reliable MMC operation under SM voltage sensor faults. Hence, this underscores the need for robust SM voltage sensors fault diagnosis and accommodation scheme to maintain system reliability and performance. So far, limited research has focused on model-based approaches for MMC aimed at either partially reducing the number of SM capacitor voltage sensors or introducing a fault diagnosis (FD) framework. Specifically, SM capacitor voltage estimation using arm voltage and arm current sensors is presented in [8], while [9] proposes a method for estimating SM voltages from system dynamics with fewer measurements within the MMC. However, the complete elimination of sensor requirements and their susceptibility to faults remains overlooked in the existing literature. In [10], a unified diagnosis and resilient control for grid current and DC voltage sensors using a constant threshold is presented. Although this technique is effective, it does not consider the effect of SM capacitor variations and is subject to high complexity issues. Consequently, the timely and accurate FD of multiple SM voltage sensors, including various fault types and time profiles, remains overlooked in the existing literature. Considering the aforementioned limitations, the key contribution of this paper is the development of a model-based, multiple SM capacitor voltage sensor fault diagnosis scheme for MMCs. First, an online estimator is developed for SM voltages, taking into account the effects of modeling uncertainty and sensor noise. A time-varying estimator gain matrix is derived for the estimator by minimizing the variance sum of the estimation errors, considering the dispersion of SM capacitance across the MMC arms. The diagnosis of SM voltage 2025 IEEE Kiel PowerTech | 979-8-3315-4397-6/25/$31.00 ©2025 EU | DOI: 10.1109/PowerTech59965.2025.11180612 Authorized licensed use limited to: University of Cyprus. Downloaded on November 10,2025 at 10:21:27 UTC from IEEE Xplore. Restrictions apply. sensor faults is then performed using the derived analytical redundancy relationship (ARRs), established through pairs of residual signals, calculated by using the online estimators and adaptive thresholds, designed by considering the converter’s real time operating conditions. If specific ARRs are violated, the presence and location of the corresponding faulty sensors are directly identified. Finally, the proposed SM sensor FD scheme is validated through a simulation based investigation. II. PROBLEM FORMULATION A. MMC system description This section presents the architecture of a grid-connected MMC with a constant DC source vdc for medium voltage applications, as shown in Fig. 1. The MMC comprises series connected identical SMs in a half-bridge configuration, as illustrated in Fig. 2. The connection between the MMC and the main grid is made through a series inductance Lsand resistance Rs, which typically includes the leakage inductance of the isolation transformer when galvanic isolation is required. Each leg of the MMC contains a lower and an upper arm, each consisting of N−1active SMs connected in series with an arm resistance Rmand arm inductor filter Lm, with redundancy provided by one additional SM per arm. Each SM is in a halfbridge configuration comprised of two IGBTs S1and S2with a parallel capacitor Cacts as an energy storage element and two anti-parallel diodes D1and D2. The capacitor voltage can be bypassed or inserted into the arm by controlling the SM switches. A bleeder resistor, Rb, is used to discharge the SM capacitor when the MMC is in deactivated mode. The total number of SM sensors per arm is Nu,l=N−1, where the subscripts uand lstands for upper and lower arm of the MMC, respectively. 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Figure 2: Schematic of an MMC with half-bridge submodules. phase p, where p∈{a,b,c}. The total number of SM sensors per branch (including both upper and lower arms) is given by 2N−2. These sensors measure the individual SM capacitor voltages vp cj, where j∈{1,2,...,N−1}corresponds to upper arm and j∈{N,N +1,...,2N−2}corresponds to the lower arm in an MMC with M=2N+1 voltage levels. The MMC operation is regulated by an advanced MMC controller (see section IV-A), which uses feedback from sensors installed in each SM, each arm, and on the DC and AC sides of converter. The dynamics of the SM capacitor voltage vp cj(t)for each phase p, with a total of N−1SMs per arm, are given by ˙vp cj(t)=dvp cj(t) dt =ip u,l(t) Cj ,(1) where ip u,l=ip lor ip uare the lower and upper arm currents of each phase pand Cjis the jth SM capacitance. Note that this study omits the arm notation and derives an expression for N−1SMs, for instance, j∈{1,2,...,N −1}in the upper arm only1. The SM capacitor voltages, by applying the Backward Euler discretization, can be discretized as vp cj(k+1)=vp cj(k)+Δt·1 Cj ·ip u,l(k),(2) where krepresents the discrete-time sample and Δt=ts denotes the sampling time or the step size. Further, (2) is a linear time-invariant system, representing one of the SMs, as the switching states effects are not considered. Moreover, the arm voltage, considering all SMs in each arm, can be determined by incorporating the effect of switching states sp u,l, where sp u:= [sp 1,s p 2,...,s p N−1]represents upper arm’s switching states vector, and sp l:= [sp N,s p N+1,...,s p 2N−2] 1The SM voltage estimation is identical for both the lower and upper arms and can be performed separately. Thus, to avoid ambiguity, the arm notation is omitted in the remainder of the derivation. Authorized licensed use limited to: University of Cyprus. Downloaded on November 10,2025 at 10:21:27 UTC from IEEE Xplore. Restrictions apply. represents lower arm’s switching states vector, as given by, vp(k+ 1)=vp(k)+ diag[sp u,l(k)] ·¯ c·ip u,l(k)·ts +ηp(sp u,l(k),ip u,l(k)),(3) yp c(k+ 1) = ¯ C1·diag[sp u,l(k)] ·vp(k) + np c(k),(4) where vp(k) = vp c1, vp c2, . . . , vp cj⊤is the state vector of SM voltages, sp u,l(k)is the switching states vector responsible for SM capacitor charging and discharging, ¯c =h1 C1,1 C2, . . . , 1 Cji⊤ is the SM capacitance vector, ηp(sp u,l(k),ip u,l(k)) = diag(sp u,l(k)).∆¯c.ts.ip u,l(k)is the parametric uncertainty vector with ∆¯c is the SM capacitance variation vector, yp c(k)is the measurement vector of the upper and lower arm voltage sensors, as the arm voltage measurement accounts for in series combination of all the activated SMs (which can also be calculated by the summation of the activated SM voltages states), ¯ C1=I1×q01×q 01×qI1×q∈R2×2q, with q=N−1, is the output measurement matrix, and np c= [nvu(k)nvl(k)]⊤is the random measurement noise vector of the arm voltage sensors. Note that, the uncertainty vector in (3) incorporates the deleterious effects of capacitance variations, often stemming from the manufacturing parametric variations and the SM capacitor degradation. B. State-space representation of SM voltages In this section, a discrete-time state-space representation of the SM capacitor voltages in a three phase M-level MMC, obtained from the system’s dynamic equations described in (3)-(4), is presented. The state-space presentation of the proposed model, accounting for sensor noise, uncertainty, and input/output disturbances, can be expressed as ¯ v(k+ 1) = ¯ A1¯ v(k) + ¯ B1u1(k) + w1(k),(5) yc(k) = ¯ C1·S(k)·¯ v(k) + nc(k),(6) where ¯ v(k) := [vu(k)vl(k)]⊤∈R2q, with q=N−1, is a state vector with vu(k)=[vc1(k), vc2(k), . . . , vcq(k)]⊤ representing upper arm voltages and vl(k) = [vcN(k), vcN+1 (k), . . . , vc2q(k)]⊤representing lower arm voltages, u1(k)⊤:= [ iu(k)il(k)]∈R2is an input vector, w1(k)⊤:= ηu(k)ηl(k)∈R2qis an uncertainty vector with ηu(k) = [ηc1(k), . . . , ηcq(k)]⊤and ηl(k) = [ηcN(k), . . . , ηc2q(k)]⊤for upper and lower arms, respectively, y⊤ c:= vu(k)vl(k)∈R2is an output vector comprised of upper and lower arm voltage measurements, as it accounts for series combination of all activated SMs, nc(k)⊤ :=[nvu(k)nvl(k)]∈R2is noise vector of arm voltages. The system dynamic matrix represented by ¯ A1∈R2q×2q, bi-linear dynamics vector represented by ¯ B1∈R2q×2, and the switching states of upper and lower arm represented by S∈R2q×2qare given by ¯ A1=Iq×q0q×q 0q×qIq×q,¯ B1=diag[su].¯c.ts0q×1 0q×1diag[sl].¯c.ts, S(k)⊤=diag[su(k)]q×1diag[sl(k)]q×1 The state space representation for SM voltages is similar for p-phases, thus, we have omitted the pnotation. III. FRAMEWORK FOR FAULT DIAGNOSIS OF SM SENSORS In this section, a model-based SM voltage sensor FD framework is presented, considering the discrete-time state space representation of the dynamic system (5)-(6). For design purposes, this paper relies on the following key assumptions. Assumption 1: The uncorrelated white noise in the SM voltage sensors is uniformly bounded for all k, such that np vcj(k)≤¯np vcj, where j∈[1,2, . . . , q]for the upper arm modules and j∈[N, N + 1, . . . , 2q]for lower arm modules of an M-level MMC, with Nrepresenting the total number of SMs per arm. The known constant bounds ¯np vcjare selected from the manufacturing tolerances of SM voltage sensors. Assumption 2: The unknown uncertainty w1(k)≤¯ w1is uniformly bounded for all k. The selection of a constant bound for ¯ w1is based on the manufacturing tolerance associated with the parametric variation of SM capacitors. Assumption 3: The input vector u1(k)(i.e., arm currents iu,l(k)) and the arm voltages yc(k)are available and accurate. A. State estimation of SM capacitor voltages The state estimator for the proposed system in discrete-time, defined by (5)-(6), is developed based on the accurate external measurements of the switching-state signals, the upper and lower arm currents, and the state voltages of individual SMs. The design of state estimator design is presented as follows, ˆ ¯ v(k+ 1) = ¯ A1ˆ ¯ v(k) + ¯ B1u1(k) + L1(k) [yc(k)−ˆ yc(k)] , (7) ˆ yc(k) = ¯ C1·S(k)·ˆ ¯ v(k).(8) where ˆ ¯ v(k)∈Rqis the estimated vector of the state ¯ vwith ˆ ¯ v(0) = 0,u1(k)is the measurable control input vector, L1(k) is the time-varying estimator gain matrix, and yc(k)∈R2is the output upper and lower arm voltage measurement vector, and ˆ yc(k)∈R2is the estimation of output vector yc(k). It is worth noting that estimator’s performance will be ineffective if direct measurements from 2qSM voltage sensors are used. B. SM sensor fault diagnosis scheme The SM sensor fault diagnosis scheme is based on the design of ARR sets for each SM capacitor voltage. Each ARR consists of a pair of residual and its corresponding adaptive threshold, derived using estimator presented in (7)- (8). The violation of ARRs directly indicates the presence and location of the corresponding faulty sensors. This approach is applicable for diagnosing potential faults in single or multiple SM sensors, encompassing different fault types (e.g, scaling and offset faults) and various time profiles (e.g, abrupt and incipient faults). However, this work does not focus on accurately identifying the structure and magnitude of these potential SM voltage sensor faults. The following sections detail the design of the aforementioned SM sensor FD scheme. 1) Residual generation: The residual signals for SM capacitor voltages are calculated by taking the difference between the SM voltage sensor measurements vector and their estimated states, resulting in ey(k) = y(k)−¯ Cˆ ¯ v(k),(9) Authorized licensed use limited to: University of Cyprus. Downloaded on November 10,2025 at 10:21:27 UTC from IEEE Xplore. Restrictions apply. where y(k)∈R2qis the output vector representing the measurements of SM voltages, ¯ C=I2q×2qis the state selection matrix, and ˆ ¯ v(k)∈R2qis the SM voltage state estimation vector. The SM voltage state estimation vector, obtained from (7), is use for the SM sensor FD. The fault diagnosis process accounts for the switching signals responsible for charging and discharging of SM capacitors. The deleterious effects of these switching signals can be mitigated by considering measurements from the SM voltage sensors vector, given by y(k) = ¯ C¯ v(k) + n(k) + Ff(¯v(k)) (10) where ¯ Crepresents state selection matrix, F∈I2q×2qis the fault distribution matrix of SM voltage sensors having full rank, n(k) := [nvc1(k), nvc2(k), . . . , nvc2q(k)]⊤∈R2q represents SM voltage sensor’s noise vector, and f(¯v(k)) := [fvc1(vc1(k)),fvc2(vc2(k)), . . . , fvc2q(vc2q(k))]⊤∈R2qis an unknown permanent sensor fault vector comprising SMs of the upper and lower arms. The sensor fault modeling can be structured by multiplying an unknown fault function, represented by ϕj(vcj(k))=φsjvcj(k)+φcj, by the time profile of the fault, β(k−kf;ρ)=1−exp(−ρ(k−kf)), as given by fvcj (vcj(k)) = β(k−kf;ρ)ϕj(vcj(k)),(11) where the unknown fault occurrence time is denoted by kf, the constant offset value resulting from the bias sensor fault is denoted by φcj, the constant scaling factor introduced by the multiplicative sensor fault is denoted by φsj, and the time evolution rate is denoted by ρ>0, which can characterize (i) an abrupt fault when ρ→ ∞, and (ii) an incipient fault when ρ<∞[11]. Using (10) in (9), the residual vector becomes ey(k) = ¯ Ce¯ v(k) + n(k) + Ff(¯ v(k)),(12) where e¯ v(k)= ¯ v(k)−ˆ ¯ v(k) = [evc1(k)evc2(k),. . ., evc2q(k)]⊤ corresponds to the estimation error vector, ¯ v(k)is represented by the state vector, ˆ ¯ v(k)is represented by the corresponding estimation vector. Moreover, ey(k) = [eyc1(k)eyc2(k). . . , eyc2q(k)]⊤corresponds to the residual vector for each SM voltage j. Using (5) together with (10) and ˆ y(k) = ¯ Cˆ ¯ v(k)in (7), instead of yc(k)and ˆ yc(k), to account for faulty SM measurements, the estimation error becomes e¯ v(k+ 1) = ¯ A1−L1(k)¯ Ce¯ v(k) + w1(k)−L1(k)n(k) −L1(k)Ff(¯ v(k)).(13) where L1(k)is the time-varying estimator gain matrix, derived by designing a minimum variance filter to account for uncertainty and measurement noise under healthy sensor measurements. The computation of the estimator gain matrix, as used in (7) and (13), is given as follows. 2) Time-varying estimator gain: The time-varying estimator gain, aimed at optimizing the estimator’s performance in the presence of sensor measurement noise and uncertainty, can be obtained by minimizing the trace of the estimation error covariance matrix, as given by J(k+ 1) = E[e¯ v(k+ 1)e¯ v(k+ 1)⊤= Tr(P1(k+ 1)),(14) where the estimation error covariance matrix is represented by P1(k+ 1) = E[e¯ v(k+ 1)e¯ v(k+ 1)⊤]≥0, the traces of a matrix is denoted by Tr(.). Further, the expression for the estimation error covariance matrix and the time-varying estimator gain can be derived in a manner similar to the method described in [12], and can be written as follows, L1(k) = ¯ A1¯ P1(k)¯ C⊤[¯ C¯ P1(k)¯ CT+R1]−1,(15) ¯ P1(k+ 1) = ¯ A1P1(k)¯ A⊤ 1−L1(k)¯ C¯ P1(k)¯ A⊤ 1+Q.(16) where the covariance matrix of uncertainty and uncorrelated sensor noise are represented by Q=E[w1(k)w1(k)⊤]≥0 and R1=E[n(k)n(k)⊤]≥0, respectively. 3) Adaptive threshold computation: The estimation error norm can be computed by considering the estimation error in (13) and the time-varying estimator gain matrix in (15), and by applying the Triangular and Holder’s inequalities under healthy measurement condition (i.e., f(¯ v(k)) = 0,∀k > 0). This leads to the following expression |e¯ v(k)| ≤ αk 1|¯ e¯ v(0)|+ k−1 X j=0 αk−1−j 1[|w1(j)|+|L1(j)||n(j)|], (17) where |¯ A1−L1(k)¯ C| ≤ α1≤1is obtained through desired pole (eigenvalue) placement, and ¯ e¯ v(0) represents the vector of initial state estimation. The norm of residual signals under healthy condition is obtained by following (10), which gives |ey(k)|≤¯ Cαk 1¯ v+¯ C k−1 X j=0 αk−1−j 1[|w1(j)|+|L1(j)||n(j)|]+n(k), (18) where ¯ v=¯ e¯ v(0) is the initial state estimation vector with ¯ v= vc1, vc2, . . . , vc2q⊤. Further, by applying Assumptions 1 and 2, the estimation error vector bound under healthy condition, as denoted by ¯ e¯ v(k), can be defined as ¯ e¯ v(k) = αk 1¯ v+ k−1 X j=0 αk−1−j 1(¯ w1+|L1(j)|¯n),(19) where ¯n = [¯nvc1(k),¯nvc2(k), . . . , ¯nvc2q(k)] is the noise-bound vector of the SM voltage sensors, obtained based on Assumption 1. The adaptive threshold, ¯ ey(k), which represents the upper bound of the residual vector, is determined by applying (12) in (19), resulting in the following expression ¯ ey(k) = ¯ Cαk 1¯ v+¯ C k−1 X j=0 αk−1−j 1¯ w1+|L1(j)|¯n+¯n.(20) C. Fault detection logic for SM voltage sensors In this subsection, the decision logic for sensor fault detection is presented, which involves evaluating the ARRs, denoted by E. The formulation of each ARR is defined as follows, E:|ey(k)| ≤ ¯ ey(k).(21) The inequality in (21) remains valid under the healthy SM sensors. However, a violation of the inequality indicates the presence of a sensor fault, which triggers the activation of a binary detection decision signal, denoted by D. The decision signal can be described by the following boolean function. Dj(k) = 0,if k < kd 1,otherwise ,(22) where the boolean signal Djcorresponds to the jth row of the observed diagnosis vector Dthat is formed Authorized licensed use limited to: University of Cyprus. Downloaded on November 10,2025 at 10:21:27 UTC from IEEE Xplore. Restrictions apply. from the vector of detection decision signals D(k) = D1(k)D2(k), . . . , Dj(k)⊤, where j∈ {1,2, . . . , 2q}, and kd= min k:|ey(k)| − ¯ ey(k)>0is the time instance of fault detection. The presence of single or multiple faults in the SM voltage sensors (i.e., in the jth SM of each arm at p-phase/s) can only affect the residual signal of the corresponding jth SM for the p-phase/s. Thus, this results in an ARR violation of the corresponding jth SM for the affected phase/s only. Consequently, when such an ARR violation is detected in the jth SM sensor, the detection signal goes to “1”, indicating the presence of a fault in the particular SM sensor. If no ARR violation occurs, the signal remains at “0”. D. Fault isolation decision logic for SM voltage sensor The detection method in our particular case is sufficient to directly isolate SM voltage sensor faults, enabling accurate identification of the specific faulty SM voltage sensor by analyzing the observed diagnosis pattern. This is because the ARRs of the SM sensors, denoted by Ein (21), are directly affected by the measurements from the SM sensor y(k). Thus, when a fault occurs in the jth SM voltage sensor, it triggers the corresponding isolation decision signal, which is identical to Dj, since ARRs of the particular faulty SM voltage sensor are activated. For instance, a fault in SM voltage sensor 1, i.e., fvcj with j= 1, only affects evc1(k), setting the isolation decision signal D1to one (i.e., D1←1) when the ARR presented in (21) is violated. Similarly, multiple SM voltage sensor faults will violate multiple ARRs, causing the corresponding ARRs to be set to “1”, indicating exact location of faulty SM sensors. IV. SIMULATION RESULTS A. Description of MMC system For validation purposes, the proposed SM fault diagnosis scheme is evaluated under single as well as multiple SM voltage sensor faults that occur simultaneously or successively in a three-phase, 26-levels, grid-connected MMC of 1 MVA power rating, considering the capacitor voltage variation effects. A detailed simulation model in Fig. 1 and internal schematic in half-bridge configuration presented in Fig. 2 for SMs, has been developed in MATLAB/Simulink considering a discrete-time fixed-step solver. The key design and tuning parameters of grid-connected MMC are summarized in Table I. In the simulation, the covariance matrices of sensor measurement noise and uncertainty are approximated by Monte Carlo simulation and set to R1= 0.0322I2×2and Q= 0.32I52×52, respectively. The proposed SM FD framework is digitally integrated in discrete time within the embedded MMC controller. The MMC controller design follows a grid-following approach that incorporates the following key components: (i) a synchronization unit (DNαβ-PLL) that estimate the frequency, phase angle, and amplitude of the positive sequence of the grid voltage in a synchronize manner [13], (ii) an advance current controller to ensure high-quality current injection even under unbalanced grid fault events [12], [13], (iii) an advanced PQ controller to regulates the injection of active power (P) based on a reference value, and reactive power (Q) to fulfill the grid regulations and support grid voltage during shortcircuit event [11], [13], (iv) a circulating current controller Table I: Design and tuning parameters of the MMC system along with proposed submodule sensor FD scheme Parameters Simulation Number of power modules/arm N= 26 Rated Power (P), Transformer X/R ratio 1 MVA, 15 Phase-to-ground AC voltage (vrms) 6 kV DC-link voltage (vdc) 15 kV Line frequency 50 Hz Transformer X/R ratio 15 Nominal SM voltage (vnom c) 578 V SM capacitance (C) 10 µF Bleeder resistance (Rb) 1 MΩ Arm inductance (Lm), Arm resistance (Rm) 3 mH, 0.1 Ω Load resistance (Rs), Load inductance (Ls) 0.2 Ω, 4 mH Controller sampling time (ts) 100 µs MMC controller parameters: −DNαβ-PLL Kp= 92,Ki= 2.35×10−3 −Current Controller Kp= 2.5,Ki= 909 SM voltage sensor fault diagnosis: ∆Cj= 7% of C, ¯vcj= 949 V,¯nvcj= 0.01¯vcj to achieve precise decoupling and suppression of unwanted circulating currents, thereby improving system stability and ensure a fast dynamic response [14], and (v) a DC-link voltage controller to stabilize the DC-link voltage, ensuring balanced power flow between the AC and DC sides [15], (vi) the nearest level control (NLC) based modulation scheme [16] is designed to minimizes the capacitor voltage ripple by selecting the nearest voltage level to the reference waveform. A sorting algorithm balances the capacitor voltages by evenly distributing the arm current across all SMs, ensuring uniform voltage ripple and minimal deviation among them. This study assumes that faults occur only in the SM voltage sensors, which can lead to abnormal gate signal generation, indicating faulty or anomalous operation of SM voltage sensors. The effective operation of the MMC controller has been verified under various operating conditions, including a step change of 0.3 MW in active power, a step change of 0.2 MVAr in reactive power, and intense harmonic distortion on grid voltage (with 5%of the fifth and seventh-order harmonics). B. Simulation based validation This section validates the performance of the SM fault diagnosis scheme in a simulation setup, considering multiple simultaneous and successive SM voltage sensor faults in phase A alongside a random 7% of dispersion in SM capacitance across MMC arms, as shown in Fig. 3. The investigation is particularly focused on the diagnosis of (i) an incipient offset fault in j= 1 SM sensor of the upper arm with a time evolution rate of ρvc1= -5 and a constant offset magnitude of φvc1= 50 V at t= 0.2 sec, (ii) simultaneous abrupt offset faults (i.e., ρ→ ∞) in the j= 4 SM sensor with a magnitude of φvc4= 200 V and in the j= 26 SM sensor with a magnitude of φvc4= 300 V, both occurring at t= 0.3 sec, followed by (iii) successive abrupt offset faults in the lower arm SM sensors of j= 27,j= 28,j= 51 and j= 52 that occur simultaneously, each exhibiting a constant offset magnitude of 400 Vat t= 0.4 sec, as shown in Fig. 3. The results show that an incipient offset fault in phase A of j= 1 SM sensor activates the corresponding sensor’s ARR, resulting in the activation of isolation decision signal Authorized licensed use limited to: University of Cyprus. Downloaded on November 10,2025 at 10:21:27 UTC from IEEE Xplore. Restrictions apply. Figure 3: Effectiveness of proposed SM fault diagnosis in MMC is shown in Fig. (a)-(p), where the absolute value of residuals ea vcj (k) (solid blue lines) and the adaptive thresholds ea ycj (k)(orange solid lines) for jSM voltage sensor of phase A are presented. The detection decision signals of jSM sensors Djare indicated by the right vertical axis and activate when a fault is detected. D1without affecting the other ARRs, as shown in Fig. 3 (a). Similarly, the simultaneous abrupt offset faults at j= 4 and j= 26 lead to their particular SM’s ARR violation without affecting other, as indicated by the activation of corresponding isolation decision signals (red-dashed lines) in Fig. 3 (d)and Fig. 3 (h)at t= 0.3 sec. Finally, the simultaneous abrupt offset faults in the multiple, lower-arm SM sensors j= 27,28,51,52 also activates their respective isolation decision signals, as shown in Fig. 3 (i), Fig. 3 (j), Fig. 3 (o), and Fig. 3 (p), respectively at t= 0.4 sec. Hence, the indication of isolation decision signals Dj(k)by red dashed lines alerts the operator for preventive maintenance. Consequently, early diagnosis of SM voltage sensor faults is crucial, as undetected faults can compromise the reliability and safety of the MMC. V. CONCLUSION This work presents a model-based sensor fault diagnosis scheme for the submodule capacitor voltage sensor, taking into account the SM capacitor voltage model and the random 7% SM capacitance variation in each arm of MMC. The SM fault diagnosis scheme is developed based on an online estimation scheme having a time-varying estimation gain matrix to accurately detect both abrupt and slow-varying (incipient) SM sensor faults, the formulation of analytical redundancy relations, and fault detection and isolation decision logic. 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