Emiliano Lorini

On the logical foundations of moral agency *

The aim of this work is to provide a logical analysis of moral agency. Although this concept has been extensively studied in social philosophy and in social sciences, it has been far less studied in the field of deontic logic and multi-agent systems (MASs). We discuss different aspects of moral agency such as the distinction between desires and moral values and the concept of moral agent. All these concepts are formalized in a variant of STIT logic with explicit actions.

Introduction

Although the concepts of morality and moral agency have been extensively studied in social philosophy and in social sciences, they have been far less studied in the areas of multi-agent systems (MASs) and normative multi-agent systems (NorMAS). Some works have been done on the extension of the BDI (Belief, Desire, Intention) model with normative concepts such as obligation [START_REF] Broersen | Goal generation in the BOID architecture[END_REF][START_REF] Governatori | BIO logical agents: Norms, beliefs, intentions in defeasible logic[END_REF], but none of them have really focused on the integration of moral aspects into the architecture of a cognitive agent. Developing formal models of cognitive agents integrating a moral dimension is a promising research avenue for these two areas. Indeed, as shown by social scientists [START_REF]Moral sentiments and material interests[END_REF][START_REF] Fehr | Theories of fairness and reciprocity: Evidence and economic applications[END_REF], decisions of human agents are often affected by moral sentiments and moral concerns (e.g., concerns for fairness or equity). Therefore, to take the presence of moral attitudes into account becomes extremely important when developing formal and computational models of social procedures to be applied to human societies, and when developing logical models of artificial agents which are expected to interact with human agents (e.g., trading agents, recommender systems, tutoring agents, etc.). The aim of this paper is to propose a logical framework in which different aspects of moral agency can be formalized such as the distinction between desires and moral attitudes and the concept of moral agent.

The rest of the paper is organized as follows. Section 2 establishes the conceptual basis of the logical analysis of moral agency developed in the second part of the paper, whereas in Section 3 the logic DL-MA (Dynamic Logic of Mental attitudes and joint Actions) is presented. As we will show, DL-MA can be seen as a variant of STIT logic -the logic of "Seeing To It That" [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF][START_REF] Horty | Agency and Deontic Logic[END_REF] -with explicit actions. In the second part of the paper (Section 4), the logic DL-MA is used to develop a logical analysis of the different aspects of moral agency discussed in Section 2.

2 Moral agency: conceptual basis Some background and clarifications of the notion of moral agency are needed in order to ground the logical analysis presented in Section 4 on a solid conceptual basis.

Desires vs. moral values A model of moral agency should be able to explain the two different origins of an agent's motivations. Some motivations originate from the agents's desires. A desire can be conceived as an agent's attitude consisting in an anticipatory mental representation of a pleasant state of affairs (representational dimension of desires) that motivates the agent to achieve it (motivational dimension of desires). In this perspective, the motivational dimension of an agent's desire is realized through its representational dimension. For example when an agent desires to be at the Japanese restaurant eating sushi, he imagines himself eating sushi at the Japanese restaurant and this representation gives him plesaure. This pleasant representation motivates him to go to the Japanese restaurant in order to eat sushi. Agents are motivated not only by their desires but also by their moral values. Moral values, and more generally moral attitudes (ideals, standards, etc.), originate from an agent's capability of discerning what from his point of view is (morally) good from what is (morally) bad. If an agent has a certain ideal ϕ, then he thinks that the realization of the state of affairs ϕ ought to be promoted because ϕ is good in itself. A similar distinction has also been made by philosophers and by social scientists. For instance, Searle [START_REF] Searle | Rationality in Action[END_REF] has recently proposed a theory of how an agent may want something without desiring it and on the problem of reasons for acting based on moral values and independent from desires. In his theory of morality [START_REF] Harsanyi | Morality and the theory of rational behaviour[END_REF], Harsanyi distinguishes a person's ethical preferences from her subjective preferences and argues that a moral choice is a choice that is based on ethical preferences. 1The distinction between desires and moral values allows to identify two different kinds of moral dilemmas. The first kind of moral dilemma is the one which originates from the logical conflict between two moral values. The paradigmatic example is the situation of a soldier during a war. As a member of the army, the soldier feels obliged to kills his enemies, if this is the only way to defend his country. But, as a catholic, he thinks that human life should be respected. Therefore, he feels morally obliged not to kill other people. The other kind of moral dilemma is the one which originates from the logical conflict between desires and moral values. The paradigmatic example is that of Adam and Eve in the garden of Eden. They are tempted by the desire to eat the forbidden fruit and, at the same time, they have a moral obligation not to do it.

Moral agents An important distinction for a logical theory of morality is self-regarding agent versus moral agent. A purely self-regarding agent is an agent who acts in order to maximize the satisfaction of his own desires, while a purely moral agent is an agent who acts in order to maximize the fulfilment of his own moral values. In other words, if an agent is purely self-regarding, the utility of this act for him coincides with the personal good the agent will obtain by performing this action -where the agent's personal good coincides with the satisfaction of the agent's own desires -. If an agent is purely moral, the utility of this act for him coincides with the moral good the agent will promote by performing this action -where the agent's promotion of the moral good coincides with the accomplishment of his own moral values -. The notion of self-regarding agent should not be confused with the rationality assumption of classical decision and game theory: according to classical decision and game theory, individuals are rational in the sense that they maximize their utility. The notions of self-regarding agent and moral agent are not in contradiction with this assumption. We can safely say that a given agent acts to maximize his utility even even though he does not act to maximize the satisfaction of his own desires (as he also cares about the fulfilment of his own moral values). Therefore, the person's act is moral even though the person is still acting to maximize her utility. Of course, purely self-regarding agents and purely moral agents are just extremes cases. An agent is more or less moral depending on whether the utility of a given option for him is more or less affected by his moral values. More precisely, the higher is the influence of the agent's moral values in evaluating the utility of a given decision option, more moral the agent is. The extent to which an agent's utility is affected by his moral values can be called degree of moral sensitivity. 2

Logical framework

In the following sections the logic DL-MA (Dynamic Logic of Mental attitudes and joint Actions) is presented. DL-MA is a modal logic which supports reasoning about actions and capabilities of agents and of coalitions of agents. DL-MA also allows to describe epistemic states of agents as well as their desires and moral values. We first present the syntax and the semantics of DL-MA (Sections 3.1, 3.2 and 3.3). A complete axiomatization of the logic is given in Section 3.4. As we will show in Section 3.5, DL-MA is tightly related to STIT logic, as it can be seen as a variant of STIT with explicit actions.

Syntax

Assume a countable set of atomic propositions Atm = {p, q, . . .}, a finite set of agents Agt = {i 1 , . . . , i |Agt| }, a finite set of atomic action types Act = {a, b, . . .}, and a finite set of natural numbers Num = {x ∈ N : 0 ≤ x ≤ max}, with max ∈ N \ {0}. We assume that the set Act includes the (in)action skip, i.e., the action of doing nothing.

We define Prop to be the set of propositional formulas, that is, the set of all Boolean combinations of atomic propositions.

For each agent i ∈ Agt, the set Act i = {a i : a ∈ Act} identifies the set of agent i's actions. 2 Agt * = 2 Agt \ {∅} is the set of non-empty sets of agents, also called coalitions. Elements of 2 Agt * are denoted by symbols H, J, . . . Let JAct = {{a i1 , . . . , a i |Agt| } : a i k ∈ Act i k for all i k ∈ Agt} be the set of all possible joint actions of all agents. One might think of JAct as the set of all possible strategy profiles in the game theoretic sense. Just as in game theory we suppose that at a given time point every agent performs exactly one action, and that all actions of different agents occur in parallel. Elements of JAct are denoted by symbols δ, δ , δ , . . . We let δ i denote the element in δ corresponding to agent i. Given a coalition H and a joint action δ of all agents, δ H = {a i : i ∈ H and a i ∈ δ} is coalition H's part in the joint action δ. Let JAct H = {δ H : δ ∈ JAct} be the set of all possible joint actions of coalition H. Finally, let JAct * be the set of all (possibly infinite) sequences of joint actions. Elements of JAct * are denoted by symbols , , , . . . For every 1 , 2 ∈ JAct * , we write 1 2 to mean that either 1 = 2 or 1 is an initial subsequence of 2 , i.e., there is 3 ∈ JAct * such that 2 = 1 ; 3 . The empty sequence of joint actions is denoted by nil . Infinite sequences of joint actions are called histories. Let Hist be the set of all histories. Elements of Hist are denoted by symbols h, h , . . . The language of DL-MA is defined by the following grammar in Backus-Naur Form:

::= δ | ; ϕ ::= p | occ( ) | des i,h | idl i,h | ¬ϕ | ϕ 1 ∧ ϕ 2 | [[δ]]ϕ | ϕ | K i ϕ
where p ranges over Atm, i ranges over Agt, δ ranges over JAct, ranges over JAct * and h ranges over Num. The other Boolean constructions , ⊥, ∨, → and ↔ are defined from p, ¬ and ∧ in the standard way.

DL-MA has special atomic formulas of three different kinds. The atomic formulas occ( ) represent information about occurrences of joint action sequences. The formula occ( ) has to be read "the joint action sequence is going to occur".

The other atomic formulas des i,h and idl i,h are used to rank the histories that an agent can imagine at a given world according to their desirability degree and to their ideality degree for the agent. Desirability captures the quantitative dimension of desires (i.e., how much a given history promotes the satisfaction of the agent's desires), whereas ideality captures the quantitative dimension of moral values (i.e., how much a given history promotes the fulfilment of the agent's moral values). Formula des i,h has to be read "the current history has for agent i a degree of desirability equal to h" while formula idl i,h has to be read "the current history has for agent i a degree of ideality equal to h". The following two abbreviations express respectively that "the current world has for agent i a degree of desirability equal to or higher than h" and "the current world has for agent i a degree of ideality equal to or higher than h":

des i,≥h def = h≤k≤max des i,k idl i,≥h def = h≤k≤max idl i,k
The logic DL-MA has three kinds of modal operators:

[[δ]],
and K i . is the historic necessity operator of STIT logic. The formula ϕ has to be read "ϕ is true in all histories passing through the current moment". We define ♦ to be the dual of , i.e., ♦ϕ def = ¬ ¬ϕ where ♦ϕ has to be read "ϕ is true in at least one history passing through the current moment".

[[δ]] is a dynamic operator describing the fact that if the joint action δ is performed then it will lead to a state in which a given state of affairs holds. In particular, [[δ]]ϕ has to be read "if the joint δ is performed, then ϕ will be true after its execution". We define δ to be the dual of

[[δ]], i.e., δ ϕ def = ¬[[δ]
]¬ϕ, where δ ϕ has to be read "the joint δ is performed and ϕ will be true after its execution".

Finally, K i is a modal operator characterizing the concept of ex ante (or choiceindependent) knowledge in the sense of Aumann & Dreze [START_REF] Aumann | Rational expectations in games[END_REF] (see also [START_REF] Roy | Epistemic logic and the foundations of decision and game theory[END_REF]). The formula K i ϕ has to be read "agent i knows that ϕ is true independently from his current choice" or "agent i thinks that ϕ is true for any choice he could have made". The dual of the operator K i is denoted by K i , i.e., K i ϕ def = ¬K i ¬ϕ. Aumann & Dreze distinguish ex ante knowledge from interim knowledge. Ex ante knowledge characterizes an agent's knowledge assuming that no decision has yet been made by him, whereas interim knowledge characterizes an agent's knowledge assuming that the agent has made his decision about which action to take, but might still be uncertain about the decisions of others. The concept of interim knowledge is expressed in DL-MA by the following operator K * i :

K * i ϕ def = ai∈Acti (choose(a i ) → K i (choose(a i ) → ϕ))
where K * i ϕ has to be read "the agent i knows that ϕ is true given his current choice". The following abbreviations will be useful to axiomatize DL-MA. For any coalition H, joint action sequence and joint action δ H we define:

choose( ,δ H ) def = δ∈JAct:δ H ⊆δ occ( ;δ) able( ,δ H ) def = ♦choose( ,δ H )
choose( ,δ H ) has to be read "at the end of the joint action sequence , the agents in H will choose the joint action δ H ", whereas able( ,δ H ) has to be read "the agents in H will be able to perform the joint action δ H , at the end of the joint action sequence ". For notational convenience, when = nil , we write choose(δ H ) instead of choose(nil ,δ H ) and able(δ H ) instead of able(nil ,δ H ). Besides, we simply write choose( ,a i ) instead of choose( ,{a i }), able( ,a i ) instead of able( ,{a i }), choose(a i ) instead of choose({a i }) and able(a i ) instead of able({a i }).

Action description

Similarly to Situation Calculus [START_REF] Reiter | Knowledge in action: logical foundations for specifying and implementing dynamical systems[END_REF], in DL-MA actions are described in terms of their positive and negative effect preconditions. In particular, we introduce two functions:

γ + : Agt × Act × Atm -→ Prop γ -: Agt × Act × Atm -→
Prop mapping agents, actions and atomic propositions to propositional formulas. The formula γ + (i, a, p) describes the positive effect preconditions of action a performed by agent i with respect to p, whereas γ -(i, a, p) describes the negative effect preconditions of action a performed by agent i with respect to p. Formula γ + (i, a, p) represents the conditions under which agent i will make p true by performing action a, if no other agent interferes with i's action; while γ -(i, a, p) represents the conditions under which agent i will make p false by performing action a, if no other agent interferes with i's action. We assume that "making p true" means changing the truth value of p from false to true, whereas "making p false" means changing the truth value of p from true to false. We make the following coherence assumption:

(COH γ ) for every i ∈ Agt, a ∈ Act and p ∈ Atm, γ + (i, a, p) and γ -(i, a, p) must be logically inconsistent.

COH γ ensures that actions do not have contradictory effects. As to the (in)action skip of doing nothing, we assume that for every agent i ∈ Agt, γ + (i, skip, p) = γ -(i, skip, p) = ⊥, i.e., an agent cannot change the truth value of p by doing nothing.

Semantics

The semantics of DL-MA is a possible world semantics with accessibility relations associated with each modal operators, with functions for desirability and ideality, and with a function designating the history starting in a given world.

Definition 1 A DL-MA model is a tuple M = W, H, ≡, {E i : i ∈ Agt}, κ des , κ idl , V
where:

• W is a set of states (or worlds),

• H is a total function H : W -→ Hist,

• ≡ and every E i are equivalence relations between states in W ,

• κ des : W × Agt -→ Num and κ idl : W × Agt -→ Num are total functions mapping worlds and agents to natural numbers in Num,

• V : W -→ 2 Atm is a valuation function.
As usual p ∈ V(w) means that proposition p is true at world w.

For every world w ∈ W , H(w) identifies the history starting in w. For notational convenience, for all ∈ JAct * , i ∈ Agt and a i ∈ Act i we write ;a i H(w) if there is δ ∈ JAct such that a i ∈ δ and ;δ H(w).

≡-equivalence classes are called moments. If w and v belong to the same moment (i.e., w ≡ v), then the history starting in w (i.e., H(w)) and the history starting in v (i.e., H(v)) are said to be alternative histories (viz., histories starting at the same moment). The concept of moment is the one used in STIT logic [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF][START_REF] Horty | Agency and Deontic Logic[END_REF] and, more generally, in the Ockhamist theory of time [START_REF] Thomason | Combinations of tense and modality[END_REF][START_REF] Zanardo | Branching-time logic with quantification over branches: The point of view of modal logic[END_REF].

The equivalence relations E i are used to interpret the ex ante epistemic operators K i . They can be viewed as functions from W to 2 W . Therefore, we can write E i (w) = {v ∈ W : wE i v}. The set E i (w) is the agent i's information state at world w: the set of worlds that at w agent i considers epistemically possible independently from his current choice or, more shortly, agent i's set of epistemic alternatives at w. As E i is an equivalence relation, if wE i v then agent i has the same information state at w and v.

The functions κ des and κ idl represent respectively desirability gradings and ideality gradings of the possible worlds and are used to interpret the atomic formulas des i,h and idl i,h . κ des (w, i) = h means that according to the agent i the world w has a degree of desirability h, whereas κ idl (w, i) = h means that according to the agent i the world w has a degree of ideality h.

We impose the following constraints on DL-MA models. For all w ∈ W , δ ∈ JAct, ∈ JAct * , i ∈ Agt and a i ∈ Act i we have: (C1) if for all i ∈ Agt there is u i such that w ≡ u i and ;δ i H(u i ), then there is u such that w ≡ u and ;δ H(u);

(C2) if there is v such that w ≡ v and ;a i H(v) then, for all u such that wE i u, there is z such that u ≡ z and ;a i H(z);

(C3) if there is v such that w ≡ v and ;a i H(v), then there is u such that wE i u and ;a i H(u).

According to the Constraint C1, if every individual action in a joint action δ can be chosen at the end of the joint action sequence , then the individual actions in δ can be chosen simultaneously at the end of . The Constraint C1 is a variant of the assumption of independence of agents of STIT logic. More intuitively, this means that agents can never be deprived of choices due to the choices made by other agents. The Constraint C2 is a basic assumption about agents' knowledge over their abilities: if an agent i can perform the action a at the end of the joint action sequence , then he knows this. In other words, an agent has perfect knowledge about his ability to perform a given action. The Constraint C3 characterizes the basic property of ex ante knowledge: if an agent i can perform the action a at the end of the joint action sequence , then there is a history that the agent considers possible independently from his current choice in which he performs the action a at the end of the joint action sequence . In other words, for every action that an agent is able to perform, there is a history that the agent considers possible in which he performs this action. The rules defining truth conditions of DL-MA formulas are the standard ones for Boolean constructions p, ¬ and ∧ plus:

• M, w |= occ( ) iff H(w) • M, w |= des i,h iff κ des (w, i) = h • M, w |= idl i,h iff κ idl (w, i) = h • M, w |= [[δ]]ϕ iff if M, w |= occ(δ) then M δ , w |= ϕ • M, w |= ϕ iff M, v |= ϕ for all v such that w ≡ v • M, w |= K i ϕ iff M, v |= ϕ for all v such that wE i v
where model M δ is defined according to Definition 2 below.

Definition 2 (Update via joint action) Given a DL-MA model M = W, H, ≡, {E i : i ∈ Agt}, κ des , κ idl , V , the update of M by δ is defined to be M δ = W δ , H δ , ≡ δ , {E δ i : i ∈ Agt}, κ δ des , κ δ idl , V δ where for all i ∈ Agt and h ∈ Hist:

W δ = {w ∈ W : M, w |= occ(δ)} H δ (w) = h if H(w) = δ ;h for some δ ∈ JAct ≡ δ = ≡ ∩(W δ × W δ ) E δ i = E i ∩ (W δ × W δ ) κ δ des (w, i) = κ des (w, i) κ δ idl (w, i) = κ idl (w, i) V δ (w) = (V(w) \ {p : ∃a i ∈ δ such that M, w |= γ -(i, a, p) and ∃b j ∈ δ such that M, w |= γ + (j, b, p)})∪ {p : ∃a i ∈ δ such that M, w |= γ + (i, a, p) and ∃b j ∈ δ such that M, w |= γ -(j, b, p)}
The performance of a joint action δ modifies the physical facts via the positive effect preconditions and the negative effect preconditions, defined in Section 3.2 (see the definition of V δ ). In particular, if there is an action in the joint action δ whose positive effect preconditions with respect to p hold and there is no other action in the joint action δ whose negative effect preconditions with respect to p hold, then p will be true after the occurrence of δ; if there is an action in the joint action δ whose negative effect preconditions with respect to p hold and there is no other action in the joint action δ whose positive effect preconditions with respect to p hold, then p will be false after the occurrence of δ. Besides, the occurrence of the joint action δ makes the current history advance one step forward (see the definition of H δ ). As to the equivalence relations ≡ and E i for historic necessity and ex ante knowledge, they are restricted to the set of worlds in which the joint action δ occurs (see the definitions of ≡ δ and E δ i ). Finally, the joint action δ does not modify the agents' desirability and ideality gradings over the histories (see the definitions of κ δ des , κ δ idl and W δ ). As stated by the following proposition, the update via a joint action preserves the constraints on DL-MA-models.

Proposition 1 If M is a DL-MA-model then M δ is a DL-MA-model too. PROOF. As an example, we prove that the model M δ satisfies the Constraint C1 after assuming that M is a DL-MA model. Let us suppose that for all i ∈ Agt there is u i such that w ≡ δ u i and ;δ i H δ (u i ). The latter implies that for all i ∈ Agt there is u i such that w ≡ u i and δ; ;δ i H(u i ). As M is a DL-MA model which satisfies the Constraint C1, it follows that there is u such that w ≡ u and δ; ;δ H(u). The latter implies that there is u such that w ≡ δ u and ;δ H δ (u).

Axiomatization

Our logic DL-MA has so-called reduction axioms. These axioms allow to eliminate all the dynamic operators [[δ]] from formulas. That elimination provides an axiomatics.

Proposition 2

The following formulas are DL-MA valid for every i ∈ Agt, a i ∈ Act, δ, δ ∈ JAct and , ∈ JAct * :

occ( ) → δ∈JAct occ( ;δ) (OneJAct) occ( ;δ) → ¬occ( ;δ ) if δ = δ (UniqueJAct) occ( ) → occ( ) if (SubSeqJAct) h∈Num des i,h (ComplDes)
des i,h → ¬des i,k if h = k (UniqueDes) h∈Num idl i,h (ComplIdl) idl i,h → ¬idl i,k if h = k (UniqueIdl) ( i∈Agt choose( ,δ i )) → occ( ;δ) (IndepAgt) able( ,a i ) → K i able( ,a i ) (KnowCan) able( ,a i ) → K i choose( ,a i ) (ExAnteKnow) Proposition 3
The following equivalences are DL-MA valid for all p ∈ Atm, i ∈ Agt, δ ∈ JAct, ∈ JAct * and h ∈ Num:

[[δ]]¬ϕ ↔ (occ(δ) → ¬[[δ]]ϕ) [[δ]](ϕ ∧ ψ) ↔ ([[δ]]ϕ ∧ [[δ]]ψ) [[δ]]p ↔ (occ(δ) → (( ai∈δ γ + (i, a, p) ∧ bj ∈δ ¬γ -(j, b, p))∨ (p ∧ ai∈δ ¬γ -(i, a, p)) ∨ (p ∧ ai∈δ γ + (i, a, p)))) [[δ]]occ( ) ↔ (occ(δ) → occ(δ; )) [[δ]]des i,h ↔ (occ(δ) → des i,h ) [[δ]]idl i,h ↔ (occ(δ) → idl i,h ) [[δ]] ϕ ↔ (occ(δ) → (occ(δ) → [[δ]]ϕ)) [[δ]]K i ϕ ↔ (occ(δ) → K i (occ(δ) → [[δ]]ϕ))
Let L-MA be the fragment of the logic DL-MA without dynamic operators [[δ]]. that is, let the language of L-MA be the set of formulas defined by the following grammar:

ϕ ::= p | occ( ) | des i,h | idl i,h | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ | K i ϕ
where p ranges over Atm, i ranges over Agt, ranges over JAct * and h ranges over Num. As the rule of replacement of equivalents preserves validity, the equivalences of Proposition 3 together with the rule of replacement of equivalents allow to reduce every DL-MA formula to an equivalent L-MA formula. Call red the mapping which iteratively applies the above equivalences from the left to the right, starting from one of the innermost modal operators. red pushes the dynamic operators inside the formula, and finally eliminates them when facing an atomic formula.

Proposition 4 Let ϕ be a formula in the language of DL-MA. Then

1. red(ϕ) has no dynamic operators [[δ]] 2. red(ϕ) ↔ ϕ is DL-MA valid 3. red(ϕ) is DL-MA valid iff red(ϕ) is L-MA valid.
PROOF.[Sketch] The first item is clear. The second item is proved using Proposition 3 (and the rule of replacement of equivalents). The last item follows from the second item and the fact that DL-MA is a conservative extension of L-MA.

PROOF.[Sketch]

To prove soundness is just a routine exercise. The completeness proof proceeds as follows. By standard canonical model argument, it is routine to show that the axioms and rules of inference of the normal modal logic S5 for the operator and for every epistemic operator K i together with the principles in Proposition 2 and all principles of classical propositional logic provide a complete axiomatization for L-MA. Now, suppose ϕ is DL-MA valid. Then red(ϕ) is valid in L-MA due to Proposition 4. By the completeness of L-MA, red(ϕ) is also provable there. DL-MA being a conservative extension of L-MA, red(ϕ) is provable in DL-MA, too. As the reduction axioms and the rule of replacement of equivalents are part of our axiomatics, the formula ϕ must also be provable in DL-MA.

We write ϕ if ϕ is a DL-MA-theorem.

Relationships between DL-MA and STIT

As pointed out in the introduction, DL-MA can be seen as a variant of STIT with explicit actions: while in STIT an action is identified with the result brought about by a coalition (i.e., in STIT one can only express that a given coalition H sees to it that ϕ), in DL-MA an action is identified both with the result brought about by the coalition and with the means used by the coalition to bring about the result (i.e., in DL-MA one can also express that a given coalition H sees to it that ϕ by choosing the joint action δ H ). 3 In DL-MA we can express different concepts of agency which have been studied in the context of STIT theory [START_REF] Belnap | Facing the future: agents and choices in our indeterminist world[END_REF][START_REF] Horty | Agency and Deontic Logic[END_REF].

For instance, the so-called 'Chellas' operator CStit H of STIT can be defined in DL-MA as follows:

CStit H ϕ def = δ H ∈JAct H (choose(δ H ) ∧ δ Agt\H ∈JAct Agt\H (occ(δ H ∪ δ Agt\H ) → ϕ))
This means that the coalition H sees to it that ϕ if and only if, the agents in H choose some joint action δ H such that, no matter what the agents outside H choose, if the agents in H choose δ H then ϕ will be true. The so-called 'deliberative' operator DStit H can be defined in DL-MA as follows:

DStit H ϕ def = CStit H ϕ ∧ ¬ ϕ
DL-MA integrates a temporal dimension which allows to express that a given coalition H sees to it that ϕ is true in the next state of the system. In particular, in DL-MA one can simulate the so-called XSTIT operator XStit H proposed by Broersen [START_REF] Broersen | A logical analysis of the interaction between 'obligation-to-do' and 'knowingly doing[END_REF]:

XStit H ϕ def = CStit H Xϕ
where X is the operator 'next' of linear temporal logic (LTL) which is defined as:

Xϕ def = δ∈JAct δ ϕ

Moral agency: a logical formalization

In what follows the logic DL-MA is applied to the formalization of the different aspects of moral agency discussed in Section 2. Section 4.1 provides a logical formalization of desires and moral values. In Section 4.2 we define the concept of moral sensitivity as well as a concept of preference based on desires and moral values.

Desires and moral values

For every agent i ∈ Agt, we define two types of dyadic operators, one for desirability and the other for ideality:

ψ ≤ des i ϕ def = h∈Num ( K i (des i,≥h ∧ ψ) → K i (des i,≥h ∧ ϕ)) ψ ≤ idl i ϕ def = h∈Num ( K i (idl i,≥h ∧ ψ) → K i (idl i,≥h ∧ ϕ))
ψ ≤ des i ϕ has to be read "ϕ is for agent i at least as desirable as ψ" and ψ ≤ idl i ϕ has to be read "ϕ is for agent i at least as ideal as ψ". Corresponding strict orderings are defined in the expected way as follows:

ψ < des i ϕ def = (ψ ≤ des i ϕ) ∧ ¬(ϕ ≤ des i ψ) and ψ < idl i ϕ def = (ψ ≤ idl i ϕ) ∧ ¬(ϕ ≤ idl i ψ).
As the following proposition highlights the comparative statements ψ ≤ des i ϕ and ψ ≤ idl i ϕ might also be read as "for every epistemically possible ψ-state there is an epistemically possible ϕ-state which is at least as desirable" and "for every epistemically possible ψ-state there is an epistemically possible ϕ-state which is at least as ideal".

Proposition 5 For every i ∈ Agt we have:

• M, w |= ψ ≤ des i ϕ if and only if for all v ∈ E i (w), if M, v |= ψ then there is u ∈ E i (w) such that κ des (v, i) ≤ κ des (u, i) and M, u |= ϕ, • M, w |= ψ ≤ idl i ϕ if and only if for all v ∈ E i (w), if M, v |= ψ then there is u ∈ E i (w) such that κ idl (v, i) ≤ κ idl (u, i) and M, u |= ϕ.
As pointed out by [START_REF] Liu | Changing for the better: Preference dynamics and agent diversity[END_REF], there is no consensus in the literature on how preferential statements between formulas should be defined. In [START_REF] Van Benthem | Everything else being equal: a modal logic for ceteris paribus preferences[END_REF] other kinds of preference comparisons between formulas (i.e., between sets of states) are defined. For instance, the previous ∀∃-reading of preference statements should be distinguished from a ∀∀-reading ("for every ψ-state and for every ϕ-state the ϕ-state is at least as desirable/ideal as the ψ-state") and a ∃∃-reading ("there are a ψ-state and a ϕ-state such that the ϕ-state is at least as desirable/ideal as the ψ-state"). A logical analysis of such alternative readings of preference statements is postponed to future work.

Proposition 6 captures some central properties of the operators ≤ des i and ≤ idl i . Proposition 6 Let x ∈ {des, idl}. Then:

ψ ≤ x i ψ (6a) ((ϕ 1 ≤ x i ϕ 2 ) ∧ (ϕ 2 ≤ x i ϕ 3 )) → (ϕ 1 ≤ x i ϕ 3 ) (6b) (ϕ 1 ≤ x i ϕ 2 ) ∨ (ϕ 2 ≤ x i ϕ 1 ) (6c) ⊥ ≤ x i (6d) if ϕ → (ψ 1 ∨ ... ∨ ψ s ) then (ϕ ≤ x i ψ 1 ) ∨ ... ∨ (ϕ ≤ x i ψ s ) (6e)
PROOF. We prove theorem (6c) with x = des as an example.

¬(ϕ 1 ≤ x i ϕ 2 ) is equivalent to h∈Num ( K i (des i,≥h ∧ ϕ 1 ) ∧ K i (des i,≥h → ¬ϕ 2 )) which in turn im- plies k∈Num ( K i (des i,≥k ∧ ϕ 2 ) → K i (des i,≥k ∧ ϕ 1 )). The latter is equivalent to (ϕ 2 ≤ x i ϕ 1 ). We have proved that ¬(ϕ 1 ≤ x i ϕ 2 ) → (ϕ 2 ≤ x i ϕ 1 ) which is equivalent to (ϕ 1 ≤ x i ϕ 2 ) ∨ (ϕ 2 ≤ x i ϕ 1 )
. Theorems (6a)-(6c) highlight that ≤ des i and ≤ idl i are total preorders. Theorems (6b)-(6d) are the three fundamental principles of Lewis's conditional logic [START_REF] Lewis | Counterfactuals[END_REF].

Moral sensitivity and preference based on desires and moral values

We extend the logic DL-MA with special constructions of the form moralSensit(i,m) which has to be read "agent i's degree of moral sensitivity is equal to m" with m ∈ Num. We call DL-MA + the resulting logic. A DL-MA + model is a tuple M, S where M is a DL-MA model and S is a total function: S : W × Agt -→ Num capturing the moral sensitivity of an agent at a given state. We assume that an agent is aware of his current degree of moral sensitivity, that is, for every i ∈ Agt and w ∈ W we suppose that:

(C4) if S(w, i) = m then, for all v such that wE i v, S(v, i) = m.
Constructions moralSensit(i,m) are interpreted by means of the function S as follows:

M, w |= moralSensit(i,m) if and only if S(w, i) = m It is straightforward to adapt the proof of Theorem 1 in order to prove that the logic DL-MA + is completely axiomatized by the axioms and rules of inference of the logic DL-MA plus the following axiom schemas:

m∈Num moralSensit(i,m) (ComplMoral) moralSensit(i,m) → ¬moralSensit(i,l) if m = l (UniqueMoral) moralSensit(i,m) → K i moralSensit(i,m) (KnowMoral)
We use the degree of moral sensitivity as a parameter for calculating the utility of a given history for an agent. Definition 3 (Utility) Given a DL-MA + model M = W, H, ≡, {E i : i ∈ Agt}, κ des , κ idl , V, S , the utility for agent i of the history starting in the world w, denoted by κ util (w, i), is defined as follows: κ util (w, i) = S(w, i) × κ idl (w, i) + (max -S(w, i)) × κ des (w, i) Moreover, we define UScale = {x : ∃h 1 , h 2 , h 3 ∈ Num such that x = h 1 × h 2 + (max -h 1 ) × h 3 } to be the agents' utility scale. 4According to Definition 3, the utility of a history is a function of both the degree of desirability and the degree of the ideality of the history. Degree of moral sensitivity captures the extent to which the utility of a given history is affected by moral values: the higher is the agent's moral sensitivity and the higher is the influence of the degree of ideality in determining the utility of the history; the lower is the agent's moral sensitivity and the higher is the influence of the degree of desirability in determining the utility of the history.

The next step in the analysis is to define the concept of preferred history, as a history that an agent envisages and that has a maximal degree of utility for him. More formally: Definition 4 (Preferred histories) Given a DL-MA + model M = W, H, ≡, {E i : i ∈ Agt}, κ des , κ idl , V, S , agent i's set of preferred histories at w, denoted by P i (w), is defined as follows:

P i (w) = argmax v∈W :wEiv κ util (v, i)
We use the notion of preferred history to define the concept of preference. We say that an agent prefers ϕ to be true if and only if, ϕ is true in all histories that he prefers. More generally, we assume that each agent envisages a given set of possible histories and he evaluates which are the best ones among them. This leads to a realistic notion of preference in the sense that an agent's set of preferred histories is a subset of the set of histories that the agent envisages.

K i ¬util i,y ∧ K i (util i,x → ϕ)) where util i,x def = k,l,m∈Num:x=m×l+(max-m)×k (moralSensit(i,m) ∧ des i,k ∧ idl i,l )
Our operator of preference corresponds to Cohen & Levesque's goal operator [START_REF] Cohen | Intention is choice with commitment[END_REF]. However, differently from DL-MA, their logic does not explain the relationships between goals and desires, and between goals and moral values. Proposition 8 captures some basic properties of our preference operator.

Proposition 8 For every i ∈ Agt we have:

(Pref i ϕ ∧ Pref i ψ) → Pref i (ϕ ∧ ψ) (8a) ¬(Pref i ϕ ∧ Pref i ¬ϕ) (8b) if ϕ then Pref i ϕ (8c) Pref i ϕ → K i Pref i ϕ (8d) ¬Pref i ϕ → K i ¬Pref i ϕ (8e) K i ϕ → Pref i ϕ (8f) ((¬ϕ < des i ϕ) ∧ moralSensit(i,0)) → Pref i ϕ (8g) ((¬ϕ < idl i ϕ) ∧ moralSensit(i,max)) → Pref i ϕ (8h)
Theorems (8a) and (8b) together with the rule of inference (8c) highlights that Pref i is a KD normal modal operator. Theorems (8d) and (8e) are properties of positive and negative introspection for preferences. Theorem (8f) is the syntactic counterpart of the realism principle for preferences discussed above. A similar property is satisfied by Cohen & Levesque's goal operator. Finally, theorems (8g) and (8h) highlight the logical relationship between desires and preferences, and moral values and preferences.

According to (8g), if an agent i has a minimal degree of moral sensitivity and considers ϕ strictly more desirable than ¬ϕ, then i prefers ϕ to be true. According to (8h), if an agent i has a maximal degree of moral sensitivity and considers ϕ strictly more ideal than ¬ϕ, then i prefers ϕ to be true. The last aspect of moral agency we consider is the relationship between preferences and choices. To this aim we need to introduce a notion of preference-based rationality.

We say that a given agent i is rational, denoted by Rat i , if and only if for every action a, if i prefers to do a and knows that is able to do it, then he decides to do a; and if he prefers not to do a, then he does not decide to do a. In other words, an agent is rational if his decision to perform a given action is completely determined by his own preferences:

Rat i def = a∈Act (((Pref i choose(a i ) ∧ K i able(a i )) → choose(a i ))∧ (Pref i ¬choose(a i ) → ¬choose(a i )))
The following two theorems, which follow from theorems (8g) and (8h) in Proposition 8, explain how desires and moral values motivate an agent to perform a given action: ((¬choose(a i ) < des i choose(a i )) ∧ moralSensit(i,0) ∧ Rat i ∧ K i able(a i )) → choose(a i ) ((¬choose(a i ) < idl i choose(a i )) ∧ moralSensit(i,max) ∧ Rat i ∧ K i able(a i )) → choose(a i )

According to the preceding two theorems, if agent i is rational, has a minimal/maximal degree of moral sensitivity, considers the situation in which he performs action a strictly more desirable/ideal than the situation in which he does not perform a and knows that he is able to do a, then i decides to do a.

Conclusion

We have devised a logic which supports reasoning about actions and capabilities of agents and coalitions, epistemic states of agents as well as their desires and moral values. We have used it to provide a formal analysis of the concept of moral agency.

Directions of future work are manifold. For instance, there are important aspects of moral agency that have not been addressed in this work and that we intend to study in the future. One of them is the concept of moral emotion [START_REF] Haidt | The moral emotions[END_REF]. Moral emotions such as guilt, moral pride and reproach are emotions which are based either on the fulfillment or on the violation of an agent's moral values by the agent himself or by another agent. Another issue we plan to investigate, and which has been briefly mentioned in Section 2, is the relationships between an agent's moral values and external norms (e.g., obligations, prohibitions, etc.). As to the logical part, we have provided a complete axiomatization of the logic DL-MA. Future work will be devoted to prove its decidability.

Acknowledgements

This research has been supported by the French ANR project EmoTES "Emotions in strategic interaction: theory, experiments, logical and computational studies", contract No. 11-EMCO-004-01.

Theorem 1 2 • 3 •

 123 The validities of DL-MA are completely axiomatized by • all principles of classical propositional logic • axiomatization of the normal modal logic S5 for the historic necessity operator • axiomatization of the normal modal logic S5 for each epistemic operator K i • the schemas of Proposition the reduction axioms of Proposition the rule of replacement of equivalents from ψ 1 ↔ ψ 2 infer ϕ ↔ ϕ[ψ 1 /ψ 2 ]

  Definition 5 (Preference, Pref i ) At world w agent i prefers ϕ to be true, i.e., M, w |= Pref i ϕ, if and only if M, v |= ϕ for all v ∈ P i (w).As the following proposition highlights, the concept of preference semantically defined in Definition 5 is syntactically expressible in the logic DL-MA + . For all i ∈ Agt we have M, w |= Pref i ϕ if and only if M, w |= x∈UScale ( K i util i,x ∧ y∈UScale:y>x

	Proposition 7

An alternative to Harsanyi's dual theory of morality is Sen's theory[START_REF] Sen | Rational fools: a critique of the behavioral foundations of economic theory[END_REF]. According to Sen, moral attitudes are kind of meta-level attitudes. In particular, moral judgments are rankings of preference rankings. In this sense, differently from Harsanyi's theory, in Sen's theory subjective preferences and ethical preferences are not at the same level. A comparison between these two theories of morality goes beyond the objectives of the present work.

For a similar idea in current economic models of morality see, e.g.,[START_REF] Battigalli | Guilt in games[END_REF][START_REF] Alger | Homo moralis: preference evolution under incomplete information and assortative matching[END_REF].

See also[START_REF] Herzig | A dynamic logic of agency I: STIT, abilities and powers[END_REF] for a variant of STIT with explicit actions.

Note that UScale is finite because Num is finite.


Grégory Bauw
email: gregory_bauw@ens.univ-artois.fr

Bertrand Cassoret

Raphaël Romary

Olivier Ninet

Damper winding for noise and vibration reduction of induction machine under sinusoidal conditions

Keywords: Induction machine, magnetic noise, harmonics, auxiliary windings, capacitors, damper, sinusoidal

The flux density harmonics in the air-gap of AC machines have many origins. They generate electromotive forces and current harmonics, Maxwell forces, noise and vibrations. The principle of the presented method is to use a new auxiliary stator winding, connected to capacitors. The induced current harmonics generate flux density components in opposition to the initial ones, creating a damping effect. The study presented in this paper concerns an induction machine and particularly focus on flux density harmonics generated by slotting effects. The method allows to reduce the sound pressure level main lines of about 15dBA. The vibrations are reduced too. Moreover the stator current harmonics are attenuated, and the power factor is significantly improved.

Introduction

Magnetic noise and vibrations of AC rotating machines come from flux density harmonics in the air-gap of the machine. Often, machines are noisy when they are fed by a PWM inverter which generates a lot of harmonics, but a machine can be noisy even when fed by sine voltages. A lot of previous works explain how to design a silent machine or how to reduce noise and vibrations using active reduction methods [START_REF] Cassoret | Magnetic noise reduction of induction machines[END_REF]- [START_REF] Timar | Noise and vibration of electrical machines[END_REF]. This paper deals with a method to reduce flux density harmonics and noise using a damper circuit when the machine is fed by the electrical network. The main principle is to connect capacitors to a new three-phase auxiliary winding. In a first time, origins of flux density harmonics and magnetic noise in induction machines are reminded. Then the method will be presented, explaining how to design the damper circuit. Finally, some experimentations validate the procedure on an 11kW induction machine under sinusoidal conditions.

Origins of noise and vibrations

Origins of harmonics in AC machines

The rotating field in the air-gap of AC machines is never perfectly sine, indeed flux density harmonics can be generated by several sources such as spatial distribution of the coils in a finite number of slots (space harmonics), variable thickness of the air-gap [START_REF] Cassoret | Magnetic noise reduction of induction machines[END_REF], [START_REF] Hesse | Air-gap permeance in doubly-slotted asynchronous machines[END_REF], [START_REF] Besnerais | Prediction of audible magnetic noise radiated by adjustable-speed drive induction machines[END_REF] (permeance harmonics), eccentricity [START_REF] Ayari | Effects of the airgap eccentricity on the SRM vibrations[END_REF], magnetic saturation or voltages harmonics due to the power supply [START_REF] Belmans | Electro-mechanical analysis of the audible noise of an inverter-fed squirrel-cage induction motor[END_REF].

The expression of the radial component bh (α,t) of a flux density harmonic characterized by its order h, its angular position in the air-gap α, its amplitude h B ˆ, its pulsation ωh and its phase ψh, in the air-gap of a 2p poles machine, is:

𝑏 ℎ (𝛼, 𝑡) = h B ˆcos(𝜔 ℎ 𝑡 -ℎ𝑝𝛼 -𝜓 ℎ ) (1)
h = 1 corresponds to the fundamental component which angular frequency is that of the network. The amplitude and frequency of harmonics are largely in relation to the number of stator and rotor slots [START_REF] Jordan | Geräuscharme elektromotoren[END_REF], [START_REF] Timar | Noise and vibration of electrical machines[END_REF], [START_REF] Alger | The nature of induction machines[END_REF].

There is an infinity of flux density harmonics in the air-gap. Each of them generates an induced electromotive force (e.m.f.) at stator and rotor frames, which induces current harmonics and generates other flux density harmonics. Fig. 1 shows the interaction between stator and rotor, and the generation of the flux density harmonics. The stator current 𝑖 𝑠 , at the supply frequency, generates a lot of flux density components: -a 𝑏 1

1 fundamental component at the supply frequency with the 2p pole number (h = 1), -several 𝑏 ℎ harmonics at different 𝜔 ℎ angular frequencies and pole numbers (h  1). Each flux density component induces in each conductor an electromotive force and so a current. Some 𝑏 ℎ flux density harmonics at 𝜔 ℎ angular frequency are generated by the fundamental current at 𝜔 angular frequency, so they can't influence it but they generate 𝑖 ℎ 𝑠 and 𝑖 ℎ 𝑟 current harmonics at stator and rotor coils. Those currents partially mitigate the 𝑏 ℎ flux density harmonics which generated them.

Maxwell forces, noise and vibrations

The flux density in the air-gap creates forces between stator and rotor. Three kinds of forces appear:

-Tangential forces which create torque and rotor rotation.

-Magnetostrictive forces, negligible for rotating machines.

-Radial forces, expressed by the Maxwell relation. The radial component of the flux density b (T) in the air-gap of a magnetic circuit of surface S (m²), creates a force f (N) which tends to attract stator and rotor. Noting µ 0 the vacuum magnetic permeability (4π 10 -7 H/m), the force can be expressed as:

𝑓 = 𝑏²𝑆 2µ 0 (2)
Because of the multiplicity of the flux density harmonics, many pressures waves occur and multiply the vibrations sources.

( )

0 2 2 ) , ( ) , ( f      = t b t h ( )                      -  -  - -  -  +  +  -  + -  +  +  -  -  +  =    )) ( p ) ' h h ( t ) cos(( )) ( p ) ' h h ( t ) cos(( B B ˆ) 2 hp 2 t 2 cos( 1 2 B 21 ' h h ' h h ' h h ' h h h ' h ' h h h h h 2 h 0 (3)
So non-stationary pressure components are expressed by:

𝑓(𝛼, 𝑡) = F ˆcos(𝜔 𝑓 𝑡 -𝑚𝛼 -𝜓 𝑓 ) (4)
Where 𝑚 is the mode number, F ˆ the amplitude of the pressure in N/m², 𝑓 𝑓 the frequency (𝜔 𝑓 = 2𝜋𝑓 𝑓 ). 𝑚 characterizes the number of maximal and minimal attraction points between the stator and the rotor. The pressure waves rotate at 𝜔 𝑓 /𝑚 angular speed, they generate variable attractions between the stator and the rotor and then vibrations and magnetic noise.

Principle of the passive noise reduction method

Supplementary stator winding and capacitors

This method needs an auxiliary stator winding. In the ideal case, it can be wounded in the same slots as the initial one, creating secondary stator coils similar to the initial stator ones. For example, Fig. 2 shows a classical three-phase winding with one pole pair and 4 slots per pole and per phase. The principle consists in creating another similar stator winding superimposed to the initial one, star or delta connected, in short circuit via three capacitors as shown in Fig. 3. The current in this auxiliary winding is noted 𝑖 𝑎 . The rotor is in short circuit as usually for induction machines. The flux density fundamental induces a current in the auxiliary windings at the same frequency than the stator. For this low frequency, the capacitors have a high impedance, so the induced currents are small and don't disturb the machine torque.

The equivalent single-phase diagram relative to the fundamental variables is presented in Fig. 4 where 𝑟 𝑠 , 𝑙 𝑠 , 𝑟 𝑎 ′ , 𝑙 𝑎 ′ , 𝑟 𝑟 ′ and 𝑙 𝑟 ′ are the resistance and leakage inductance respectively of the stator, auxiliary and rotor windings. 𝑍 µ is the classical magnetizing impedance, and 𝑠 is the slip. The symbol ′ means that the considered values are brought back to the stator, depending on the number of turns.

Compared to the classical equivalent diagram of an induction machine, a new auxiliary branch appears with the capacitor 𝐶 𝑎 ′ , and a new inductance 𝑙 𝑠𝑎 which represents the flux in common between stator and auxiliary windings [START_REF] Kron | Equivalent circuits of electric machinery[END_REF][START_REF] Kouki | Double star induction machine modelling: Impact of the stator mutual leakage inductance[END_REF].

Principle of harmonic reduction

The flux density components induce current harmonics in the auxiliary windings. For the high frequency harmonics, these induced currents can easily circulate and generate flux density components in opposition to those which create them, according to the Lenz law. However, the created components must have the same polarity than the original ones. Fig. 5 illustrates the phenomena. The 𝑏 ℎ flux density component which generates 𝑖 ℎ 𝑎 has a different frequency from the supplying current. So a fictitious current can be considered: 𝑖 ℎ 𝑓𝑖𝑐𝑡 is the current at 𝜔 ℎ angular frequency which would generate 𝑏 ℎ . For the harmonics, it can be assumed that the network is a short circuit and the slip is equal to 1. According to these hypothesis, the equivalent single phase diagram for the harmonics given in Fig. 6 can be used. The auxiliary branch creates an important damping effect because the value of 𝐶 𝑎 ′ can be chosen so that a resonance exists with the inductances at a given frequency. In those conditions, the voltage 𝑒 ℎ is minimized. The latter is proportional to the flux density harmonic that we want to reduce.

The best value of the capacitors depends on the inductances values. A large range of capacitors values is suitable: if the resonance frequency is slightly below the unwanted frequencies, the impedance of the auxiliary branch will be small for a large frequency range.

Experimental results

Used machine

The used induction machine has the following characteristics: 11kW -50Hz -380/660V -23/13Ap = 2 pole pairs -𝑁 𝑡 𝑠 = 48 stator slots -𝑁 𝑡 𝑟 = 32 rotor slots. The theoretical analysis [START_REF] Cassoret | Magnetic noise reduction of induction machines[END_REF], [START_REF] Jordan | Geräuscharme elektromotoren[END_REF] shows that flux density harmonics exist at the frequency 50(1 + 𝑘

𝑁 𝑡 𝑟 (1-𝑠) 𝑝 )
where k is an integer (𝑘 = 0; ±1; ±2; ±3…). Neglecting the slip 𝑠, the frequencies of the important flux density harmonics are 750Hz, 850Hz, 1550Hz, 1650Hz, 2350Hz, 2450Hz, etc. Consequently, current harmonics circulate at those frequencies in the stator windings.

The magnetic noise and vibrations, due to the Maxwell forces, are essentially due to the combination of those harmonics with components at the supply frequency. So, according to equation ( 3), magnetic noise and vibrations exist at frequencies near to 700Hz, 800Hz, 900Hz, 1500Hz, 1600Hz, 1700Hz, 2300Hz, 2400Hz, 2500Hz, etc. The acoustic spectrum shows important noise components at 800 and 900Hz, in accordance to the theory; this is shown in Fig. 7 where the acoustic pressure is measured with a microphone at 1 meter. The global level is 74.8dBA at no-load. There is no auxiliary winding in this machine but all the windings connections are accessible outside the machine. As there are 4 slots per pole and per phase, it was decided to use an half of the windings (2 slots per pole and per phase) to supply the machine by the network, and the other half as auxiliary windings.

Experimental noise and vibrations reduction

As only half of the windings is connected to the supply (star), the 50Hz phase to phase voltage U is reduced to 325V in order to have a current value (15A) similar to the no-load fundamental current with a classical power supply. At no-load, the rotating speed is 1496rpm. In a first time, there is no impedance connected to the three auxiliary windings, allowing us to measure the induced e.m.f. spectrum. Those e.m.f. harmonics are in relation to the flux density harmonics in the air-gap. It can be seen in Fig. 8 that induced e.m.f. harmonics appear at 750Hz, 850Hz, 1550Hz and 1650Hz as shown in theory; other harmonics exist because only an half of the windings are used. Fig. 8. Induced e.m.f. at the free winding (2 slots per pole and phase). Three 120µF capacitors have been connected on the three free windings (2 slots per pole and phase). This value of capacitor, determined experimentally, corresponds to the optimum for noise and vibrations reduction. The sound pressure level at 900Hz decreases from 70 to 52dBA as illustrated in Fig. 9 which can be compared to Fig. 7. Moreover, the global acoustic pressure level decreases from 74.8 to 69.9dBA. The vibrations are measured too with an accelerometer placed on the stator, which confirm the acoustic measures: the vibrations at 800 and 900Hz decrease respectively from 0.104 and 0.965m.s -2 to 0.070 and 0.213m.s -2 . The current harmonics flowing in the stator windings are reduced too.

Other advantages 4.3.1. Influence on the power factor

Without capacitors, the machine requires a reactive power 𝑄. With auxiliary windings and capacitors, at the supplying frequency, a small current circulates in the capacitors. It produces some reactive power 𝑄 𝑐 that the network has not to provide. Its value can easily be estimated by relation [START_REF] Hesse | Air-gap permeance in doubly-slotted asynchronous machines[END_REF], taking into account the winding turn ratio.

𝑄 𝑐 = ( 𝑧 𝑎 𝑧 𝑠 𝑈) 2 𝐶𝜔 (5)
In order to minimize the stator current and to maximize the power factor of the studied machine where 𝑧 𝑠 = 𝑧 𝑎 , the capacitor should produce 𝑄 𝑐 = 𝑄 = 3800VAR. As 𝑈 = 325V and the frequency is 50Hz, the value of the capacitor should be 115μF. In this case, it appears that the capacitor value is almost the same that the one used to minimize noise and vibrations. With the used 120μF capacitors, the auxiliary current at 50Hz is 7.6A, 𝑄 𝑐 = 3982VAR, so the stator fundamental current at no-load decreases from 14.8 to 9.1A and the power factor at no-load increases from 0.2 to 0.3.

Influence on the stator current and torque harmonics

As the flux density harmonics are mitigated by the auxiliary windings and the capacitors, the induced e.m.f. and the stator current harmonics are mitigated too; except for the line at 250Hz. This is illustrated by Fig. 10 which shows the supply current spectrum without and with capacitors. As the torque harmonics are in relation to the flux density harmonics, they can be reduced too.

Conclusion

It is usual to connect some capacitors in order to compensate the reactive power absorbed by induction machines. Another possibility is to connect capacitors on a new auxiliary three-phase winding which can be wounded into the same slots than the initial windings. By this way, the power factor of the machine is improved and the stator current harmonics are attenuated. Moreover, this damper circuit considerably mitigate flux density harmonics in the air-gap, reduce noise and vibrations, and can attenuate stator current harmonics. The drawback is to design the stator of the machine with supplementary windings, but those one can be small. No electronic is used and the damping effect is permanent.

Fig. 1 .

 1 Fig. 1. Interactions between stator and rotor.

Fig. 2 .

 2 Fig.2. Three-phase winding with one pole pair and 4 slots per pole and phase.

Fig. 3 .

 3 Fig. 3. Representation of the machine with auxiliary windings.

Fig. 4 .

 4 Fig. 4. Equivalent single-phase diagram.

Fig. 5 .

 5 Fig. 5. Interactions between stator, rotor and auxiliary windings.

Fig. 6 .

 6 Fig. 6. Equivalent single-phase diagram for the harmonics.

Fig. 7 .

 7 Fig. 7. Acoustic pressure level of the no-load machine at 1 meter.

Fig. 9 .

 9 Fig. 9. Acoustic pressure level of the no-load machine at 1 meter, with three 120µF capacitors star connected.

Fig. 10 .

 10 Fig. 10. Stator current spectrum without and with capacitors.

Acknowledgements

This work has been achieved within the framework of CE2I project (Convertisseur d'Energie Intégré Intelligent). CE2I is co-financed by European Union with the financial support of European Regional Development Fund (ERDF), French State and the French Region of Hauts-de-France.

The authors would like to thank SATT Nord-de-France (Société d'Accélération du Transfert de Technologie) that has patented this invention [12].


Gregory Bauw
email: gregory_bauw@ens.univ-artois.fr

Bertrand Cassoret

Olivier Ninet

Raphael Romary

Design of an induction machine with damper windings for noise and vibrations reduction

Keywords: Design, Induction machine, Damper, Noise and vibrations reduction, PWM. Paper type -Research paper

Purpose -

The paper presents a design method for induction machines including a three-phase damper winding for noise and vibrations reduction. Design/methodology/approach -In a first part, the principle of the damper winding is recalled. The second part presents the iterative design method which is applied on a 4kW PWM-fed induction machine in order to study the impact of the additional winding on the geometry. In a third part, the finite element method is used to validate the designed geometry and highlight the harmonic flux density reduction. Finally, some experimental results are given. Findings -The study shows that the impact of the additional three-phase winding on the geometry and weight of the machine is low. Moreover, the proposed noise reduction method allows one to reduce the total noise level of a PWM-fed induction machine up to 8.5dBA. Originality/value -The originality of the paper concerns the design and characterization of a three-phase damper winding for a noiseless induction machine. The principle of this proposed noise reduction method is new and has been patented.

I. INTRODUCTION

Induction machine design has been studied for decades as a field in its own right, and the construction rules are nowadays well known [START_REF] Boldea | The induction machines design handbook[END_REF][START_REF] Pyrhonen | Design of rotating electrical machines[END_REF][START_REF] Toliyat | Handbook of electric motors[END_REF]. The environmental context encourages manufacturers to design high efficiency electric motors in keeping with increasingly strong constraints. Thus, more recently, acoustic standards have added a new constraint on the design process. The main international acoustic standard (IEC 60034-9) (Rotating electrical machines -Part 9: Noise limits) specifies a total sound power level (SWL) limit depending on the speed of the machine.

When the machine is fed by sine voltage, magnetic noise can be predicted and anticipated in the design process by choosing an adequate number of stator/rotor slots or by skewing the rotor for example. If the machine is noisy despite the respect of the construction rules, it is possible to reduce noise and vibrations using active reduction methods [START_REF] Ojeda | Piezoelectric actuator design and placement for switched reluctance motors active damping[END_REF][START_REF] Pellerey | Active reduction of electrical machines magnetic noise by the control of low frequency current harmonics[END_REF]. When the machine is fed by a PWM inverter, noise is often corrected after the manufacturing by using an antiharmonic filter or by modifying the PWM strategy [START_REF] Brudny | Method for controlling the PWM switching: Application to magnetic noise reduction[END_REF][START_REF] Gabsi | Noise reduction of switched reluctance machine[END_REF][START_REF] Zhang | A new vibration reduction control strategy of switched reluctance machine[END_REF].

In this context, we propose a method to include a passive noise reduction solution for PWM supply in the design process. Numerical applications concern a 4kW induction machine which targeted characteristics are: 1500rpm -400V -8.8A -50Hz -3 phasespower factor: 0.79.

II. PRINCIPLE OF THE DAMPER WINDING

The proposed noise reduction method is a three-phase damper winding [START_REF] Cassoret | Moteur électrique à enroulement auxiliaire isolé[END_REF] wound into the stator slots and superimposed to the initial winding as shown in Figure 1. This additional winding, following the same path and having the same number of turn than the stator winding, is connected to three capacitors of suitable values in order to create a resonance around the switching frequency of the PWM inverter. At these frequencies, the voltage harmonics are important. They are responsible of flux density harmonics, and consequently at the origin of magnetic noise. The main principle is similar to that of a classical LC filter but in our case the inductance is integrated to the machine. Previous work [START_REF] Bauw | Damper winding for noise and vibration reduction of PWM-fed induction machines[END_REF] shows that the capacitors mainly resonate with the leakage inductances of the machine. Experimentations on a PWM-fed 4kW prototype including damper windings allowed to reduce the overall sound pressure level (SPL) by more than 8.5dBA in some cases. Vibrations were also considerably reduced.

III. DESIGN METHOD

A. Context

The proposed study is based on a classical iterative design method of induction machine [START_REF] Boldea | The induction machines design handbook[END_REF] which has to be adapted to take into account the space required by the damper winding in the stator slots. The difficulty is that this additional space is linked to the value of the current in the damper, itself linked to the impedance of the machine and the chosen capacitors.

The first step is to design a reference machine without damper windings to provide a comparison point. The computations using the characteristics defined in the introduction lead to a stator outside diameter 𝐷 𝑜𝑢𝑡 of 185.4mm and a stack length 𝐿 of 129mm. The stator has 36 trapezoidal slots and the squirrel-cage of the rotor has 28 bars. The current density in the stator winding is chosen at 4.5A/mm² as recommended by [START_REF] Boldea | The induction machines design handbook[END_REF]. Parameters of the classical single-phase equivalent diagram of the machine have been calculated analytically and are given in Table I. The goal of the design method is to minimize the increase of the diameter 𝐷 𝑜𝑢𝑡 of the machine with damper windings while keeping the same flux density in the yoke and the same performances. We assume that the damper does not impact the stator inner diameter and the rotor geometry.

B. Design of the machine with damper windings

Empirically, the maximum current value in the damper windings is first set to 20% of the rated stator current, which corresponds to 1.76A for the studied machine. Experience and experimentations on existing prototypes have shown that this value is coherent. The current density in the damper windings is the same to that of the stator. The design process has to be done again increasing the stator slot depth and so the outside diameter. The slot depth is increased while keeping the tooth width constant. Consequently, the outside diameter 𝐷 𝑜𝑢𝑡 is increased to 192mm.

C. Single-phase equivalent diagram

Since the geometry is now fully designed, analytical calculations of resistances and leakage inductances of the machine [START_REF] Lipo | Introduction to AC machine design[END_REF] allow one to establish the parameters of the equivalent diagram of Figure 2. 𝑙 𝑠 is the leakage inductance of the stator winding; 𝑙′ 𝑎𝑧 is a mutual inductance relative to the common flux between the stator and the auxiliary which does not link the rotor winding (Bauw et al., 2018a;[START_REF] Bauw | Equivalent circuit of PWM-fed induction machine with damper winding for noise and vibration reduction[END_REF]. 𝑙 𝑎𝑐 ′ corresponds to the leakage flux of the auxiliary which does not link any other winding, this flux is essentially composed of coil-end leakage. Analytical values of these parameters are given in Table II. One can notice that the sum of 𝑙 𝑠 and 𝑙′ 𝑎𝑧 gives a close approximation of the value of 𝑙 𝑠 given in Table I. Indeed, when the damper windings is not connected, the auxiliary branch of the equivalent diagram is open and the inductances 𝑙 𝑠 and 𝑙′ 𝑎𝑧 are then in series to give the classical induction machine diagram.

D. Capacitor value and auxiliary current

We assume that the machine is fed by a PWM inverter which minimum and maximum switching frequencies are respectively 3kHz and 8kHz. For each switching frequency, the harmonic voltage content can be experimentally measured. In our case, the main voltage harmonics are respectively 3100 and 8100Hz and their amplitudes are both 49V. Figure 3 shows the peak flux density 𝐵 ̂ in the air-gap at these frequencies as a function of the capacitor value, calculated from (1) where 𝐿 µ is the magnetizing inductance, 𝐼 µ the magnetizing current for the considered harmonic, 𝑝 the number of pole pairs, 𝑛 the number of turns per phase, 𝐷 𝑖𝑠 the stator inner diameter and 𝐿 the stack length. The value 𝐶 𝑎 ′ = 0 corresponds to the amplitude of the flux density harmonic when the damper has no influence. An optimal capacitor value for both operating point would be of 1µF, but taking into account the accuracy of leakage inductances estimation, there is a risk of amplification of the harmonic flux density. Therefore, we will choose the value of 2µF. For this particular capacitor value, considering every harmonic components of the 3kHz PWM voltage supply, the computed RMS current in the damper windings is 0.75A. The current value for higher switching frequency is lower because the nature of the damper winding's impedance is mainly inductive.

𝐵 ̂ = √2 𝐿 µ 𝐼 µ 𝑝 𝑛 𝐷 𝑖𝑠 𝐿 (1)

E. Optimization of the geometry

Since the calculated current value in the damper windings is lower than the estimated value of section III.B, the design can be optimized. By choosing a maximum current of 1A in the damper, the outside diameter of the machine can be reduce from 192 to 189mm. 𝐷 𝑜𝑢𝑡 is thus increased by less than 2% compared to the reference machine without damper windings as shown in Figure 4. The design process has been finally done again in order to verify that the current value in the damper and the optimum value of capacitor of the optimized machine are not changed.

F. Weight criterion

Let 𝐷 𝑜𝑢𝑡 and 𝐷 𝑜𝑢𝑡 ′ respectively be the outside diameters of the reference and optimized machine with damper windings, 𝐿 the stack length of both machines and 𝜌 𝑖𝑟𝑜𝑛 the density of the iron. The added iron weight due to the increase of the outside diameter can be easily calculated from:

𝑚 𝑖𝑟𝑜𝑛 = 𝜌 𝑖𝑟𝑜𝑛 𝜋𝐿 [( 𝐷 𝑜𝑢𝑡 ′ 2 ) 2 -( 𝐷 𝑜𝑢𝑡 2 ) 2 ]
(2)

Considering a density of the iron equal to 7860kg/m 3 , the weight of the magnetic core should be increased by only 1kg. The copper weight of the damper winding can be calculated considering 𝜌 𝑖𝑟𝑜𝑛 the density of the copper, 𝑙 𝑐 the coil-end length, 𝑆 the cross section of the wire, 𝑁 the number of turns per phase of the winding and 𝑞 the number of phases: 𝑚 𝑐𝑜𝑝𝑝𝑒𝑟 = 2 𝜌 𝑖𝑟𝑜𝑛 (𝐿 + 𝑙 𝑐 ) 𝑆 𝑁 𝑞

(3)

In our case, the damper winding represents a weight of 1.512kg. The total weight due to the addition of the damper winding is then 2.512kg, which is small as an induction machine of this size usually weighs about 40kg. This aspect can be particularly attractive for embedded applications giving the size and weight of a traditional anti-harmonic filter. The optimized geometry of the machine with damper windings obtained from the analytical design has been simulated on the FEM software Flux2D as shown in Figure 5. The coupling circuit in Figure 6 allows us to integrate the two three-phase windings, their resistances and coil-end inductances are calculated analytically. The stator winding is fed by a balanced system of voltage harmonics which amplitudes are frequency dependant. The damper windings are short-circuited via three 2µF capacitors. The rotor is a squirrel-cage rotor which ring impedances are calculated analytically.

IV. FINITE ELEMENT METHOD VALIDATION

A. Used geometry and coupling circuit

B. Current value in the capacitors

Table III compares the current value in the capacitors calculated analytically from the equivalent diagram of Figure 2 and by finite element method for different harmonics. The chosen voltage harmonics have been obtained experimentally and correspond to the most significant ones of a PWM inverter when the switching frequency is set to 3kHz. Results show a good correlation between analytical and numerical models, which confirms that the analytical computation of the leakage inductances has been done accurately. Moreover, one can notice that the RMS value of the capacitor current calculated by FEM with the four studied harmonics is 0.69A, which is close to the value of 0.75A obtained from the equivalent diagram in section III.D. We can therefore neglect the contribution of the fundamental and other harmonics on the current value.

C. Validation of the harmonic flux density reduction in the air-gap

Figure 7 shows the normal flux density in the air-gap without and with damper windings as a function of the position on a line plotted in the middle of the air-gap. The frequency is 3100Hz and corresponds, in our case, to the main voltage harmonic for a switching frequency of 3kHz. One can observe that the harmonic flux density at the origin of noise and vibrations is drastically reduced thanks to the damper windings. It also confirms that the capacitor value of 2µF is well chosen. A simulation has been done using three 4µF capacitors and give satisfactory results too but a slightly higher capacitor current value.

V. EXPERIMENTATIONS

In order to demonstrate the effectiveness of the proposed reduction method, experiments have been done on a 4kW prototype of induction machine with damper windings which characteristics are the same as the designed machine introduced previously. Nevertheless, the design method used by the manufacturer must be different than the one presented here so the geometry of the prototype is slightly different. Consequently, current value in the capacitors will not be compared to the theory. The machine is PWM-fed and the switching frequency is set to 4kHz. The noise level spectrum has been measured with a microphone at 1 meter of the machine in a semi-anechoic room. The A-weighting is used in order to take into account the sensibility of the human ear. Figure 8 shows the noise level spectrum of the machine without damper windings, the total noise level is 63.3dBA. With damper windings connected to three 4µF capacitors, the main lines around 8kHz are drastically reduced as shown in Figure 9. The total noise level of the machine with damper windings is 56.2dBA, so the reduction is about 7dBA in this case. Vibrations have been measured on the same machine using accelerometer and show significant reduction too.

CONCLUSION

In this paper, an iterative design method for induction machines including a passive noise reduction solution is proposed. It allows one to optimally design the geometry of a noiseless machine minimizing the impact of the damper windings in terms of size and weight of the magnetic core. Finite element method and experimental results validate the design process and the effectiveness of the proposed noise reduction method.

Fig. 1 .

 1 Fig.1. Induction machine with damper windings.

Fig. 2 .

 2 Fig.2. Single-phase equivalent diagram of the machine with damper windings.

Fig. 3 .

 3 Fig.3. Peak flux density in the air-gap with (solid line) and without (dashed line) damper windings as a function of the capacitor 𝐶 𝑎 ′ for 𝑓 = 3100Hz and 8100Hz.

Fig. 4 .

 4 Fig.4. Comparison of the geometries: reference machine (dashed line) / optimized machine with damper windings (solid line).

Fig. 5 .

 5 Fig.5. Used geometry of the 4kW induction machine with damper windings for FEM simulation (Flux2D).

Fig. 6 .

 6 Fig.6. Coupling circuit of the 4kW induction machine with damper windings for FEM simulation (Flux2D).

Fig. 7 .

 7 Fig.7. Normal flux density in the air-gap without (dashed line) and with (solid line) damper windings as a function of the position for 𝑓 = 3100Hz and 2µF capacitors.

Fig. 8 .

 8 Fig.8. Noise level spectrum without damper windings for 𝑓 𝑃𝑊𝑀 = 4kHz.

Fig. 9 .

 9 Fig.9. Noise level spectrum with damper windings for 𝑓 𝑃𝑊𝑀 = 4kHz and 4µF capacitors.

TABLE I PARAMETERS

 I OF THE SINGLE-PHASE EQUIVALENT DIAGRAM OF THE REFERENCE MACHINE

	𝑟 𝑠 (Ω)	0.56
	𝑙 𝑠 (mH)	9.3
	𝐿 µ (mH) ′ (Ω) 𝑟 𝑟 𝑙 𝑟 ′ (mH)	221.5 0.986 10.1

TABLE II PARAMETERS

 II OF THE SINGLE-PHASE EQUIVALENT DIAGRAM OF THE MACHINE WITH DAMPER (𝐼 𝑎 = 0.2 𝐼 𝑠 )

	𝑟 𝑠 (Ω)	0.56
	𝑙 𝑠 (mH)	4.9
	𝑙′ 𝑎𝑧 (mH)	5.7
	𝐿 µ (mH) ′ (Ω) 𝑟 𝑟 ′ (mH) 𝑙 𝑟 ′ (Ω) 𝑟 𝑎 𝑙 𝑎𝑐 ′ (mH)	215.4 0.986 10.1 5.61 2.6

ACKNOWLEDGEMENTS

This work has been achieved within the framework of CE2I project (Convertisseur d'Energie Intégré Intelligent). CE2I is cofinanced by European Union with the financial support of European Regional Development Fund (ERDF), French State and the French Region of Hauts-de-France.

The authors would like to thank SATT Nord-de-France (Société d'Accélération du Transfert de Technologie) that has patented this invention (Cassoret and Romary, 2016).