Liudmila Tumash
email: liudmila.tumash@gipsa-lab.fr

Carlos Canudas De Wit

Maria Laura

Member, IEEE Delle Monache

Fellow, IEEE Carlos Canudas-De-Wit
email: carlos.canudas-de-wit@gipsa-lab.grenoble-inp.fr

M L Delle Monache
email: ml.dellemonache@inria.fr

Boundary Control Design for Traffic with Nonlinear Dynamics

Keywords: boundary control problem, Hamilton-Jacobi formulation, LWR model, partial differential equations, traffic control. I

subject. Some examples of control with application to traffic can be found in [START_REF] Gugat | Optimal control for traffic flow networks[END_REF], [START_REF] Carlson | Local feedback-based mainstream traffic flow control on motorways using variable speed limits[END_REF]. Traffic flow control does not require changes in the topology of urban traffic network and can be applied directly provided that there is some knowledge about current traffic situation. Being modeled by conservation laws, traffic systems are usually controlled by variable speed limit [START_REF] Zhang | Coordinated variable speed limit, ramp metering and lane change control of highway traffic[END_REF], or from the boundary, i.e. either from entry or exit of the road, e.g. ramp metering [START_REF] Reilly | Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering[END_REF]. Classical techniques widely used for control of conservation laws (without model reduction from PDE to ODE) are backstepping [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF], Lyapunov-based [START_REF] Blandin | Boundary stabilization of the inviscid burgers equation using a Lyapunov method[END_REF] and optimal control methods [START_REF] Ulbrich | A Sensitivity and Adjoint Calculus for Discontinuous Saolutions of Hyperbolic Conservation Laws with Source Terms[END_REF], [START_REF] Gugat | Optimal control for traffic flow networks[END_REF], [START_REF] Jacquet | Optimal control of scalar one-dimensional conservation laws[END_REF] that are used to achieve minimization of total travel time and of fuel consumption or maximization of the vehicle outflow.

It is quite challenging to control hyperbolic conservation laws, since discontinuities (shocks) emerge in finite time even for smooth initial data [START_REF] Evans | Partial Differential Equations[END_REF]. The classical control methods mentioned above are not always well suited to handle shocks, since they require the knowledge on the internal shock dynamics. However, shock dynamics inside the domain were tracked, e.g., in [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF], where the weak formulation and the Rankine-Hugoniot relation were used to stabilize the solution of the Burgers equation to a constant equilibrium. In a recent work [START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF] the problem of boundary control of solutions with jump discontinuities has been considered. In both [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF], [START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF], the desired state was stationary and the Lyapunov methods were applied. In [START_REF] Donadello | Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions[END_REF] the exact controllability of solutions to conservation laws to space-and time-dependent trajectories has been studied. Nevertheless, the problem of stabilizing a state with shocks to space-and time-dependent trajectories that may also contain shocks has never been considered before.

In situations where control design becomes too tedious due to shocks, one can alternatively use the Hamilton-Jacobi formulation that is an integral form of the LWR PDE. The solution of a H-J PDE does not contain any shocks, only discontinuities in its derivatives. The state represents a cumulated vehicle count, which is a Lipschitz continuous function that can be evaluated directly from the boundary or initial conditions [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i,ii and iii[END_REF]. The existence of such a function guarantees the conservation of the number of vehicles even if shocks arise. The solution of a H-J PDE can be obtained by solving a simple minimization problem, therefore it does not require a model discretization providing exact results. Thus, several computational algorithms based on H-J formulation of LWR model have been developed. A variational formulation of kinematic waves was first proposed in [START_REF] Daganzo | A variational formulation of kinematic waves: basic theory and complex boundary conditions[END_REF], [START_REF] Daganzo | On the variational theory of traffic flow: Well-posedness, duality and applications[END_REF], who showed that every well-posed traffic problem with a concave flowdensity relation can be solved as a set of shortest paths.

The solution of Hamilton-Jacobi PDE can be equivalently obtained using the viability framework [START_REF] Joseph | Explicit formula for the solution of convex conservation laws with boundary condition[END_REF], [START_REF] Joseph | Solution of convex conservation laws in a strip[END_REF]. This framework is based on using Lax-Hopf formula that exploits the structure of a dynamic programming problem and the solution is obtained as the minimum of all valid paths [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory[END_REF]. In [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Computational methods[END_REF] they develop a computational method of H-J PDE for any piecewise affine initial condition. The Lax-Hopf algorithm for computation of a solution on a single link for any concave fundamental diagram has been addressed in [START_REF] Mazaré | Analytical and grid-free solutions to the Lighthill-Whitham-Richards traffic model[END_REF]. Its improved version with a lower computational time has been proposed in [START_REF] Simoni | A Fast Lax-Hopf formula to solve the Lighthill-Whitham-Richards traffic flow model on networks[END_REF]. Optimal control methods for a traffic network based on viability framework are proposed in [START_REF] Li | Optimal Control of Scalar Conservation Laws Using Linear/Quadratic Programming: Application to Transportation Networks[END_REF], [START_REF] Li | Optimal traffic control in highway transportation networks using linear programming[END_REF], where they demonstrate how the new method has reduced the computational complexity. The framework has also been used to develop a convex optimization approach to reduce the fuel consumption in [START_REF] Zu | Convex Optimization for Energy-Efficient Traffic Control[END_REF]. Also a recent work [START_REF] Bekiaris-Liberis | Nonlinear bilateral outputfeedback control for a class of viscous Hamilton-Jacobi PDEs[END_REF] considered a H-J PDE with viscous term that allowed them to perform a feedback linearisation, which enabled tracking a desired timedependent state on some fixed space point.

In this paper, we derive an explicit solution of a Hamilton-Jacobi PDE with Hamiltonian corresponding to the triangular flux function. Its solution corresponds to the cumulated vehicle count function in time on a single traffic road. The main contributions of this paper are the following:

• Tracking space-and time-dependent trajectory: we extend the results presented in [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF], [START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF] and present a controller able to drive a state with shocks to any time-and space-dependent vehicle density that also may contain shocks. This is the first boundary controller in the traffic community able to solve such general tasks. Moreover, comparing to [START_REF] Bekiaris-Liberis | Nonlinear bilateral outputfeedback control for a class of viscous Hamilton-Jacobi PDEs[END_REF], our analysis is done for the original LWR system without linearisation and viscosity term. • The solution of H-J PDE is used to provide conditions on when and which control can be applied: we have considered a general case, when it is not guaranteed that supply at the entrance and demand at the exit are large enough, i.e., it might not always be possible to impose any control. We prove in this work that even in that case the goal will be achieved. For this, we have formulated weak boundary conditions in terms of control restriction functions, see more details in Section III.

This paper is organized as follows. First, we state the problem of finding a boundary control law for a traffic system. In Section II we introduce a relation between LWR and H-J formulations and explicitly calculate the solution of H-J PDE asymptotic in time for the triangular flux function. Then, the problem is stated in H-J formulation. Section III contains the main result (the control laws) and the proof. In Section IV we consider a numerical example verifying our controller. Finally, the concluding remarks are given in Section V.

B. Problem Statement

The LWR model is based on the conservation of the number of vehicles and can be expressed as

∂ρ(x, t) ∂t + ∂Φ(ρ(x, t)) ∂x = 0, ∀(x, t) ∈ [0, L] × R + (1)
where ρ(x, t) : [0, L] × R + → R + indicates the traffic density, and the flux function Φ(ρ(x, t)) : [0, ρ max ] → R + is a Lipschitz continuous and concave function that admits a unique maximum φ max (capacity) attained at ρ c (critical density), while Φ(0) = Φ(ρ max ) = 0 (ρ max is the "jam density"). The LWR model is based on an empirically established law [START_REF] Greenshields | A study of traffic capacity[END_REF] that relates density ρ(x, t) and flow φ(x, t) as Φ(ρ(x, t)) = φ(x, t), known as a fundamental diagram (FD). We distinguish two density regimes: the traffic has a positive kinematic wave speed if ρ ∈ [0, ρ c ] (free-flow regime), and the wave speed is negative if ρ ∈ (ρ c , ρ max ] (congested regime). Here we use a triangular (bilinear) FD, see Figure 1, proposed by Daganzo [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory[END_REF]:

Φ(ρ) = v f ρ, ρ ∈ [0, ρ c ] -w(ρ -ρ max ), ρ ∈ (ρ c , ρ max ] (2)
where v f is the free-flow speed (wave moving forwards), and -w is the congestion speed (wave moving backwards). Let us fix an initial condition ρ 0 (x) ∀x ∈ [0, L], boundary flows u in (t) and u out (t) ∀t ∈ R + (which we treat as control variables), and introduce an initial boundary value problem:

ρ Φ ρ c φ max 0 ρ max v f -w
Σ =      ∂ t ρ(x, t) + ∂ x Φ (ρ(x, t)) = 0, Φ (ρ(0, t)) = φ in (t), Φ (ρ(L, t)) = φ out (t), ρ(x, 0) = ρ 0 (x). (3)
Inflows φ in (t) and outflows φ out (t) are defined ∀t ∈ R + as

φ in (t) = φ in (u, S) = min {u in (t), S(ρ(0, t))} , φ out (t) = φ out (D, u) = min {D(ρ(L, t)), u out (t)} , (4)
with S(ρ(0, t)) the supply and D(ρ(L, t)) the demand functions at the right and left boundary, respectively, given by

S(ρ) = min {w(ρ max -ρ), v f ρ c } , D(ρ) = min {v f ρ, v f ρ c } . (5)
Note that the boundary conditions (4) are given in terms of flows. This is equivalent to the weak boundary conditions formulation [START_REF] Colombo | On the modeling and management of traffic[END_REF] in terms of densities given in [START_REF] Strub | Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling[END_REF]. Thus, the problem given by ( 3) and ( 4) is well-posed. Now we define the desired space-and time-varying density ρ d (x, t) that should be achieved with the help of boundary control. In order to be admissible ρ d (x, t) ∈ R + must be a weak entropy solution of the following system:

Σ d =        ∂ρ d (x, t) ∂t + ∂Φ (ρ d (x, t)) ∂x = 0, Φ (ρ d (0, t)) = φ in d (t), Φ (ρ d (L, t)) = φ out d (t), ρ d (x, 0) = ρ 0 d (x), (6)
where inflows and outflows in the desired system must satisfy φ in d (t) ≤ S(ρ d (0, t)) and φ out d (t) ≤ D(ρ d (L, t)).

Thus, in scope of this paper, our objective is to find ∀t ∈ R + boundary control laws u in (t) and u out (t) such that the density follows a desired trajectory ρ d (x, t) as t → ∞.

Finally, throughout this paper we assume the following: Assumption 1. The initial conditions have left the system, thus, the solution of Σ is determined by the boundaries only.

Assumption 2. There exists ε > 0 such that φ in and φ out from Σ satisfy the following inequalities in its time average:

t+T t φ in (τ )dτ ≤T φ max -ε and t+T t φ out (τ )dτ ≤ T φ max -ε, where t > 0 and T = min L v f , L ω .
Thus, flows can not hold their maximal values during the time interval given by T . This assumption is needed to prove exponential convergence to a desired profile, see details in the proof of Theorem 1.

Remark 1. Note that if Assumption 2 is satisfied, then by taking t ≥ t min , where t min is defined as

t min = min L v f , L w   1 +     1 ε L 0 (ρ max + ρ c )       , (7)
Assumption 1 holds trivially, as it is shown in Appendix II.

II. PRELIMINARIES

As mentioned above, discontinuities may emerge in solutions to conservation laws in finite time even for smooth initial conditions. Note that control enters the system through the minimum function [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF]. If density would be in one regime only, then it would be possible to reduce (4) to φ in (t) = u in (t) and φ out (t) = u out (t). However, to solve control problems for a traffic state in a mixed regime, we must handle (4). This might be problematic for an LWR PDE system due to the necessity to track shock positions.

Thus, we will equivalently use the Hamilton-Jacobi formulation, which gives a closed-form solution corresponding to a minimum function, and which allows to analyse weak boundary conditions explicitly. In particular, we will be able to analyse when and for how long prescribed control values are accepted by the system. Being an integral form of LWR, the solution of Hamilton-Jacobi PDE has no shocks.

A. Link Between H-J and LWR Formulation

We introduce the cumulative vehicle count also called the Moskowitz function M (x, t) [START_REF] Moskowitz | Discussion of freeway level of service as influenced by volume and capacity characteristics by dr drew and cj keese[END_REF], [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i,ii and iii[END_REF]. It should be interpreted as a cumulated (counted) number of vehicles at location x and at time t. The density ρ(x, t) and the flow Φ (ρ(x, t)) can be computed from M (x, t) as

ρ(x, t) = - ∂M (x, t) ∂x , Φ(ρ(x, t)) = ∂M (x, t) ∂t . (8)
Intuitively, one can see a relation between HJ and LWR by performing a formal computation, namely by taking the derivative of the density w.r.t. time and the derivative of the flow w.r.t. space (assuming both of them being continuous) and by using [START_REF] Joseph | Explicit formula for the solution of convex conservation laws with boundary condition[END_REF], and thus obtaining the LWR PDE:

∂ρ(x, t) ∂t + ∂Φ (ρ(x, t)) ∂x = 0 ⇔ ∂ 2 M (x, t) ∂x∂t - ∂ 2 M (x, t) ∂t∂x = 0.
The rigorous relation was shown in [START_REF] Joseph | Explicit formula for the solution of convex conservation laws with boundary condition[END_REF]. Further, we can rewrite Φ (ρ(x, t)) = φ(x, t) using (8) as

∂M (x, t) ∂t -Φ - ∂M (x, t) ∂x = 0, (9)
which is a Hamilton-Jacobi PDE, and Φ plays the role of a Hamiltonian. Its solution corresponds to the cumulative vehicle count M (x, t). Note that the LWR PDE can be obtained if ( 9) is differentiated w.r.t. space and expressed in terms of density. Note that (9) depends only on derivatives of M (x, t). Therefore, for any solution M (x, t) adding any constant M 0 gives also a solution M (x, t) + M 0 . This is obvious, since we can start numeration of cars from any particular number.

The existence of M (x, t) itself guarantees the conservation of vehicle numbers. A shock corresponds to a discontinuity in the first derivative of M (x, t), then the conservation equations are still valid if M (x, t) is continuous across the shock path.

B. General Solution of H-J

The H-J PDE (9) can be solved analytically using initial condition function M Ini (x), upstream M Up (t) and downstream M Down (t) boundary condition functions. Note that the boundary conditions should be consistent with the weak boundary conditions formulation (3)-( 4).

Thus, let us introduce the following Cauchy problem for the

HJ PDE ∀(x, t) ∈ [0, L] × R + :              ∂M (x, t) ∂t -Φ - ∂M (x, t) ∂x = 0, M (x, 0) = M Ini (x), M (0, t) = M Up (t), M (L, t) = M Down (t). (10)
The relation of M Up (t) and M Down (t) with φ in (t) and φ out (t) are studied in detail in Appendix A.

For convenience, let us introduce the value condition function c(x, t) : Dom(c) → R + , where Dom(c) = ({0, L} × R + ) ∪ ((0, L) × {0}), which aggregates the initial and boundary conditions of [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i,ii and iii[END_REF] (see [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory[END_REF]):

c(x, t) =      M Ini (x) t s = 0, M Up (t) x s = 0, M Down (t) x s = L, (11)
where x s and t s are space and time points in space and time, respectively, related to the start of travelling.

To obtain a closed-form solution of [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i,ii and iii[END_REF], one should treat [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i,ii and iii[END_REF] as a constrained optimization problem [START_REF] Daganzo | A variational formulation of kinematic waves: basic theory and complex boundary conditions[END_REF]. Let us introduce a Legendre-Fenchel transform of the flow Φ(ρ) as

L(v): ∀v ∈ [-w, v f ], L(v) = sup ρ∈[0,ρmax] (Φ(ρ) -vρ). (12)
Let us also introduce T = t -t s . The closed-form solution to [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i,ii and iii[END_REF] is known as Lax-Hopf formula, which reads

M (x, t) = inf{c(x -T v, t -T ) + T L(v)} s.t. (v, T ) ∈ [-w, v f ] × R + and (x -T v, t s ) ∈ Dom(c). (13)
Thus, M (x, t) is determined by the infimum of the infinite number of functions of the value condition (see also [START_REF] Aubin | Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints[END_REF]).

C. Explicit Solution for t → ∞

Our aim here is to obtain an explicit solution of (10) for a triangular FD (2) using [START_REF] Aw | Resurrection of "second order models" of traffic flow?[END_REF]. The calculations are presented in Appendix A. The final solution M (x, t) is obtained for large enough time, which equivalently means that the effect of initial conditions should have left the system. If Assumptions 1 and 2 hold, then the time can be estimated as in Appendix B:

∀(x, t) ∈ [0, L] × [t min , +∞) : M (x, t) = min t-x v f 0 φ in (τ )dτ + L 0 ρ 0 (s)ds, t-L-x w 0 φ out (τ )dτ + ρ max (L -x) . (14)

D. Problem Statement in H-J Formulation

We define the desired Moskowitz function (MF) similar to [START_REF] Daganzo | A behavioral theory of multi-lane traffic flow. Part I: long homogeneous freeway sections[END_REF]:

∀(x, t) ∈ [0, L] × [t min , +∞) : M d (x, t) = min t-x v f 0 φ in d (τ )dτ + L 0 ρ d0 (s)ds, t-L-x w 0 φ out d (τ )dτ + ρ max (L -x) . ( 15
)
Note that both M (x, t) and M d (x, t) are defined up to a constant since the starting point for the numeration of cars can be arbitrary. Therefore, we will state our problem in Hamilton-Jacobi formulation as a pointwise convergence of Moskowitz functions M (x, t) to M d (x, t) as t → ∞.

Problem 1. Given a desired state ρ d (x, t) solving Σ d , find control laws u in (t) and u out (t) for system Σ such that

∃M 0 ∈ R : ∀x ∈ [0, L] lim t→∞ (M (x, t) -M d (x, t)) = M 0 .
Remark 2. Physically, we need to control inflow and outflow such that all the "excess" cars (given by the integral difference in ρ 0 (x) and ρ d0 (x)) leave the domain, and then inflows and outflows match the desired ones. When the goal is achieved, the following conditions hold for ∀t ≥ t min :

(i) φ in (t)=φ in d (t), (ii) φ out (t)=φ out d (t), (iii) t-x v f 0 (φ in (τ ) -φ in d (τ )) dτ + ( 16
) L 0 (ρ 0 (s) -ρ d0 (s))ds = t-L-x w 0 (φ out (τ ) -φ out d (τ )) dτ.
The derivation of these conditions is given in Appendix III.

III. CONTROL LAW DESIGN

Theorem 1. Assume system Σ for which Assumptions 1 and 2 hold with the MF solution given by (14), and system Σ d for which Assumption 1 holds with the MF solution [START_REF] Ulbrich | A Sensitivity and Adjoint Calculus for Discontinuous Saolutions of Hyperbolic Conservation Laws with Source Terms[END_REF]. Then a control law that achieves the goal stated in Problem 1 is given by

(1) u in (t) = φ in d (t) -ke(t), (2)
u out (t) = φ out d (t) + ke(t), t ∈ R + where e(t) = L 0 (ρ(s, t) -ρ d (s, t)) ds and k > 0. (17)
Proof. This proof consists of five subsections: it is shown that e(t) goes to zero as t → ∞ in subsections 1) -3), and then we show that this is enough for the convergence of Moskowitz functions in subsections 4) and 5).

Error e(t) is defined as the difference in the overall number of cars in the real and the desired systems. Using (51) for M (0, t) we can rewrite the definition of error as

e(t) = M (0, t) -M d (0, t) + t 0 (φ out d (τ ) -φ out (τ )) dτ.
Now we can rewrite it using (52) for M (0, t) as

e(t) = t 0 (φ in (τ ) -φ out (τ ) + φ out d (τ ) -φ in d (τ )) dτ + L 0 (ρ 0 (s) -ρ 0 d (s)) ds. ( 18
)
Error dynamics are the time derivative of ( 18)

ė(t) = φ in (t) -φ out (t) -φ in d (t) + φ out d (t). ( 19
)
The main difficulty with the control of Σ is that φ in is not always equal to u in (and similarly φ out ), i.e., in some cases we lose the ability to impose any control u in (t) or u out (t) on the boundaries. Let us investigate this problem in more details.

1) Analysis of Flow Restrictions: By definition (8) inflows and outflows are time derivatives of M (0, t) and M (L, t), respectively. First, we focus on the inflow at the upstream boundary defined as φ in (t) = ∂M (0, t)/∂t. Let us express M (0, t) by taking the time integral of φ in (t) and get

M (0, t) = t 0 φ in (τ )dτ + M (0, 0), (20)
where M (0, 0) = L 0 ρ 0 (s)ds by (51) for (x, t) = (0, 0).

At the same time, by ( 14) for x = 0 we obtain

M (0, t) = min t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds, t-L ω 0 φ out (τ )dτ + Lρ max . (21)
Combining ( 20) with ( 21), we obtain a minimum problem:

t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds = min t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds, t-L ω 0 φ out (τ )dτ + Lφ max . (22)
From ( 22) one can see that the following must hold

t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds ≤ t-L ω 0 φ out (τ )dτ + Lφ max . ( 23
)
In case of equality in [START_REF] Strub | Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling[END_REF], we must provide that the righthand term grows more quickly than the left-hand term. Thus, by taking time derivative we obtain φ in (t) ≤ φ out (t -L w ). All this can be combined in the following property. We define a control restriction function for the downstream boundary

g in (t) = t-L ω 0 φ out (τ )dτ +Lρ max - t 0 φ in (τ )dτ - L 0 ρ 0 (s)ds,
which represents the difference between two arguments of the minimum. By [START_REF] Strub | Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling[END_REF] we obtain that g in (t) ≥ 0 always. Moreover, using the definition of φ in (t) given by (4), the condition on inflow restriction can be formulated as:

g in (t) > 0 : φ in (t) = u in (t), g in (t) = 0 : φ in (t) = min u in (t), φ out t - L w . ( 24
)
Note that the notation of control restriction should be understood as saturation control, with φ out t -L w being the saturation time-varying threshold.

Similarly, we proceed with M (L, t) to analyse φ out (t) for the downstream boundary, and get its control restriction function:

g out (t) = t-L v f 0 φ in (τ )dτ + L 0 ρ 0 (s)ds - t 0 φ out (τ )dτ,
and the following condition on outflow restriction

g out (t) > 0 : φ out (t) = u out (t), g out (t) = 0 : φ out (t) = min u out (t), φ in t - L v f . ( 25
)
Any boundary control can be imposed if g in > 0 and g out > 0.

Defining R(t) = L 0 ρ(s, t)ds and R 0 = R(0), and using the equality of ( 51) and ( 52)

for x = 0 we obtain ∀t ∈ R + R(t ) = R 0 + t 0 φ in (τ )dτ - t 0 φ out (τ )dτ. (26)
Thus, the control restriction functions can be rewritten as

g in (t) = Lρmax -R(t ) - t t-L w φ out (τ )dτ - t t φ in (τ )dτ, g out (t) = R(t ) - t t-L v f φ in (τ )dτ - t t φ out (τ )dτ. (27)
Note also that inflows and outflows are bounded, i.e. φ in ≤ φ max and φ out ≤ φ max , where φ max = v f ρ c . To find the time interval, when no control law can be imposed, we set g in (t) = 0 and then express R(t ) from ( 27):

R(t ) = Lρmax - t t-L w φ out (τ )dτ - t t φ in (τ )dτ ≥ Lρmax - t t-L w φmaxdτ - t t φmaxdτ = Lρmax - L w φmax = Lρc.
The same steps are performed for g out (t) = 0, and we get

g in (t) = 0 ⇒ R(t ) ≥ Lρ c ∀t ∈ t - L w , t , g out (t) = 0 ⇒ R(t ) ≤ Lρ c ∀t ∈ t - L v f , t . (28)
This means that not any control law can be imposed at the inflow boundary only if during the preceding time interval t -L w , t the mean density was bigger than the critical density (and inversely for the outflow boundary).

By Assumption 2, setting g in (t) = 0 in (27

) implies R(t) ≥ Lρ c + ε and R(t -L/w) ≥ Lρ c + ε, as well as g out (t) = 0 implies R(t) ≤ Lρ c -ε and R(t -L/v f ) ≤ Lρ c -ε.
Let us consider (28) to investigate whether it is possible that control can not be imposed at both boundaries simultaneously. We pick up some time point t such that g in (t) = 0 and some [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory[END_REF]. This is a contradiction, since satisfying both R(t) ≥ Lρ c + ε and R(t) ≤ Lρ c at the same t is impossible. Thus, the time point t when g out (t ) = 0 can occur at least after the interval L/v f after g in (t) = 0.

time point t ∈ [t, t + L/v f ] with g out (t ) = 0. As written above, g in (t) = 0 implies R(t) ≥ Lρ c + ε. However, t ∈ [t -L/v f , t ], thus R(t) ≤ Lρ c for g out (t ) = 0 by
Moreover, if t ≥ t + L/v f and g out (t ) = 0, then R(t -L/v f ) ≤ Lρ c -ε. The maximal inflow is φ max , therefore the difference in the integral densities R(t -L/v f ) -R(t) can be passed at least in time t -L/v f -t ≥ 2ε/φ max . Performing the same analysis for the inverse case, we conclude that

g in (t) = 0 ⇒ g out (t ) > 0∀t ∈ t, t + L v f + 2ε φmax , g out (t) = 0 ⇒ g in (t ) > 0∀t ∈ t, t + L w + 2ε φmax . (29)
Thus, it is impossible for two boundaries to be unable to accept the control simultaneously, and the periods of "uncontrollability" are separated in time by at least

L v f + 2ε φmax or L w + 2ε φmax . 2)
Dynamics of e(t): At each moment either one of control restriction functions or none of them is zero [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Computational methods[END_REF]. Hence, we separate the dynamics of e(t) into three possible cases.

1. Assume both g in (t) > 0 and g out (t) > 0. Then φ in (t) = φ in d (t) -ke(t) and φ out (t) = φ out d (t) + ke(t). According to [START_REF] Daganzo | A variational formulation of kinematic waves: basic theory and complex boundary conditions[END_REF], the error dynamics are given by

ė(t) = -2ke(t), (30)
and, thus, e(t) converges exponentially to zero.

2. Assume g in (t) = 0. Then φ in (t) ≤ φ in d (t) -ke(t) and φ out (t) = φ out d (t) + ke(t), which means ė(t) ≤ -2ke(t). (31)
Thus, e(t) > 0 implies even faster convergence to zero. If e(t) < 0, such dynamics can diverge from zero. However, it is possible to show that after a period of not being able to impose any control, the error will not be further away from zero than at the beginning of the period. Consider the desired control restriction function for the upstream boundary g in d (t):

g in d (t) = Lρ max -R d (t ) - t t-L w φ out d (τ )dτ - t t φ in d (τ )dτ ≥ 0, for t ∈ t - L w , t .
Using e(t ) = R(t ) -R d (t ), we obtain

g in d (t) -g in (t) = t t-L w (φ out (τ ) -φ out d (τ ))dτ + t t (φ in (τ ) -φ in d (τ ))dτ + e(t ) ≥ 0. (32)
ė(t) ≥ -2ke(t). ( 34
)
Using the same analysis as above for e(t) > 0, we obtain

e(t) + e t - L v f ≤ 0. ( 35
)
3) Proof that e(t) converges to zero: In the table below we have summarized three regimes of error dynamics:

Summary of error regimes Regime 1 g in (t) > 0, ė(t) = -2ke(t) g out (t) > 0 Regime 2 g in (t) = 0, ė(t) ≤ -2ke(t) g out (t ) > 0 ∀t ∈ t, t + L v f + 2ε φmax Regime 3 g in (t) > 0, ė(t) ≥ -2ke(t) g out (t ) = 0 ∀t ∈ t, t + L w + 2ε φmax
The regimes can alternate as depicted in Figure 2. In this part of the proof we will show that only positive error can enter regime 2 at some time, while it can enter regime 3 only being negative. Thus, if the error is positive in regime 1 (green circle), it either remains in regime 1 forever, where the exponential convergence to zero is guaranteed by [START_REF] Carlson | Local feedback-based mainstream traffic flow control on motorways using variable speed limits[END_REF], or it enters regime 2 (blue circle). Then, being positive, by [START_REF] Colombo | On the modeling and management of traffic[END_REF] the error converges to zero even faster than in regime 1. However, it can also become negative, and in this case the error might diverge from zero. Nevertheless, the divergence from zero can last only for a bounded time interval, and the absolute value of the error term can not exceed its value some time ago (see [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]). As this happens, the error enters again regime 1 as a negative term. It either stays there forever, or switches to regime 3, where it goes to zero even more quickly by [START_REF] Li | Optimal Control of Scalar Conservation Laws Using Linear/Quadratic Programming: Application to Transportation Networks[END_REF]. The rest can be described in a symmetric manner. Recall also that regimes 2 and 3 are always separated by at least L v f + 2ε φmax or L w + 2ε φmax . Further, we provide a strict proof of the exponential convergence of the error term to zero.

Imagine a time axis split into three types of intervals corresponding to three different regimes as shown in Figure 3. Recalling that regimes 2 and 3 cannot occur in a row, we can observe, e.g., a sequence like this: 12121313121.... Thus, it is possible that after regime 2 and then regime 1, the regime 2 comes again, since nothing prohibits g in (t) to become zero again almost immediately. We denote this sequence of regimes as "g in (t) = 0 sometimes" which is defined as the largest row of regimes 1 and 2 that starts and ends with regime 2 and does not contain any interval of regime 3. The same we can do with the regime sequence containing regimes 1 and 3 and call it "g out (t) = 0 sometimes". These sequences "g in (t) = 0 3. A possible error behaviour e(t) (thick black line). From left to right: divergence for "gout(t) = 0 sometimes" (in orange); exponential convergence in regime 1 (in green); fast convergence for "g in (t) = 0 sometimes" (in blue); then divergence for e(t ) < 0 ∀t ∈ [t i , ti ]; exponential convergence in regime 1. Blue empty circles are related to [START_REF] Reilly | Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering[END_REF].

sometimes" and "g out (t) = 0 sometimes" alternate strictly, always having regime 1 between them. Finally, for a time interval corresponding to the regime (or regime sequence) with index i we can define entrance time t i and exit time ti . By [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Computational methods[END_REF] we see that t

i -ti-1 ≥ L ω + 2ε φmax if the regime i is "g in (t) = 0 sometimes", and t i -ti-1 ≥ L v f + 2ε φmax if the regime i is "g out (t) = 0 sometimes".
Let us fix i corresponding to "g in (t) = 0 sometimes" (the other case is symmetric). First of all, by (33) we obtain

g in (t i ) = 0 and g in ( ti ) = 0. Therefore e(t i ) + e t i - L w ≥ 0 (36)
and e( ti ) + e ti -L w ≥ 0.

It is clear that t i -L w ≥ ti-1 , which means that the dynamics of e in the interval [t i -L w , t i ] are exponential. Thus, both e(t i ) and e t i -L w have the same sign, and by (36) they are both positive. A similar analysis can be done for the regime sequence "g out (t) = 0 sometimes", which means that from g out (t) = 0 to g in (t) = 0 the error term is positive and from g in (t) = 0 to g out (t) = 0 the error term is negative (and thus e( ti ) < 0). Consequently, inside each regime sequence i there should be a time point t i , when e(t i ) = 0. Now, by [START_REF] Reilly | Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering[END_REF] and using that e( ti ) is negative, we see that e ti -L w ≥ 0, which means that ti -L w ≤ t i (see Figure 3). On the time interval [ ti-1 , t i ] it is clear that the convergence is exponential (regime 1). On the time interval [t i , t i ] the dynamics are ė ≤ -2ke, and e(t) ≥ 0.

Time point ti -L w ∈ [ ti-1 , t i ], therefore e ti - L w ≤ e( ti-1 ) exp -2k( ti-L w -ti-1) .
We can write the same for the absolute values,

e ti - L w ≤ |e( ti-1 )| exp -2k( ti-L w -ti-1) .
Now from [START_REF] Reilly | Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering[END_REF] and the fact that e( ti ) < 0 we see that |e( ti

)| ≤ e ti -L w , thus |e( ti )| |e( ti-1 )| exp -2k( ti-L w -ti-1)
.

We can write titi-1

≥ t i -ti-1 ≥ L ω + 2ε φmax , which yields ti -L w -ti-1 ti -ti-1 = 1 - L w ti -ti-1 ≥ 1 - L w L ω + 2ε φmax = 2ε φmax L ω + 2ε φmax = 2εω Lφ max + 2εω
.

The time interval in the exponential function can be bounded

ti - L w -ti-1 ≥ 2εω Lφ max + 2εω
( titi-1 ) , which finally leads to

|e( ti )| ≤ |e( ti-1 )| exp -2k( 2εω Lφmax+2εω )( ti-ti-1) , (38)
which proves the exponential convergence to zero of e(t).

4) Proof of Integral Inflow Convergence:

To proceed we need to show that the integral of difference of inflows in the real and in the desired system has a limit, as it is required for the convergence of the Moskowitz functions as stated in Problem 1 (see next subsection, where the existence of this limit is used for the introduction of constant M 0 ):

∃ lim t→∞ t 0 (φ in (τ ) -φ in d (τ ))dτ. ( 39
)
By the Cauchy criterion for the convergence of functions, it suffices to show that

lim t1,t2→∞ t2 t1 (φ in (τ ) -φ in d (τ ))dτ = 0 ( 40
)
for any t 1 and t 2 , where t 2 > t 1 . First, we find an upper bound for this limit. By combining [START_REF] Aubin | Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints[END_REF] with ( 17) we obtain

φ in ≤ φ in d -ke(t), thus, we can write ∀t 1 , t 2 → ∞ t2 t1 (φ in (τ ) -φ in d (τ ))dτ ≤ -k t2 t1 e(τ )dτ → 0, which implies lim t1,t2→∞ t2 t1 (φ in (τ ) -φ in d (τ ))dτ ≤ 0. ( 41
)
Now let us estimate the lower bound for the limit [START_REF] Zhang | Coordinated variable speed limit, ramp metering and lane change control of highway traffic[END_REF]. Thereby, we distinguish two possible cases: 1) Assume g in (t) > 0. In this case:

t t-L w (φ in (τ ) -φ in d (τ ))dτ ≥ -k t t-L w |e(τ )| dτ.
2) Assume g in (t) = 0. Using (32) for t = t -L w , we get

t t-L w (φ in (τ ) -φ in d (τ ))dτ ≥ -e(t - L w ) ≥ -e(t - L w ) .
The combination of two latter expression yields

t t-L w (φ in (τ ) -φ in d (τ ))dτ ≥ -k t t-L w |e(τ )| dτ -e(t - L w ) (42)
Now let us divide the time interval [t 1 , t 2 ] into subintervals of length L/w. Thus, ( 42) can be rewritten for a larger time interval as

t2 t1 (φ in (τ ) -φ in d (τ )) dτ ≥ -k t2 t1 |e(τ )| dτ - t 2 -t 1 L/w n=0 e t 1 + n L w .
where the sum goes over intervals of size L/w, i.e., t 1 , t 1 + L/w, . . . , t 2 . If we take the time limit of the latter expression for t 1 , t 2 → ∞, both right-hand terms converge to zero, as for the sum term we can apply the integral test for convergence:

lim t1,t2→∞ t2 t1 (φ in (τ ) -φ in d (τ )) dτ ≥ 0. ( 43
)
The combination of ( 43) and (41) provides that the limit is zero, which proves the existence of the limit of the difference integral between inflows in the real and desired systems.

M 1 (x, t) = R 0 -R 0 d + t-x v f 0 (φ in (τ ) -φ in d (τ ))dτ, (44) M 2 (x, t) = t-L-x w 0 (φ out (τ ) -φ out d (τ ))dτ. ( 45
)
Notice that M 1 (x, t) and M 2 (x, t) correspond to the left and to the right side of (16) (iii), which must hold for large t. First of all, using e 0 = R 0 -R 0 d and the dynamics of e(t) given by ( 19), we obtain that ∀x ∈ [0, L]

lim t→∞ (M 1 (x, t) -M 2 (x, t)) = ∞ 0 (φ in (τ ) -φ in d (τ ) -φ out (τ ) + φ out d (τ ))dτ + e 0 = lim t→∞ e(t) = 0. (46)
Moreover, as it was proven in the previous subsection [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF], M 1 (x, t) has a limit. Therefore, we can define

M 0 := lim t→∞ M 1 (x, t),
and by (46) we get

lim t→∞ M 2 (x, t) = M 0 .
We can also define the error terms as

M1 (x, t) = M 1 (x, t) - M 0 and M2 (x, t) = M 2 (x, t) -M 0 .
Finally, recall that the desired Moskowitz function given by ( 15) can be expressed as

M d (x, t) = min M Up d (x, t), M Down d (x, t) ,
and using ( 14), ( 44) and ( 45) we obtain

M (x, t) = min{M Up d (x, t) + M 1 (x, t), M Down d (x, t) + M 2 (x, t)}, or M (x, t) = min{M Up d (x, t) + M1 (x, t), M Down d (x, t) + M2 (x, t)} + M 0 .
Minimum is a continuous function on both arguments, thus we obtain for t → ∞ that:

M (x, t) → M d (x, t) + M 0 ∀x ∈ [0, L], (47)
as stated in Problem 1.

After we have solved the problem stated in Hamilton-Jacobi formulation, let us establish the link to LWR formulation. Lemma 1. Problem 1 is equivalent to the integral convergence of densities over arbitrarily small intervals, i.e., ∀a, b:

0 ≤ a < b ≤ L we obtain lim t→∞ b a (ρ(s, t) -ρ d (s, t)) ds = 0,
where a and b can be arbitrarily close points in space.

Proof. By (51) we can write

M (a, t) -M (b, t)= L a ρ (s, t) ds- L b ρ (s, t) ds = b a ρ(s, t)ds, (48)
and

M d (a, t) -M d (b, t) = b a ρ d (s, t) ds. (49)
For Remark 3. Notice that pointwise convergence of two functions does not imply the convergence for their derivatives in any of L p norms. However, we reach the equality of densities over arbitrarily small intervals, which means that the state approximates the desired trajectory as time goes to infinity.

x = a and x = b in Problem 1 we get M (a, t) → M d (a, t) + M 0 , M (b, t) → M d (b, t) + M 0 . This is equivalent to M (a, t) -M (b, t) → M d (a, t) -M d (b, t),
Remark 4. Note that Assumption 2 is non-limiting. Indeed, it requires that the flow integral over time T is less than its maximum value by at least ε, which is always possible, except when vehicles enter and leave the system at maximum rate during T. Obviously, in this case, it is also possible to reach the goal. The need to use Assumption 2 comes from the fact that at maximum flow by [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory[END_REF] both g in (t) = 0 and g out (t) = 0 for the same t, which means that with the slightest fluctuation a boundary becomes "uncontrollable". Thus, the state with maximum flows at both boundaries during time T is "unstable", and therefore it is impossible to prove the exponential convergence of the error term e(t).

A. Numerical Scheme

We illustrate the efficiency of feedback control terms ( 17) by driving a state being inially completely in a traffic jam to a desired space-and time-dependent vehicle density being in a mixed regime. The numerical scheme we use is a standard Godunov scheme [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF] with a space interval [0, L] divided into n = 500 cells, i.e. ∆x = L/n, and a time discretization step ∆t chosen such that the Courant-Friedrich-Lewy condition is satisfied [START_REF] Courant | On the Partial Difference Equations of Mathematical Physics[END_REF]. This condition is necessary to prevent the violation of information flow.

The integral corresponding to the feedback term is computed using the Riemann summation over cells i ∈ {2..n -1}. Recall that in general the minimum is not always resolved to the benefit of the control variable. In this section we demonstrate the efficiency of using a state feedback for a road that is initially empty at the entry and in the traffic jam at the exit. Thus, both boundaries accept controls, i.e. ϕ in (t) = u in (t) and ϕ out (t) = u out (t).

For the simulation we set the following parameters (which are taken from the real traffic data [START_REF] De Wit | Grenoble Traffic Lab: An Experimental Platform for Advanced Traffic Monitoring and Forecasting [Applications of Control[END_REF]): v f = 16.67 m/s, w = 7.14 m/s, ρ max = 0.181, veh/m, ρ c = 0.054 veh/m, L = 1000 m, and for initial time the system is given as

ρ 0 (0 ≤ x < 1/4L) = 0, and ρ 0 (1/4L ≤ x ≤ L) = ρ max .
Thus, we will consider the system being almost completely in the traffic jam as initial condition.

As a target state we consider a vehicle density in a mixed case regime, whose evolution is given by "ghost" cells (which are set by copying the value from the neighbour cell) with ρ d0 (t) = 0.04+0.04 sin (t/8) and ρ d L (t) = 0.1+0.06 sin (t/4) ,

The boundary values are prescribed in terms of densities, which are transformed into flows by using the supply-demand formulation [START_REF] Moskowitz | Discussion of freeway level of service as influenced by volume and capacity characteristics by dr drew and cj keese[END_REF]. We will demonstrate, how the feedback term given by -ke(t) for the upstream and +ke(t) for the downstream boundary improves the result and provides the asymptotic convergence targeting the desired profile. Two control strategies are compared:

1

)

B. Results

In Figure 4a) the evolution of desired vehicle density being in a mixed regime is illustrated. The results of achieving this state with and without feedback are shown on the plots below 4 (c) -(f). Thereby, the left column shows the evolution of state, whereas on the right column one can see the evolution of the absolute difference between the real and the desired states. Plots (c) and (d) illustrate the result if no feedback is applied on the boundaries, while plots (e) and (f) depict the situation if feedback with gains k = 0.1 is applied. The corresponding error behaviour for different gains is shown on plot (b).

Comparing the plots we can see that the control including the feedback term performs considerably better. Without feedback the congested regime almost completely occupies the domain as time runs, while the feedback term makes the system approach the desired state after the time inferior to the minimal controllability time, which is

T c = L v f + L |w| = 200.5s.
Note that an open-loop control (such as applying absorbing boundary conditions until the road becomes empty, and then applying desired inflows and outflows) will not achieve the goal at all due to the difference in initial densities [START_REF] Greenberg | Congestion on multilane highways[END_REF].

IV. CONCLUSIONS

We have designed boundary control laws in order to track a target space-and time-dependent trajectory. Physically, the real and the desired states correspond to the vehicle density. Both states are the solutions of LWR PDE with triangular fundamental diagram and are allowed to be in the mixed regime. This means that our controller allows us to drive any state, being in a partly congested partly free-flow regime or even being completely congested, to some desired state being in any regime we want. Achieving such a goal is possible using a feedback term, which physically corresponds to the difference between the number of vehicles in our system and in the desired one multiplied by some controller gain. This result was obtained by using the explicit solution formula to H-J, which enabled us to formulate and analyse the control restriction functions. The explicit solution was obtained using the properties of triangular FD due to the convenient shape of its Legendre transform. The numerical example verified the results and illustrated that feedback is absolutely necessary to achieve the goal.

For the future studies it might be challenging to extend this result to a large-scale, i.e., controlling the vehicle density being in any regime in the whole city. Moreover, it could be interesting to derive an explicit solution for a H-J PDE for other shapes of fundamental diagram.

APPENDIX I SOLUTION OF HAMILTON-JACOBI EQUATION

For a triangular FD the convex transform of FD [START_REF] Evans | Partial Differential Equations[END_REF] yields

L(v) = φ max -ρ c v ∀v ∈ [-w, v f ]. (50)
Being an integral function, M (x, t) is defined up to a constant. Let us choose it such that M (L, 0) = 0. From the geometrical interpretation and by integrating (8), we can equivalently write

∀(x, t) ∈ [0, L] × R + M (x, t) = t 0 φ out (τ )dτ + L x ρ(s, t)ds, (51)
Using these functions and the Lax-Hopf formula (13), we can restate the solution to (10) as a minimum of three new functions M Up (x, t), M Down (x, t) and M Ini (x, t) depending on both space and time, which are solutions to the Lax-Hopf formula associated only to one boundary or initial condition. For example, M Up (x, t) is obtained by

M Up (x, t) = inf{c(x -T v, t -T ) + T L(v)} s.t. (v, T ) ∈ [-w, v f ] × R + and (x -T v, t -T ) ∈ {0} × R + , (56)
and the same formula yields M Down (x, t) for (

x-T v, t-T ) ∈ {L} × R + and M Ini (x, t) for (x -T v, t -T ) ∈ [0, L] × {0}.
By properties of infimum the original Lax-Hopf formula can thus be rewritten as M (x, t) = min (M Ini (x, t), M Up (x, t), M Down (x, t)) . (57)

Now we calculate all these functions explicitly. Note that we consider the solutions only for large time. Thus, in the following we will write the solutions only for

t ≥ max x v f , L -x w . (58)

A. Upstream Boundary Condition

The solution M Up (x, t) is related to the number of vehicles that originate from the boundary x s = 0 and start travelling at t s := t -T . By looking at [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory[END_REF], we establish that c(x -T v, t -T ) = M Up (t -T ) in (56). The equality to zero of the first argument of c is provided by T

= x v , where v ∈ [-w, v f ].
Since T is positive by definition, the minimal value of the time interval should be T min = x v f , which is consistent with (58). Using (50), this yields the following problem:

M Up (x, t) = inf T ∈[ x v f ,+∞] [M Up (t -T ) + T φ max ] -xρ c .
Using (53), the infimum problem reads

M Up (x, t) = inf T ∈[ x v f ,+∞]   t-T 0 φ in (τ )dτ + T φ max   + L 0 ρ 0 (s)ds -xρ c ,
which by using T φ max = T t-T φ max dτ can be rewritten as

M Up (x, t) = inf T ∈[ x v f ,+∞]   t t-T (φ max -φ in (τ )) dτ   + L 0 ρ 0 (s)ds + t 0 φ in (τ )dτ -xρ c ,
The infimum is provided by T * = T min = x v f , i.e. the correct solution is related to the path along which the vehicle has the maximal velocity. Substituting T = T min , we get

M Up (x, t) = t-x v f 0 φ in (τ )dτ + L 0 ρ 0 (s)ds. (59)

B. Downstream Boundary Condition

Further, we need to calculate M Down (x, t), which is related to the downstream boundary x s = L. The calculation is done performing the same steps as for M Up (x, t), and we obtain

M Down (x, t) = t-L-x w 0 φ out (τ )dτ + ρ max (L -x). (60)

C. Initial Condition

Finally, we calculate the function M Ini (x, t) determined by the vehicle with known label at t s = 0 (55). The equality to zero of the time argument in the value condition function is provided by t = T . This yields

M Ini (x, t) = inf v∈[-w,v f ] [M Ini (x -tv) -tρ c v] + tφ max .
Using the definition of M Ini (x) from (55), we obtain

M Ini (x, t) = inf v∈[-w,v f ]   L x-tv ρ 0 (s)ds -tρ c v   + tφ max .
We decompose the integral as -ρ c ds:

M Ini (x, t) = L x ρ 0 (s)ds + tφmax + inf v∈[-w,v f ] x x-tv (ρ 0 (s) -ρc)ds.
We define y = x -tv and take the infimum over y: Note that the coordinates can not lie outside the interval [0, L]: To obtain a unique solution, we insert the results from (59), (60) and (61) in (57), and we get the minimum function. However, we still need to solve one infinum problem due to (61), which requires information on crossings of the critical value by the density at initial condition. In (61) the only time depending term is tφ max , thus dM Ini /dt = φ max . Taking the time derivative for other terms yields dM Up /dt = φ in (tx/v f ) and dM Down /dt = φ out (t-(L-x)/w). By φ(ρ(•, •)) ≤ φ max we see that except for some special cases M Ini grows faster than the others. In Appendix II we have estimated the minimal time t min (67) needed for initial conditions to leave the system Thus, M Ini can be excluded from the minimum operator ∀t ∈ [t min , +∞): M (x, t) = min

M Ini (x, t) =
M Ini (x, t) =

APPENDIX II TIME FOR NEGLECTING THE INITIAL CONDITIONS

We want to find the minimal time t min , after which the initial conditions have definitely left the system, i.e., ∀(x, t) ∈ [0, L] × [t min , +∞): M Ini (x, t) ≥ M Up (x, t) or M Ini (x, t) ≥ M Down (x, t). Let us first establish the time for which M Ini (x, t) ≥ M Up (x, t), then we will do the same for M Down (x, t), and then the final value will be the minimum from of cases.

Thus, using M Up (x, t) from (59) and M Ini (x, t) from (61)

with tφ max = (

) 62
We can estimate an upper bound of (62) by using:

x v f φ max ≥ 0, - Further, we use Assumption 2 to make the following bound:

t-x v f 0 (φ max -φ in (τ )) dτ ≥ t -x v f T ε,
where ε > 0 and T = min L v f , L w . Thus, we can further bound (63) by:

M Ini (x, t) -M Up (x, t) ≥     t -L v f T     ε - L 0 (ρmax + ρc) ds (64)
Finally, we can determine for which t the expression (64) is non-negative:

t ≥ L v f +     1 ε L 0 (ρ max + ρ c )     min L v f , L w (65)
Following the same steps, we obtain a similar result for t such that M Ini (x, t) -M Down (x, t) ≥ 0:

t ≥ L w +     1 ε L 0 (ρ max + ρ c )     min L v f , L w (66)
The combination of (65) and (66) yields the lowest time bound for the initial conditions to leave the system:

t min = min L v f , L w   1 +     1 ε L 0 (ρ max + ρ c )       . ( 67
)

APPENDIX III NECESSARY CONDITIONS FOR TRACKING DESIRED STATE

In accordance with Problem 1, we should find u in (t) and u out (t) such that the asymptotic equality of M (x, t) and M d (x, t) up to some constant M 0 is guaranteed. For the equality of two min functions ( 14) and [START_REF] Ulbrich | A Sensitivity and Adjoint Calculus for Discontinuous Saolutions of Hyperbolic Conservation Laws with Source Terms[END_REF], it is suffices to provide the equality of their arguments, thus ∀ (x, t) ∈ (68) Firstly, by taking the time derivative of (68), we see that in the steady-state φ in (t) ≡ φ in d (t) and φ out (t) ≡ φ out d (t).

Secondly, by expressing M 0 from both parts of (68), we obtain the necessary condition [START_REF] Wong | A multi-class traffic flow model: an extension of LWR model with heterogeneous drivers[END_REF] to track ρ d .

Fig. 1 .

 1 Fig. 1. Triangular Fundamental Diagram: free-flow regime (in green) and congested regime (in red).

3 Fig. 2 .

 32 Fig. 2. Diagram of regimes illustrating how they can alternate. Arrows denote possible regime switches. FC, CC and FF are used to denote regimes on the boundaries, where F stays for free-flow and C for congested regime.

5 )

 5 Proof of Convergence of Moskowitz Functions: Finally, let us define two auxiliary Moskowitz functions

  which by (48) and (49) we rewrite as b a ρ (s, t) ds → b a ρ d (s, t) ds.

  (a) ρ d (x, t) (b) ||ρ -ρ d || L1 (c) ρ(x, t), no feedback (d) |ρ -ρ d |, no feedback (e) ρ(x, t), k = 0.1 (f) |ρ -ρ d |, k = 0.1

Fig. 4 .

 4 Fig. 4. (a) Desired profile in space-time, (b) L 1 error as a function of time for different control gains. Spatio-temporal evolution of the density (left) and of the absolute difference between the real and the target state (right) for: (c),(d) k = 0; (e),(f) k = 0.1.

  t)ds. (52) Now we can determine the value condition function which implies the calculation of M Up (t), M Down (t) and M Ini (x): M Up (t) = c(0, t) = t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds. (53) M Down (t) = c(L, t) = t 0 φ out (τ )dτ. (54) M Ini (x) = c(x, 0) = L x ρ 0 (s)ds.

  ρ 0 (s)ds, and the second term as -tρ c v = x x-tv

L x ρ 0

 0 (s)ds + tφmax + inf y∈[x-tv f ,x+tw] x y (ρ 0 (s) -ρc)ds.

  (s)-ρc)ds. (61) D. Unique Solution M(x, t)

φ

  out (τ )dτ + ρ max (L -x) .

φ

  max dτ + x v f φ max , we can write ∀(x, t) ∈ [0, L] × max x v f , L-x w , +∞ that M Ini (x, t) -M Up (x, t) = (s) -ρ c ) ds.

0 (

 0 (s) -ρ c ) ds ≥ -L 0 ρ c ds,and, thus, we getM Ini (x, t) -M Up (x, t) ≥ -φ in (τ )) dτ -L ρ max + ρ c ) ds (63)

[ 0 ,φ

 0 L] × [t min , +∞) we get out d (τ )dτ + ρ max (L -x) + M 0 = t-L-x w 0 φ out (τ )dτ + ρ max (L -x).

  We substitute t = t to get the first inequality, and then t = t -L w to get the second one. The sum of these inequalities yields Assume g out (t) = 0. Then φ out (t) ≤ φ out d (t) + ke(t) and φ in (t) = φ in d (t) -ke(t), which yields

	e(t) + e t -	L w	≥ 0.	(33)
	3.

Using the properties φ in (t) ≤ φ in d (t) -ke(t) and φ out (t) = φ out d (t) + ke(t), we obtain e(t ) + k t t-L w e(τ )dτ -k t t e(τ )dτ ≥ 0.

  No feedback is performed, i.e. only u in (t) = φ in d (t) and u out (t) = φ out d (t). 2) Feedback terms u in (t) = φ in d (t) -ke(t), u out (t) = φ out d (t)+ke(t) are applied.

This work is a part of Scale-FreeBack project that received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement N 694209).

L.


Kelly Ribeiro De Faria

Daniel Sadarnac

Charif Karimi

Larbi Bendani

Gang Yang

Design of an Unbalanced High Power Three-Winding Planar Transformer for Electric Vehicle Application

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In order to increase converter's power density, planar transformer has become an interesting solution; due to its singularity and manufacturing process, it becomes a key component in power converter design. In addition, to reduce the number of components and increase the power density of Electric Vehicle power systems, multi-functional converters have become a frequent demand. One of the recent designs for EV power systems is the use of a three-port converter used to integrate parts of the on-board charger (OBC) used to charge the propulsion battery; and the low/high voltage DC/DC converter (LDC) used to charge the auxiliary low voltage battery. One of the examples is presented in [START_REF] Ma | An integrated design of power converters for electric vehicles[END_REF], where the power flow between the ports is controlled as in the Three-port active bridge converter. Normally, for these applications, the OBC is designed for 3.3 kW or 11 kW, while the LDC usually works with 2.2 kW or 5 kW. The focus of this paper is to use the planar transformer technology to design a three-winding transformer that integrates both structures as represented in Fig. 1. Due to the high difference between the voltage and power levels into the ports, the design of the transformer requires parallel winding to support the high current in the low voltage side.

In addition, nowadays the converters are being controlled at higher frequencies and an accurate prediction of power loss is critical for the application. In [START_REF] Abdi | Investigation of Current Sharing in Paralleled Winding at High Frequency Transformers[END_REF], the current sharing among parallel winding has been discussed and shows the importance of a balanced distribution to reduce power loss and temperature. The current balancing can be improved by using interleaved configurations, however, due to the minimum insulation thickness between low and high voltage sides required for Electric Vehicle applications, interleaved configurations associated with a high number of turns ratio could increase the difficulty in the use of planar transformers. In this paper, three different design configurations are proposed for the three-winding planar transformer, operating for the OBC and the LDC converters. The three configurations are proposed to reduce the cost and facilitate the final manufacturing process of the unbalanced transformer. In order to estimate a more precisely power loss on the conductor layers, a complement to the commonly used calculation method is presented in this text to help in the selection of the transformer with the best performance, taking care of the parallel winding current distribution problem. The method presented here is easily integrated into a program routine and can be applied for more than two-winding transformers or in the presence of an air gap. This paper is organized as follows: Section 2 presents the calculation method to estimate the power loss distribution on the conductor layers of a planar structure. In section 3, is presented the design of the three-winding planar transformer for OBC and LDC for a specific power flow condition and the power losses are estimated by the method presented in section 2. In section 4, the results obtained are compared with Finite Element Analysis (FEA) results and an optimal configuration is identified.

The analytic approach to estimate power loss in the planar transformer

When it is necessary to use parallel-connected layers, by considering only Dowell assumptions [START_REF] Dowell | Effects of Eddy currents in transformer windings[END_REF], the conductor power loss is not well estimated. This occurs because the total current flowing in each conductor is considered unaltered and this is not true for parallel-connected layers. When the frequency increases the current sharing between the layers needs to be identified before the analysis of power losses. This difficulty becomes more evident in a more than two windingtransformer, as in the case of a three-winding transformer, or when the winding currents are not in phase.

Most of the analysis of power loss in planar transformer, considers that the frequency will only alter the current distribution across the conductors, but not the total current flowing in each conductor.

Due to the difficulty in to estimate the current distribution, normally the problem is addressed to Finite Element Simulations. In [START_REF] Asensi | Automatized Connection of the layers of planar transformers with parallel windings to improve the component behavior[END_REF] an analytic method is proposed to estimate power losses for parallel winding connections and improve the transformer configuration, but it needs to use FEA at first to determine the self and mutual impedances. In [START_REF] Margueron | Complete analytical calculation of static leakage parameters[END_REF] a method to determine static leakage inductances is presented and in [START_REF] Ouyang | Calculation of leakage inductance for high-frequency transformers[END_REF] the frequency influence appears in the current distribution inside the copper layers, but since it is applied to a planar inductance design, it does not have the problem of parallel-connected layers or different phases.

The method presented here can be solved very quickly by a program routine and allows further improvements to an optimized version. It is a continuation of the work presented in [START_REF] Sadarnac | Du composant magnétique à l'électronique de puissance -Analyse, modélisation, conception, dimensionnement des transformateurs, inductances, convertisseurs, Ellipses[END_REF].

The method can be explained by using the magnetic structure from Fig. 2. The N's conductor layers are distributed in the schematic around the magnetic central-leg. A small air gap is added to the structure. Some assumptions are made: The magnetic flux strength is perpendicular to the central leg of the magnetic core; the core permeability is infinite and the gap is small to prevent that the field lines go too far from the central leg. At each conductor layer is considered a current ik and a magnetic flux produced by this current, ØK. Outside the conductive region, a magneto motive force (MMF), Jk, is created and it is the image of the total current flowing into that conductor slice. φK is the magnetic flux in the insulation layers between the conductors. The total magnetic flux just after the conductor number k is represented by 𝜓k and is the contribution of all the layers placed on its left side. In Fig. 3, we have a single layer divided into subsections. In each sub-section flows a current portion Δi(x). The voltage drop on the x-axis between the portions x and x+Δx is the equivalent variation of the magnetic flux, ϕ(x+Δx) and ϕ(x):

Δi (x + Δx) -𝛥𝑖 (𝑥) = 1 𝑟 𝑑[𝜙(𝑥+𝛥𝑥)-𝜙(𝑥)] 𝑑𝑡 (1)
𝐽 = ∫ 𝐵 𝜇 𝑜 𝑑𝑦 = ∫ Δ∅ 𝜇 𝑜 (Δ𝐿 Δ𝑥) 𝑑𝑦 = Δ∅ 𝜇 𝑜 Δ𝑥 ∫ 𝑑𝑦 Δ𝐿 (2)
The resistance of each portion of the copper layer can be written as:

1 𝛥𝑟 = ∫ Δ𝑥 𝜌 𝛥𝐿 𝑑𝑦 = Δ𝑥 𝜌 ∫ 𝑑𝑦 Δ𝐿 (3)
Applying the Eq. (3) in Eq. ( 2) we obtain the magnetic flux that depends on the resistance:

∑ Δ∅ = 𝐽 𝜇 𝑜 ∑ Δ𝑥 ∫ 𝑑𝑦 Δ𝐿 = 𝐽 𝜇 𝑜 Δ𝑥 2 𝜌 𝑟 (4)
The Eq. ( 1) and Eq. ( 4) are equivalents:

𝑑𝜙 𝑑𝑡 = 𝑑𝐽 𝑑𝑡 𝜇 𝑜 Δ𝑥 2 𝑟 𝜌 = 𝑟 [𝛥𝑖(𝑥 + 𝛥𝑥) -𝛥𝑖(𝑥)] (5)
Supposing sinusoidal current waveform, the solution for the MMF, JK, is obtained from Eq. ( 6):

𝑗𝜔 𝐽 𝜇 𝑜 𝜌 = [𝛥𝑖(𝑥 + 𝛥𝑥) -𝛥𝑖(𝑥)] 𝛥𝑥 2 = 𝑑 2 𝐽 𝑑𝑥 2 (6)
J = J Ak . 𝑒 (1+𝑗)𝑥 𝑥 𝑜 + 𝐽 𝐵𝑘 . 𝑒 - (1+𝑗)𝑥 𝑥 𝑜 , 𝑥 𝑜 =√ 2𝜌 𝜇 𝑜 𝜔 (7)
The coefficients, JAk and JBk in Eq. ( 7), are determined by the conditions at x=0 and x=e, Jk and Jk+1, respectively, where 𝛼=(1+j)/xo.

J Ak = 𝐽 𝐾+1 -𝐽 𝑘 . 𝑒 -𝛼𝑒 𝑒 𝛼𝑒 -𝑒 -𝛼𝑒 J Bk = -𝐽 𝑘 𝑒 𝛼𝑒 -𝐽 𝑘+1 𝑒 𝛼𝑒 -𝑒 -𝛼𝑒 (8)
The voltage drop in each portion of the conductor layer can now be calculated:

v k = 𝑗𝑤 𝜓 𝑘 + 𝑟 𝑑𝐽 𝑑𝑥 (𝑥=𝐽 𝑘 -𝐽 𝑘+1 ) (9)
v k = 𝑗𝑤 𝜓 𝑘 + 𝑟 𝛥𝑥 𝛼 (𝐽 𝐴𝑘 -𝐽 𝐵𝑘 ) (10)
The total magnetic flux in the center of the magnetic core will be the contribution of the magnetic flux flowing between the conductors, 𝜑k, and flowing in the conductors, ϕk.

ψ k = ∑ ∅ 𝑛 𝑁 𝑛=𝑘 + ∑ 𝜑 𝑛 𝑁+1 𝑛=𝑘+1 (11)
ϕ k = 𝜇 𝑜 𝐿 𝑐 𝛼 ℎ . 𝑒 𝛼𝑒 + 𝑒 -𝛼𝑒 -2 𝑒 𝛼𝑒 -𝑒 -𝛼𝑒 (𝐽 𝑘 + 𝐽 𝑘+1 ) (12)
φ k = 𝜇 𝑜 𝐿 𝑐 𝛼 ℎ . 𝑒𝑖 𝑘 𝐽 𝑘 , ( 13
)
Where eik is the insulation thickness. In Eq. ( 9) we see that the voltage drop in a single layer is the contribution of the magnetic fluxes ϕk and 𝜑k, flowing in the other conductors and in the insulating surfaces, plus the contribution that becomes of its own current distribution, which is also the contribution of the skin effect. The expressions in Eq. ( 7), (11), ( 12) and (13) show the linear dependence of the voltage difference created in a single layer, vk, and the MMF, JK, generated by each conductor portion. Now we only have to find the solution of the system with the linear equations. The system has N voltages vk linked to the (N+1) MMF, Jk. By using the information about the winding connections and the winding electrical conditions, as voltage or current, we can find the solution associated to the linear system. There are necessary (Nw) conditions to describe the winding connections, such as voltage and current, where Nw is the number of windings. The indication of voltage and current for a same winding can also contain a phase between them different from zero.

For turns that are series connected, the total current does not change from one layer to another.

For winding where is necessary to include parallel connections, the voltage drop between x=[0: e] is the same for all the layers. The last condition is that the MMF=0 at the end of the layout arrangement is zero, J1=0.

A matrix (2N+1) x (2N+1) is then obtained relating JK and 𝜓k, using Eq. ( 9) and Eq. ( 11).

[

ψ 1 ψ k … ] = [ 𝜇 𝑜 𝐿 𝑐 𝛼 ℎ . 𝑒𝑖 1 … ] [ 𝐽 1 𝐽 𝑘 … ] (14)
A routine program in Matlab consisting in to inform the characteristics of the layer conductors, the space between conductors, dimensions of the magnetic material, frequency, the arrangement of the layers and the electrical conditions was used to verify the power loss of the transformers proposed in section 3.

Design of the three-winding planar transformer

The application addressed in this section concerns the design of a three-winding planar transformer as an integrated part of a three-port converter that integrates the OBC and LDC converters, as it has been discussed in section 1.

In this context, the nominal condition of the transformer operation is the case when the electrical grid through the Power Factor Corrector (PFC) charges both batteries from Fig. 1, VHV and VLV. VDC provides 5 kW for VHV and 2 kW for VLV as can be seen in Tab. 1. The converter operation is not detailed in this text, but it is considered that the switching frequency is kept constant at 300 kHz for all the ports. The E planar core shape is a plausible choice since it allows higher power density integration. In addition, at Fsw=300 kHz, the 3F36 and the 3F4 from Ferroxcube are the candidates pre-selected. The closest magnetic core is selected for a maximum current density of Ji=20 A/mm 2 Three proposed arrangements are presented in Fig. 4. The three configurations have the same conductor layout for both high voltage sides. The maximum PCB thickness allowed in this configuration is 3.2 mm. The transformer parameters are presented in Tab. 2. The design consists of 3 turns by layer connected in series for port 1 (P), then 8 layers also series connected to attend the turns ratio. For port 2 (S), the same configuration is used. The port 3 (T) can have between 5 or 8 layers connected in parallel to handle the high current. In the 1° configuration, the insulation thickness can be reduced because the low voltage side does not participate in the interleaving layout. This only occurs because the highest insulation thickness is always required between the layers P and T or between the layers S and T. Considering first, a linear behavior for the magnetic field inside the conductive layers and second, the current equally distributed between the parallel-connected layers, the magneto-motive force is drawn in Fig. 4. We see in the first configuration, at the low voltage side, that the parallel layers seem to behave as a conductor block with a big surface area with a substantial influence of the skin effect. The unbalanced magnetic field profile generates a very poor current distribution on the parallel-connected layers.

In the 2° configuration, to improve the magnetic field distribution, the layers T are interleaved between P and the S type layers, but its fabrication process and mechanical assembling are more difficult. Due to the difficulty in the mechanical assembling, the 3° configuration is proposed, where the T layers are set to be the external layers of each PCB to make easier the assembling process. The last proposition is supposed to represent a midterm between the first and second arrangements considering efficiency and fabrication complexity. The estimated MMF is illustrated on the left side of each configuration for the conditions in Tab. 1. Note that the first configuration seems to have the poorest performance and the last one seems to be acceptable.

Using the analytic method to estimate power losses

Now we are going to use the analytic procedure described in section 2.

Using the analytic approach, first we need to define the connection conditions. Only three conditions sets the three-winding transformer, voltage or currents, vk or Jk and these conditions do not need to be defined for all the windings. For example, we can define the voltage and current for the first winding and the current for the second winding only, as in Tab. 3.

The current in the first winding and in the secondary winding does not change from one layer to another since they are series-connected. However, for the third winding, the voltage is the one kept the same for all the layers. With that information, 25 equations relate the voltage and the MMF, Eq. ( 9) and Eq. ( 11). The magnetic flux density and the current density of each conductor layer in the first arrangement of Fig. 4 is presented in Fig. 5.

The magnetic flux density contains real and imaginary parts and the x-axis represents the layer order, from the bottom to the top layer. Note that the current in the parallel layers T is concentrated in the internal layers. Figures 6 and7 present the results for the fullinterleaved and partial-interleaved arrangements.

As we can see from the results, the partialinterleaved arrangement has higher power loss than the first configuration.

Finite-elements analysis simulations

In order to validate the results obtained, Finite Elements Analysis is now used to confirm the profile obtained with the analytic method.

Since in the analytic study, the magnetic field intensity is considered as being perpendicular to the central-leg of the magnetic core, a 2D-FEA analysis will be enough to verify the profile of the frequency dependent magnetic flux and current density inside the conductor layers. The software used is the FEMM. The same layout configurations described before and the electrical conditions are indicated in the analysis. The distances between the conductor layers are configured and the frequency is set at 300 kHz. Figure 8 shows the magnetic field intensity and current density for the non-interleaved arrangement. Figures 9 and 10 present the results for the full interleaved and the partial-interleaved arrangements, respectively. We can see that the magnetic field profile obtained with the simulation is very similar to the results obtained with the analytic approach. In the three evaluated configurations, we see that the second arrangement has the best profile for current distribution and reduced power losses. The third configuration, however, has the poorest performance. When the method for power loss calculation considers a linear magnetic field variation inside the conductor layers and that the total current in the conductor is unaltered for parallel-connected layers, the third configuration seems to have an acceptable performance. Nevertheless, in reality, if we consider the frequency dependency for the magnetic lines inside the conductors and that the current is not homogeneous distributed between the parallel layers, the third configuration has higher power losses, as can be seen in Fig. 11. These results are correctly estimated with the proposed modified calculation and by FEA. The analytic method offers a simple, precise and fast solution to the problem and can be used in a pre-design phase for example to compare different arrangements as it has been made in this paper. The method also allows the determination of a physical equivalent model for a transformer with N windings. For example, as it was made in section 3, the application of some conditions in the windings allows determine an impedance matrix or the leakage flux between them by powering up each of the windings while keeping the others open. In this paper, the design of a three-winding planar transformer for EV application was presented. The transformer is studied as an important part of the integrated system for OBC and LDC converters, responsible for the power transfer between high and low voltage batteries in the EV. Three different configurations were proposed to make easier its fabrication. Owing to the fact that is necessary parallel-connected layers to handle the high current on the low voltage side, a modified analytic method was studied to determine the current and the power loss distribution. The method allows to quickly highlight the drawbacks of the planar magnetic design. In the application discussed here, it helped in the selection of the configuration with the best performance. The results obtained with the analytic methodology and compared with results from FEA show that the second configuration presents a better power loss distribution for this specific application.

Fig. 1 :

 1 Fig. 1: Three-port converter integrating part of the OBC and the LDC converters in EV.

Fig. 2 :Fig. 3 :

 23 Fig. 2: The planar structure used as reference

Tab. 1 :

 1 Design requirements for the application example. Switching frequency required at 300 kHz.

Fig. 4 :

 4 Fig. 4: Transformer arrangement propositions and magneto-motive force estimated with approximations for the current distribution and magnetic field. A) Noninterleaved. B) Full-interleaved. C) Partial-interleaved.

Fig. 5 :

 5 Fig. 5: Magnetic flux density and current density for the non-interleaved arrangement. Red: imaginary parts.

Fig. 6 :

 6 Fig. 6: Magnetic flux density and current density for the full-interleaved arrangement.

Fig. 7 :

 7 Fig. 7: Magnetic flux density and current density for the partial-interleaved arrangement.

Fig. 8 :

 8 Fig. 8: Magnetic flux density and current density for the non-interleaved arrangement obtained with FEA.

Fig. 9 :

 9 Fig. 9: Magnetic flux density and current density for the full-interleaved arrangement obtained with FEA.

Fig. 10 :

 10 Fig. 10: Magnetic flux density and current density for the partial interleaved arrangement obtained with FEA.

Fig. 11 :

 11 Fig. 11: Total power loss obtained with the analytic method and with the simplification of considering a linear distribution of the magnetic flux on the conductor layers.