François Pacaud

Pierre Carpentier

Jean-Philippe Chancelier

Michel De Lara

Euro e) p d t

Keywords:

In this study, a microgrid with storage (battery, hot water tank) and solar panel is considered. We benchmark two algorithms, MPC and SDDP, that yield online policies to manage the microgrid, and compare them with a rule based policy. Model Predictive Control (MPC) is a well-known algorithm which models the future uncertainties with a deterministic forecast. By contrast, Stochastic Dual Dynamic Programming (SDDP) models the future uncertainties as stagewise independent random variables with known probability distributions. We present a scheme, based on out-of-sample validation, to fairly compare the two online policies yielded by MPC and SDDP. Our numerical studies put to light that MPC and SDDP achieve significant gains compared to the rule based policy, and that SDDP overperforms MPC not only on average but on most of the out-of-sample assessment scenarios.

Optimization of a domestic microgrid equipped with solar panel and battery: Model Predictive Control and Stochastic Dual Dynamic Programming approaches 1 Introduction

Background introduction

A microgrid is a local energy network that produces part of its energy and controls its own demand. Such systems are complex to control because, on the one hand, of the different stocks and interconnections, and, on the other hand, of electrical demands and weather conditions (heat demand and renewable energy production) that are highly variable and hard to predict at local scale (see [START_REF] Olivares | Trends in microgrid control[END_REF][START_REF] Morstyn | Control strategies for microgrids with distributed energy storage systems: An overview[END_REF] for a panorama of the challenges faced when controlling microgrids).

Thermal comfort (virtual price) A p Surface of solar panel (m We consider here a domestic microgrid equipped with a battery, an electrical hot water tank and a solar panel, as in Figure 1. The microgrid is connected to an external grid to import electricity when needed. The battery stores energy when external grid prices are low or when the production of the solar panel is above the electrical demand. The house's envelope also plays the role of heat storage. As a consequence, the system has four stocks to store energy: a battery, a hot water tank, and two passive stocks being the house's walls and inner rooms. Two kinds of uncertainties affect the system: the electrical and domestic hot water demands are not known in advance; the production of the solar panel is substantially perturbed by the variable weather nebulosity.

We aim to compare two classes of algorithms to tackle uncertainties in a microgrid Energy Management System (EMS). The Model Predictive Control (MPC) algorithm (or its stochastic variant, Stochastic Model Predictive Control) relies on a mathematical representation of the future uncertainties as deterministic forecasts; then, MPC computes decisions online as solutions of a deterministic multistage optimization problem. Stochastic Dual Dynamic Programming (SDDP) relies on a mathematical representation of the future uncertainties as stagewise independent random variables with known probability distributions; then, SDDP computes offline a set of value functions by backward induction, and computes online decisions as solutions of a single stage stochastic optimization problem, using the value functions. We present a fair comparison of these two algorithms, and highlight the pros and cons of both methods.

Literature review

Optimization and energy management systems EMS are integrated automated tools used to monitor and control energy systems. In [START_REF] Olivares | Trends in microgrid control[END_REF], the authors give an overview of the use of optimization methods in the design of EMS. The MPC algorithm [START_REF] Garcia | Model predictive control: theory and practice-a survey[END_REF] and its stochastic variant, Stochastic Model Predictive Control (SMPC) [START_REF] Mesbah | Stochastic model predictive control: An overview and perspectives for future research[END_REF], have been widely used to control EMS. We refer the reader to [START_REF] Širokỳ | Experimental analysis of model predictive control for an energy efficient building heating system[END_REF] for applications of MPC in buildings. In [START_REF] Holjevac | Adaptive control for evaluation of flexibility benefits in microgrid systems[END_REF], the authors use MPC for the optimal control of a domestic microgrid, and investigate how to balance the uncertainties of renewable energies with the microgrid. In [START_REF] Olivares | Stochastic-predictive energy management system for isolated microgrids[END_REF], the authors apply SMPC to the management of an isolated microgrid, and highlight the benefit of this method; an application of SMPC in buildings is presented in [START_REF] Appino | On the use of probabilistic forecasts in scheduling of renewable energy sources coupled to storages[END_REF]; a variant based on robust optimization is proposed in [START_REF] Paridari | Robust scheduling of smart appliances in active apartments with user behavior uncertainty[END_REF].

Stochastic optimization

At local scale, electrical demand and production are highly variable, especially as microgrids are expected to absorb renewable energies. This leads to pay attention to stochastic optimization approaches [START_REF] Lara | Optimization Methods for the Smart Grid[END_REF]. Stochastic optimization has been widely applied to hydrovalleys management [START_REF] Pereira | Multi-stage stochastic optimization applied to energy planning[END_REF]. Other applications have arisen recently, such as integration of wind energy and storage [START_REF] Haessig | Energy storage control with aging limitation[END_REF] or insulated microgrids management [START_REF] Heymann | A stochastic continuous time model for microgrid energy management[END_REF][START_REF] Alasseur | Regression Monte Carlo for microgrid management[END_REF].

Stochastic Dynamic Programming (SDP) [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF] is a general method to solve stochastic optimal control problems. In energy applications, a variant of SDP, Stochastic Dual Dynamic Programming (SDDP), has demonstrated its adequacy for large scale convex applications. SDDP was first described in the seminal paper [START_REF] Pereira | Multi-stage stochastic optimization applied to energy planning[END_REF]; we refer to [START_REF] Shapiro | Analysis of stochastic dual dynamic programming method[END_REF] for a generic description of the algorithm and its application to the management of hydrovalleys; a proof of convergence in the linear case is given in [START_REF] Philpott | On the convergence of stochastic dual dynamic programming and related methods[END_REF], and in the convex case in [START_REF] Girardeau | On the convergence of decomposition methods for multistage stochastic convex programs[END_REF]. Recent articles have applied SDDP to the management of energy systems. In [START_REF] Bhattacharya | Managing energy storage in microgrids: A multistage stochastic programming approach[END_REF], numerical experiments show that SDDP yields better results than a myopic policy. In [START_REF] Papavasiliou | Application of stochastic dual dynamic programming to the real-time dispatch of storage under renewable supply uncertainty[END_REF], SDDP is applied to the dispatch of energy inside the German national grid, under time correlated uncertainties; the authors observe that SDDP achieves better performances than those of a deterministic-based policy. Other Approximate Dynamic Programming algorithms have been designed to tackle different stochastic optimal control problems like, for instance, incorporating probability constraints [START_REF] Balata | Statistical learning for probability-constrained stochastic optimal control[END_REF].

Main contributions and structure of the paper

We provide a rigorous mathematical formulation of the optimal management of a domestic microgrid -equipped with a battery, an electrical hot water tank and a solar panel, and connected to an external supply network -under stochasticity of demand and of renewable energy production. To manage the microgrid, we design online policies using two different algorithms, MPC and SDDP. Then, we develop a fair benchmark methodology to compare the two algorithms, based on a realistic use case. The comparison reveals that SDDP overperforms MPC not only on average but, interestingly, on most of the out-of-sample assessment scenarios.

The paper is organized as follows. In Sect. 2, we detail the modeling of a small residential microgrid and formulate a mathematical multistage stochastic optimization problem. Then, we outline the two algorithms, MPC and SDDP, in Sect. 3. Finally, in Sect. 4 we detail the benchmark methodology and we provide numerical results on the systematic comparison of MPC, SDDP and a rule based policy. Sect. 5 concludes and the Appendix 6 provides details on the physical equations of the microgrid.

Optimization problem statement

We consider the optimal management of a microgrid which, as depicted in Figure 1, consists of a single house equipped with a battery and an electrical hot water tank. An electrical heater can produce heat in winter, and a solar panel can produce energy locally. The decision maker aims at minimizing the energy bill -that is, the cost of the possible recourse energy supplied by the external network -while satisfying the energy demands (hot water and electricity), and ensuring a minimal thermal comfort.

In this section, we write up a multistage stochastic optimization problem. As decisions are taken at discrete time steps, we start by discretizing the time interval in §2.1. Then, we introduce the uncertainties in §2.2, the controls and the stocks in §2.3 and §2.4. We detail the nonanticipativity constraints in §2.5, the bounds constraints in §2.6 and the objective function in §2.7. Finally, we formulate a multistage stochastic optimization problem in §2.8.

f h f b d el f ne d hw f t φ pv

Decisions are taken at discrete time steps

The EMS takes decisions every 15 minutes to control the system. We consider a time interval ∆ = 15mn, a time horizon T 0 =24h and a number T =T 0 /∆=96 of time steps. We adopt the following convention for discrete time processes: for each time step t ∈ {0, 1, • • • , T }, x t denotes the value of the variable x at the beginning of the interval [t∆, (t + 1)∆). Otherwise stated, we will denote by [t, t + 1) the continuous time interval [t∆, (t + 1)∆). 1

Modeling uncertainties as random variables

Because of their unpredictable nature, the EMS cannot anticipate the values of electrical and thermal demands, nor the production of the solar panel. We choose to model these quantities as random variables over a probability space (Ω, F, P). We adopt the following convention: a random variable will be denoted by an uppercase bold letter Z , and its realization for a given outcome ω ∈ Ω will be denoted in lowercase z = Z (ω). We denote by D el t the electrical demand, D th t the hot water demand and Φ pv t the production of the solar panel, all being real-valued random variables. For each time step t = 1, . . . , T , we define the (uncertainty) 1 Here, by using the notation [t, t + 1), we mean the time interval between t and t + 1, excluding t + 1. Indeed, we denote a time interval between two decisions by [t, t + 1), and not by [t, t+1], to indicate that a decision is taken at the beginning of the time interval [t, t + 1), and that a new one will be taken at the beginning of the time interval [t+1, t+2), and that these two consecutive intervals do not overlap. random variable

W t = (D el t , D th t , Φ pv t ) . (1)
The uncertainty W t is a multivariate random variable taking values in W t = R 3 .

Modeling controls as random variables

As decisions depend on the previous uncertainties, controls are random variables. At the beginning of the time interval [t, t + 1), the EMS takes three decisions:

• F b t , how much energy to charge in/discharge from the battery, • F w t , how much energy to store in the electrical hot water tank, • F h t , how much energy to inject in the electrical heater, during the time interval [t, t + 1). For each time step t = 1, . . . , T , we define the decision multivariate random variable, taking values in U t = R 3 , as

U t = (F b t , F w t , F h t ) . (2)
During the time interval [t, t+1), the EMS imports an energy quantity F ne t+1 from the external network in order to fulfill the load balance equation

F ne t+1 + Φ pv t+1 = F b t + F w t + F h t + D el t+1 , (3)
whatever the demand D el t+1 and the production of the solar panel Φ pv t+1 , unknown at the beginning of the time interval [t, t + 1). On the left-hand side of Equation ( 3), the load consists of • F ne t+1 , the energy surplus or shortage (when F ne t+1 < 0, one wastes the surplus; when F ne t+1 > 0, one imports the shortage from the network), • Φ pv t+1 , the production of the solar panel, all during the time interval [t, t + 1). On the right-hand side of Equation ( 3), the electrical demand is the sum of

• F b
t , the energy exchanged with the battery, • F w t , the energy injected into the electrical hot water tank, • F h t , the energy injected in the electrical heater, • D el t+1 , the electrical demands (lightning, cooking...), aggregated in a single demand, all during the time interval [t, t + 1).

Later, we will aggregate the solar panel production Φ pv t+1 with the demands D el t+1 in Equation ( 3), as these two quantities appear only by their difference.

States and dynamics

The state is the multivariate random variable

X t = (B t , H t , θ w t , θ i t ) , (4)
which consists of the stocks B t in the battery and H t in the electrical hot water tank, plus the two temperatures (θ w t , θ i t ) of the thermal envelope. Thus, the state random variable X t takes values in X t = R 4 .

The discrete time dynamics f t : X t × U t × W t+1 → X t+1 describes the time evolution

X t+1 = f t X t , U t , W t+1 (5)
of the state, where f t is a piecewise linear function (a property that will prove important for the SDDP algorithm), that corresponds to the integration of the continuous dynamics ( 21)-( 22)-( 24) given in Appendix 6. We suppose that we start from a given state x 0 ∈ X 0 = R 4 , thus adding the constraint X 0 = x 0 .

Nonanticipativity constraints

The future realizations of uncertainties are unknown. Thus, decisions at time step t are functions of previous history only, that is, the information collected between step 0 and step t. Such a constraint is encoded as an algebraic constraint, using the tools of Probability theory [START_REF] Kallenberg | Foundations of Modern Probability[END_REF], in the so-called nonanticipativity constraints written as

σ(U t ) ⊂ F t , (6a)
where σ(U t ) is the σ-algebra generated by the random variable U t and

F t = σ(W 1 , • • • , W t )
is the σ-algebra generated by the previous uncertainties (W 1 , . . . , W t ). If Constraint (6a) holds true, the Doob Lemma [START_REF] Kallenberg | Foundations of Modern Probability[END_REF] ensures that there exists a measurable function

π t : W 1 × • • • × W t → U t such that U t = π t (W 1 , . . . , W t ) . (6b)
This is how we turn an (abstract) algebraic constraint into a more practical functional constraint. The function π t is called a policy.

Bounds constraints

The stocks B t in the battery and H t in the tank are bounded:

b ≤ B t ≤ b , 0 ≤ H t ≤ h . (7)
At time step t, the control F b t must ensure that the next state B t+1 is admissible, that is, satisfies b ≤ B t+1 ≤ b, which, by time discretization of [START_REF] Balata | Statistical learning for probability-constrained stochastic optimal control[END_REF], is equivalent to the two inequalities

2 b ≤ B t + ∆ ρ c (F b t ) + + 1 ρ d (F b t ) -≤ b , (8)
with ρ c and ρ d being the charge and discharge efficiencies of the battery. Thus, the constraints on F b t depends on the stock B t . The same reasoning applies for the tank energy F w t and the stock H t . Furthermore, we set bound constraints on controls:

-f b ≤ F b t ≤ f b , 0 ≤ F w t ≤ f w t , 0 ≤ F h t ≤ f h t . (9)
Finally, the load-balance equation ( 3) also acts as a constraint on the controls. We gather all these constraints into an admissible subset U ad t (X t ) of U t = R3 , depending on the current state X t , giving the constraint 3

U t ∈ U ad t (X t ) . ( 10
)
Note that we do not enforce any explicit bounds on the inner temperature. Instead, we choose to add a penalization term in the objective function if the temperature is below a given threshold, as explained below.

Objective function

At time step t, the instantaneous cost L t :

X t × U t × W t+1 → R aggregates two different costs as in the formula L t (x t , u t , w t+1 ) = p e t × max{0, f ne t+1 } + p d t × max{0, θ i t -θ i t } , (11)
where f ne t+1 is a function of (u t , w t+1 ) by ( 3), and θ i t is part of the state x t in (4). First, one pays a unitary price p e t to import electricity from the external network between step t and t + 1; hence, the electricity cost is equal to p e t × max{0, f ne t+1 }. Second, if the indoor temperature is below a given threshold, we penalize the induced discomfort with a cost p d t × max{0, θ i t -θ i t }, where p d t is a virtual price of discomfort (we choose not to penalize the temperature in case it is above a given threshold, as we do not consider any air conditioning in this study). The cost L t is a convex piecewise linear function, a property that will prove important for the SDDP algorithm. To ensure that stocks are not empty at the final time step T , we add a convex piecewise linear final cost K : X T → R of the form

K(x T ) = κ × max{0, x 0 -x T } , ( 12
)
where κ is a positive penalization coefficient (calibrated by trial and error).

As decisions U t and states X t are random, the costs L t (X t , U t , W t+1 ) and K(X T ) become also random variables. We choose to minimize the expected value of the daily operational cost, that is,

E T -1 t=0 L t (X t , U t , W t+1 ) + K(X T ) , (13)
yielding the expected value of a convex piecewise linear cost.

Stochastic optimal control formulation

Finally, the EMS problem is written as a stochastic optimal control problem min

X,U E T -1 t=0 L t (X t , U t , W t+1 ) + K(X T ) , (14a)
s.t. X t+1 = f t X t , U t , W t+1 , X 0 = x 0 , t = 0, . . . , T -1 , (14b)
U t ∈ U ad t (X t ) , t = 0, . . . , T -1 , (14c) σ(U t ) ⊂ F t , t = 0, . . . , T -1 . (14d)
Problem [START_REF] Alasseur | Regression Monte Carlo for microgrid management[END_REF] expresses that the microgrid manager aims to minimize the expected value of the costs while satisfying the dynamics, the control bounds and the nonanticipativity constraints.

Resolution methods

The exact resolution of Problem ( 14) is out of reach in general. We propose two different algorithms that provide policies4 π t :

W 1 × • • • × W t → U t that
map available information w 1 , . . . , w t at step t to a decision u t . In §3.1, we start by presenting how to design management policies with the MPC algorithm. Then, we depict SDDP-based policies in §3.2. Both methods use the dynamics f t in (5), the constraints sets U ad t (•) in [START_REF] Lara | Optimization Methods for the Smart Grid[END_REF], and the cost functions L t in [START_REF] Pereira | Multi-stage stochastic optimization applied to energy planning[END_REF] and K in (12).

Model Predictive Control (MPC)

Be it ordinary or stochastic, MPC is a classical algorithm that is commonly used to solve stochastic optimization problems. Regarding MPC, we follow [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF]. At time step t, we consider a deterministic forecast (w t+1 , . . . , w T ) of the future uncertainties (W t+1 , . . . , W T ) (see §6.4 for more details) and we solve the following deterministic problem, where the state x t is given:

min (ut,••• ,u T -1 ) T -1 j=t L j (x j , u j , w j+1 ) + K(x T ) , (15a)
s.t. x j+1 = f j x j , u j , w j+1 , j = t, . . . , T -1 , (15b)
u j ∈ U ad j (x j ) , j = t, . . . , T -1 . (15c)
Then, we retrieve the optimal decisions (u t , . . . , u T -1 ) and only keep the first decision u t to control the system at the beginning of the time interval [t, t + 1). This procedure is then restarted at step t + 1. Thus, MPC solves an optimization problem at each time step, with a time span going from the current time step t to the final time step T . Then, at the next time t + 1, the optimizer updates the scenario (w t+1 , . . . , w T ) to take into account the latest observation made at t + 1.

Stochastic Dual Dynamic Programming (SDDP)

As said in §1.2, SDDP is an algorithm widely used to optimize energy systems. The SDDP algorithm provides, at each time step, a value function and an online control policy. Whereas the value functions are computed offline (hence with offline data), online control policies are computed taking into account an online probability distribution on the next period noise.

Dynamic Programming and Bellman principle

The Dynamic Programming method [START_REF] Bellman | Dynamic Programming[END_REF] provides solutions of Problem ( 14) as state feedbacks π t : X t → U t (these feedbacks are optimal when the noise process is made of independent random variables). Dynamic Programming makes use of a sequence of value functions, obtained offline by setting V T (x T ) = K(x T ) and by solving backward in time the recursive functional equations

V t (x t ) = min u∈U ad t (xt) W t+1 L t (x t , u, w t+1 )+ V t+1 f t (x t , u, w t+1 ) µ of t+1 (dw t+1 ) , (16)
where µ of t+1 is a (offline) probability distribution on W t+1 . Once these functions obtained, we compute a decision at time step t as a state feedback π t (x t ) ∈ arg min

u∈U ad t (xt) W t+1 L t (x t , u, w t+1 )+ V t+1 f t (x t , u, w t+1 ) µ on t+1 (dw t+1 ) , (17)
where µ on t+1 is an online probability distribution on W t+1 . This method proves to be optimal when the random variables W 1 , . . . , W T are stagewise independent and when µ on t = µ of t is the probability distribution of W t .

Description of Stochastic Dual Dynamic Programming

Dynamic Programming suffers from the well-known curse of dimensionality [START_REF] Bellman | Dynamic Programming[END_REF]: its numerical resolution fails for state dimension typically greater than 4 when value functions are computed on a numerical grid. When uncertainties are stagewise independent random variables, costs L t and K are convex and dynamics f t are linear, the SDDP algorithm provides a solution to Problem [START_REF] Alasseur | Regression Monte Carlo for microgrid management[END_REF] where the value functions are represented by convex polyhedral functions [START_REF] Girardeau | On the convergence of decomposition methods for multistage stochastic convex programs[END_REF]. Most achievements of SDDP have been obtained in the linear case -when all the costs are linear -as each subproblem ( 16) can be solved with the simplex algorithm. However, the extension to the generic convex case has not been studied extensively.

SDDP provides an outer approximation of the value function V t in [START_REF] Shapiro | Analysis of stochastic dual dynamic programming method[END_REF]. Provided a bundle of k supporting hyperplanes {(λ j t , β j t )} j=1,••• ,k (with, for each j = 1, • • • , k, λ j t ∈ R 4 and β j t ∈ R), the corresponding outer approximation V k t is given by

V k t (x t ) = min t∈R t , (18a)
s.t. λ j t , x t + β j t ≤ t , ∀j = 1, • • • , k . (18b)
SDDP takes as input an initial value function V 0 t+1 (usually -∞) and an initial state x 0 . In the algorithm, we assume that the probability distributions have finite support {w 1 t+1 , . . . , w S t+1 }. The offline distribution µ of t+1 is then written in the form µ of t+1 = S s=1 p s t+1 δ w s t+1 , where δ w s t+1 is the Dirac measure at w s t+1 and p 1 t+1 , . . . , p S t+1 are probability weights. Then, each iteration k of SDDP encompasses two passes.

• During the forward pass, we draw a scenario w k 1 , . . . , w k T of uncertainties, and obtain a state trajectory x k t t=0•••T along this scenario as follows. Starting from initial state x 0 , we compute x k t+1 from x k t in an iterative fashion: i) we obtain a control u k t at time step t, using the available V k t+1 function, by

u k t ∈ arg min u∈U ad t (xt) S i=1 p i t+1 L t (x k t , u, w i t+1 ) + V k t+1 f t (x k t , u, w i t+1 ) , (19a)
and ii), we set x k t+1 = f t (x k t , u k t , w k t+1 ) where f t is the piecewise linear dynamics in (5). • During the backward pass, we update the approximated value functions V k t t=0,••• ,T backward in time along the trajectory x k t t=0,••• ,T . At time step t, we solve the problem

k+1 t = min u∈U ad t (xt) S i=1 p i t+1 L t (x k t , u, w i t+1 ) + V k+1 t+1 f t (x k t , u, w i t+1 ) , (19b)
and we obtain a new cut (λ k+1 t , β k+1 t ) where λ k+1 t is a subgradient of the optimal cost function (19b) evaluated at the point We use the stopping criterion introduced in [START_REF] Shapiro | Analysis of stochastic dual dynamic programming method[END_REF] to stop SDDP once the gap between the upper and lower bounds is lower than 0.1 %. In practice, the subproblems (19a) and (19b) are linear, and can be solved efficiently with any linear programming solver. At iteration k, the lower bound is given directly by V 0 (x 0 ). The upper bound is computed by running N sim forward passes of SDDP on a fixed set of scenarios w 1 , • • • , w N sim ; the costs associated with each scenario are then averaged to get the upper bound (the method is thus akin to a Monte Carlo simulation). We obtain a sequence {V t } t=0,••• ,T of functions, that are lower approximations of the original Bellman functions.

x t = x k t , λ k+1 t ∈ S i=1 p i t+1 ∂ x L t (x k t , u k+1 t , w i t+1 ) + ∂ x V k+1 t+1 f t (x k t ,

Obtaining online controls with SDDP

We obtain an online policy by means of the following procedure:

• approximated value functions V t are computed offline with the SDDP algorithm (see §3.2),

• the approximated value functions V t are then used to compute online a decision at any time step t for any state x t as follows.

We compute the SDDP policy π sddp t by

π sddp t (x t ) ∈ arg min u∈U ad t (xt) S i=1 p i t+1 L t (x t , u, w t+1 ) + V t+1 f t (x t , u, w t+1 ) , (20)
which corresponds to replacing the value function V t+1 in Equation ( 17) with its approximation V t+1 . The decision π sddp t (x t ) is used to control the system between steps t and t + 1. Then, we solve Problem (20) at step t + 1.

Discussion

In this section, we have introduced two methods to design management policies, the first one based on MPC, the second on SDDP. Both methods differ on how they model the future uncertainties. In SDDP, one represents the future as a sequence of independent random variables, whereas in MPC it is with a deterministic forecast. It remains now to compare the policy π sdpp with the policy π mpc .

In the next Sect. 4, we compare the performances of π mpc and π sddp on a set of assessment scenarios, using a simulator. We depict in Figure 2 the flow chart of the simulation procedure. The devised policies are used to compute a decision u t at each time step t, using the information already available at step t. Then, by comparing the total costs obtained and repeating the procedure on a bundle of scenarios, we are able to draw conclusions about the respective performances of each policy.

Numerical results

In §4.1, we describe a case study. In §4.2, we develop a protocol to fairly compare the MPC and SDDP algorithms altogether with a rule based policy, then discuss the results

Assessment scenarios Simulation

x t+1 = f t (x t , u t , w t+1 ) X t X t+1 Policy u t = π t (x t ) • Objective • Admissible constraints • Uncertainty model at time t u t x t x t x t+1 w t+1

Case study

Settings We aim to solve the stochastic optimization problem ( 14) over one day, with 96 time steps. The battery size is 3 kWh, and the hot water tank has a capacity of 120 l. We suppose that the house has a surface A p = 20 m 2 of solar panel at disposal, oriented south, and with a yield of 15%. We penalize the recourse variable F ne t+1 in [START_REF] Pereira | Multi-stage stochastic optimization applied to energy planning[END_REF] with on-peak and off-peak tariff, corresponding to Électricité de France's (EDF) individual tariffs. The house's thermal envelope corresponds to the French RT2012 specifications [START_REF][END_REF]. Meteorological data comes from Météo France measurements corresponding to the year 2015.

Implementing the algorithms We implement MPC and SDDP in Julia 0.6, using JuMP [START_REF] Dunning | JuMP: A modeling language for mathematical optimization[END_REF] as a modeler, StochDynamicProgramming.jl as a SDDP solver, and Gurobi 7.02 [START_REF] Optimization | Gurobi Optimizer Reference Manual[END_REF] as a LP solver. All computations run on a Core i7 2.5 GHz processor, with 16GB RAM.

Rule based method

We choose to compare the MPC and SDDP algorithms with the following basic decision rule: the battery is charged whenever the solar production Φ pv is available, and discharged to fulfill the demand if there remains enough energy in the battery; the tank is charged (F w t > 0) if the tank energy H t is lower than H 0 , the heater F h t is switched on when the temperature is below the setpoint θ i t , and switched off whenever the temperature is above the setpoint plus a given margin.

Benchmark

Demand scenarios Scenarios of electrical and domestic hot water demands, at time steps evenly spaced every ∆ = 15 minutes, are generated with the software StRoBe [START_REF] Baetens | Modelling uncertainty in district energy simulations by stochastic residential occupant behaviour[END_REF]. In Figure 3, 100 scenarios of electrical and hot water demands are displayed. We observe almost null demand during the night, and demand peaks around midday and 8 pm; peaks in hot water demand correspond to showers. We aggregate the electrical demand D el minus the production Φ pv of the solar panel in a single variable D el , so that we consider only two uncertainties W t = (D el t , D th t ).

Building offline probability distributions for SDDP

We use the optimization scenarios to build marginal probability distributions µ of t that will feed the SDDP procedure in (19a)-(19b). In order to obtain a discrete probability distribution at each time t, we use a Lloyd-Max quantization scheme [START_REF] Lloyd | Least squares quantization in PCM[END_REF] to compute S representative points from the N opt optimization scenarios.

Out of sample assessment of policies

To obtain a fair comparison between SDDP and MPC, we use an out-of-sample validation. We generate 2,000 scenarios of electrical and hot water demands, and we split these scenarios in two distinct parts: the first N opt = 1, 000 scenarios are called optimization scenarios, and the remaining N sim = 1, 000 scenarios are called assessment scenarios.

First, during the offline phase, we use the optimization scenarios to build models for the uncertainties, under the mathematical form required by each algorithm (see Sect. 3). Second, during the online phase, we use the assessment scenarios to compare the policies produced by these algorithms. At time step t during the assessment, the algorithms cannot use the future values of the assessment scenarios, but can take advantage of the observed values up to t to update their statistical models of future uncertainties. Our method differs from [START_REF] Bhattacharya | Managing energy storage in microgrids: A multistage stochastic programming approach[END_REF][START_REF] Papavasiliou | Application of stochastic dual dynamic programming to the real-time dispatch of storage under renewable supply uncertainty[END_REF] where scenarios serve to fit a probability distribution which is used, on the one hand, to design the SDDP algorithm and, on the other hand, to generate new scenarios to assess the performances of SDDP.

MPC procedure

Electrical and thermal demands are naturally correlated in time [START_REF] Widén | A high-resolution stochastic model of domestic activity patterns and electricity demand[END_REF]. To take into account such a dependence across the different time steps, we choose to model the pro-cess W 1 , . . . , W T with an auto-regressive (AR) model and use the deterministic trend to yield the forecast in [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF]. We detail the overall procedure in §6.4.

SDDP procedure

Computing value functions offline We fit a sequence of finite probability distributions {µ of t } t=0,••• ,T using only the optimization scenarios. Then we compute a set of value functions with the procedure described in §3.2.

Using value functions online Once the value functions have been obtained by SDDP, we compute online decisions at step t with Equation ( 20), using a finite online probability distribution µ on t , fitted with both the optimization scenarios and the past of the current assessment scenario (respecting thus the nonanticipativity constraint).

Assessing on different meteorological conditions

We assess the algorithms on three different days, with different meteorological conditions (see Table 2). Therefore, we use three distinct sets of N opt + N sim scenarios for demands (with N opt the number of optimization scenarios and N sim the number of assessment scenarios), one for each typical day.

Date

Temp. ( These three different days correspond to different heating needs. During Winter day, the heating is maximal, whereas it is medium during Spring day and null during Summer day. The production of the solar panel varies accordingly.

Comparing the algorithms performances

During assessment, we use MPC (see [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF]) and SDDP (see [START_REF] Papavasiliou | Application of stochastic dual dynamic programming to the real-time dispatch of storage under renewable supply uncertainty[END_REF]) policies to compute online decisions along N sim assessment scenarios. Then, we compare the average electricity bill obtained with these two policies and with the rule based policy. The assessment results are given in Table 3, where means and standard deviation σ are computed with the N sim = 1, 000 scenarios; the notation ± corresponds to the 95% confidence interval ±1.96 σ √ N sim .

Regarding mean electricity bills, we observe, on the three last rows in Table 3, that MPC and SDDP yield much better results than the rule based policy. On Winter day, SDDP is slightly better than MPC; on Spring day, they are equivalent; on Summer day, SDDP is able to almost halve the mean costs yielded by MPC (in both case, these costs are low). 4 displays the histogram of the difference between SDDP and MPC total costs during Summer day. In this way, for each scenario (and not only in the mean), we can measure the discrepancy between SDDP and MPC performances. We observe that SDDP is better than MPC for about 93% of the scenarios. The distribution in Figure 4 exhibits a heavy tail that reveals the superiority of SDDP on extreme scenarios. Thus, not only SDDP achieves better performance than MPC in the mean, but also for the vast majority of scenarios. Similar analyses hold for Winter and Spring days.

Comparing the algorithms running times

In terms of numerical performance, it takes less than one minute to compute approximated Bellman functions V t as in §3.2 with SDDP on a particular day. Then, the online computation of a single decision takes 1.5 ms on average, compared to 0.5 ms for MPC. Indeed, MPC is favored by the linearity of the optimization Problem [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF], whereas, for SDDP, the higher the quantization size S of the online probability distribution µ on t , the slower is the online resolution of Problem [START_REF] Papavasiliou | Application of stochastic dual dynamic programming to the real-time dispatch of storage under renewable supply uncertainty[END_REF], but the more accurate µ on t is. MPC's offline resolution time is equal to 0s, as MPC is not an algorithm based on cost-to-go functions, and as we do not include the offline time devoted to find coefficients of an AR process. As we are not in a distributed setting, the computation time is not a challenge here, and computing the value functions or the online decisions is a doable task for a central controller.

Analyzing the use of storage capacities

We analyze now the trajectories of stocks in assessment during Summer day, where heating is off and production of the solar panel is nominal at midday.

Figure 5 displays the state of charge of the battery along a subset of assessment scenarios, for SDDP and MPC. We observe that SDDP charges earlier the battery at its maximum. On the contrary, MPC charges the battery later and does not use the full potential of the battery 5 . The two algorithms discharge the battery to fulfill the evening demands. We notice that each trajectory exhibits a single cycle of charge/discharge, thus preserving the battery's aging. Figure 6 displays the charge of the domestic hot water tank along the same subset of assessment scenarios. We observe a similar behavior as for the battery trajectories: SDDP uses more the electrical hot water tank to store the excess of PV energy, and the level of the tank is greater at the end of the day than in MPC. This analysis suggests that SDDP makes a better use of storage capacities than MPC.

battery to its maximum level. Indeed, its forecast only comprises the average energy demand; the algorithm is filling the battery only to satisfy this average demand (in a sense, we are "overfitting" the average scenario). Hence, MPC is unable to anticipate a demand higher than usual, leading to an increase use of the recourse variable (the importation from the external grid) when MPC encounters an unexpected scenario. SDDP does not fall into this pitfall, as it considers a broader range of demands at each time step with its quantized probability laws (and not just the average demand). This pushes SDDP to fill the battery to its maximum level, in anticipation of potential high demands during the evening.

Results

We assess SDDP and MPC with different level of uncertainties, that is, with increasing values of σ T . The costs correspond to the management costs to operate the microgrid during a particular day in Summer, where the production of the solar panel is nominal. The detailed results are given in Figure 8, which shows the evolution of the performance of the two algorithms as a function of the level σ T of uncertainty. The costs of each algorithm are obtained via Monte-Carlo simulation, with 10,000 assessment scenarios. We observe that MPC's cost increases quicker than SDDP's cost when the level of uncertainty increases. This second use case allows to quantify the sensitivity of the algorithms with respect to the level of uncertainty. It seems that SDDP behaves better than MPC when facing high uncertainties.

Conclusion

We have presented the optimal management of a domestic microgrid, and have compared different management policies (the core of an Energy Management System) under uncertainties. Our results show that the two optimization-based policies (MPC and SDDP) outperform the proposed rule based policy in terms of money savings. Furthermore, SDDP outperforms MPC during Winter and Summer days -and displays similar performance as MPC during Spring day. Even when SDDP and MPC exhibit close average performances, a comparison scenario by scenario shows that SDDP beats MPC most of the time (more than 90% of scenarios during Summer day). Thus, SDDP proves better than MPC to manage uncertainties in our study, although MPC displays also good performances. SDDP makes better use of storage capacities too.

What if we had compared SDDP not only with "ordinary" MPC but with its stochastic variant, Stochastic Model Predictive Control (SMPC)? A systematic analysis done in [START_REF] Le Franc | EMSx: A numerical benchmark for energy management systems[END_REF], using a large dataset of microgrids, reveals that our conclusions remain valid: algorithms based on the offline computation of cost-to-go functions (SDP, SDDP) outperform lookahead algorithms (MPC, SMPC).

Our study can be extended in different directions. First, we could mix SDDP and MPC to grasp the benefits of these two algorithms. Indeed, SDDP is designed to handle the uncertainties variability but fails to capture the time correlation (as it relies on an assumption of stagewise independence), whereas ordinary MPC ignores the uncertainties variability, but considers time correlation by means of a future scenario. Second, we have extended this study in [START_REF] Carpentier | Mixed spatial and temporal decompositions for large-scale multistage stochastic optimization problems[END_REF], where we applied decomposition methods to optimize microgrids comprising several buildings connected together. Finally, a comparison of MPC and SDDP with novel methods based on reinforcement learning could also be of interest.

Figure 1 :

 1 Figure 1: Electrical microgrid

Figure 2 :

 2 Figure 2: Flow chart of the simulation procedure

Figure 3 :

 3 Figure 3: Electrical (left) and domestic hot water (right) demand scenarios

Figure 4 :

 4 Figure 4: SDDP total costs minus MPC total costs during Summer day

Figure 5 :

 5 Figure 5: Battery charge trajectories for SDDP (left) and MPC (right) during Summer day

Figure 8 :

 8 Figure 8: Evolution of the performance of MPC and SDDP as a function of the level σ T of uncertainty

Table 1 :

 1 2 ) ρ c Battery charge rate ρ d Battery discharge rate β h Conversion yield for hot water tank Decision variables and uncertainties (Ω, F, P) Probability spaceB X t × U t × W t+1 → X t+1 Linear dynamics L t : X t × U t × W t+1 → R Convex operational cost K : X T → R Convex final cost π t : X t → U t Control policy Nomenclature 2

	D el t	Electrical demand (kW)
	D th t Φ pv t	Hot water demand (kW) Production of the solar panel (kW)
	θ i t	Inner temperature ( • C)
	θ w t	Walls temperature ( • C)
	θ e t	Outdoor temperature ( • C)
	F b t F w t F h t t , F h t , F w U t = (F b t ) Controls Energy exchanged with the battery (kW) Energy injected in the hot water tank (kW) Energy injected in the electrical heater (kW) W t = (D el t , D th t , Φ pv t ) Uncertainties X t = (B t , H t , θ w t , θ i t ) States
		Mappings
	f t :

t Level of energy in battery (kWh) H t Level of energy in hot water tank (kWh)

Table 2 :

 2 Different meteorological conditions

	• C) PV Production (kWh)

Table 3 :

 3 Comparison of SDDP, MPC and rule based policiesFigure

		SDDP	MPC	rule based policy
	Offline time	50 s	0 s	0 s
	Online time	1.5 ms	0.5 ms	0.005 ms
	Electricity bill (e)
	Winter day	4.38 ± 0.02 4.59 ± 0.02	5.55 ± 0.02
	Spring day	1.46 ± 0.01 1.45 ± 0.01	2.83 ± 0.01
	Summer day	0.10 ± 0.01 0.18 ± 0.01	0.33 ± 0.02

We have used the notation f + = max{0, f } and f -= max{0, -f }.

More formally, for each time step t = 1, . . . , T , U ad t is a nonempty set-valued mapping U ad t : X t ⇒ U t .

See §2.5 on how the algebraic nonanticipativity constraint (14d) can be turned into a more practical functional constraint, making it possible to search for solutions that are policies.

In Figure5, we observe that SDDP charges the battery to its maximum level during day-time, when the solar panels are producing electricity. Then, the battery is fully discharged during the evening to satisfy the local demand when the energy prices are high (evening peak). On the contrary, MPC does not charge the

AppendixIn this Appendix, we depict physical equations of the energy system in Figure1. These equations are naturally written in continuous time t. We model the battery and the hot water tank with stock dynamics, and the dynamics of the house's temperatures with an electrical analogy.

Here, we compare the performances of the two algorithms (MPC and SDDP) when the level of uncertainty increases.

Uncertainty model

We suppose that, at each time step t, the random variable Φ pv t (the photovoltaic production in Equation ( 1)) is given by

, where µ t is deterministic (it corresponds to the forecasted value of Φ pv t ) and where ε t is a zero-mean Gaussian random variable, with standard-deviation σ t = σ 0 +(σ T -σ 0 )t/T increasing linearly over time (where σ 0 is the initial standard-deviation and σ T the final standard-deviation). We define the level of uncertainty through the value of σ T : the greater it is, the more difficult it becomes to predict the value of the random variable Φ pv t . With this setting, we have that E(Φ pv t ) = µ t and Var(Φ pv t ) = µ 2 t σ 2 t . In Figure 7, we display scenarios that are realizations of sequences Φ pv 0 , . . . , Φ pv T . for production of solar panel Φ pv . These scenarios are generated with σ 0 = 0 and σ T = 0.2.

Energy storage

We consider a battery, whose state of charge at time t is denoted by b(t). The battery dynamics is given by the differential equation

with ρ c and ρ d being the charge and discharge efficiency and f b (t) denoting the energy exchange with the battery.

Electrical hot water tank

We use a simple linear model for the electrical hot water tank dynamics. The enthalpy balance equation writes

where

is the electrical energy used to heat the tank, satisfying

• d th (t) is the domestic hot water demand,

• β h is a conversion yield.

Thermal envelope

We model the evolution of the temperatures inside the house with an electrical analogy: we view temperatures as voltages, walls as capacitors, and thermal flows as currents. A model with 6 resistances and 2 capacitors (R6C2) proves to be accurate to describe small buildings [START_REF] Berthou | Development and validation of a gray box model to predict thermal behavior of occupied office buildings[END_REF]. The model takes into account two temperatures:

• the wall's temperature θ w (t),

• the inner temperature θ i (t).

Their evolution is governed by the two following differential equations

where we denote

• the energy injected in the heater by f h (t),

• the external temperature by θ e (t),

• the radiation through the wall by Φ ext (t),

• the radiation through the windows by Φ int (t).

The time-varying quantities θ e (t), Φ int (t) and Φ ext (t) are exogenous. We denote by R i , R s , R m , R e , R v , R f the different resistances of the R6C2 model, and by c i , c m the capacities of the inner rooms and the walls. We denote by γ the proportion of heating dissipated in the wall through conduction, and by (1 -γ) the proportion of heating dissipated in the inner room through convection. We detail the numerical values in Table 4.

8.30 × 10 7 SI c m 5.85 × 10 6 SI Table 4: Numerical values for the electrical analogy

MPC

Building offline an AR model for MPC We fit an AR(1) model using the optimization scenarios (we do not consider higher order lags for the sake of simplicity). For i ∈ {el, hw}, the AR model writes

where the nonstationary coefficients (α i t , β i t ) are, for any time step t, solutions of the leastsquare problem

The points (d


Xiaokang Zhang
email: xiaokang.zhang@insa-lyon.fr

Modeling, Control and Experimental Evaluation of Multi-functional Converter System

Xiaokang Zhang, Student Member, IEEE, Jean-Yves Gauthier, Xuefang Lin-Shi, Romain Delpoux, and Jean-Franc ¸ois Tr égou ët

Abstract-The multi-functional converter system (MFCS) is a special topology of three-phase AC motor drives, which inherently integrates a DC-voltage regulation function. Firstly, in order to explain the mechanism of the MFCS, an average model is derived. An equivalent circuit is developed, which explicitly reveals that the MFCS is equivalent to a cascaded DC/DC converter-DC motor drive system. Secondly, the inverter's mean duty-cycle is identified as an independent control degree of freedom for the DC-bus voltage. Then, the influence of pulse width modulation (PWM) on the DC-bus voltage is studied. It concludes that space vector PWM is invalid for the MFCS. To address this problem, the zero-sequence voltage injection PWM (ZSVIPWM) is proposed and actually sinusoidal PWM is a special case of the ZSVIPWM. Thirdly, an overall control strategy is proposed for the MFCS, which includes not only the vector control for the motor but also a closedloop control for the DC-bus voltage. Finally, comparative experiments with the standard topology are carried out on a permanent magnet synchronous machine prototype to verify these theoretical analyses and the control strategy. Inverter pole-voltages in d-q-0 coordinate.

i A , i B , i C Motor phase-currents in A-B-C coordinate. i d , i q , i 0 Motor phase-currents in d-q-0 coordinate. i N , i bus Neutral current, DC-bus current. L d , L q , L 0 d-axis, q-axis, zero-sequence inductance. R Phase-resistance. ω e , ω, θ e Rotor electrical speed, rotor mechanical speed, electrical position. T e Electromagnetic torque. ψ f Permanent magnet flux linkage. α X Duty-cycle of inverter's upper switch, (X ∈ A, B, C). α d , α q , α 0 The values of α A , α B , α C converted to d-q-0 coordinate. α h

Inverter's mean duty-cycle. C DC-bus capacitance. ω n , T n , P n Motor rated speed, rated torque, rated power. p Motor pole pairs. f , T s Switching frequency, PWM period.

I. INTRODUCTION

D UE to the advantages of high power density, high ef- ficiency and wide speed regulation range, three-phase PMSM drives have been investigated for many years and extensively applied to a variety of industrial applications, such as machine tools, robotics, EVs, HEVs and aerospace [START_REF] Lara | Effects of rotor position error in the performance of field-oriented-controlled PMSM drives for electric vehicle traction applications[END_REF]- [START_REF] Yang | Disturbance/uncertainty estimation and attenuation techniques in PMSM drives-a survey[END_REF]. The three-phase two-level VSI is the standard topology (see Fig. 1(a)) to drive a three-phase PMSM thanks to its compact structure. Generally speaking, the DC-source is supplied from the front end of inverter, and the motor's neutral point is unused [START_REF] Yan | PWM-VSI fault diagnosis for a PMSM drive based on the fuzzy logic approach[END_REF]. Regarding the DC-source with limited voltage, such as the battery pack of EVs and HEVs whose voltage class is insufficient to meet the drive requirement, extra DC/DC converters are typically employed to boost the battery pack voltage and to exchange the energy between the battery pack and DC-bus [START_REF] Cheng | An integrated electrified powertrain topology with SRG and SRM for plug-in hybrid electrical vehicle[END_REF]. Apparently, the use of extra DC/DC converters sacrifices the volume, weight and cost of the whole system.

A PMSM B C C i A i B i C + - u in N C + - u bus + - u in i N i bus (a) (b) o o S A S B S C + - u bus (¡ A ) (¡ B ) (¡ C ) S A S B S C S' A S' B S' C S' A S' B S' C i bus A B C i A i B i C N PMSM
The MFCS presented in [START_REF] Nakai | Overview of multi-functional converter systems[END_REF]- [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF] gives a promising solution for the aforementioned issue because it has a free DC-voltage regulation function. As shown in Fig. 1(b), the MFCS evolves from the standard topology with the same components. But its DC-source is connected to the neutral point of the motor. With a zero-sequence voltage compensation strategy [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF], the MFCS can drive the motor and regulate the DC-bus voltage simultaneously without physical DC/DC converters. The essence of the MFCS is utilizing the zero-sequence inductance of the motor and the inverter to indirectly construct an equivalent DC/DC converter function. Therefore, the MFCS can provide a compact and cost-saving solution for certain motor drives, especially the electric propulsion system of EVs and HEVs.

However, the zero-sequence voltage compensation strategy proposed in [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF] is an open-loop control manner for the DC-bus voltage. The compensated voltage is derived from a steady-state equation in which the differential term of zerosequence current is neglected. Therefore, in some transient states such as acceleration, deceleration and sudden loadchange conditions, this strategy is insufficient to maintain a constant DC-bus voltage. In other words, the non-constant and fluctuant DC-bus voltage will deteriorate the motor performance, produce larger torque ripple and increase the overvoltage risk on the semi-conductor devices. Although the authors in [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF] further proposed a two-motor type MFCS by connecting the DC-source between the neutral points of two motors for suppressing current ripple and improving the DC-voltage regulation ratio, it also suffers from the openloop control strategy for the DC-bus voltage. In addition, only steady-state experiments were presented in [START_REF] Nakai | Overview of multi-functional converter systems[END_REF]- [START_REF] Moriya | A novel multi-functional converter system equipped with input voltage regulation and current ripple suppression[END_REF]. In order to improve the dynamic performance of the MFCS, a closed-loop control for the DC-bus voltage is necessary. However, it is not easy to find a control DOF for the DCbus voltage because the switching states of the inverter are determined by the control of the motor. Recently, [START_REF] Gauthier | Voltage boost by neutral point supply of AC machine[END_REF] and [START_REF] Zhang | Neutral point supply scheme for PMSM drive to boost DC voltage[END_REF] respectively studied the similar idea as the MFCS on a three-phase induction machine and a three-phase PMSM. Unfortunately, neither of them proposed a closed-loop control strategy for the DC-bus voltage. By connecting the battery pack to the neutral point of a traction motor, [START_REF] Chiang | DC/DC boost converter functionality in a three-phase indirect matrix converter[END_REF] proposed a low-cost electric propulsion system for HEVs. Compared with the MFCS, this scheme employs an auxiliary chopper so that a control DOF for the neutral current is provided. Some remarkable experimental results with dynamic tests were presented in [START_REF] Chiang | DC/DC boost converter functionality in a three-phase indirect matrix converter[END_REF]. Similarly, the improved MFCS with galvanic isolation for EVs was proposed in [START_REF] Plesko | Novel concepts for integrating the electric drive and auxiliary dc-dc converter for hybrid vehicles[END_REF]- [START_REF] Plesko | Novel modulation concepts for a drive-integrated auxiliary dc-dc converter for hybrid vehicles[END_REF] by adding auxiliary choppers. However, it is not an ideal solution to improve the dynamic performance of the MFCS by using auxiliary choppers because it will increase the volume, weight and cost of the whole system, which seems to go against the original intention of the MFCS.

There is not much literature focusing on the topic of the MFCS and few deeper study is achieved. Besides, the comparative studies with the standard topology are also missing to distinguish the merits and drawbacks of the MFCS. Therefore, this paper intends to comprehensively explore the mechanism of the MFCS and its overall control strategy. The paper is organized as follows: In Section II, a brief description of the MFCS and its control challenge are introduced. An average model of the MFCS is built. Furthermore, an equivalent circuit is developed which clearly explains the intrinsically integrated DC-voltage regulation function. The inverter's mean dutycycle is identified as an independent control DOF for the DC-bus voltage. In Section III, the influence on the DCbus voltage caused by PWM behaviors is discussed. The ZSVIPWM is proposed to replace SVPWM and SPWM. Based on the ZSVIPWM, an overall control strategy is proposed in Section IV. In Section V, comparative experiments are performed. Finally, the main results are summarized in the Conclusion.

II. ANALYTICAL MODELING

The MFCS consists of a DC-bus capacitor, a three-phase two-level VSI, a three-phase PMSM and a DC-source which is connected to the neutral point 'N', as shown in Fig. 1(b). S A , S B , S C and S A , S B , S C are the complementary PWM signals. Looking from left to right, the MFCS is a motor drive in which the neutral point 'N' is clamped to u in . Looking from right to left, the MFCS is like a three-phase DC/DC converter but the motor's back-EMFs are the disturbances of u in . Normally, for three-phase AC motor drives, the switching states of the inverter are determined by SVPWM or SPWM according to motor control algorithms. For three-phase DC/DC converters, the interleaved PWM with 120 • phase-shifted are typically used [START_REF] Choi | A three-phase interleaved DC-DC converter with active clamp for fuel cells[END_REF]. As for the MFCS, the key point is to find the independent control DOFs respectively for controlling the motor and the DC-bus voltage only by one inverter.

A. Average Model

The average modeling method is adopted in this paper. Defining the average value of a variable x(t) in one PWM period (T s ) as

x(t) = 1 T s t t-Ts x(τ )dτ (1)
When considering an ideal inverter, voltage drop and dead time are neglected. The average voltage equation of the MFCS can be formulated by

  u AO u BO u CO   =   α A α B α C   u bus =   u AN u BN u CN   +   u in u in u in   (2)
The average current equations of the MFCS can be formulated by

i bus = C du bus dt = -   α A α B α C   T   i A i B i C   = -α T ABC i ABC (3)
i A + i B + i C + i N = 0 (4)
where α T ABC is the duty-cycle vector; i ABC represents the phase-current vector.

By using the Park transformation (T p )

T p = 2 3     cos θ e cos θ e -2π 3 cos θ e + 2π 3 -sin θ e -sin θ e -2π 3 -sin θ e + 2π 3 1 2 1 2 1 2     (5)
(2) can be transformed from A-B-C to d-q-0 coordinates, as

  u dO u qO u 0O   =   α d α q α 0   u bus =   u d u q u 0   +   0 0 √ 3 u in   (6)
In addition, the mathematical model of the PMSM in d-q-0 coordinate can be formulated by

  u d u q u 0   =   R 0 0 0 R 0 0 0 R     i d i q i 0   +   L d 0 0 0 L q 0 0 0 L 0   d dt   i d i q i 0   + ω e   -L q i q L d i d + ψ f 0   (7
) Substituting ( 7) into ( 6), the 0-axis term of (6) becomes

α 0 u bus = u 0 + √ 3u in = Ri 0 + L 0 di 0 dt + √ 3u in (8)
According to the Park transformation (T p ), i 0 = (i

A + i B + i C )/ √ 3.
Combining this with (4), it gives i 0 = -i N / √ 3. Therefore, (8) can be reformulated as

u in = R 3 i N + L 0 3 di N dt + α 0 √ 3 u bus (9)
Because α A , α B and α C are strictly limited between 0 and 1, the value of

α 0 / √ 3 = (α A + α B + α C )/3
is also restricted between 0 and 1. For convenience, the inverter's mean dutycycle (α h ) is defined as

α h = (α A + α B + α C ) 3 = α 0 √ 3 (10)
Substituting ( 10) into ( 9), [START_REF] Nakai | Multi-functional converter systems (MFCS) using 2 neutral-points[END_REF] becomes

u in = R 3 i N + L 0 3 di N dt + α h u bus (11)
Equation ( 11) is similar to the average model of a singlephase DC/DC converter, with R/3 as the parasitic resistance, L 0 /3 as the filtering inductance and α h as the duty-cycle of the chopper. Therefore, from the average model point of view, the MFCS integrates the possibility of regulating the DC-bus voltage. By omitting R/3, u bus ≈u in /α h in steady-state. In order to clearly detail the correlations between the DC-bus voltage and the motor, an equivalent circuit of the MFCS will be developed in the following.

B. Equivalent Circuit

Conducting the following transformations to (3), i bus becomes

i bus = -α T ABC (T -1 p T p )i ABC = -(α T ABC T -1 p )(T p i ABC ) = -((T -1 p ) T α ABC ) T (T p i ABC ) = -(T p α ABC ) T (T p i ABC ) = -   α d α q α 0   T   i d i q i 0   = -α d i d -α q i q -α 0 i 0 = -α d i d -α q i q + α h i N (12)
It is known that FOC makes it feasible to control an AC motor as a separately excited DC motor by decoupling the torque and flux linkage. Therefore, the PMSM is equivalent to a DC motor in d-q coordinate. By defining a virtual chopper as 'C1' and two virtual H-bridges respectively as 'H1' and 'H2', the MFCS can be converted to a cascaded DC/DC converter-DC motor drive system, as shown in Fig. 2. The PWM signals of the virtual switches S vn (n = 1, 2, 3, 4, 5) are complementary with those of S vn . α h represents the duty-cycle of S v1 , α d represents the difference between the duty-cycles of S v3 and S v4 , and α q represents the difference between the duty-cycles of S v2 and S v5 . The explanations of the equivalent circuit are as follows:

According to Kirchhoff's Voltage Law, [START_REF] Gauthier | Voltage boost by neutral point supply of AC machine[END_REF] is visualized by the contour 'K' in Fig. 2. In the same way, the relationships of u d = α d u bus and u q = α q u bus in (6) are visualized by the double H-bridges circuit 'H1' and 'H2'. The current relation in [START_REF] Zhang | Neutral point supply scheme for PMSM drive to boost DC voltage[END_REF] is visualized on the node 'P' according to Kirchhoff's Current Law. e d and e q are respectively equal to -ω e L q i q and ω e (L d i d + ψ f ) according to [START_REF] Nakai | Overview of multi-functional converter systems[END_REF].

Consequently, the equivalent circuit explicitly reveals an EDC function exists in the MFCS. The PMSM can be seen as an equivalent load of the EDC. Meanwhile, it also reveals that the motor and the EDC are uncoupled in d-q-0 coordinate. Hence, α h can be an independent control DOF of the EDC.

III. ZSVIPWM

Thanks to its high output quality, SVPWM is chosen preferentially in many PMSM drives [START_REF] Chen | Extension of space-vector-signalinjection based MTPA control into SVPWM fault-tolerant operation for five-phase IPMSM[END_REF]. The modulation ratio m (fundamental voltage amplitude to carrier amplitude) of SVPWM can reach 1.15 in the absence of overmodulation [START_REF] Bowes | The relationship between space-vector modulation and regular-sampled PWM[END_REF] [START_REF] Singh | A new SVPWM technique to reduce the inductor current ripple of three-phase z-source inverter[END_REF]. For the MFCS, the inverter's mean duty-cycle α h should be analyzed before using PWM technologies since the existence of the EDC. According to the SVPWM algorithm, the evolution of α h with m in one fundamental period is shown in Fig. 3 [START_REF] Shu | An efficient SVPWM algorithm with low computational overhead for three-phase inverters[END_REF]. It can be seen that for a given m, α h is almost a triangular wave. Its mean value is 0.5 and its frequency is three times the frequency the fundamental wave. The amplitude of α h increases with the increasing of m. When m = 1.15, the peak and valley points of α h reach around 0.66 and 0.34 respectively. According to the EDC phenomenon, the DC-bus voltage will be fluctuant with the same frequency of α h if SVPWM is employed in the MFCS. Especially, the oscillation amplitude will be (1/0.34 -1/0.66)u in ≈ 1.43 u in when m = 1.15. Such large oscillations will deteriorate the motor performance undoubtedly. Thus, SVPWM is inapplicable for the MFCS. To address this problem, the ZSVIPWM will be proposed in the following.

A. Zero-sequence Voltage and Duty-cycle Calculations

For the MFCS, a certain amount of zero-sequence voltage is necessary. The requisite zero-sequence voltage (u ZS ) can be expressed by

u ZS = (u AN + u BN + u CN ) 3 = u 0 √ 3 (13)
According to ( 8) and ( 10)

u ZS = u 0 √ 3 = α h u bus -u in (14)
Therefore, the actual phase-voltage reference u XN * can be expressed as

u XN * = u XN 1 * + u ZS = u XN 1 * + α h u bus -u in (15)
Because of the clamped neutral point, the positive and negative amplitudes of carrier signals in the MFCS are respectively u bus -u in and -u in , as shown in Fig. 4(a). According to the regular-sampling method (see Fig. 4(b)), the duty-cycle α X is derived as

α X = δ/2 T s /2 = u XN * + u in u bus = α h + u XN 1 *
u bus [START_REF] Plesko | Novel modulation concepts for a drive-integrated auxiliary dc-dc converter for hybrid vehicles[END_REF] in which δ is the high-level time of S X in one PWM period. Consequently, ( 16) is the final algorithm of the ZSVIPWM. Especially, when α h = 0.5, (16) corresponds to the algorithm of SPWM. However, SPWM can merely regulate u bus to 2 times of u in in an open-loop way. The proposed ZSVIPWM can flexibly adjust u bus to different values in a closed-loop way thanks to the EDC.

B. DC-source Utilization Ratio (DCSUR)

Due to the EDC, the DC-bus voltage u bus can reach higher value than u in . But the reachable AC-voltages for the motor are bounded. The step-up ratio (r 0 ) of the EDC and the DCSUR (r 1 ) are respectively defined as The zero-sequence voltage ratio (r 2 ) is defined as

r 0 = u bus u in r 1 = u XN 1amp u in (17)
r 2 = |u ZS | u in (18)
In order to guarantee u XN * within the linear modulation region, u XN amp is limited by the smaller one between u in and u bus -u in . Thus, u XN 1amp is also limited. As shown in Fig. 5(a), when u bus > 2u in (i.e. r 0 > 2), the carrier signal is asymmetrical because u bus -u in is larger than u in . Meanwhile, u XN * shows a downward offset because of the injected zero-sequence voltage u ZS . It is obvious that the limit of u XN amp is u in . Therefore, the limit of u XN 1amp is u in -|u ZS | (i.e. r 1 = 1 -r 2 ). Similarly, r 1 can be analyzed in the same way when u in < u bus ≤ 2u in (i.e. 1 < r 0 ≤ 2).

The analysis results are summarized as follows: when r 0 > 2, r 1 = 1 -r 2 ; when 1 < r 0 ≤ 2, r 1 = r 0 + r 2 -1. For the sake of explanations, the curves of r 1 are visualized in Fig. 5

(b).

For a certain r 2 , r 1 maintains 1 -r 2 when r 0 ≥ (2 -2r 2 ). r 1 decreases with the decrease of r 0 when 1 < r 0 < (2 -2r 2 ). Therefore, the largest DCSUR is 1 -r 2 and the point 'a' (r 0 = 2 -2r 2 ) is the optimal operating point for the whole system. When r 2 reduces to r 2 , r 1 becomes r 1 . In this case, the largest DCSUR increases to 1 -r 2 and the best operating point moves to 'b'. Further, when r 2 reduces to 0, r 1 becomes r 1 . In this case, the largest DCSUR is 1 and the best operating point moves to the point '2' (u bus = 2u in ).

In steady-state, the differential term of (11

) L0 3 di N dt = 0. Combining with (14), u ZS is equal to -Ri N
3 . This means |u ZS | is equal to the voltage drop in R/3. Generally speaking, the stator resistance of a PMSM is much less than its inductive reactance [START_REF] Plesko | Novel concepts for integrating the electric drive and auxiliary DC-DC converter for hybrid vehicles[END_REF]. Therefore, r 2 is small. If r 2 is omitted, the largest DCSUR of the MFCS approximates to 1 and the maximum effective step-up ratio r 0 of the EDC is 2. According to the definition of DCSUR, the DCSUR of the standard topology is only 0.58 by using SVPWM [START_REF] Bowes | The relationship between space-vector modulation and regular-sampled PWM[END_REF].

IV. CONTROL OF THE MFCS

According to the proposed ZSVIPWM algorithm in ( 16), the fundamental voltage u XN 1 * and α h are respectively the independent control DOFs for the motor and the EDC. Thus, it is feasible to control the motor and the EDC simultaneously by employing a FOC controller and a common DC/DC converter controller.

A. Overall Control Strategy

The overall control strategy of the MFCS is divided into three parts, as shown in Fig. 6. Firstly, the FOC controller includes two current-loops and one speed-loop. The PI controller of speed-loop outputs the q-axis current reference i q * . For a surface-mounted PMSM without flux-weakening demand, the d-axis current reference i d * = 0. The voltage references u d * and u q * are obtained by the two PI controllers of current-loops with decoupling compensation. By inverse Park transformation, the fundamental voltage references are obtained. Secondly, a cascaded PI controller is used for the EDC. The PI controller of voltage-loop outputs the neutral current reference i N * and the PI controller of current-loop outputs the inverter's mean duty-cycle α h . After obtaining u XN 1 * (X ∈ A, B, C) and α h , the duty-cycles α A , α B and α C are calculated according to proposed ZSVIPWM algorithm in [START_REF] Plesko | Novel modulation concepts for a drive-integrated auxiliary dc-dc converter for hybrid vehicles[END_REF]. Finally, the PWM signals S A , S B , S C and S A , S B , S C are generated by comparing these duty-cycles with a triangular signal in the microprocessor.

N C + - u bus ¡ ¢ PMSM u in + -

B. Small-Signal Model of the EDC

According to Fig. 2, the PMSM can be seen as an equivalent load (R Load ) of the EDC. Therefore, Fig. 2 can be simplified to Fig. 7. D represents the duty-cycle of S v1 and α h = 1 -D by neglecting the dead time. Without considering the losses, R Load can be calculated approximatively as

R Load ≈ u * 2 bus P n ( 19
)
The equilibrium states (D s and i N s ) of the EDC are

       D s = 1 - u in u * bus i N s = u * 2 bus u in R Load (20)
Therefore, the small-signal model of the EDC can be built as

i N ubus =     0 D s -1 L 0 /3 1 -D s C -1 R Load C     ĩ N ũ bus +     u * bus L 0 /3 -i N s C     D (21)
According to the small-signal model, the system transfer functions can be derived as

H1(s) = ĩ N (s) D(s) = u * bus L 0 /3 s + u * bus R Load (L 0 /3)C + (1 -D s ) (L 0 /3)C i N s s 2 + 1 R Load C s + (1 -D s ) 2 (L 0 /3)C ( 22
)
H2(s) = ũ bus (s) D(s) = -i N s C s + (1 -D s ) (L 0 /3)C u * bus s 2 + 1 R Load C s + (1 -D s ) 2 (L 0 /3)C (23) + - u in C i N R/3 L 0 /3 S v1 O + - u bus S' v1 R Load (D) (¡ h )
(s) = ũ bus (s) ĩN (s) = H2(s) H1(s) (24)
According to H1(s) and H3(s), the PI parameters for the current-loop and voltage-loop of the EDC can be tuned.

V. EXPERIMENTAL RESULTS

Experiments are carried out on a surface-mounted PMSM (whose neutral point is available) to verify the theoretical analyses, ZSVIPWM and overall control strategy, as well as to assess the MFCS. The parameters of the PMSM drive are:

u bus = 30 V, C = 510 uF, P n = 52.5 W, T n = 125 mN•m, ω n = 4000 rpm, R = 0.6 Ω, L d = L q = 1.1 mH, L 0 /3 = 0.26 mH, p = 4, ψ f = 0.0056 Wb, f = 20 kHz.
According to the analyses in Section III, the maximum effective step-up ratio of the EDC approximates to 2. Therefore, u in is chosen as 15 V (half of u bus ). For contrast, the similar experiments are implemented on the PMSM fed by the standard topology, in which a 30 V u in is required. The parameters of the FOC controllers in both topologies are identical. The algorithms are implemented on a dSPACE DS1202 MicroLabBox platform and the experiment setup is illustrated in Fig. 8. The torquemeter and hysteresis brake used in this test-bench are MAGTROL TM-303 and HB-450 respectively.

For the MFCS, the DC-bus voltage needs to be raised to 30 V before starting the motor. As shown in Fig. 9, u bus is 15 V before the startup. When the EDC runs, u bus is boosted to 30 V with 5 V overshoots and it tends to stable state rapidly. This demonstrates the EDC phenomenon of the MFCS. For the results of the MFCS, it can be seen that u bus is boosted at 30 V stably while α h is around 0.47 (considering the voltage drop in R/3, α h is not exactly 0.5). This verifies the correctness of the theoretical analyses presented in this paper. The rotor speed ω remains stable at 4000 rpm with slight fluctuations of 11 rpm (0.27%). The electromagnetic torque T e maintains 125 mN•m stably with 10 mN•m (8%) fluctuations. For the standard topology, a stable 30 V DCsource is connected to the DC-bus directly. The ripple of u bus is only caused by the charge-discharge of the DC-bus capacitor in different switching states. However, for the MFCS, u bus has to be controlled at 30 V by the EDC function. The ripple of u bus results from not only the charge-discharge of the DC-bus capacitor but also the control errors of the EDC. Therefore, it can be seen that the voltage ripple of the MFCS is a little larger than that of the standard topology (3 V to 2 V). Even so, the motor performance (speed and torque ripple) of both topologies is similar.

A. Steady-state Performance

In order to evaluate the characteristics of voltages and currents, the real-time datum recorded by the Tektronix oscilloscope MDO3000 (which can record 1 M points in 40 ms period) are shown in Fig. 11(a) and (c). Firstly, in Fig. 11(a), it can be seen that the mean values of i B and i C are respectively -1.58 A and -1.49 A. This means the phase-currents of the MFCS shift downward because of the zero-sequence current. The offset value approximates to -1/3 the mean value of i N (4.46 A). Comparatively speaking, the phase-currents of the standard topology are symmetrical whose mean values are near 0 A, as shown in Fig. 11(c). Secondly, Fig. 11(a) shows u CN of the MFCS only has two voltage-levels, u bus -u in (15 V) and -u in (-15 V) because of the clamped neutral point. u CN of the standard topology has five voltage-levels ± 1 3 u bus (±10 V), 0 V and ± 2 3 u bus (±20 V) thanks to the floating neutral point, as shown in Fig. 11(c).

The FFT analyses of u CN , i B and i C in both topologies are shown in Fig. 11 (b) and (d). Firstly, the fundamental voltage (FV) amplitude and fundamental current (FI) amplitude are respectively 11.94 V and 3.57 A in the MFCS while they are respectively 11.59 V and 3.37 A in the standard topology. The fundamental power factor cosφ is about 0.88 in the MFCS while it is 0.89 in the standard topology. Comparatively speaking, these fundamental characteristics of both topologies are similar. Secondly, for the MFCS, the zero-sequence current flowing into the motor coils undoubtedly produces more losses in the motor. In order to calculate the total active power of the motor, an easy method of accumulating the power generated by all the harmonics (including zero-sequence and fundamental components) is adopted. The result indicates that the total active power of the motor in the MFCS increases by 6.02 W compared to that in the standard topology. Therefore, it can be concluded that the increased motor losses in the MFCS is about 11.5% of P n at rated condition. The increased copper losses caused by the zero-sequence current are just the power consumed in R/3, which can be calculated by i N 2 R 3 = 3.98 W (7.6% to P n ). Therefore, the increased iron losses are about 2.04 W (3.9% to P n ). Although the motor losses of the MFCS increase, they can be mitigated by proper design of motor winding layout with larger L 0 . For example, [START_REF] Hackner | Optimization of the winding arrangement to increase the leakage inductance of a synchronous machine with multi-functional converter drive[END_REF] and [START_REF] Hackner | Optimization of the winding arrangement to increase the zero-sequence inductance of a synchronous machine with multifunctional converter drive[END_REF] presented that a motor with a two-layer fractional slot winding has bigger leakage inductance than the motor with side-byside placed winding. Thirdly, the 20 kHz harmonic voltage (20kHV) amplitude of u CN in the MFCS is larger than that in the standard topology (9.48 V to 0.37 V). Thus, it can be seen that the 20 kHz harmonic current (20kHC) amplitude of i C in the MFCS is also larger than that in the standard topology (0.68 A to 0.003 A). However, the 20kHC in the MFCS does not affect the electromagnetic torque because the 20kHCs of i B and i C almost have the same amplitude (0.68 A to 0.64 A) and phase position (58.47 • to 59.70 • ). In other words, the 20kHCs do not generate harmonic torque. Therefore, although the phase-current ripple of the MFCS is larger, the torque performance can be maintained.

B. Dynamic Performance

The dynamic tests including speed regulation and sudden loading/deloading are shown in Fig. 12. On the one hand, ω * is set to 1000 rpm before 2 s, and it is 2000 rpm between 2 and 6 s, after which it is reset to 1000 rpm. On the other hand, the load torque is 40 mN•m before 10 s. Then it is set to 120 mN•m at 10 s and reduced to 40 mN•m again at 15 s.

In Fig. 12(a), it can be seen that u bus of the MFCS can remain relatively stable at 30 V during the acceleration, deceleration, loading and deloading processes. Specifically, when the motor accelerates suddenly, it means the equivalent load of the EDC increases instantly. In order to withstand the voltage sag, the controller of the EDC decreases α h sharply so that i N increases quickly for providing more power for the system. Consequently, u bus shows 3 V fluctuations at the beginning of the acceleration then it tends to be stable with 2 V voltage ripple (VR). For the standard topology in the same condition, u bus also shows a 2 V VR when the motor accelerates instantly. During the deceleration process, the motor becomes a generator. In order to prevent the voltage rise, the controller of the EDC increases α h sharply so that i N decreases to negative values for bleeding the increasing power stored in the DC-bus capacitor. Therefore, u bus scarcely appears overshoot in the MFCS. Without the EDC function, u bus of the the standard topology shows a 3 V overshoot.

For both topologies, the rotor speed can track the reference well without overshoot in the acceleration and deceleration processes. When the load is applied and removed, the speed shows slight dip (40 rpm) but it turns to stable state quickly. The torque ripple (TR) is around 11 mN•m in both topologies. Comparatively speaking, the motor dynamic performance of the MFCS is same as that of the standard topology.

In general, the main characteristics of the MFCS compared to the standard topology are summarized in Table. I.

VI. CONCLUSION

In this paper, the comprehensive studies of the MFCS are presented. Thanks to a rigorous modeling, the inverter's mean duty-cycle is identified as an independent control DOF for the EDC. It makes possible to control the EDC in a closed-loop way without auxiliary choppers. The proposed overall control strategy is simple, in which the FOC can still be used without modifications to control the motor and a cascaded PI controller is sufficient to control the EDC. The convincing experiments verify the effectiveness of the proposed control strategy. Although the motor losses increase, the MFCS provides a promising solution for AC motor drives with lower cost and smaller volume when the DC-source is limited.

The main contributions of this paper are summarized as follows: 1) An equivalent circuit is developed to clearly reveals the mechanism of the MFCS.

2) The inverter's mean duty-cycle is identified as an independent control DOF for the EDC.

3) The ZSVIPWM is proposed for the MFCS to replace SPWM and SVPWM . 4) An overall control strategy is proposed for the MFCS, in which the motor and the EDC can be controlled independently.

Fig. 1 .

 1 Fig. 1. PMSM drive topologies. (a) The standard topology. (b) The MFCS.

Fig. 2 .

 2 Fig. 2. Equivalent circuit of the MFCS.

3 mFig. 3 .

 33 Fig.3. The inverter's mean duty-cycle α h under SVPWM.

Fig. 4 .

 4 Fig. 4. The schematic diagram of the proposed ZSVIPWM. (a) Carrier comparison. (b) Regular-sampling method.

0Fig. 5 .

 5 Fig. 5. The DCSUR analyses of the MFCS. (a) When (r 0 > 2). (b) Analysis results.

Fig. 6 .

 6 Fig.6. The overall control strategy of the MFCS.

Fig. 7 .

 7 Fig. 7. The simplified equivalent circuit of the MFCS.

H3

Fig. 10

 10 Fig. 10 illustrates the steady-state performance respectively conducted by the MFCS and standard topology when the motor operates at rated condition (4000 rpm and 125 mN•m).

Fig. 8 .Fig. 9 .

 89 Fig. 8. Experiment setup.

Fig. 10 .

 10 Fig. 10. Steady-state performance (speed, torque and DC-bus voltage) at rated condition. (a) The MFCS with 15 V DC-source. (b) The standard topology with 30 V DC-source

FC( 3 .Fig. 11 .

 311 Fig. 11. Steady-state performance (neutral current, phase-currents, phase-voltages and FFT analyses) at rated condition. (a), (b) The MFCS with 15 V DC-source. (c), (d) The standard topology with 30 V DCsource.

Fig. 12 .

 12 Fig. 12. Dynamic performance. (a) The MFCS with 15 V DC-source. (b) The standard topology with 30 V DC-source.

  XN 1amp Amplitude of u * XN 1 . u d , u q , u 0 Motor phase-voltages in d-q-0 coordinate. u ZSMotor zero-sequence voltage. u AO , u BO , u

	u XN , u * XN Motor actual phase-voltage and phase-voltage
		reference, (X ∈ A, B, C).
	u * XN 1 u XN amp	Fundamental component of u * XN . Amplitude of u * XN .
	u

CO

Inverter pole-voltages in

A-B-C coordinate. u dO , u qO , u

0O

TABLE I THE

 I MAIN CHARACTERISTICS OF THE MFCS COMPARED TO THE STANDARD TOPOLOGY

	Characteristic	Evaluation
	Speed control performance	Similar
	Torque ripple performance	Similar
	DCSUR	Larger (1 vs 0.58)
	Integrated EDC function	YES vs NO
	Current ripple	Bigger
	Losses	Bigger (11.5% for the test PMSM)