Optimization Methods to Reduce Capacitor Stress in Modular Multilevel Converters

Modular multilevel converters consist of many capacitor half-bridge or full-bridge converter cells. Stresses on the capacitors result from peak capacitor voltage, capacitor voltage ripple and current ripple. Three new tractable convex optimisation problems are presented that reduce peak capacitor voltage and limit rms arm current using injected zero-sequence voltages, circulating currents and selecting the initial stored capacitor energy. These optimisation problems are compared to assess their performance in terms of key indicators which include factors related to capacitor stress, capacitor and semiconductor ratings, and power loss: peak capacitor voltage, capacitor voltage ripple, capacitor current ripple, average and rms arm current. The optimisation methods permit designers to understand the trade-offs in achieving the limits imposed by device specifications and power losses.


I. INTRODUCTION
A high efficiency converter with medium and high power applications is the modular multilevel converter (MMC) [1]- [4]. A circuit schematic of a three-phase MMC is shown in Fig. 1. Each MMC leg (a,b,c) contains two arms (upper u and lower l), each of which consists of serially connected submodules (SMs) which are full-bridge capacitor cells (or alternatively half-bridge cells [5]).
An important consideration in the design of MMCs is to limit the stress on the components. Current ratings of semiconductors and capacitors require limits on the maximum arm current. Power losses and thermal stresses on semiconductors are related to rms currents [6], [7]. The stress on capacitors depend upon the peak capacitor voltage [8], [9]. Selection of semiconductor voltage specification is based upon the maximum peak capacitor voltage [10]. Other stresses on capacitors are caused by ripple voltage and ripple current [9], [11].
To decrease stress on capacitors, papers in the literature proposed methods that determine the zero-sequence voltages and/or circulating currents to reduce peak-to-peak voltage The associate editor coordinating the review of this manuscript and approving it for publication was Giambattista Gruosso .
ripple [12]- [21] or peak capacitor voltage [22]. Brief summaries of some of these papers are given below: Exhaustive search methods exist that minimise peak-topeak capacitor voltage ripple using circulating currents [12], [13]. The disadvantages of exhaustive search techniques relative to optimisation-based solvers are time-inefficiency and the solution's dependence on quantisation of optimisation parameters. In [12], the magnitude and phase angle of a second-order and a fourth-order circulating current harmonic are exhaustively sought to find optimal solutions for three optimisation problems: the minimisation of capacitor voltage ripple; the minimisation of a weighted sum of capacitor voltage ripple and rms current, and; the minimisation of capacitor voltage ripple subject to a constraint on rms current. The authors of [13] present an exhaustive search of the same circulating current harmonic parameters as those considered in [12]. The method presented in [13] also minimises capacitor voltage ripple, however the authors utilise a different model for capacitor voltage that is proposed in [23]. The technique studied in [13] was designed to overcome the drawbacks of another method to reduce capacitor voltage ripple, shown in [14], in which an approximate expression for capacitor voltage ripple is minimised using circulating currents. Neither [13] or [14] include any other component or system stress related constraints, such as constraints on peak capacitor voltage.
The authors of [22] show a technique that is used to determine only the second order circulating current that decreases peak capacitor voltage. Namely, an exhaustive approach is used to find the phase angle and magnitude of the circulating current such that the capacitor voltage has a zero first-order derivative at a certain point in time. This method is used as a means to avoid excessive capacitor voltages that stress the MMC components. However, no constraints are considered to enforce limits on power loss and stress on MMC components.
The authors of [15], [16] suggest analytical methods that determine circulating currents, under conditions in which a zero-sequence voltage is injected, that remove low-frequency capacitor energy fluctuation terms. This method is utilised as an approach to reduce capacitor voltage ripple relative to the case in which no harmonic voltages and currents are injected. However, there are no constraints imposed on the selection of circulating currents that limit the increases in peak or rms arm current.
The authors of [17] proposed an optimisation problem that minimises the rms value of circulating currents subject to a low-frequency capacitor voltage term equal to zero. This equality constraint is proposed as an approach to reduce capacitor voltage ripple. Again, the method in this paper does not include other constraints that set upper bounds on quantities that relate to component stress, such as peak capacitor voltage, peak arm voltages or peak arm currents.
The authors of [18], [19] consider optimisation problems that minimise objective functions that contain a linear combination of variables that influence capacitor stress and MMC power losses. A two-step linear programming method is considered in [18] that uses auxiliary optimisation parameters that provide upper and lower bounds to stored capacitor energy, an upper bound on average current, and upper bounds on harmonic zero-sequence voltage magnitudes. A linear combination of these auxiliary variables form the objective functions considered in [18] for each linear program step. The primary purpose of the technique studied in [18] is to reduce capacitor voltage ripple subject to constraints on peak arm voltage, peak arm current and average arm current. However, there was no method to minimise the peak capacitor voltage, which is a key metric in determining overall capacitor and semiconductor stress. The Matlab function linprog, together with the interior-point-legacy algorithm [24], [25] are used to solve each optimisation step. A nonlinear objective function that consists of a weighted-sum of capacitor energy ripple, rms current and rate of change of arm voltages and currents is considered in [19], which was solved using a nonlinear Matlab solver function fmincon.
In this paper, new and tractable convex optimisation problems are proposed that allow designers to understand the trade-offs in achieving the limits imposed by device specifications and power losses, and thus permit informed decisions in the design phase. Well-established and reliable optimisation algorithms can be used to solve the problems presented in this paper. The parameters over which the optimisation problems are solved include zero-sequence voltages, circulating currents and initial capacitor energy. In the proposed optimisation problems, constraints are included that are necessary and sufficient to impose upper limits on peak arm currents, peak rms arm currents and peak arm voltages. These constraints can be tightened and loosened to explore the design space and the trade-offs in terms of key performance indicators: peak capacitor voltage, peak-to-peak capacitor ripple voltages and currents, rms and average arm current. These performance indicators are selected because they are important contributors to capacitor stress [8], [9], [11] and MMC power losses [7].
The key contributions of this paper are: 1. A detailed formulation of a simplified MMC model with all assumption clearly described. 2. Three new and tractable two-step convex optimisation problems that reduce peak capacitor voltage using injected zero-sequence voltages, circulating current and initial capacitor energy. 3. The introduction of an optimisation constraint which imposes an upper limit on rms arm current. 4. The solutions to each of the three two-step optimisation problems are compared in terms of key performance indicators related to capacitor stress and power losses to illustrate, for the first time, the trade-offs achievable with optimised MMC designs.
The paper is structured as follows. The MMC model is derived in Section II. Novel two-step convex optimisation methods are described in Section III. The methods for solving the convex optimisation problems and their results are considered in Section IV. Summary remarks are given in Section V.

II. MODELLING
In this section, the MMC arm voltages, current and capacitor energy are modelled in terms of the nominal (operational) condition parameters, injected zero-sequence voltage VOLUME 8, 2020 and circulating currents parameters, and MMC components parameters. Tellegen's theorem is used to derive a model for the stored capacitor energy in an arm as a function of the arm voltage and arm current. Similar stored capacitor energy relationships have been reported in [9], [12], [18], [20], [22], however alternative assumptions are used and it is not clear how these models are related to the one proposed in this paper. Experimental validation of the model is presented in Section III-C. The optimisation criteria and constraints in the optimisation problems, which are introduced in Section III, are defined in terms of this model's voltages, currents and energies.

A. ARM VOLTAGES AND CURRENTS
With reference to Fig. 1, let v pq (t) and i pq (t) denote voltages and currents in an arm, where p ∈ {a, b, c}, and q ∈ {u, l}. The voltages and currents in the upper and lower arms of the MMC, v pu (t), i pu (t), v pl (t) and i pl (t), are defined in (1)- (4).
where ω 0 is the fundamental frequency, T 0 = 2π/ω 0 and The arm voltages and currents have been formulated generically to include nominal voltage v Npq (t) and current i Npq (t), injected voltages, v IVpq (t), and circulating currents, i CCpq (t); which can be created in an MMC [12]- [21], [26], [27]. The T 0 -periodic arm voltages and currents are formulated to be three-phase time offset by T 0 /3. In (1)-(4) the harmonics due to PWM switching in the SMs are assumed to be negligible and the arm inductances are assumed to be small so that the arm voltages required to create the arm currents are insignificant. In the general case the harmonics injected in arm voltages and currents in (1)-(4) are assumed to be harmonics of the fundamental ω 0 up to some maximum N H as defined by the sets V and I.
The arm voltage and currents are re-written in vector form in (8)- (11), where f p (t), x v and x i are defined in the Appendix A.
v pu (t,

B. STORED CAPACITOR ENERGY
The circuit in Fig. 2 shows the serially connected submodules in any arm pq. In Fig. 2, the rest of the MMC circuit is represented by the one-port element characterised by the relationship of arm voltage to arm current, v pq (t, x v ) and i pq (t, x i ) as defined in (8) to (11). integral of (12), it follows that: The parameter α is introduced to allow for a time shift that exists, for the cases of interest in the paper, between the upper and lower arm voltage and current functions in the same phase, as defined in (16).
The value of α is determined in Section III-B7 for a special set of V and I that are of interest in this paper. From (13) and the voltage-current relationship in a capacitor, it follows that: (17) It is assumed that there are balanced capacitor voltages and equal submodule capacitor currents in an arm, i.e. v Control strategies for achieving the balance described in (18) have been proposed in the literature [23]. Given every SM capacitor has the same capacitance, C SM , the subscript j is dropped from (17) to obtain: Consider the definitions in (20)- (22).
The time-average, defined in (24), of the function e Cpq (t + T pq ) is assumed to be equal in all the arms. This assumption is stated in (25). An experimental method to enforce (25) is described in [26].
Using the arm voltage and current relationships defined in (1)-(4) and (16), it can be shown that (25) implies (26). The derivation is detailed in Appendix C.
x e C0 pq (t 0 + T pq ) = x e C0 au (t 0 ) It follows from (26), that the e Cpq (t + T pq ) in the MMC arms are time-offset in the same manner as the voltages and currents as defined in (1)-(4), (16), as described in (27). The derivation of (27) is detailed in Appendix C.
An interpretation of (27) is: the behaviour of e Cpq (t +T pq ) for all pq is known if e Cpq (t + T pq ) for any pq is known. As explained in Section III-B7, for a special subset of voltage and current harmonics, only one of the six arm's voltage, current and energy functions needs to be considered when solving the optimisation problems presented in this paper.
The upper arm of leg a, p = a and q = u, is arbitrarily selected hereafter and for simplicity, the subscript au is dropped from the notation. Substituting T pq = T au = 0, (8) and (9) into (23), it can be shown that e C (t) in arm au can be written as in (28). Furthermore, it is made explicit that e C (t) is a function of the x v , x i and x e C0 in (28).
where i (t), v (t) and P(t) are defined in the Appendix A1.

III. OPTIMISATION PROBLEM FORMULATION
The optimisation problems proposed in this paper minimize the peak capacitor voltage subject to constraints, using the optimisation parameters that are controllable in the MMC: x v , x i and x e C0 . The objective function, constraints and the proposed two-step optimisation problems are detailed in this section.

A. OBJECTIVE
Stresses on a capacitor arise from large peak voltages. In this paper, the objective function, , of the optimisation problems is devised such that peak capacitor voltages are minimized, thereby decreasing stress on capacitors. Thus the objective function given by (29) is proposed.
For correct operation of the diode/switch assembly, the capacitor voltages need to be always positive, i.e. v C (t) ≥ 0, [29]. It follows from (29) and v C (t) ≥ 0 that the objective function defined by (30) is equivalent to that defined by (29).
The objective function (30) avoids any issues associated with the square root operator which would arise in (29).

B. CONSTRAINTS
This section introduces the constraints that are utilized in the optimisation problems.

1) PEAK ARM CURRENT CONSTRAINT
The allowable peak arm current is dependent on the rating of the MMC IGBTs/semiconductors. Thus the magnitude of the current needs to be constrained below some specified maximum, as defined in (31).
The two linear inequality constraints on the current in (32) are equivalent to the required constraint defined by (31).
Thermal limits of semiconductors and power loss considerations are addressed by imposing an upper bound on arm rms current. It can be shown that the rms current in an arm, i rms (x i ) can be written in terms of nominal and injected circulating currents as given in (33).
It is clear from (33) that the injection of circulating currents increases the arm rms current relative to nominal conditions, that is Thus, as defined in (36) a parameter I RMSF ≥ 1, is introduced to define an upper bound on the rms current. Note that the bound imposes a maximum relative to the nominal rms current.
Root-mean-square quantities are positive, and therefore deriving the root-mean-square current from the mean-square quantity involves a monotone transformation. Thus the constraint defined by (37) is equivalent to the constraint defined by (36).
Equation (37) is a direct way on placing an upper bound on the arm rms current as opposed to adding a term to the objective function [12], [19], which would require a decision on the weight to be placed on this term.

3) MINIMUM ARM VOLTAGE CONSTRAINT
For correct operation of MMC submodules implemented as unipolar (half-bridge) converter submodules, the arm voltages need to be greater than or equal to zero. Thus, the con- It is not difficult to show that the more convenient form (39) is equivalent to (38).
Note that bipolar (full-bridge) converter submodules can function as unipolar submodules, but the reverse is not true.

4) PEAK ARM VOLTAGE CONSTRAINT
Similar to peak currents, another constraint is introduced to place an upper limit on peak arm voltage as defined in (40).
Given the inclusion of the constraint in (38), it is clear that (40) is equivalently achieved using (41).
Furthermore, the linear inequality constraint on the arm voltage in (42) is equivalent to the required constraint defined by (41).

5) MINIMUM CAPACITOR STORED ENERGY REQUIREMENT
Due to the half-bridge structure of submodules, (43) needs to hold during the operation of an MMC [8], [9].
It can be shown that (44) is equivalent to the desired constraint in (43). where The definition of E MINpq (t, x v ) for any arm is given in To simplify notation, the inequality (44) can be replaced by the equivalent inequality in (46). where The definition of ε Cpq (t, x v , x i , x e C0 ) for any arm is given in (110). Note that the constraint defined by (46) can be revised to include a margin by introducing a positive parameter on the right-hand side of the inequality.

6) MINIMUM INITIAL CAPACITOR ENERGY
Given the definition of x e C0 in (22), it is clear that It can be shown the constraint in (48) is met upon enforcing the constraint in (46). Therefore (48) is met without explicitly including it as a constraint in the formulation of the optimisation problems.

7) HARMONIC SELECTION
The notation used in this paper considers injection of all integer harmonics from 2 to N H into arm voltages and currents as shown in (76). Generally, MMCs are operated such that a special subsets of voltage and current harmonics, V s ⊆ V and I s ⊆ I, are injected.

a: EQUALITY CONSTRAINTS
A method using equality constraints is defined to enforce the selection of harmonic numbers in V s and I s in the optimisation problems: Square matrices B v and B i can be defined such that the equality constraints defined by (49) and (50) ensure that any harmonic in x v and x i that is an element of V c s ∩ V or I c s ∩ I, respectively, is set to zero and the desired harmonic selection is achieved. The equality constraints defined by (49) and (50) appear in the optimisation problems as required.

: SELECTION OF VOLTAGE AND CURRENT HARMONICS
Special subsets of the injected voltages and currents defined by (6) and (7) are identified. Let Z ++ be the positive integers and N V , N I ∈ Z ++ , N V , N I ≤ N H . Let the special subsets of V and I, V s and I s , be defined as: For voltage harmonics in V s , the injected voltages v IVpq (t) are zero-sequence voltages, which are triplen harmonics utilised since they do not appear in the line-to-line output voltages [18]. Further, for current harmonics in I s , which are even non-triplen harmonics, it can be shown that the sum of arm currents add to I dc , that is: and therefore the dc-link current, I dc shown in Fig. 1 is not altered by injection of circulating currents. For the special subsets V s and I s , it is shown in Appendix B that α = T 0 /2. Substituting α = T 0 /2 into (16) and (27) Given the result in (54) and the definition of E MINpq (t, x v ) in (106), leads to (57).
The relationships in (54)-(57) are termed here as the symmetric properties of the circuit functions. For voltage and current harmonics in V s and I s , it can be shown that e Cpq (t, It can be shown that e Cpq (t, are T 0 -periodic, three-phase time-offset and symmetric. The derivation is detailed in the Appendix C, Appendix D and Appendix E. In view of the properties established for the circuit functions in the arms, the circuit function v pq (t, x v ), i pq (t, x i ), VOLUME 8, 2020 e Cpq (t, x v , x i , x e C0 ) and E MINpq (t, x v ) for any arm can be determined from the corresponding circuit functions in arm au. This important result implies only the circuit parameters in one arm needs to be treated in the optimisation problem. For this reason, only the upper arm of phase a is considered.

C. MODEL VALIDATION
The MMC model presented in Section II contains a number of assumptions that permits the formulation of tractable optimisation problems. It is important to check that this idealised model is a good approximation of the a real MMC system. The voltage, current and energy waveforms that are derived from the model are computed in Matlab. These waveforms are compared to those generated in an MMC experimental setting to establish the validity of the model. In this comparison, the nominal and injected voltage and current harmonics are the same in the model and three-phase MMC experimental setup. The C eff used in the model is set to the value used in the experimental setup (see Table 1), that is: The fundamental frequency used in the Matlab-computed model and experimental setup is 10π rad/s (5Hz) which is a typical low speed operational condition in drive applications, where injection of harmonic components is often utilised to prevent excessive capacitor voltage peaks [4]. The nominal conditions and injected voltage and current parameters are presented in Table 2. Experimental results were obtained using the setup described in [30], with system parameters shown in Table 1. Details of the signal processor and control structure are provided in [30]. The experimental setup controls the currents and average submodule capacitor voltage, v C,ave ave t [v C (t)], using the current control technique presented in [26].
The arm experimental energy waveforms are presented in Fig. 3. The waveforms in Fig. 3 are periodic and threephase time offset, thus adhering to the properties shown earlier in Section III-B7. The arm au experimental and model energy waveforms are shown in the top-left of Fig. 4. The superscript mo describes the result of the (Matlab-computed) model of the MMC discussed in this paper. The resultant model waveforms are computed for arm au only since the waveforms of other arms are simply time-offset with respect to arm au. The superscript ex describes the result obtained from the experimental setup. The resultant E MIN (t) and e C (t) align closely from the model and experimental results. Note that the model waveforms have been time-shifted to adjust for sampling offset with the experimental waveforms. There is close agreement between the mo and ex results. The small discrepancies arise from the voltage drops across the semiconductors, arm inductances and resistances that were not modelled.
The resultant arm voltages and currents for the upper arms across the three phases are presented in the right two subplots of Fig. 4. The results of the model are superimposed on these experimental results. Again, close alignment of the arm voltage and current of the modelling and experimental results occurs. As expected, voltages and currents are periodic and three-phase time-offset. The nonnegative constraint (39) ensures nonnegative arm voltages and this means that bipolar (full-bridge) converters submodules are not required in the proposed approach. Rather, unipolar (half-bridge) converters can be used to implement these optimisation results if desired. It is clear from the bottom-left of Fig. 4 that each submodule capacitor voltage in an arm is are closely balanced throughout the experiment. These preliminary results provide evidence that the modelling assumptions are acceptable.

D. PROPOSED OPTIMISATION PROBLEMS STEPS
This subsection presents the ''steps'' which form part of the two-step optimisation methods proposed in this paper.
It follows from (28) that objective function in (30) is a bilinear function of injected voltage and current harmonics, and is in general nonconvex. A drawback of nonconvex objective functions is that finding a minimum does not guarantee that it is the global minimum. A benefit of convex optimisation problems is that widely-available solving techniques can consistently find optimal values to such problems [31]. Further, the minimum of convex optimisation is the global minimum.
Two-step optimisation problems that involve subsets of the optimisation variables (x v , x i and x e C0 ) are proposed such that each optimisation step involves a convex optimisation problem. The optimisation problem in each step is tractable and can be solved using known methods. Four optimisation steps are proposed: LPv, QPv, LPi and QPi. Their details are discussed subsequently. As will be evident, each of the proposed optimisation steps involves either the voltage  parameters (LPv and QPv) or the current parameters (LPi and QPi) but not both.
All functions considered in the objective and constraints are T 0 -periodic and thus every optimisation problem is solved over the interval t ∈ [t 0 , t 0 + T 0 ).
It is noted that if the injection of voltage harmonics is considered undesirable, then the LPv or QPv steps can be omitted, and the voltage harmonic coefficients can be set to zero.

1) QPv
QPv is the optimisation problem in (60). The optimisation problem minimizes e C (t, x v , x i , x e C0 ) over the optimisation variables x v and x e C0 for a fixed value for x i that is denotedx i . From (28) it is clear that for a givenx i , that e C (t, From (85) it obvious that ε C (t, x v ,x i , x e C0 ) in (60) is a quadratic function of x v . Note that QP is an abbreviation given to the names of problems that can be formulated as a quadratic program.

2) LPv
Another optimisation problem step which is solved over voltage harmonics, x v , is proposed. An approach to reduce (30) is to minimize the maximum of E MIN (t, x v ), which is a function of only the optimisation variables, x v . The optimisation problem, LPv, defined by (61) is introduced.
It can be shown that the optimisation problem in (62) is equivalent to the optimisation problem in (61) because the objective function in (62) is related to the objective function in (61) by a monotone transformation, (·) 2 , over the domain of feasible solutions [32].
Note that LP is an abbreviation given to the names of problems that can be formulated as a linear program.

3) LPi
The optimisation problem, LPi, is defined by (63). In this problem e C (t, x v , x i , x e C0 ) is minimized with respect to the VOLUME 8, 2020 optimisation variables x i and x e C0 for a givenx v .

4) QPi
Finally, the optimisation problem QPi is proposed. Compared to LPi the key addition in the optimisation problem QPi defined in (64) is the inclusion of a constraint on the rms current. From (37) it is clear that this is a quadratic constraint.

E. PROPOSED TWO-STEP OPTIMISATION PROBLEMS
The names given to each two-step optimisation problem indicates the which optimisation problems steps are solved and their sequence. For example LPvLPi solves LPv first, followed by LPi second. The three two-step optimisation problems proposed in this paper are: 1. LPvLPi. 2. LPvQPi. 3. QPvQPi.
The selection of fixed value parameters in each step of the two-step optimisation problems are described as follows. For each two-step problem, the first step selects the fixed values to be zero, whereas the second step selects the fixed values to be the optimal values of the first step. That is, for LPv, the fixed values,x e C0 andx i , are selected to be zero. For QPv x i is selected to be zero. For LPi and QPi,x v is selected to be the optimal x v that results from LPv or QPv.

IV. OPTIMISATION IMPLEMENTATION AND RESULTS
In this section, methods are presented for the simplification, computation and comparison of the optimisation problems proposed in Section III-D.

A. DIMENSION REDUCTION
The linear equality constraints defined in (49) and (50) in each of the defined optimisation problems have a special structure which sets undesired voltage and current harmonics to zero. This structure can be exploited to reduce the dimension of the parameter vector in each of the optimisation problems. It can be shown, that there exists matrices P s,v and P s,i such that: where the resulting parameter vectors of z v and z i contain only the injected voltages and currents of the desired sets V s and I s , respectively. The dimensions of the vectors in (65) are reduced with respect to In this paper two voltage and four current harmonics are injected, V 3g , V 3h , V 9g , V 9h and I 2g , I 2h , I 4g , I 4h , I 8g , I 8h , I 10g , I 10h , and it is found that z v ∈ R 4 and z i ∈ R 8 . It can be shown that the range-space of P s,v and P s,i is equal to the nullspace of B v and B i , respectively. Because of this property, the equality constraints defined by matrices B v and B i are removed without affecting the optimal solution [31] when considering the optimisation problems in terms of the reduced dimension parameter vectors, z v and z i .

1) PROPOSED OPTIMISATION STEPS OF REDUCED DIMENSION
Each proposed optimisation step is re-written in terms of the reduced dimension parameter vectors, z v and z i . This reduced dimension vector form is considered hereafter and is used to generate the results in Section IV-H.
The optimisation problems defined so far involve objective functions and constraints over continuous time and hence are in general infinite dimensional problems. Discretisation of the functions in the time domain is one approach to generating finite dimensional opimization problems that approximate the infinite dimensional problems. The discretisation approach adopted in this paper is to consider uniformly sampled values over one period. The selection of appropriate sampling interval and the effect of this simple discretisation approach is an important area that requires in-depth investigation but is beyond the scope of this paper. For simplicity, the continuous time notation is retained with the understanding that t belongs to a finite set of values {t 0 , where f s is the sampling frequency. As such, the optimisation problems in (66)-(69) can be solved using known numerical algorithms.

C. ALGORITHM
There exists multiple Matlab functions to solve the linear optimisation problems LPv and LPi. Examples include, fmincon, fminimax and linprog. Matlab functions which can be used to solve QPv and QPi include fmincon and fminimax.
The fmincon Matlab function and interior-point algorithm [33]- [35] are selected to study the solutions of the three twostep optimisation problems. The optimisation tolerances on optimality and constraints are the same for each two-step optimisation problem considered. By using the same Matlab function and algorithm for each problem ensures the stopping criteria is consistent, which gives an effective means for comparative studies.
To solve using fmincon, the optimisation problems need to be transformed into the equivalent epigraph form, which is possible given the optimisation problems are convex [31]. This is achieved by introducing an auxiliary variable to the optimisation problems [31].

D. OPTIMISATION OVER A RANGE OF POWER FACTORS
The optimisation problems are solved over a range of power factors by considering the nominal operational parameters in (70).
where θ V ∈ [−π, π]. Using (70), the ac components (subscript ∼) of the nominal voltages and currents are v N∼ (t) = sin( Therefore, the optimisation problems can be solved over a range of power factors. Hereafter, the functions which depend on θ V are made explicit in the arguments.

E. PARAMETERS FOR NUMERICAL OPTIMISATION
The peak arm current upper bound, I MAX , is selected to be: The upper bound, V MAX , is set to be large to keep the peak arm voltage constraint inactive and therefore exclude the effects of V MAX as a factor upon comparing the different optimisation problems.

F. NOMINAL CONDITION PARAMETERS
In the nominal conditions, there are no injected voltages or currents, z v = z i = 0. The nominal parameter x e C0 N needs to be selected to satisfy the constraint in (46). In order to compare the results of the nominal conditions to those generated from the optimisation problems, a selection process is defined for the nominal x e C0 N . Because the focus on this paper is on peak capacitor voltage reduction, it follows that the lowest possible x e C0 N that satisfies the condition in (46) is defined by (74).
This is because the definition of x e C0 N in (74) implies that the minimum of the nominal ε C , ε C (t, z v = 0, z i = 0, x e C0 N ), is zero, that is: The result in (75) means that using (74) for nominal conditions creates an capacitor energy waveform which touches E MIN (t, z v = 0).

G. PERFORMANCE INDICATORS
The performance indicators are defined in Table 4. Each of the performance indicators are key sources of capacitor stress [8], [9], [11] and MMC power losses [7] which are considered VOLUME 8, 2020 to select component ratings for MMCs. Trends in these performance indicators give designers an overview of the tradeoffs among power losses and stresses on components. Thê notation denotes a variable that is evaluated at the optimal parametersẑ v ,ẑ i andx e C0 that result from numerically solving the relevant two-step optimisation problems.  . The optimal rms current at each I RMSF and θ V which results from QPvQPi using parameters in Table 3 at C eff = 3 mF. The nominal rms current is the black line. FIGURE 6. The optimal peak capacitor voltage at each I RMSF and θ V which results from QPvQPi using parameters in Table 3 at C eff = 3 mF.

H. RESULTS COMPARISON
All three two-step optimisation problems are solved numerically for each θ V and I RMSF defined in Table 3. The resulting performance indicators for each optimisation problems at every θ V and I RMSF are termed here as the optimisation results.
Examples of optimisation results generated using QPvQPi are illustrated in 3D plots in Fig. 5 and Fig. 6. Fig. 5 illustrates the rms current of QPvQPi, which approaches to the nominal rms current for all θ V as the upper FIGURE 7. The optimal third harmonic voltage coefficient of Cosine for each I RMSF and θ V which results from QPvQPi using parameters in Table 3 at C eff = 3 mF.

FIGURE 8.
The optimal second harmonic voltage coefficient of Sine for each I RMSF and θ V which results from QPvQPi using parameters in Table 3 at C eff = 3 mF.
bound on rms current is tightened (decreased). This is the result of defining the rms current constraint limit relative to the rms of the nominal current. It is clear from Fig. 6 that the cost incurred for reducing rms currents is the increase in peak capacitor voltages.
Examples of optimal injected voltage and current harmonic parameters,V 3g ∈ẑ v andÎ 2h ∈ẑ i are shown in Fig. 7 and Fig. 8, respectively. It is clear that as rms current constraint is loosened (increased), the optimal parameterÎ 2h increases. Together, Fig. 6 to Fig. 8 illustrate the variation of performance indicators and optimisation parameters with varying constraint, I RMSF , and power factor, θ V .
The optimisation results, which can be shown as 3D plots in Fig. 5 and Fig. 6, are condensed as 2D plots by applying the operator max θ V (·). This is for convenience of assessing overall performance in terms of the maximum, which is an important indicator for design purposes.
All 2D plots show that the LPvQPi optimisation results approach those of LPvLPi as I RMSF is loosened. This occurs because the only difference between LPi and QPi is the rms current constraint. The removal of the rms current constraint from QPi, which is effectively achieved by loosening I RMSF , gives the same results as LPi. LPi does not have a constraint on rms current, and therefore it is independent of I RMSF , as shown by constant LPvLPi lines in the 2D plots. For the same reason, the nominal results are constant in the 2D plots. Fig. 9 to Fig. 13 present the optimisation results for C eff =3mF. Fig. 9 shows the peak capacitor voltage performance indicator as a function of I RMSF . These results reflect the optimal objective criteria as the upper bound of the rms current constraint is varied. All optimisation problems give significanltly lower peak capacitor voltage compared to the nominal case. This demonstrates the effectiveness of the optimisation problems, in particular the objective function, in decreasing capacitor stress caused by peak voltage.
The peak capacitor voltage ripple, arm rms and average current, are plotted against I RMSF in Fig. 10, Fig. 11 and Fig. 12, respectively. The cost of reducing arm rms current are an increase in peak capacitor voltage and peak capacitor voltage ripple. However, the benefit is a decrease in average arm current and, obviously, rms current. Lower rms and average currents give lower power losses [6], [7], [36]. These figures illustrate the trade-offs in terms of increases in the capacitor stresses and lower power losses. Further, it is clear that the I RMSF is effective in limiting the rms currents which increase with circulating current injection.
The capacitor current ripple performance indicator for C eff =3 mF is presented in Fig. 13. All optimisation problems result in lower capacitor current ripple with respect to the nominal, which is favourable for reducing capacitor stress. The lower capacitor stress from a combination of lower peak capacitor voltage and current ripple, along with limits on rms and average arm currents, exhibit the benefits of LPvQPi and QPvQPi. Further, QPvQPi is particularly useful when considering power losses and reducing peak capacitor voltage for all I RMSF .  Table 3 and C eff = 3 mF.
The optimisation results for a larger C eff (6 mF) are presented in Fig. 14 to Fig. 16. Both peak capacitor voltage and capacitor voltage ripple decrease with larger capacitance when compared to Fig. 9 and Fig. 10.  Table 3 and C eff = 3 mF.

FIGURE 11
. Optimal arm rms current as a function of I RMSF , using parameters in Table 3 and C eff = 3 mF.

FIGURE 12.
Optimal arm average current as a function of I RMSF , using parameters in Table 3 and C eff = 3 mF.
The capacitor current ripple is increased with respect to those presented with C eff = 3mF. In addition, for C eff = 6mF, the nominal capacitor current ripple is lower than those resulting from the optimisation problems. VOLUME 8, 2020 FIGURE 13. Optimal capacitor current ripple as a function of I RMSF , using parameters in Table 3 and C eff = 3 mF.

FIGURE 14.
Optimal peak capacitor voltage as a function of I RMSF , using parameters in Table 3 and C eff = 6 mF.

1) KEY OUTCOMES
The key outcomes of the applying the two-step optimisations are summarized as follows: 1) The peak capacitor voltages are significantly reduced with respect to the nominal for each two-step optimisation problem presented. 2) The QPi optimisation step has demonstrated its ability to constrain arm rms current and lower average current, which are important for power loss and thermal considerations. 3) For a lower capacitance value, lowering rms current does not come at a cost of capacitor voltage and current ripple that are larger than the nominal. 4) Loosening rms upper bound is beneficial in reducing capacitor stress in terms of peak and ripple capacitor voltages. However there is a trade-off since rms and average arm currents increase. 5) Holistically, LPvQPi and QPvQPi are good candidates for reducing peak capacitor voltages.
These results highlight the trade-offs which occur when optimising for lower capacitor stress and power losses. The key outcomes give designers a perspective on how the circuit parameters behave when adjusting constraints over a range of operating points.  Table 3 and C eff = 6 mF.  Table 3 and C eff = 6 mF.

V. CONCLUSION
This paper proposed three two-step convex optimisation problems which serve as tools for designers to reduce peak capacitor voltage and limiting rms currents in MMCs based on a simplified circuit model of an MMC. Each has been compared in terms of performance indicators which reflect capacitor stresses and power losses. The proposed methods have been shown to clearly identify that a trade-off that exists between capacitor stress and power losses.
Note that it is possible to consider variants of the two-step optimisation problems considered. For example, the order of QPvQPi can be changed to QPiQPv, or even alternation optimisation can be considered for the former or latter.

A. VECTOR FORM OF ARM VOLTAGE AND CURRENT
The vector forms of arm voltages and currents are defined. Let the vectors containing the harmonic coefficients for voltages, x v , and currents, x i , be defined in (76). The vectors each contain coefficients for both Sine and Cosine terms for all harmonics from 2 to some integer N H , where N H ≥ 2.

1) CIRCUIT FUNCTIONS IN ARM AU
For simplicity, at times the au notation is dropped when considering circuit functions of the upper arm of leg a, and thus the voltage and current in arm au are: where T p = T q = T pq = 0 and f(t) f a (t). Other key functions are defined in (80)-(83).
Using these defined functions, it can be shown that the vector form of key circuit functions e C (t, x v , x i , x e C0 ) and ε C (t, x v , x i , x e C0 ) are given by (84) and (85).
The T 0 /2-delayed voltage across any of the upper arms is The harmonic numbers in V s , m, are odd, it follows that: The T 0 /2-delayed current in any of the upper arms is (103) Equation (103) implies (104).
It can be shown that v pq (τ, x v )i pq (τ, x i ) contains dc components, and sinusoids with frequencies of ω 0 and integer multiples of ω 0 . Therefore, (104) holds if and only if the dc component of v pq (τ, x v )i pq (τ, x i ) is zero. It can be shown that the dc component of v pq (τ, x v )i pq (τ, x i ) is equal to: V dc I dc /6+ V 1g I 1g /2 + V 1h I 1h /2, and thus e Cpq (t + T pq ) is T 0 -periodic if and only if: The definition of E MINpq (t, x) is:

1) THREE-PHASE TIME-OFFSET
It can be shown that the relationships between E MINpu (t, x v ) and E MINpl (t, x v ), with respect to E MINau (t, x v ) are given by: Substituting T p for each p ∈ {a, b, c}, it is clear that E MINpq (t, x v ) are three-phase time-offset in the upper arms as well as the lower arms.
2) T 0 -PERIODICITY It can be shown that E MINpq (t, x v ) can be described as a constant plus a sum of sinusoids. The greatest common frequency divisor of the sinusoids in E MINpq (t, x v ) is the fundamental frequency, ω 0 , and therefore E MINpq (t, x v ) is T 0 -periodic.
At times the arguments x v , x i and x e C0 pq are dropped from ε Cpq (t, x v , x i , x e C0 pq ) to simplify notation. The definition of ε Cpq (t) is given below:
2) PERIODICITY FOR V s ⊆ V AND I s ⊆ I Under the condition in (105), and because E MINpq (t, x v ) it T 0periodic, it follows from (110) that ε Cpq (t) is T 0 -periodic. The key arm voltage, current and energy relationships when considering the specific voltage and current harmonics defined for V s and I s are summarised below. All relationships are defined with respect to the function of arm au.
CHRISTOPHER D. TOWNSEND (Member, IEEE) received the B.E. and Ph.D. degrees in electrical engineering from the University of Newcastle, Australia, in 2009 and 2013, respectively. Subsequently, he spent three years working at ABB Corporate Research, Sweden, working on next-generation high-power converter technologies. Since then, he has held various postdoctoral research positions including at the University of New South Wales, Australia; The University of Newcastle, Australia; and Nanyang Technological University, Singapore. In 2019, he joined the Department of Electrical, Electronic and Computer Engineering, The University of Western Australia, as a Senior Lecturer. He has authored more than 60 published technical papers and has been involved in several industrial projects and educational programs in the field of power electronics. His research interests include topologies and modulation strategies for multilevel converters applied in power systems, renewable energy integration, and electric vehicle applications. He is a member of the IEEE Power Electronics and Industrial Electronics societies.  VOLUME 8, 2020