João Victor

Matos Farias
email: joaofarias@ufmg.br

Luc-André Grégoire
email: luc.andre.gregoire@gmail.com

Member, IEEE Allan Fagner Cupertino
email: afcupertino@ieee.org

Member, IEEE, Seleme Heverton Augusto Pereira
email: heverton.pereira@ufv.br

Isaac Seleme
email: seleme@cpdee.ufmg.br

Member, IEEE Maurice Fadel
email: maurice.fadel@laplace.univ-tlse.fr

V M Farias

L.-A Grégoire

A sliding-mode observer for MMC-HVDC systems: fault-tolerant scheme with reduced number of sensors

Keywords: Sliding-mode observer, modular multilevel converter, high-voltage direct current, fault-tolerant scheme

The modular multilevel converter (MMC) has been widely adopted in different applications, such as high-voltage direct current (HVDC), electric drives and static synchronous compensators. Despite the application, MMC realization requires a high number of power electronic components, sensors, and communication cables. This paper proposes a sliding-mode observer for capacitor voltages of the MMC, using the measurements of arm currents and arm voltages. A linear corrector within the hysteresis band is proposed to reduce the chattering in the observer dynamics. Moreover, guidelines for tuning the observer gain are presented. The SM capacitance value is included as a parameter uncertainty of the observer. The proposal reduces the requirements of voltage sensors and optical links in MMC and allows fault-tolerant operation. The proposed observer performance is evaluated through a simulation of 100 MVA three-phase MMC-HVDC during initialization, steady-state, and submodule failures. The effect of the switching and sampling frequencies on the observer performance is also evaluated. Finally, experimental results validate the proposal in a downscale 3 kVA single-phase MMC prototype.

I. INTRODUCTION

S INCE the modular multilevel converter (MMC) conception [START_REF] Marquadt | Stromrichterschaltungen mit verteilten energiespeichern[END_REF], significant effort has been made to address the relevant issues to improve the operation and control of this converter. Mainly due to features such as modularity, scalability, inherent fault tolerance and presence of dc-link, the MMC has been widely adopted for applications in high-voltage direct current (MMC-HVDC) systems [START_REF] Hagiwara | Control and experiment of pulse width modulated modular multilevel converters[END_REF]. A real MMC-HVDC system employs hundreds of cascaded submodules (SMs) per converter arm to withstand dc-link voltages up to ± 800 kV [START_REF] Li | Feasibility and reliability analysis of lcc dc grids and lcc/vsc hybrid dc grids[END_REF]. Also, MMC have been employed in variable-speed electric drives, static synchronous compensators and battery energy storage.

The high number of SMs increases the amount of electronic components, such as semiconductor switches and capacitors. In addition, the conventional control techniques employed in this converter are based on a high number of sensors and communication cables associated to the capacitor voltage measurements [START_REF] Debnath | Operation, control, and applications of the modular multilevel converter: A review[END_REF]. Therefore, concerns regarding reliability have arisen and some MMC fault-tolerant methods have been proposed in the literature [START_REF] Liu | Fault diagnosis and fault-tolerant control of modular multi-level converter high-voltage dc system: A review[END_REF].

The MMC fault tolerance is mainly based on 3 steps: the detection of the faulty SM, the SM fault isolation and the post-fault control scheme. The faulty SM is often isolated by the SM bypass structure [START_REF] Farias | On converter fault tolerance in mmc-hvdc systems: A comprehensive survey[END_REF]. Therefore, a redundancy strategy at the SM level could be used to ensure converter operation without affecting the overall performance [START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF]. Moreover, the post-fault operation can be achieved via conceiving observer-based control schemes [START_REF] Goerges | Control of a modular converter having distributed energy stores with the ad of an observer for the currents and an estimating unit for the intermediate circuit energy[END_REF]. The main advantage in using methods based on capacitor voltage estimation is the drastic reduction in the amount of voltage measurements [START_REF] Yufei | Capacitor voltage predictor-corrector balancing approach with single sensor for single-phase modular multilevel converter[END_REF], which reduces the MMC cost and hardware complexity. This approach is especially important in HVDC systems, and can also be extended for other medium-and high-voltage MMC applications. Most observer-based control schemes rely on the capacitor current measurement, as well as knowledge of the converter parameters or their estimation [START_REF] Wang | Grouping capacitor voltage estimation and fault diagnosis with capacitance self-updating in modular multilevel converters[END_REF]. Furthermore, the sliding-mode approaches can be applied to deal with parameter uncertainties [START_REF] Chakraborty | Capacitor voltage estimation of mmc using a discrete-time sliding mode observer based on discrete model approach[END_REF].

Silva et al. propose a discrete sliding-mode observer to avoid sending all capacitor voltage measurements to the central processor, which reduces the requirements of the communication system [START_REF] Da Silva | Discrete-time sliding-mode observer for capacitor voltage control in modular multilevel converters[END_REF]. A state observer based capacitor-voltage-balancing method is presented in [START_REF] Hu | State observer based capacitor-voltage-balancing method for modular multilevel converters without arm-current sensors[END_REF], which allows for compacting the entire system and overall cost savings. Abushafa et al. propose an observer based on weighted recursive least square (RLS). The SM capacitor voltage estimation is performed through only one voltage sensor per MMC arm [START_REF] Abushafa | Capacitor voltage estimation scheme with reduced number of sensors for modular multilevel converters[END_REF]. A Kalman-filter-based estimation of SM voltages is presented in [START_REF] Islam | Arm-sensorless sub-module voltage estimation and balancing of modular multilevel converters[END_REF]. The voltage measurement in the MMC arms is avoided by adding a correction factor to the previous observed capacitor voltages. Other observer method based on sample delay compensation is proposed in [START_REF] Rong | A new voltage measure method for mmc based on sample delay compensation[END_REF]. This method replaces the measurements of each SM, measuring a set of SMs within the MMC arm. Abdelsalam et al. propose an integrated control strategy with fault detection and tolerant control capability. The capacitor voltage estimation is based on adaptive linear neuron and RLS algorithms, which requires only three sensors per MMC phase, one for the phase voltage and two for the arm inductor voltages [START_REF] Abdelsalam | An integrated control strategy with fault detection and tolerant control capability based on capacitor voltage estimation for modular multilevel converters[END_REF]. However, the aforementioned references do not present experimental verification of sliding-mode observer during SM fault conditions.

In this way, this paper proposes a sliding-mode observer for MMC-HVDC systems, using the measurements of arm currents and arm voltages. A previous study with real-time simulation concerning this proposed observer has been presented in [START_REF] Grégoire | High reliability observers for modular multilevel converter capacitor voltage evaluation[END_REF]. In view of the points aforementioned and the previous work, the following contributions of this work can be highlighted:

• The observability and stability of the proposed observer are proven; • A linear corrector is proposed to reduce the chattering in the observer dynamics; • Guidelines for tuning the observer gain are presented.

• The fault-tolerant operation is evaluated through simulated and experimental results, based on typical redundancy factors at the SM level applied to commercial HVDC systems (i.e., 6-12%); • The effect of the switching and sampling frequencies on the observer performance is analyzed, in addition to the variation of the SM capacitance value. This paper is outlined as follows. Section II presents the MMC switched dynamics, besides the average behavior for the reduced-order model. Section V presents considerations on the observability for both the linearized system and the nonlinear model. The proposed sliding-mode observer is introduced in Section III, presenting a Lyapunov stability and the convergence proof. The proposed observer performance is evaluated through the simulation results for a 100 MVA three-phase MMC-HVDC in Section VII. In Section VIII, the experimental validation of a 3 kVA single-phase MMC is performed. The conclusions are stated in Section IX.

II. MMC MODEL

Figure 1 shows a three-phase MMC composed of three legs, each one containing an upper and a lower arm. Each MMC arm is composed of an arm inductor L, an arm inductor resistance R, N non-redundant submodules and N R redundant submodules. These SMs are in the half-bridge configuration, composed of two IGBTs S 1 and S 2 , two antiparallel connected diodes D 1 and D 2 , and a capacitor C. Moreover, R b represents the bleeder resistor, responsible for discharging the SM capacitor when the MMC is turned off. There is usually a bypass structure S T in parallel with the SM bypassing it in case of fault. Moreover, the inrush current during energization is limited by pre-charge resistances R ac . The pre-charge is controlled by the circuit breakers K a1 and K a2 .

Four sensors are shown per MMC phase, two for the arm currents i arm,u and i arm,l , and two for the arm voltages v arm,u and v arm,l . Furthermore, the per-pole dc voltages are described by v dc,u and v dc,l , while v refers to the SM capacitor voltage. It is important to remark that, for the proposed methodology, the SM capacitor voltages are estimated by the observers, instead of measured by sensors.

Two control loops are usually applied in the MMC-HVDC control: the output current and circulating current controls. The output current is commonly used to control the active power flow or the stored energy in the MMC, considering the HVDC systems composed by two ac stations [START_REF] Bergna | A generalized power control approach in abc frame for modular multilevel converter hvdc links based on mathematical optimization[END_REF]. Regarding the circulating current control, the external loop has been used to balance and equally distribute the internal energy of the MMC [START_REF] Adam | Modular multilevel inverter: Pulse width modulation and capacitor balancing technique[END_REF]. When the active redundancy with load-sharing effect is considered, the SM capacitor voltage reference is described by [START_REF] Farias | On the redundancy strategies of modular multilevel converters[END_REF]:

V * = V dc N T , (1)
where N T = N + N R is the total number of SMs per MMC arm. According to the modulation strategy, an additional individual balancing control loop can be necessary [START_REF] Hagiwara | Control and experiment of pulse width modulated modular multilevel converters[END_REF]. Moreover, the inner loop is responsible for the suppression of the harmonics of the circulating current during asymmetric operation [START_REF] Adam | Modular multilevel inverter: Pulse width modulation and capacitor balancing technique[END_REF]. The circulating current is calculated per MMC phase, as given by:

i c = i u + i l 2 . (2)
For the sake of simplicity, the subscripts u and l are omitted, since the equations for the upper or lower arms are equivalent. Therefore, the MMC switched dynamics of a generic MMC arm can be described by:

      vs 1 . . . vs N is arm       = A nl (q)       0 • • • 0 q1 C . . . . . . . . . . . . 0 • • • 0 q N C -q1 L . . . -q N L -R L             v s 1 . . . v s N i s arm       +        0 . . . 0 0.5v s dc -v s ac L        , (3)
where v s n are the instantaneous (switched) voltages of the SM capacitors and q n are the gate signals applied to the SM upper-switch S 2 for n = 1, • • • , N . Moreover, i s arm are the instantaneous arm currents of the upper or lower arms and v s ac is the instantaneous ac grid voltage. Under balanced conditions, the instantaneous per-pole dc voltage can be described by 0.5v s dc . The gate signals q n determine the SM states, as well as the insertion of the SM capacitors into the MMC arm. Moreover, during the switching period T s , q n = 1 for t ≤ kT s + t n k-on and q n = 0 for t n k-on < t ≤ (k + 1)T s , where t n k-on is the k-th switching time for the n-th SM, with k = 0, 1, • • • , N , the index number of the switching period in an N switching event series. Furthermore, the system outputs can be described by:

y v s arm i s arm = C nl (q) q 1 • • • q N 0 0 • • • 0 1       v s 1 . . . v s N i s arm       , (4)
where v s arm are the sum of the instantaneous SM capacitor voltages of the upper or lower arms. Therefore, the dynamics of the MMC output variables v s arm and i s arm are described by (3) and (4). Indeed, these are the input variables of the state observer adopted in this work.

III. PROPOSED OBSERVER

A capacitor voltage observer is adopted based on the MMC switched model described above. Furthermore, aiming at robustness with respect to parameter uncertainty, the observed voltage value can be adjusted using a sliding-mode corrector. The capacitor voltage observer proposed is given by:

vn =                      Estimate i arm q n C nom + Sliding-mode corrector λ q n sgn (e) if |e| ≥ V th , Estimate i arm q n C nom + linear corrector λ V th q n e if |e| < V th , (5)
where λ is the observer gain, C nom is the nominal SM capacitance, V th is the hysteresis band voltage and the error is given by e = v arm -v arm . The arm voltage measurement v arm and the arm current measurement i arm are the observer inputs.

As noted, within a hysteresis band (-V th < e < V th ), the sliding-mode observer is a standard proportional type. Indeed, when this error tends to zero, a reduction in the correction effect is expected avoiding chattering in observer dynamics. The introduction of the linear corrector in the hysteresis band is evaluated in Section VII of this work, where V th = 0.02V * is chosen. Furthermore, the function sgn in (5) returns either -1, 0, or 1. As noted, such function is multiplied by λ and also by the SM upper-switch gate signals q n , ensuring the correction is applied only to the SMs used to compute the error. Under such conditions, the proposed observer presents fast convergence, as well as interesting dynamic features, which are evaluated in the simulated and experimental results of the next sections.

As noted, the proposed observer in ( 5) needs a duty cycle signal applied to each SM, i.e., the MMC must be in controlled mode (CM). However, there is no duty cycle signal applied to both switches of the SM during the initialization of the converter. Under such conditions, i arm flows through the antiparallel diodes of each SM. A positive arm current charges the SM capacitor through the antiparallel diode D 2 . A negative arm current polarizes the lower antiparallel diode D 1 , bypassing the SM. This converter operation is the well-known natural rectifying mode (NRM). Therefore, to obtain the right behavior of the proposed observer during NRM, q n must be modified by the following switching function:

q n = 1, if i arm > 0, 0, if i arm ≤ 0. (6)

IV. STABILITY AND ROBUSTNESS PROOFS

The observed SM capacitor voltages are given in (5) as a summation of integrals in each switching interval T s . However, the dynamics of the MMC switched model tends to a response of an averaged nonlinear model. Thus, the stability and robustness proofs of the observer are described in the next sections, considering the MMC averaged nonlinear model.

A. Averaged Nonlinear Model

A nonlinear state-space model of the averaged system is derived as follows:

      v1 . . . vN iarm       = A nl (g)       0 • • • 0 g1 C . . . . . . . . . . . . 0 • • • 0 g N C -g1 L . . . -g N L -R L             v 1 . . . v N i arm       +       0 . . . 0 0.5v dc -vac L       , (7)
where g n are the SM duty cycles of the upper or lower arm for n = 1, • • • , N , i.e., g n = 1 Ts t t-Ts q n (τ )dτ , which provides the fraction of the switching period when the capacitor is inserted into the MMC arm. v n , i arm , v arm , v dc and v ac are the average values in the switching interval, defined as:

v n =
y v arm i arm = C nl (g) g 1 • • • g N 0 0 • • • 0 1       v 1 . . . v N i arm       , (8)
which describes a nonlinear system, since matrices A nl (g) and C nl (g) in ( 7) and ( 8) are functions of the duty cycles g n .

B. Stability Proof

The differential equations of the SM capacitor voltages are obtained from [START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF] as:

vn = i arm g n C , ( 9
)
where C is the actual value of the SM capacitance.

Considering that C = C nom and that e = v arm -varm = N n=1 g n v n -N n=1 g n vn , the observer error ṽn = v n -vn , is obtained by ( 9) and ( 5), as follows:

vn = vn -vn =        -λ g n sign N n=1 g n ṽn , if |e| ≥ V th , -λ V th g n N n=1 g n ṽn . if |e| < V th . (10) Let, v = [v 1 , • • • , v N ] T , v = [v 1 , • • • , vN ] T , s = v -v = ṽ = [ṽ 1 , • • • , ṽN ] T , (11)
be an N dimension vector, which defines a surface s = 0. Moreover, the Lyapunov function candidate is described by:

V lf = 1 2 s T s. (12)
Therefore, the time derivative of V lf is:

Vlf = s T ṡ =        -λ N n=1 g n ṽn ≤ 0, if |e| ≥ V th , -λ V th N n=1 g n ṽ2 n ≤ 0, if |e| < V th . (13
) Thus, equation ( 13) guarantees stability and asymptotic convergence of the observer for any λ > 0.

C. Robustness

The observed voltages based on sliding-mode observers present robustness to parameter uncertainties (e.g., SM capacitance values). To evaluate the observer robustness, the differential equations of the observed SM capacitor voltages for the error outside the hysteresis band (|e| ≥ V th ) can be obtained from [START_REF] Liu | Fault diagnosis and fault-tolerant control of modular multi-level converter high-voltage dc system: A review[END_REF] as:

vn = i arm g n C nom + λ g n sign (e) . ( 14
)
Under such conditions, the derivative of the Lyapunov function of [START_REF] Hu | State observer based capacitor-voltage-balancing method for modular multilevel converters without arm-current sensors[END_REF] for the error outside the hysteresis band is given as:

-λ N n=1 g n ṽn + i arm N n=1 g n ṽn ≤ (-λ+| σ|) N n=1 g n ṽn ≤ 0, (15) where = 1 C -1
Cnom is the worst-case discrepancy in C, and σ > |i arm | is an upper bound of MMC arm current i arm . The error is confined within the hysteresis band for λ > | σ|, as obtained from [START_REF] Islam | Arm-sensorless sub-module voltage estimation and balancing of modular multilevel converters[END_REF]. Assuming λ * = | σ|, the observer gain should satisfy λ > λ * to guarantee stability under uncertainty, which can be rewritten as:

λ > λ * = i arm 1 C - 1 C nom . ( 16
)
In addition, the upper bound criteria for the observer gain can be determined by the effect caused by the discretized controller, since the observer correction is applied in the sampling time causing oscillation in the estimation. Figure 2 shows the observer performance for different observer gains, considering 1 pu of active power injected into the grid from t = 100 ms. For the sake of simplicity, only one SM capacitor voltage in the upper arm of MMC phase A is shown. As noted in Fig. 2 (a), the measured voltage reaches the reference in about 10 ms considering 10λ, where λ = 76000 is employed according to [START_REF] Rong | A new voltage measure method for mmc based on sample delay compensation[END_REF] considering the parameters of MMC simulation in Tab. II. Moreover, a small gain increases the required time for convergence. It is important to remark that for the observer gain λ = 76000, the steady-state is achieved in about 100 ms with low oscillation, as illustrated in detail in Fig. 2 (b). In the next sections, optimal λ is founded empirically based on the proposed guideline for tuning the observer gain.

V. SYSTEM OBSERVABILITY

The electrical dynamics of the MMC, given by (3), is both nonlinear and discontinuous due to the bilinear and switching characteristics of the system. Nevertheless, the average model dynamics of the MMC, given by [START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF], is smooth and can be considered to estimate the average unmeasured variables, such as SM capacitor voltages. Therefore, a local stability for the linearized system is analyzed in the next sections.

A. Linearized System Model

Aiming to define the system local observability, a linearization is carried out around an operating point (X, U ), and a small-signal linear model is obtained, with

X = [V 1 , • • • , V N , I arm ] T , U = [ḡ 1 , • • • , ḡN , I arm ] T and Y = [V arm , I arm ] T . Let x = [v 1 , • • • , v N , i arm ] T be the state vector of the nonlinear system, u = [g 1 , • • • , g N , i arm ] T the input vector, y = [v arm , i arm ]
T the measured output. Furthermore, x s = x -X, u s = u -U and y s = y -Y are the small-signal state, input and output vectors, respectively, of the linearized system around an operating point. Such operating point is given by a constant state vector X, constant duty cycles U , and a constant output Y , as given by:

ẋs = A l ∂F (x, u) ∂x X,U x s + B l ∂F (x, u) ∂u X,U u s , y s = C l ∂G (x, u) ∂x X,U x s + D l ∂G (x, u) ∂u X,U u s , (17)
where F (x, u) = A nl (g)x and G (x, u) = C nl (g)x.

Moreover, considering u s = [g s1 , • • • , g s N , i sarm ] T , the linear matrices of the small-signal model in [START_REF] Abdelsalam | An integrated control strategy with fault detection and tolerant control capability based on capacitor voltage estimation for modular multilevel converters[END_REF] are given by:

A l =       0 • • • 0 ḡ1 C . . . . . . . . . . . . 0 • • • 0 ḡN C -ḡ1 L . . . -ḡ N L -R L       B l =        Iarm C • • • 0 0 . . . . . . . . . . . . 0 • • • Iarm C 0 -V1 L . . . -V N L -1 L        , (18)
C l = ḡ1 • • • ḡN 0 0 • • • 0 1 D l = V 1 • • • V N 0 0 • • • 0 0 . (19)

B. Linear System Observability

As noted, the state vector x is not globally observable, i.e., the rank r of the observability matrix is r = 2 < N :

O = C l C l A l • • • C l A N -1 l T .
Nevertheless, the individual capacitor voltages v n for n = 1, • • • , N are observable locally, considering the remaining capacitor voltages as perturbations. The resulting individual small-signal dynamics is given by ( 20) and [START_REF] Farias | On the redundancy strategies of modular multilevel converters[END_REF].

ẋsn vsn isarm = An 0 ḡn C -ḡn L -R L v sn i sarm + Bn Iarm C -Vn L g sn - 0 ϕ L , (20)
y v sarm i sarm = Cn 1 0 0 1 v sn i sarm + ϕ 0 . (21)
It is important to remark that v sarm , i sarm , v sn and g sn are small signals of the arm voltage, the arm current, the n-th capacitor voltage and the n-th capacitor duty-cycle, respectively. The perturbation term ϕ is given as:

ϕ = N k=1 k =n ḡk v s k , (22)
Now, the observability matrix

O n = [C n , C n A n ]
T has rank r = 2, which defines the order of the subsystem. Then, the capacitor voltage for each SM can be estimated employing the arm current and its gate signals. Notice that 0 < g n < 1 corresponds to the duty cycle applied to the SM upper-switch (S 2 ) over the period T .

VI. DISCUSSION

The SM voltage control is generally performed by the knowledge of each SM voltage in real-time. In this way, a voltage sensor must be installed for each SM, which increases the burden on the data acquisition system, number of optical links and system costs.

In this work, only one voltage sensor is required per MMC arm. This approach does not reduce the reliability of the system, since the high number of voltage sensors and optical links of traditional MMC hardware raises concerns about random failures [START_REF] Farias | On converter fault tolerance in mmc-hvdc systems: A comprehensive survey[END_REF]. However, it is noteworthy that the measurement with only one sensor is much more impaired by a sensor failure. This issue can be solved with redundant arm voltage measurement, which is a common practice in some industry segments.

The measured arm voltages and the gate signals must be synchronized for the proper operation of the observer. In this way, the gate signals can be delayed to match the measured voltage, since the time when gate signals are produced is known, as well as the latency introduced by the measurement. Moreover, the voltage balancing algorithm presents slow time constants in comparison with the current control. In that sense, adding latency to the measurement presents few effect to the MMC control.

Another important point is the accuracy required for the arm voltage sensor. In some MMC applications, the voltage of a single SM is lower than 1% of the whole arm voltage. In this case, sensor accuracy can affect the performance of the proposed observed. This issue can be avoided by measuring a set of SMs [START_REF] Rong | A new voltage measure method for mmc based on sample delay compensation[END_REF], as shown in Fig. 1. As noted, the SMs are

Descriptions

Ref. [START_REF] Da Silva | Discrete-time sliding-mode observer for capacitor voltage control in modular multilevel converters[END_REF] Ref. [START_REF] Hu | State observer based capacitor-voltage-balancing method for modular multilevel converters without arm-current sensors[END_REF] Ref. [START_REF] Abushafa | Capacitor voltage estimation scheme with reduced number of sensors for modular multilevel converters[END_REF] Ref. [START_REF] Islam | Arm-sensorless sub-module voltage estimation and balancing of modular multilevel converters[END_REF] Ref. [START_REF] Rong | A new voltage measure method for mmc based on sample delay compensation[END_REF] Ref. [ divided into m sets with k SMs, where each set only needs one voltage sensor. In this case, the proposed observer is employed for each set of SMs, and a partial reduction in the MMC hardware complexity is reached.

The comparison among the proposed observer and other methods in the introduction section is described in Table I. A chattering reduction in the observed capacitor voltage is reached through the proposed linear corrector, as validated in the next sections. Moreover, the effect of switching and sampling frequencies on the observer is evaluated for only two frequency values in [START_REF] Da Silva | Discrete-time sliding-mode observer for capacitor voltage control in modular multilevel converters[END_REF]. In this sense, a range of typical switching frequencies applied to MMC is evaluated in this work. In addition, a guideline to evaluate the observer gain is presented. Finally, the experimental results under faulty SMs are validated by a downscale setup, which is a gap in the aforementioned references.

VII. SIMULATION RESULTS

The performance of the proposed capacitor voltage observer is evaluated for a 100 MVA three-phase MMC-HVDC. Simulations are carried out in PLECS environment and the main parameters are summarized in Table II. The converter is connected to the main grid through a three-phase transformer. The control strategy presented in [START_REF] Sharifabadi | Design, Control and Application of Modular Multilevel Converters for HVDC Transmission Systems[END_REF] and [START_REF] Amorim | Analysis of double star modular multilevel topologies applied in hvdc system for grid connection of offshore wind power plants[END_REF] is employed in the simulation, considering the phase-shifted pulse-width modulation (PS-PWM). The redundant SMs are equivalent to 10% of the non-redundant [START_REF] Farias | On the redundancy strategies of modular multilevel converters[END_REF]. Moreover, the observer is employed in closed loop, which means that SM capacitor voltages are measured only to validate the proposed observer.

A. Steady-state

In steady-state operation, the active power reference is defined to 1 pu. Figure 3 (a) shows the maximum relative error among all measured and estimated SM capacitor voltages during steady-state operation for different switching frequency. As noted, an increased switching frequency reduces the maximum relative error. Moreover, a reduced capacitance tolerance value reduces the maximum relative error. In addition, the observer is operating with and without the proposed linear corrector. As noted, the linear corrector results in a decrease in the relative error mainly for higher capacitance tolerances. Figure 3 (b) shows the proposed observer effect on the spreading of the SM capacitor voltages1 . As noted, the spreading of capacitor voltages is similar in both strategies. Therefore, the increased spreading effect at low-switching frequencies is not caused by the proposed observer. Indeed, the spreading in capacitor voltages at low-switching frequencies is an expected effect in MMC.

B. Energization

The charging resistors limit the in-rush current during the energization in NRM until the converter operates in CM. These stages can be divided as follows: i) t = 0s: the system is connected by the grid side. K a2 is closed and K a1 is opened; ii) t = 0.14s: removing the ac-side charging resistances. K a1 is closed; iii) t = 0.17s: enabling the dc-link with voltage equal to the peak of line-to-line ac voltage; iv) t = 0.18s: increasing the dc-link voltage to the rated value in ramp; v) t = 0.23s: enabling the closed-loop control; vi) t = 0.25s: enabling the ramp of 2000 MVA/s of active power injected into the main grid. For the sake of simplicity, only the upper arm of MMC phase A is analyzed. Only non-redundant SMs are considered in this case. Measurements and estimates of the SM capacitor voltages in per unit (pu) are illustrated in Figs. 4 (a) and (b), respectively. The dashed lines indicate that the SM capacitor voltages oscillate with 10% of ripple. Notice that the observed voltages are smoother than the measured ones, thus highlighting the well-known filtering feature of observers. During the NRM, the relative errors between the measured and estimated values are within 5%, as shown in Fig. 4 (c). Moreover, the relative errors are within 2% during the CM operation.

Figure 5 (a) illustrates the grid currents in pu. As observed, the grid current is limited in the NRM and tends to decrease, since the closed-loop control is disabled. After t = 0.25s, the grid current amplitude increases according to the active power reference in ramp with slope of 2000 MVA/s. Similar behavior is observed in the circulating currents, as shown in Fig. 5 (b). As observed, the circulating currents present a dc value when the active power flows through the converter. by the gate signals q = 0. Furthermore, a small-time interval is associated with detecting SM failures and transmitting this information to the observer. Within this time interval, the observer continues to estimate the faulty SM voltages, as well as the remaining SM voltages, based on the configuration previous to the failure. This leads to errors in the observed voltages of the SM in operation. A time interval of 50 ms is considered in this work. Therefore, the observer neglects the faulty SMs at t = 0.15s, and the relative errors of the remaining SMs are within 1%. Moreover, the relative error of the faulty SMs increases due to the bleeder resistor R b present in each SM.

VIII. EXPERIMENTAL RESULTS

The experimental results are obtained through a single-phase MMC prototype connected to an ac load, as shown in the illustrative photographs in Fig. 7. The main parameters are summarized in Table II. OPAL-RT Technologies real-time simulator is used to control the converter and compute the observer. The current in the ac load is controlled using a decentralized predictive control based on interleaved PWM modulation, as described in [START_REF] Seleme | Modular control with carrier auto-interleaving and capacitor-voltage balancing for mmcs[END_REF]. Similarly to simulation, SM capacitor voltages are measured only to be compared to the estimated values. Balancing control algorithm for the SM and converter regulation are achieved using the observed voltages. Furthermore, the SM capacitance values are within 5% of dispersion.

Figures 8 (a) and (b) show the measured and estimated values for the capacitor voltages from the MMC upper arm, respectively. The failure of an SM is performed at t = 0.1s. As observed, the voltage of the remaining SMs increases by around 13% to compensate the faulty SM. The voltage increase is more significant, due to the lower number of SM per arm, compared to the simulation case. As noted in Fig. 8 (c), the relative errors remain less than 5% and are mainly caused by measurement noise.

The single-phase MMC dynamics are presented in Fig. 9. Measurements and estimates of the upper-arm voltage are showed in Fig. 9 (a). The transient referring to the SM removal is extinguished after approximately 0.05s. A similar behavior is observed in the arm currents, circulating current and ac-load current showed in Figs. 9 (b) and (c), respectively. As noted, the transient operation is highlighted for the upper-arm current, since one SM failure occurred in the upper arm. Moreover, the ac-load current control presents a suitable performance during the transient operation.

IX. CONCLUSION

This work proposes a sliding-mode observer for MMC-HVDC systems, using the measurements of arm currents and arm voltages. A linear corrector is proposed within the hysteresis band to reduce the chattering in the observer dynamics. In addition, a guideline for tuning the observer gain is presented. The SM capacitance value is included as a parameter uncertainty. The observer is adapted to the NRM and CM operating modes during the converter initialization. Moreover, the stability of the observer is proven through the Lyapunov's direct method.

Simulation and experimental results demonstrate the effectiveness of the proposed observer in both dynamic and steady-state operating conditions. The observer presented adequate performance considering the removal of all redundant SMs in a single MMC arm. Furthermore, an increase in the switching and sampling frequencies reduces the relative errors among measured and estimated SM capacitor voltages.

In addition, a reduced SM capacitance tolerance reduces the relative error. Finally, the proposed observer can be implemented in the existing controller and replace the capacitor voltage measurements for its observed voltages.

Fig. 1 .

 1 Fig.1. Schematic of the three-phase MMC with half-bridge SMs.

  ẋ

Fig. 2 .

 2 Fig. 2. Evaluation of the observer performance for different λ: (a) Observed voltage; (b) Relative error. Remark: The switching frequency is fsw = 333 Hz, sampling frequency is fs = 52 kHz and observer gain is λ = 76000.

Fig. 3 .

 3 Fig. 3. Simulation results of the steady-state operation: (a) Effect of both linear corrector and capacitance tolerance on the maximum relative error among all SM voltage measurements and estimates; (b) Maximum spreading of the capacitor voltages. Remark: fs = 2N T fsw.

CFig. 4 .

 4 Fig. 4. Simulation results during energization: (a) Measured voltages; (b) Observed voltages; (c) Relative error. Base value: V dc N = 3.29 kV.

Figure 6 (Fig. 5 .Fig. 6 .

 656 Fig. 5. Simulation results during energization: (a) Grid currents; (b) Circulating currents. Base value: 0.67 kA.

Fig. 7 .

 7 Fig. 7. Photographs of a 3 kVA MMC experimental setup adapted to single-phase configuration with 9 SMs and one arm inductor per arm.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Experimental results of the SM capacitor voltages of the single-phase MMC upper arm: (a) Measured voltages; (b) Observed voltages; (c) Relative error. Base value: V dc N = 75 V.

TABLE I COMPARISON

 I OF DIFFERENT CAPACITOR VOLTAGE ESTIMATION METHODS.

  Only the HIL test platform is performed. ++, + and -represent a good, moderate and low performance, respectively.

							17]	This paper
	Reduction of chattering in observer dynamics	-	-	-	-	-	-	++
	Evaluation of the effect of switching and sampling frequencies on the observer	+	-	-	-	-	-	++
	Observability and stability proofs	Yes	Yes Yes No No No No	No	Yes 1	Yes
	1

It is important to remark that only in this case, the MMC control is carried out in both situations: with voltage measurements and with observed voltages.

ACKNOWLEDGEMENT

The authors would like to thank the Brazilian agencies CAPES, National Council for Scientific and Technological Development -CNPq (Grant 408058/2021-8), and FAPEMIG (Grant APQ-02556-21) by funding.

This study was supported by the National Council for Scientific and Technological Development -CNPq (Grant 408058/2021-8), FAPEMIG (Grant APQ-02556-21) and CAPES.

J.

João Victor Matos Farias received the B.S. degree in electrical engineering from the Federal University of Vic ¸osa (UFV), Brazil, in 2018, the M.S. degree from the Federal Center of Technological Education of Minas Gerais, Brazil, in 2019, all in electrical engineering. Currently, he is working toward the Ph.D. degree in electrical engineering from the Federal University of Minas Gerais. Since 2022, he has been with the Fraunhofer Institute for Silicon Technology ISIT, where he is currently Assistant Researcher in the area of hybrid grids and power electronics. His main research interests include modular multilevel converters, HVDC and STATCOM applications, electric drives and reliability of power converters.

Luc-André Grégoire received the B.Ing. degree from the École de Technologie Supérieure ( ÉTS), in 2008, the master's degree with the Groupe de Recherche en Électronique de Puissance et Commande Industrielle (GREPCI-ETS), while starting to work at OPAL-RT Technologies Inc., and the Ph.D. degree in real-time simulation of multirate system, using microprocessors and FPGA from ÉTS, in 2016.

During which time, he conducted an internship at the Laboratoire Plasma et Conversion D'énergie (LAPLACE), Unisersité Polytechnique de Toulouse, Toulouse, France, in 2015. He worked on multilevel power convert during his master's degree. In 2010, he developed the first modular multilevel converter (MMC) model for real-time simulation. He then spent his following year at the Supergrid Institute, Lyon, France, before returning to OPAL-RT Technologies Inc. His main research interests include real-time simulation, numerical stability, power converters, and multilevel converters.


Elie Hleihel
email: hleihel@laplace.univ-tlse.fr

Maurice Fadel
email: maurice.fadel@laplace.univ-tlse.fr

Hadi Y Kanaan
email: hadi.kanaan@usj.edu.lb

Multi-objective Optimization of a DC Microgrid with a Back-up Diesel Generator

Nowadays, microgrid applications are proliferate all around the world. Owing to many grounds, such the ease of control, the high efficiency and reliability, the improvement of power electronics devices, the rise of DC type loads and sources, etc. researchers' interest was diverted from AC to DC microgrids. Yet, on a global control and management level, several challenges are confronted. A variety of objectives can be achieved by controlling the power flow of each of the distributed energy sources. By means of this, an optimization problem is formulated and solved using heuristic methods such the genetic algorithm (GA), the particle swarm optimization (PSO), the pattern search (PS), etc. However, other techniques were exploited in the literature such the dynamic programming (DP) which is a stepby-step optimization algorithm. In this paper, a (DP) technique is applied to solve a multi-objective optimization problem. Two objectives are set: DC microgrid operation cost minimization, and pollutant gas emissions reduction. A sole cost function is established, and weights are assigned to each of the predefined goals. Besides, each objective function is detailed apart, and several constrains are set. Two simulations tests are performed to prove the convergence, and the viability of the applied (DP) technique. Finally, different weights are selected in each of simulation tests to validate the effectiveness, and robustness of the (DP) in solving such problems.

I. INTRODUCTION

Over the past few years, microgrid applications have emerged and expanded worldwide owing to several motifs such, the increasing demand for power electricity in all fields, the rising integration of renewable energy sources (RESs) as alternatives to traditional pollutant ones, the high merits of distributed energy generation over traditional centralized approach in terms of reliability, robustness, expandability, etc. [START_REF] Parhizi | State of the Art in Research on Microgrids: A Review[END_REF]. Besides, the rapid development of power electronic devices, and the considerable growth in DC-type loads, sources, and energy storage systems diverted the attention of researchers to DC microgrids [START_REF] Dragičević | DC Microgrids-Part I: A Review of Control Strategies and Stabilization Techniques[END_REF]. Despite the ease of control of a DC microgrid on a primary control level, compared to AC counterparts [START_REF] Hammerstrom | AC Versus DC Distribution SystemsDid We Get it Right?[END_REF], many challenges arise if advanced functionalities and objectives are to be achieved on the global power management level. In such instances, an optimization problem is formulated to reach certain predefined goals. The literature proposes various optimization techniques for the energy management of DC microgrids. Heuristic methods such as the genetic algorithm (GA), particle swarm optimization (PSO), pattern search (PS), simulated annealing (SA), etc. are utilized to solve these kinds of problems. In [START_REF] Baziar | Considering uncertainty in the optimal energy management of renewable micro-grids including storage devices[END_REF] an adaptive (PSO) technique is used to consider the uncertainties in the optimal energy management of a microgrid. Isolation niche immune genetic algorithm (INIGA) is proposed in [START_REF] Liao | The optimal economic dispatch of smart Microgrid including Distributed Generation[END_REF] to optimize the economic operation of the microgrid as well as to minimize toxic gas emissions. However, a multi-objective adaptive modified particle swarm optimization algorithm (AMPSO) is presented in [START_REF] Moghaddam | Multi-objective operation management of a renewable MG (micro-grid) with back-up micro-turbine/fuel cell/battery hybrid power source[END_REF] for the optimal operation of a typical microgrid with (RESs).To minimize the operation cost, and the pollutants emission, two objective functions are set along with constraints. Despite their efficiency in finding global minimums, such heuristic methods require an excessive computational burden and are time-consuming especially in problems with a high number of decision variables. Meanwhile, dynamic programming (DP) is a step-by-step optimization technique based on restrained research domains. It has been widely used in the literature and yields similar and comparable results to heuristic methods in terms of algorithm convergence, global minimum attainment, and multi-objective achievement [START_REF] Rigo-Mariani | Méthodes de conception intégrée 'dimensionnement-gestion' par optimisation d'un micro-réseau avec stockage[END_REF], [START_REF] Hankache | Gestion optimisée de l'énergie électrique d'un groupe électrogène hybride à pile à combustible[END_REF]. In this paper, a multi-objective optimization problem is formulated and solved using a (DP) algorithm. Two main objectives are set: DC microgrid operation cost minimization, and pollutant gas emissions reduction. A sole cost function is evaluated and minimized. The two mentioned goals are reached with a preference degree defined through fixed weights. These weight variables are included in the basic cost function. The rest of this paper is organized as follows: in section II the DC microgrid model is presented in detail. The optimization problem is formulated in section III. Simulation tests and results are performed in section IV. Finally, section V concludes the paper.

II. 24-HOUR DC MICROGRID MODELLING STRATEGY

The adopted DC microgrid topology is shown in Fig. 1. It consists of renewable energy sources (RESs): a solar PV array, and a wind turbine, a lithium-ion battery as an (ESS), a backup diesel generator (DG) as a pollutant source, and DC loads. The DC microgrid can operate in islanded as well as in gridconnected mode to either buy or sell energy from/to the utility grid. Next, each energy source is modeled separately.

A. PV Array

The (RESs), as non-dispatchable sources, are continuously functioning in maximum power point tracking (MPPT) mode. The PV output power function at time 𝑘 is expressed as [START_REF] Taha | An Online Energy Management System for a Grid-Connected Hybrid Energy Source[END_REF]:

𝑃 𝑃𝑉 (𝑘) = 𝑃 𝑆𝑇𝐶 × 𝐺 𝐼𝑁 (𝑘) 𝐺 𝑆𝑇𝐶 × (1 + 𝛼(𝑇 𝐶 (𝑘)-𝑇 𝑅 )) (1)
where, 𝑃 𝑆𝑇𝐶 , 𝐺 𝐼𝑁 (𝑘), 𝐺 𝑆𝑇𝐶 , 𝛼, 𝑇 𝐶 (𝑘), and 𝑇 𝑅 are respectively the maximum power at standard test conditions (STCs), the incident irradiation at time 𝑘, the irradiation at (STCs), the temperature coefficient, the cell temperature at time 𝑘, and the cell temperature at (STCs).

B. Wind turbine

As prementioned, the wind turbine (WT) is operating in MPPT mode. The (WT) output power function can be expressed as [START_REF] Suman | Optimisation of Solar/Wind/Bio-generator/Diesel/Battery Based Microgrids for Rural Areas: A PSO-GWO Approach[END_REF]:

𝑃 𝑤𝑖𝑛𝑑 (𝑘) = { 0 0 ≤ 𝑉(𝑘) < 𝑉 𝑐𝑖 𝑎. 𝑉(𝑘) -𝑏 𝑉 𝑐𝑖 ≤ 𝑉(𝑘) < 𝑉 𝛼 𝑐. 𝑉(𝑘) 3 𝑉 𝛼 ≤ 𝑉(𝑘) < 𝑉 𝑟 𝑑. 𝑉(𝑘) -𝑒 𝑉 𝑟 ≤ 𝑉(𝑘) < 𝑉 𝛽 𝑃 𝑤𝑖𝑛𝑑 𝑟 𝑉 𝛽 ≤ 𝑉(𝑘) < 𝑉 𝑐𝑜 0 𝑉(𝑘) ≥ 𝑉 𝑐𝑜 (2)
where 𝑉(𝑘), 𝑉 𝑐𝑖 , 𝑉 𝑐𝑜 , 𝑉 𝛼 , 𝑉 𝛽 , and 𝑃 𝑤𝑖𝑛𝑑 𝑟 are respectively, the wind speed at time 𝑘, the cut-in speed, the cut-out speed, the lower limit linear speed, the upper limit linear speed, and the rated wind power. 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are constants defined based on the (MPPT) characteristic curve of the (WT).

C. Lithium-ion battery dynamic model

A lithium-ion battery is selected as an (ESS) due to its compliance with such applications. The battery applied model is that figuring in the Simscape library of MATLAB/Simulink, and originally proposed in [START_REF] Tremblay | Experimental validation of a battery dynamic model[END_REF]. The discrete equation of the battery state of charge is expressed as:

𝑆𝑂𝐶(𝑘 + 1) = 𝑆𝑂𝐶(𝑘) - 𝑃 𝑑𝑖𝑠 (𝑘).𝑇 𝑠 𝜂 𝑑𝑖𝑠 .𝐶 𝑏𝑎𝑡𝑡 .𝑉 𝑏𝑎𝑡𝑡 (𝑘) - 𝑃 𝑐ℎ (𝑘).𝜂 𝑐ℎ .𝑇 𝑠 𝐶 𝑏𝑎𝑡𝑡 .𝑉 𝑏𝑎𝑡𝑡 (𝑘) (3)
where 𝑆𝑂𝐶, 𝑃 𝑑𝑖𝑠 (𝑘), 𝑃 𝑐ℎ (𝑘), 𝑉 𝑏𝑎𝑡𝑡 (𝑘), 𝐶 𝑏𝑎𝑡𝑡 , 𝜂 𝑑𝑖𝑠 , 𝜂 𝑐ℎ , 𝑇 𝑠 are respectively the battery state of charge in (%), the discharged power at time 𝑘 in (W), the charged power at time 𝑘 in (W), the battery voltage at time 𝑘 in (V), the battery rated capacity in (Ah), the discharging efficiency, the charging efficiency, and the sampling time in hours. At each time step, a function 𝑓 𝑏𝑎𝑡𝑡 is called to compute the battery electrical components following the expressions in [START_REF] Tremblay | Experimental validation of a battery dynamic model[END_REF]. Then, the battery state of charge is deduced using equation ( 3). The function script is represented in algorithm.1.

III. OPTIMIZATION PROBLEM FORMULATION

In this section, the optimization problem is formulated and solved using the (DP) algorithm. The two main objectives to be attained are: minimizing the total operation cost of the DC microgrid and minimizing the pollutant gas emissions.

A. Objective function

There are several methods to solve these kinds of optimization problems. One is to set distinct objective functions corresponding to each of the predefined goals and minimize them. Another simple way is to scalarize the set of objectives into a single function by multiplying each objective by a fixed weight. This method is known as the weighted sum method.

Besides the reduction of the complexity of the problem, it offers to the user the possibility of objectives' preference through the selected weights. Therefore, the total objective function is the sum of weighted cost functions of the total operating cost, and the pollutant gas emissions. It can be expressed as follow:

𝐽 𝑡𝑜𝑡 = 𝛾 𝑜𝑐 . 𝐽 𝑜𝑐 + 𝛾 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 . 𝐽 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 (4)
where 𝐽 𝑡𝑜𝑡 , 𝐽 𝑜𝑐 , 𝐽 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 , 𝛾 𝑜𝑐 , and 𝛾 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 are respectively the total objective function in $, the operation cost function in $, the pollutant gas emissions cost function in $, the operation cost weight, and the pollutant gas emissions weigh.

A.1. Total operating cost function

The total operating cost function is the sum of the (RES)s cost 𝐽 𝑜𝑐 = 𝐽 𝑅𝐸𝑆𝑠 + 𝐽 𝑔𝑟𝑖𝑑 + 𝐽 𝑏𝑎𝑡𝑡 + 𝐽 𝐷𝐺

(5)

1) (RES)s cost function: it consists of the operation and maintenance (O&M) costs of the (RES)s:

𝐽 𝑅𝐸𝑆𝑠 = 𝐽 𝑃𝑉 𝑂&𝑀 + 𝐽 𝑊𝑇 𝑂&𝑀 (6)
where 𝐽 𝑅𝐸𝑆𝑠 , 𝐽 𝑃𝑉 𝑂&𝑀 , and 𝐽 𝑊𝑇 𝑂&𝑀 are respectively, the (RES)s cost function in $, the (O&M) cost of the PV in $, and the (O&M) cost of the WT in $. 𝐽 𝑃𝑉 𝑂&𝑀 and 𝐽 𝑊𝑇 𝑂&𝑀 can be calculated as:

𝐽 𝑃𝑉 𝑂&𝑀 = ∑ 𝛿 𝑃𝑉 𝑂&𝑀 . 𝑃 𝑃𝑉 (𝑘). 𝑇 𝑠 𝑁 𝑘=1 (7)
𝐽 𝑊𝑇 𝑂&𝑀 = ∑ 𝛿 𝑊𝑇 𝑂&𝑀 . 𝑃 𝑊𝑇 (𝑘). 𝑇 𝑠 𝑁 𝑘=1 [START_REF] Hankache | Gestion optimisée de l'énergie électrique d'un groupe électrogène hybride à pile à combustible[END_REF] where 𝑁, 𝛿 𝑃𝑉 𝑂&𝑀 , 𝛿 𝑊𝑇 𝑂&𝑀 are respectively, the time horizon steps, the (O&M) costs per 𝐾𝑊ℎ of the PV, and the WT (𝛿 𝑃𝑉 𝑂&𝑀 = 0.0024 $/𝐾𝑊ℎ, and 𝛿 𝑊𝑇 𝑂&𝑀 = 0.0098 $/𝐾𝑊ℎ).

2) Power grid cost function: it can be expressed as the difference between the purchased energy from the utility grid and the sold energy to the utility grid over the 24 hours' time horizon.

𝐽 𝑔𝑟𝑖𝑑 = ∑ (𝛿 𝑔𝑟𝑖𝑑 𝑝𝑢𝑟 . 𝑃 𝑝𝑢𝑟 (𝑘) -𝛿 𝑔𝑟𝑖𝑑 𝑠𝑜𝑙𝑑 . 𝑃 𝑠𝑜𝑙𝑑 (𝑘)) . 𝑇 𝑠 𝑁 𝑘=1 (9)
where 𝛿 𝑔𝑟𝑖𝑑 𝑝𝑢𝑟 , 𝛿 𝑔𝑟𝑖𝑑 𝑠𝑜𝑙𝑑 , 𝑃 𝑝𝑢𝑟 (𝑘), and 𝑃 𝑠𝑜𝑙𝑑 (𝑘) are respectively the purchased electricity cost per KWh (δ 𝑔𝑟𝑖𝑑 𝑝𝑢𝑟 is the electricity pool price, and is represented in Fig. 4), the sold electricity cost per KWh (δ 𝑔𝑟𝑖𝑑 𝑠𝑜𝑙𝑑 = 0.068 $/KWh), the purchased electricity power at time 𝑘, and the sold electricity power at time 𝑘. 𝑃 𝑝𝑢𝑟 (𝑘) and 𝑃 𝑠𝑜𝑙𝑑 (𝑘) are expressed as functions of the grid power 𝑃 𝑔𝑟𝑖𝑑 (𝑘) at time k as follows:

𝑝𝑢𝑟 (𝑘) = 𝑃 𝑔𝑟𝑖𝑑 (𝑘). max(sign(𝑃 𝑔𝑟𝑖𝑑 (𝑘)),0)

𝑃 𝑠𝑜𝑙𝑑 (𝑘) = 𝑃 𝑔𝑟𝑖𝑑 (𝑘). min(sign(𝑃 𝑔𝑟𝑖𝑑 (𝑘)),0)

By using [START_REF] Suman | Optimisation of Solar/Wind/Bio-generator/Diesel/Battery Based Microgrids for Rural Areas: A PSO-GWO Approach[END_REF] and [START_REF] Tremblay | Experimental validation of a battery dynamic model[END_REF], the purchased and sold electricity at any time are deduced from the utility grid power which, reduces the number of decision variables. By referring to [START_REF] Suman | Optimisation of Solar/Wind/Bio-generator/Diesel/Battery Based Microgrids for Rural Areas: A PSO-GWO Approach[END_REF] and [START_REF] Tremblay | Experimental validation of a battery dynamic model[END_REF], 𝑃 𝑝𝑢𝑟 (𝑘) is equal to 𝑃 𝑔𝑟𝑖𝑑 (𝑘) and 𝑃 𝑠𝑜𝑙𝑑 (𝑘) is null when 𝑃 𝑔𝑟𝑖𝑑 (𝑘) is positive whereas, 𝑃 𝑝𝑢𝑟 (𝑘) is set to zero and 𝑃 𝑠𝑜𝑙𝑑 (𝑘) is equal to -𝑃 𝑔𝑟𝑖𝑑 (𝑘) when 𝑃 𝑔𝑟𝑖𝑑 (𝑘) is negative. Thereby, the grid electricity cannot be purchased and sold at the same time 𝑘.

3) Battery storage cost function: the battery lifetime is represented as the number of charges and discharges cycles. Hence, to quantify the storage operating cost, one way is to divide the battery capital cost per KWh over the number of cycles to obtain the battery operating cost per cycle. In addition, the battery degradation issue is added to the operating cost function as expressed in [START_REF] Garcia-Torres | Optimal Economical Schedule of Hydrogen-Based Microgrids with Hybrid Storage Using Model Predictive Control[END_REF]:

𝐽 𝑏𝑎𝑡𝑡 =
A piecewise approximation of the fuel consumption, proposed in [START_REF] Palma-Behnke | A Microgrid Energy Management System Based on the Rolling Horizon Strategy[END_REF], is applied. The piecewise linearized fuel consumption function is shown in Fig. 2. For more details on the approximation method, it can be referred to [START_REF] Palma-Behnke | A Microgrid Energy Management System Based on the Rolling Horizon Strategy[END_REF]. Thus, the fuel consumption cost can be expressed as:

𝐽 𝐷𝐺 𝑓𝑢𝑒𝑙 = ∑ 𝜆 𝑓𝑢𝑒𝑙 . 𝐹(𝑘). 𝑇 𝑠 𝑁 𝑘=1 ( 13
)
where 𝜆 𝑓𝑢𝑒𝑙 , 𝐹(𝑘) are respectively the price of diesel per liter (𝜆 𝑓𝑢𝑒𝑙 = 1.05 $/L), and the fuel consumption in L/h. Knowing 𝑃 𝐷𝐺 (𝑘), 𝐹(𝑘) is determined based on the plotted curve in Fig. 2. The constant parameters are defined in table 2.

The start-up cost function 𝐽 𝐷𝐺 𝑆𝑈 corresponds to the fuel consumed during the start-up phase before any power production. The start-up cost is the cost per start-up (𝜉 𝑆𝑈 = 0.011 $) times the number of start-ups over the time horizon. This can be calculated as:

𝐽 𝐷𝐺 𝑆𝑈 = 𝜉 𝑆𝑈 . ∑ 𝜎 𝐷𝐺 𝑂𝑁 (𝑘) 𝑁 𝑘=1 (14)
Fig. 2 Piecewise approximation of the (DG) fuel cosumption function where 𝜎 𝐷𝐺 𝑂𝑁 (𝑘) is a binary variable equal to one when the (DG) is turned-on at time 𝑘 and zero otherwise. 𝜎 𝐷𝐺 𝑂𝑁 (𝑘) is defined as: 𝜎 𝐷𝐺 𝑂𝑁 (𝑘) = max (sign(𝑃 𝐷𝐺 (𝑘 + 1)) -sign(𝑃 𝐷𝐺 (𝑘)), 0) (

Finally, the maintenance cost depends on the operation time of the (DG). It can be calculated as the maintenance cost per hour (𝜒 𝑀 = 0.03 $/h) times the total operating hours of the (DG):

𝐽 𝐷𝐺 𝑀 = 𝜒 𝑀 . ∑ sign(𝑃 𝐷𝐺 (𝑘)). 𝑇 𝑠 𝑁 𝑘=1 (16)

A.2. Pollutant gas emissions cost function

Fossil fuel consumption produces toxic gases such 𝑁𝑂 𝑥 , 𝐶𝑂 2 , 𝐶𝑂, and 𝑆𝑂 2 . These gas emissions are the main source of air pollution and greenhouse. The pollution aspect can be considered by introducing the quantity of emitted toxic gases in a (DG) application, (𝛼) expressed in (g/KWh), and the expenses related to environmental damages resulting from the pollutant gas emissions, (𝜇) expressed in ($/Kg) [START_REF] Moradi | Operational Strategy Optimization in an Optimal Sized Smart Microgrid[END_REF]. All parameters' values are listed in table 1. Thereby, the pollutant gas emissions cost function can be represented as:

𝐽 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 =
We take note that 𝑃 𝑃𝑉 (𝑘), 𝑃 𝑊𝑇 (𝑘), and 𝑃 𝐷𝐺 (𝑘), as energy sources, are always positive. Similarly, 𝑃 𝑙𝑜𝑎𝑑 (𝑘) is positive and set on the right side of the equation. However, 𝑃 𝑏𝑎𝑡𝑡 (𝑘), and 𝑃 𝑔𝑟𝑖𝑑 (𝑘) can either have positive or negative values. When positive, 𝑃 𝑏𝑎𝑡𝑡 (𝑘) corresponds to power charged into the battery, and 𝑃 𝑔𝑟𝑖𝑑 (𝑘) to the power sold to the utility grid. However, when they are negative, 𝑃 𝑏𝑎𝑡𝑡 (𝑘) corresponds to the power discharged from the battery, and 𝑃 𝑔𝑟𝑖𝑑 (𝑘) to the power purchased from the utility grid.

B.2. Utility grid constraints

Lower and upper bounds are fixed to limit the purchased/sold from/to the utility grid power. This can be represented as:

𝑃 𝑔𝑟𝑖𝑑_𝑚𝑖𝑛 ≤ 𝑃 𝑔𝑟𝑖𝑑 (𝑘) ≤ 𝑃 𝑔𝑟𝑖𝑑_𝑚𝑎𝑥 (20)
where 𝑃 𝑔𝑟𝑖𝑑_𝑚𝑖𝑛 , and 𝑃 𝑔𝑟𝑖𝑑_𝑚𝑎𝑥 are respectively the maximum allowable power to be sold to the utility grid (𝑃 𝑔𝑟𝑖𝑑_𝑚𝑖𝑛 = -𝑃 𝑚𝑎𝑥 _𝑠𝑜𝑙𝑑 ), and the maximum allowable power to be purchased from the utility grid (𝑃 𝑔𝑟𝑖𝑑_𝑚𝑎𝑥 = 𝑃 𝑚𝑎𝑥 _𝑝𝑢𝑟 ). 𝑃 𝑚𝑎𝑥 _𝑠𝑜𝑙𝑑 and 𝑃 𝑚𝑎𝑥 _𝑝𝑢𝑟 are positive parameters.

B.3. Battery storage constraints

To ensure an optimized battery lifetime and performance, several constraints are set. First, the battery power is restricted is terms of maximum charged/discharged power: 𝑃 𝑏𝑎𝑡𝑡_𝑚𝑖𝑛 ≤ 𝑃 𝑏𝑎𝑡𝑡 (𝑘) ≤ 𝑃 𝑏𝑎𝑡𝑡_𝑚𝑎𝑥 (21) where 𝑃 𝑏𝑎𝑡𝑡_𝑚𝑖𝑛 , and 𝑃 𝑏𝑎𝑡𝑡_𝑚𝑎𝑥 are respectively the maximum allowable power to be charged into the battery (𝑃 𝑏𝑎𝑡𝑡_𝑚𝑖𝑛 = -𝑃 𝑚𝑎𝑥_𝑐ℎ ), and the maximum allowable power to be discharged from the battery (𝑃 𝑏𝑎𝑡𝑡_𝑚𝑎𝑥 = 𝑃 𝑚𝑎𝑥 _𝑑𝑖𝑠𝑐ℎ ). 𝑃 𝑚𝑎𝑥 _𝑐ℎ and 𝑃 𝑚𝑎𝑥 _𝑑𝑖𝑠𝑐ℎ are positive parameters. Second, the battery safe operation is ensured by limiting its state of charge (𝑆𝑂𝐶) within allowable limits:

𝑆𝑂𝐶 𝑚𝑖𝑛 ≤ 𝑆𝑂𝐶(𝑘) ≤ 𝑆𝑂𝐶 𝑚𝑎𝑥 (22) where 𝑆𝑂𝐶 𝑚𝑖𝑛 , and 𝑆𝑂𝐶 𝑚𝑎𝑥 are respectively the minimum, and the maximum allowable (𝑆𝑂𝐶)s. Finally, the battery initial state at time (𝑘 = 1) should be retrieved at the end of the time horizon (𝑘 = 𝑁) to further standardize and optimize the battery performance:

𝑆𝑂𝐶(𝑘 = 1) = 𝑆𝑂𝐶(𝑁)

B.4. (DG) constraints

For proper and efficient operation, (DG) manufacturers define an output power range for generators. Then, when turned on, the (DG) operation is limited by lower and upper bounds (in this paper a range between 25%-100% of the rated power is applied). This can be expressed as: sign(𝑃 𝐷𝐺 (𝑘)). 𝑃 𝐷𝐺_𝑚𝑖𝑛 ≤ 𝑃 𝐷𝐺 (𝑘) ≤ sign(𝑃 𝐷𝐺 (𝑘)). 𝑃 𝐷𝐺_𝑚𝑎𝑥 (24)

Besides, and following the predefined objectives, the (DG), as a backup pollutant source, intercepts to directly feed the load in case of an energy deficit, and/or to charge the battery. Hence, at each time k none of (DG) produced power should be sold to the utility grid. By this, the (DG) main functionality is secured. This can be expressed as:

Fig. 1

 1 Fig.1 DC microgrid configuration

  𝛽 𝑒𝑚𝑖𝑠𝑠 . ∑ 𝑃 𝐷𝐺 (𝑘). 𝑇 𝑠 𝛽 𝑒𝑚𝑖𝑠𝑠 = (𝛼 𝑁𝑂 𝑥 .𝜇 𝑁𝑂 𝑥 +𝛼 𝐶𝑂 2 .𝜇 𝐶𝑂 2 +𝛼 𝐶𝑂 .𝜇 𝐶𝑂 +𝛼 𝑆𝑂 2 .𝜇 𝑆𝑂 2 ). 10 -3

	𝑁
	(17)
	𝑘=1
	(18)
	B. Problem constraints
	Several constraints are set to emulate a realistic and practical
	microgrid scenario. Next, constraints for each microgrid unit
	are detailed apart.
	B.1. Power balance constraint
	At each sampling time 𝑘, the generated power should be equal
	to the demanded one. This is known as the power balance
	equation:
	𝑃 𝑃𝑉 (𝑘) + 𝑃 𝑊𝑇 (𝑘)+𝑃 𝑏𝑎𝑡𝑡 (𝑘)+ 𝑃 𝑔𝑟𝑖𝑑 (𝑘)+ 𝑃 𝐷𝐺 (𝑘) = 𝑃 𝑙𝑜𝑎𝑑 (𝑘)

Table 1 .

 1 Toxic gas emissions in (g/KWh) in a (DG) application, and related environmental expenses in ($/Kg)

			Toxic gas emissions
	Emission	𝛼 𝑁𝑂 𝑥	𝛼 𝐶𝑂 2	𝛼 𝐶𝑂	𝛼 𝑆𝑂 2
	levels (g/KWh)	4.331	232.037	2.32	0.464
	Cost	𝜇 𝑁𝑂 𝑥	𝜇 𝐶𝑂 2	𝜇 𝐶𝑂	𝜇 𝑆𝑂 2
	($/Kg)	0.27	0.0012	0.022	0.12

𝑃 𝐷𝐺 (𝑘) ≤ sign(𝑃 𝐷𝐺 (𝑘)). (𝑃 𝑙𝑜𝑎𝑑 (𝑘) -𝑃 𝑃𝑉 (𝑘) -𝑃 𝑊𝑇 (𝑘) + 𝑃 𝑏𝑎𝑡𝑡 (𝑘))

Finally, Fig. 3 shows the block diagram of the (DP) adopted algorithm. First, the input data are sampled at each period 𝑇 𝑠 , and all constraints are evaluated. Second, the selected decision variables (𝑃 𝐷𝐺 and 𝑆𝑂𝐶) are meshed correspondingly to the defined sampling steps (𝛥𝑆𝑂𝐶 = 0.5%, and 𝛥𝑃 𝐷𝐺 = 55 𝑊).

It's worth mentioning that the selection of the sampling steps of each variable is instrumental, and thoroughly impacts the algorithm convergence, the simulation time, and results' accuracy. Thereby, several trials were made to define the best trade-off between the simulation time and results' precision. Samples that don't satisfy the constraints are rejected, and the remaining ones are used to build the cost and the sequence matrices. At the last stage, the optimal trajectory matrix is established next, the optimal variables, over the whole-time horizon, and the optimal total cost are deduced. For further information on (DP) algorithm, it can be referred to [START_REF] Rigo-Mariani | Méthodes de conception intégrée 'dimensionnement-gestion' par optimisation d'un micro-réseau avec stockage[END_REF][START_REF] Hankache | Gestion optimisée de l'énergie électrique d'un groupe électrogène hybride à pile à combustible[END_REF].

IV.

SIMULATION TESTS AND RESULTS

In this section, simulation tests are performed to validate the results of the proposed optimization problem. Therefore, a real case scenario is applied with precise weather conditions, residential load demand, and electricity pool price. The solar power, the wind power, the load demand, and the pool price are depicted in Fig. 4. To prove the viability of the applied (DP), two simulation tests are conducted.

Simulation test 1: in this simulation, the minimization of the total operating cost is selected, as a primary goal, over the reduction of pollutant gas emissions. This can be reached by setting a high weight for the operating cost function (𝛾 𝑜𝑐 = 0.9 and 𝛾 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 = 0.1). Simulation test 2: in this simulation, the minimization of the pollutant gas emissions is set as a preferred goal. Hence, a higher weight is attributed to the gas emissions cost function (𝛾 𝑜𝑐 = 0.02 and 𝛾 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 = 0.98). As seen, a high ratio between the two weights is set (𝛾 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑠 /𝛾 𝑜𝑐 =49) to prioritize 3. It can be seen by Fig. 5 that both SOCs trajectories are within the admissible limits, and all constraints are respected. Fig. 6.a and b show that the deficit in power demand is covered by the utility grid at low pool price hours (0:00 till 6:00, and 22:00 till 24:00), whereas it is taken over by the battery, and the (DG) at high pool price hours. On the other hand, limiting the pollutant gas emissions weight, in simulation test 1, slightly impacts the total cost function. This can be verified through the excessive pollutant gas emissions in test 1 (49.6 Kg), and high operating hours of the (DG) (3.83 hours). Oppositely, by increasing this weight in simulation test 2, the total pollutant gas emissions are greatly alleviated (25.9 Kg), and so is the usage of (DG) (2 hours). The reduction in (DG) produced power is compensated by the utility grid which operating cost increases in simulation test 2 to 95.2$ compared to 52.33 $ in simulation test 1. Finally, by favoring the pollutant gas emissions goal over the operating cost, the total cost of the system is inconsiderably increased (a difference of 2.12$ between two simulation tests). Hence, the applied (DP) doesn't compromise the objectives' achievement. In conclusion, the (DP) algorithm has a high convergence capability in multi-objective optimization problems and ensures the best tradeoff in achieving different predefined objectives.

V. CONCLUSION

In this paper, a multi-objective optimization problem is formulated and solved using the (DP) algorithm. The minimization of the total operating cost of the DC microgrid, and the reduction of the pollutant gas emissions are set as two independent objectives. To achieve these goals, a sole objective function is established, and weights are assigned to each of the predefined objectives. Several constraints, on operating units, are introduced to emulate a realistic microgrid model and scenario. To prove the viability of the (DP) algorithm in solving such optimization problem, and the effectiveness in achieving multi-objectives, two simulation tests are conducted in which different weights are attributed to each of the fixed goals. Results show that (DP) converges in both simulations and find a feasible solution with respect to all defined constraints. Moreover, it offers to the user the option of goal's preference, through weights' selection, without compromising the optimality, and feasibility of the solution.