Francois Pigache
email: pigache@leei.enseeiht.fr

Frédéric Messine

Bertrand Nogarede

Optimal Design of Piezoelectric Transformers: A Rational Approach Based on an Analytical Model and a Deterministic Global Optimization

This paper deals with a deterministic and rational way to design piezoelectric transformers in radial mode. The proposed approach is based on the study of the inverse problem of design and on its reformulation as a mixed constrained global optimization problem. The methodology relies on the association of the analytical models for describing the corresponding optimization problem and on an exact global optimization software, named IBBA and developed by the second author to solve it. Numerical experiments are presented and compared in order to validate the proposed approach.

I. Introduction

S ince the first demonstrator of piezoelectric transformer made by Rosen in 1956 [START_REF] Rosen | Ceramic transformers and filters[END_REF], the improvement of ceramics properties and various available shapes have extended the applications of these devices. Its low bulk, high efficiency and power density, high degree of insulation, and nonmagnetic properties fulfill the actual needs of the miniaturized devices. Tube photomultipliers supplies, backlight of LCD screens, cold cathode lamps, AC/DC or DC/DC converters, galvanic insulation for sensors, and gate drivers are some examples of the broad range of applications which need specific requirements.

The applications require different conditions of operating frequency, voltage ratio, acceptable temperature rise, etc. All of these properties are strongly dependent on the dimensioning, the shape, the mechanical, and the electrical properties. For example, by considering only a radial mode transformer, the design parameters are the radius, the thickness, and the number of layers of primary and secondary sides. The ceramics available on the market offer many combinations of mechanical and electrical properties which influence the global stiffness, electromechanical coupling, and the dielectric and mechanical losses. Moreover, the performances of piezoelectric transformers are strongly affected by the supply voltage, the operating frequency, and the load. Therefore, a precise dimensioning of the transformer must be chosen in accordance with the expected conditions.

As a result, compromises must be done by the choice of these different parameters in order to obtain the best performances according to the main criteria. This problem becomes quickly complex if some constraints are not fixed empirically and, in particular, when a multi-objective optimization is expected (minimal primary capacitance with a maximal efficiency, compactness, etc.). Moreover, this optimization problem groups continuous and discrete variables, corresponding to the geometrical parameters and the ceramic type.

This paper deals with a rational methodology to design radial mode transformers, dimensioned for an electronic ballast application. Our first wish is to stay free from arbitrary constraints which are not directly imposed by the schedule of conditions. Then, in addition to the constraints of the application, technical limitations of manufacturers will be taken into account in the design elaboration. The results obtained are discussed and compared with another design method deduced for the same type of applications in order to emphasize the advantage of our rational approach.

Nowadays, the design of an electromechanical actuator is understood and expressed as an inverse problem: From the characteristic values fixed by a schedule of conditions, obtain the structure, the composition, and the dimensions of the actuator.

This inverse problem of design is formulated as a mixed constrained global optimization problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ min x∈R nr ,z∈N ne , σ∈ nc i=1 K i ,b∈B n b f (x, z, σ, b) g i (x, z, σ, b) ≤ 0 ∀i ∈ {1, . . . , n g } h j (x, z, σ, b) = 0 ∀j ∈ {1, . . . , n h }, ( 1
)
where f is a real function, K i represents an enumerated set of categorical variables (the type of piezoceramic in the present study), B = {0, 1} the boolean set, and N and R the integer and real sets, respectively. In [START_REF] Fitan | The electromagnetic actuator design problem: A general and rational approach[END_REF], we solve exactly this problem by associating analytical models of electrical machines with a deterministic global optimization code, named IBBA for interval branch and bound algorithm which has been developed by the second author [START_REF] Messine | A deterministic global optimization algorithm for design problems[END_REF]. In this paper, IBBA is used for solving the inverse problem of design of a radial mode transformer. In Section II, the analytical model of a transformer is detailed. IBBA is then recalled in Section III and applied to solve problems with categorical variables. Some numerical tests are presented in Section IV in order to validate our approach. These results are compared with those presented in [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF].

II. Analytical Model

A. Radial Mode Transformer

All piezoelectric transformers (PT) are based on a double electromechanical conversion using the direct and inverse effect properties of the piezoceramics. This double electromechanical coupling comes from the mechanical association of two piezoelectric elements which compose the primary and secondary sectors. These lead-zirconatetitanate (PZT) elements are able to work according to three elementary coupling modes: longitudinal, transversal, and shear.

The radial mode transformer is made from two ringshape ceramics polarized in the thickness direction, and the mechanical stress is generated along the radius. This transformer is classically used for high-power applications (some ten watts) with relatively low voltage gain. The provided power density is approximately three times higher than for a Rosen-type transformer with the same volume [START_REF] Lin | Piezoelectric transformer characterization and application of electronic ballast[END_REF]. The ratio between the number of primary and secondary layers is an easy way to choose the ideal voltage gain N.

Fig. 1 shows the principle of electromechanical coupling and the different geometrical variables; r the radius, and t 1 n 1 and t 2 n 2 , respectively, the thickness and the number of primary and secondary layers. The optimal design problem relies on the interpretation of the PT by an electrical equivalent circuit.

B. Analytical Model of the Transformer

The characterization and simulation of piezoelectric transformers is currently based on the electrical equivalent model describing the dielectric and mechanical resonance behaviors. Around one vibratory mode, this equivalent scheme can be illustrated by using passive components and an ideal transformer; refer to Fig. 2. The input .

(n

1 t 1 +n 2 t 2 ) n 2 1 .r
Prim. coupling factor

ψ 1 = 2 √ 2π.r.n 1 d 31 S E 11 (1-σ)
Sec. coupling factor

ψ 2 = 2 √ 2π.r.n 2 d 31 S E 11 (1-σ) Ideal voltage gain N = ψ 1 ψ 2 = n 1 n 2
voltage V 1 is a sine wave in the vicinity of the resonance frequency in order to avoid the excitation of spurious vibratory modes.

All of these equivalent parameters can be expressed from the geometrical, electrical, and mechanical properties of the PT, obtained by the derivation of a physics-based equivalent model or the Hamiltonian approach. The study developed by Lin in [START_REF] Lin | Piezoelectric transformer characterization and application of electronic ballast[END_REF] is relatively complex and will not be detailed in this paper. The main equations are recalled in the Table I.

Considering only one vibratory mode, the relations in Table I are relatively simple and they are based on some main piezoelectric parameters, collected in Table II.

Another assumption is considered about the mechanical quality factor. Indeed, the transformer is composed of two ceramics, electrodes, and sometimes an insulating layer which can decrease the global mechanical quality fac- tor. In our study, gluing and assembling are considered to be perfect, and the Q m values are obtained from data of the manufacturers. The simulations of the equivalent circuit provide typical curves and illustrate the electrical behavior of the PT. First, the input impedance is essential to dimension the supply of the PT. Fig. 3 illustrates the input impedance and the phase of the classical PT. The frequency zone bounded by the resonance and the antiresonance frequencies corresponds to the zone of interest, as confirmed by Fig. 4. These two specific frequencies are defined later.

The resistive load and the operating frequency are important variables which strongly affect the performances such as the efficiency and the voltage gain. It can be noticed that the voltage gain and the output power are strongly increased for a specific operating frequency. Consequently, the choice of this operating frequency will be a criterion of optimization. Another essential characteristic is the variation of the output power and the efficiency depending on the resistive load. This load affects the resonance properties of the PT. Indeed, the resonance frequency is minimal when the secondary is short-circuited, and is slightly increased with the resistive load. As a consequence, the schemes on Fig. 5 are given as functions of the resistive load and by following the resonance point.

The maximal efficiency is reached at the point where the output power is minimal. For an optimal functioning of the PT, the dimensioning must make it possible to fit the optimal value with the rated resistive load.

This optimal efficiency point is characterized by the electrical quality factor expressed as follows [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF]:

q i ≈ R load .C d2 .2π.f ≈ 1.
(

) 2
The optimal efficiency is reached when the secondary capacitance is dimensioned in accordance with the rated load and frequency. In order to obtain the optimal electromechanical conversion, this frequency will be chosen close to the resonance fequency. From this first definition, other functions are deduced. The equivalent serial capacitance:

C eq = C d2 .N 2 .C m (1 + q 2 i ) q 2 i .C m + N 2 .C d2 (1 + q 2 i ) . ( 3
)
The resonance frequency:

f r = 1 2π L m .C eq . ( 4
)
The antiresonance frequency:

f a = 1 2π L m ( C d1 .Ceq C d1 +Ceq ) . ( 5
)
The equivalent circuit on Fig. 2 is a useful approach for simulations of electrical behavior under some acceptable assumptions. However, the technical limitations specific to the PZT ceramics are not taken into account in this linear model. These limitations, itemized below, will be included in the algorithm as constraints:

• Dielectric strength: limitation of the applied voltage, • Coercive field strength: critical voltage of depolarization, • Delamination: critical maximal stress, • Temperature rise: depolarization at the Curie point. If multilayer ceramics are used for the primary and the secondary sectors, a maximal thickness must be considered in order to avoid the delamination of the ceramics. Furthermore, the diameter of the transformer must be greater than the global thickness (D > 5T ) to support the radial coupling mode. All of these technical limitations will be considered as constraints in the optimization algorithm, and the model is developed keeping in mind these conditions.

The input and output powers (respectively, P in and P out ) are calculated for the selected rated load. From these values, the global temperature rise is obtained by the following equation:

∆Θ = P in 1 -η/100 h conv A , ( 6
)
where η is the efficiency at the rated functioning point, h conv the convection factor of ceramic (≈ 15

• C.W -1 .m -2 ),
and A the total area. This maximal temperature rise is an essential constraint for the dimensioning, to avoid the critical point of depolarization. Moreover, it depends on the input power and consequently on the input voltage (120 V in the present case).

C. Most Important Criteria to be Optimized

From the previous characteristics of electromechanical behaviors, the main criteria are deduced. First of all, the rated resistive load is deduced according to the expected application.

Maximal Efficiency:

The essential criterion for any application is to obtain the maximal efficiency for the optimal resistive load R opt . This condition means to fix the electrical quality factor near to the unit value, corresponding to (2). To reach this value, the operating frequency and the secondary dielectric capacitance are the fitting variables. After fixing the electrical quality factor, the maximal efficiency must be as great as possible; this optimal point is calculated as below:

η max = R opt R 2 d2 N 2 R d1 (R 2 m N 4 X 2 R 2 eq + (R eq + N 2 R m ) 2 )/ ((((1 + R 2 eq X 2 )(R opt + R d2 ) 2 R 2 m N 4 + R eq (2N 2 R m + R eq ) (R opt + R d2 ) 2 )(R 2 eq + (R d1 + 2R m )N 2 R eq + R m N 4 (R m + R d1 )(1 + R 2 eq X 2 )))), with R eq = R opt R d2 R opt + R d2 and X = C d2 .2π.f r . ( 7
)
Contrary to the analytical hand-optimization approach, the computational method allows taking into account the dielectric losses by [START_REF] Seo | Optimal design of contour vibration mode piezoelectric transformer[END_REF].

Electromechanical Coupling Factor:

This parameter strongly influences the power density of the transformer, according to the following proportionality,

Power Volume ∝ k 2 eff . .f r , (8)
with the permittivity and f r the resonance frequency. Furthermore, it also determines the operating frequency range where the performances are optimal correspondingly to:

k eff = (f 2 a -f 2 r ) f 2 a . ( 9
)
Hence, to obtain the best efficiency on a widest range of frequency variation, this parameter must be maximized.

Primary Dielectric Capacitance:

Depending on the converter topology selected to supply the PT, the primary capacitance is an important element of the optimal design. Indeed, this capacitance induces high reactive current which implies the oversizing of the inverter components (increasing of commutation losses). As a consequence, for a magnetic-less topology, the dielectric capacitance must be minimized. In the case of a resonant inverter, an inductance correctly dimensioned with the primary dielectric capacitance is used in order to operate at the resonance or antiresonance frequency of the PT. For this structure, the input electrical quality factor also implies the minimization of the capacitance.

Volume:

The volume can also be a criterion of optimization in order to obtain a compact solution for onboard devices. However, the dielectric losses and the internal friction of the PT induce a temperature rise which becomes critical near the Curie point. So, the minimization of the global volume must take into account the technical limit of temperature directly linked to the dissipation area.

The expected optimal design of the transformer sometimes needs to consider simultaneously several criteria. In this case, the optimization problem becomes quickly complex and an analytical method is inefficient if strong simplifications are not done.

III. The Exact Global Optimization Method: IBBA

Different methods have already been used to solve the design problem of piezoelectric transformers, such as an analytical approach [START_REF] Lin | Piezoelectric transformer characterization and application of electronic ballast[END_REF], [START_REF] Vasic | A new method to design piezoelectric transformer used in MOSFET and IGBT gate drive circuits[END_REF] or evolutionary strategy-based algorithms [START_REF] Seo | Optimal design of contour vibration mode piezoelectric transformer[END_REF]. The analytical approach is applicable if some preliminary variables are fixed (fixed operating frequency, type of ceramic, etc.) and the model is also simplified. Consequently, the arbitrary choice of these constant parameters limits the possible configurations that can satisfy the performances, especially when the consideration of several optimal criteria is required.

In comparison, the algorithmic method detailed below permits keeping a wide set of variables and thus extracting the exact optimal result amongabroadrangeofintermediate solutions that satisfy the constraints. This method is based on a deterministic approach that permits reaching the global solution for continuous problems. It relies on analytical equations, and interval arithmetic and bound and branch techniques. In addition to the optimal dimensioning, this approach permits integration of discrete variables and, in our application, allows selection of the suitable piezoelectric material type among a broad range of ceramics. In this way, the properties announced by manufacturers are collected and tested together, and the best category is kept for the optimal solution. This advantage is important compared to the simple analytical method because this last solution is generally developed for a particular ceramic and, moreover, by neglecting the dielectric losses. As a result, the maximal efficiency is optimized only by reducing the mechanical losses. These losses and the induced temperature rise are particularly nonnegligible when multilayered ceramics are used.

A. Interval Branch and Bound Algorithm and Extensions

The exact global optimization algorithm used for this work was already developed for other applications [START_REF] Fitan | The electromagnetic actuator design problem: A general and rational approach[END_REF] [8]- [START_REF] Messine | An interval branch and bound method dedicated to the optimal design of piezoelectric actuators[END_REF]. This algorithm is an extension of an interval branch and bound algorithm named IBBA; see [START_REF] Hansen | Global Optimization Using Interval Analysis[END_REF]- [START_REF] Ratschek | New Computer Methods for Global Optimization[END_REF]. It is based on interval analysis which permits computation of bounds for a continuous function over a box (i.e., an interval vector). In [START_REF] Fitan | The electromagnetic actuator design problem: A general and rational approach[END_REF], this method has been extended to deal with mixed variables: real, integer, categorical, or logical. The detailed algorithm IBBA is given in [START_REF] Fitan | The electromagnetic actuator design problem: A general and rational approach[END_REF], [START_REF] Messine | A deterministic global optimization algorithm for design problems[END_REF], and is briefly recalled here, supplemented with some particular extensions dedicated to this application.

Algorithm IBBA.

1. Set X := the initial domain in which the global minimum is sought,

X ⊆ R nr × N ne × nc i=1 K i × B n b . 2. Set f := +∞. 3. Set L := (+∞, X). 4.
Extract from L the box for which the lowest lower bound has been computed. 5. Bisect the considered box chosen by its midpoint, yielding V 1 , V 2 . 6. For j:=1 to 2 do (a) Compute v j := lower bound of f over V j . (b) Compute the lower and upper bounds for the interesting constraints over V j ; deduction steps using the constraints permit to reduce V j , [START_REF] Messine | A deterministic global optimization algorithm for design problems[END_REF], [14] (c) if f ≥ v j and no constraint is unsatisfied then Because the algorithm stops when the global minimum is sufficiently accurate less than f , and also when all of the sub-boxes Z are sufficiently small, all of the global solutions are given by the minimizers belonging to the union of the remaining sub-boxes in L, and the minimal value is given by the current minimum f . Generally, only f and its corresponding solution are considered.

• insert (v j , V j ) in L. • set f := min( f, f(m)),
The algorithm works by the four distinct following phases, which are completely detailed in the following subsections:

1. Bisection Rules: This phase is important because it determines the efficient way to decompose the initial problem into smaller ones. In our implementation, all of the components of a box are represented by real-interval vectors. Thus, attention must be paid when the components represent integer, boolean, or categorical variables.

The classical principle of bisection (in the continuous case) is to choose a coordinate direction parallel to which Z has an edge of maximal length. Then, Z is bisected normal to this direction [START_REF] Ratschek | New Computer Methods for Global Optimization[END_REF]. In our applications (the design of electromechanical devices), a lot of variables are nonhomogeneous, coming from different physical characteristics (number of primary and secondary layers, for example). Hence, the accuracy given by the designer, and represented in the algorithm by the variable , must be a real vector representing the desired precision for the solution at the end of the algorithm; k is fixed to 0, if it represents a discrete (integer, boolean, or categorical) component. Therefore, the bisection rule is modified considering continuous or discrete variables ( k = 0).

Let us denote by ω X i , ω Z i , ω Σ i , and ω B i the given weights for the real variables x i , the integer variables z i , the categorical variable σ i , and the boolean variables b i , respectively.

First, the following real values are computed:

Ω = ω X i × X i ε i , ω Z i × (|Z i | -1), ω Σ i × (|Σ i | -1), ω B i × (|B i | -1) , (10)
where the application |.| denotes the cardinal (i.e., the number of elements) of the considered discrete sets. The k th variable, which will be bisected, corresponds to the largest real value of the set Ω. Then, it is bisected as follows:

1. Z 1 := Z and Z 2 := Z 2. if ( k = 0) then (for discrete variables) (a) (Z 1 ) k := z L k , z L k +z U k 2 I (b) (Z 2 ) k := z L k +z U k 2 I + 1, z U k 3. else Z k is divided by its midpoint, this directly pro- duces Z 1 and Z 2 ,
where

Z k = [z L k , z U k ], respectively (Z 1
) k and (Z 2 ) k , denotes the k th components of Z, respectively of Z 1 and Z 2 . [x] I represents the integer part of the considered real value x.

Remark I: It is more efficient to emphasize the bisection for discrete variables because this involves a lot of important modifications of the so-considered optimization problem (1). In the following numerical examples, the weights for the discrete variable are fixed to 100.

Computation of the Bounds:

The computation of the bounds is the fundamental part of the algorithm, because all of the techniques of exclusion depend on it.

An inclusion function is an interval function such that it encloses the range of a considered function over a box Y . For a considered function f , a corresponding inclusion function is denoted by F ; hence, one has [min y∈Y f (y), max y∈Y f (y)] ⊆ F (Y ), and also, that for all Z ⊆ Y, F (Z) ⊆ F (Y ), [START_REF] Moore | Interval Analysis[END_REF].

In the case when continuous functions are considered, interval arithmetic can be directly used to construct interval inclusion functions; please refer to [START_REF] Messine | A deterministic global optimization algorithm for design problems[END_REF], [START_REF] Hansen | Global Optimization Using Interval Analysis[END_REF], [START_REF] Ratschek | New Computer Methods for Global Optimization[END_REF], [START_REF] Moore | Interval Analysis[END_REF] for more details on this subject. In our problem [START_REF] Rosen | Ceramic transformers and filters[END_REF], mixed function has to be considered, and then, new kinds of inclusion functions must be introduced. For a mixed function coming from problem (1), these inclusion functions were introduced in [START_REF] Fitan | The electromagnetic actuator design problem: A general and rational approach[END_REF] and [START_REF] Fitan | A general analytical model of electrical permanent magnet machine dedicated to optimal design[END_REF].

The following paragraphs recall the techniques used in [START_REF] Fitan | The electromagnetic actuator design problem: A general and rational approach[END_REF] to construct inclusion functions for mixed functions of type (1), by using interval analysis, for example, [START_REF] Moore | Interval Analysis[END_REF].

Let I be the set of real compact intervals [a, b], where a, b are real (or floating point) numbers. The arithmetic operations for intervals are defined as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [a, b] + [c, d] = [a + c, b + d], [a, b] -[c, d] = [a -d, b -c], [a, b] × [c, d] = [min{a × c, a × d, b × c, b × d} max{a × c, a × d, b × c, b × d}], [a, b] ÷ [c, d] = [a, b] × 1 d , 1 c if 0 ∈ [c, d]. ( 11
)
The above definitions [START_REF] Hansen | Global Optimization Using Interval Analysis[END_REF] show that subtraction and division in I are not the reverse operations of addition and multiplication. Unfortunately, the interval arithmetic does not conserve all of the properties of the standard one. The division by an interval containing zero is undefined, and then, an extended interval arithmetic has been developed; refer to [START_REF] Ratschek | New Computer Methods for Global Optimization[END_REF].

A fundamental theorem of interval analysis is that the natural extension of an expression of f into intervals, consisting of replacing each occurrence of a variable by its corresponding interval (which encloses it), and then by applying the above rules of interval arithmetic, is an inclusion function; special procedures for bounding trigonometric and transcendental functions allow the extension of this procedure to a great number of analytical functions. The proof of inclusion is given in [START_REF] Ratschek | New Computer Methods for Global Optimization[END_REF].

The bounds evaluated in such a way (by the natural extension of an expression of f ) are not always accurate in the sense where the bounds can become too large, and several other techniques are often used; refer to [START_REF] Messine | A deterministic global optimization algorithm for design problems[END_REF], [START_REF] Ratschek | New Computer Methods for Global Optimization[END_REF], [START_REF] Moore | Interval Analysis[END_REF] for a thorough survey and discussion on this subject.

For our considered design problems, the natural extension into interval has been sufficient.

Interval arithmetic is only defined for continuous real functions, and then, inclusion functions must be extended to deal with discrete variables.

For boolean and integer variables, one must just relax the fact that these variables are discrete: the discrete boolean sets {0, 1} become the continuous interval compact sets [0, 1], and the discrete integer sets {0, . . . , n}, {1, . . . , n}, or more generally {z L , z L + 1, z L + 2, . . . , z U } are relaxed by the following compact intervals: [0, n], [1, n], and [z L , z U ]. Hence, a new inclusion function concerning mixed variables-boolean, integer, and real-can then be constructed.

For categorical variables, intermediate real univariate functions must be introduced, such as a i : K k -→ R. In our practical cases, these real functions depend on only one categorical variable, but this could be easily generalized and extended. Therefore, in an expression of a considered function f , one has:

f (x, z, σ, b) = • • • * i a i (σ k ) * j • • • * k a j (σ m ) * l • • • * m a l (σ k ) . . . ,
where * i denotes an operation, for example, +, -, ×, ÷; hence, f produces a real result.

In IBBA, all categorical variables σ k are denoted by an integer number beginning at 1 to |K k |. Each of these numbers corresponds to a category previously defined.

In this study, a loop is done on each possible value of the categorical variables. Therefore a constrained global optimization problem with only real and integer variables is solved iteratively. Each time the best value for the objective function is kept.

Therefore, bounds are computed using directly the interval arithmetic tools by relaxing the boolean and integer sets as explained above.

Remark 2: There exist four other different approaches for solving this type of problem (1), detailed in [START_REF] Jeannet | Interval branch and bound algorithms for optimization problems with real and categorical mixed variables[END_REF]. They are not used in this first study of design.

Remark 3: This is the first time that IBBA has been used for solving an electromechanical design problem with a categorical variable which owns a lot of possible distinct values.

Exclusion Principle:

The techniques of exclusion are based on the fact that it is proved that the global optimum cannot occur in a box. This leads to two main possibilities by considering a sub-box denoted by X, Z, σ, B (where σ is iteratively fixed to a value):

1. The solution already found, denoted by f cannot be improved in this box: F L (X, Z, σ, B) > f; i.e., the lower bound of f over the sub-box X, Z, σ, B is greater than a solution already found; no point in the box can improve the current solution f . 2. It can be proved that a constraint could never be satisfied in the sub-box X, Z, σ, B :

G L k (X, Z, σ, B) > 0 or 0 ∈ H k (X, Z, σ, B).
In the present case, the computation of the bounds is exact and sure; no numerical error can produce wrong lower or upper bounds. Thus, the associated global optimization algorithm is exact and the global optimum is then perfectly enclosed with a given accuracy.

Improving of Bounds for η max :

The expression of η max (7) contains a lot of occurrences of some identical variables. Hence, interval analysis generally produces in such a case interval results that are too wide and without interest even if the considered box is quite small. An idea that is used here with efficiency is to construct an underestimation and an overestimation function of η max such that these two new functions own expressions that reduce the number of occurrences of the variables. Therefore the use of interval analysis over these new expressions produces sometimes better bounds than those performed using directly the expression of η max .

For η max we construct the two following estimation functions:

η over = 100R opt R m N 2 (1 + q 2 i ) + R opt , ( 12
)
with q i as the electrical quality factor explained in (2). Eq. ( 12) corresponds to the efficiency without considering the dielectric losses. Therefore, the result of η over will be always over the η max . The under estimation is formulated from an unacceptable value of electrical quality factor q i = 2,

η under = 100R opt R m N 2 (1 + 2 2 ) + R opt . ( 13
)
This reduction of bound values is computed as follows:

if η U max > η U over then η U max := η U over if η L max < η L under then η L max := η L under
This principle of reduction of bounds can be applied to all other functions. This is the first time that IBBA is used with these reduction methods, and numerical results are obtained faster than without it. The formulation of the optimization problem and the results are detailed in the next section.

IV. Results of Optimal Designs

A. Formulation of the Optimization Problem

In order to formulate the design problem, the first step is to declare the different variables which are real, integer, and of category:

• continuous variables:

the radius r, -primary and secondary layers thickness t 1 and t 2 , -resistive load R load , number of primary and secondary layers n 1 and n 2 ,

• variable of category (integer):

type of ceramic.

These variables are bounded by minimal and maximal values, defined by the schedule of conditions and the technical limitations. The most important functions are bounded, too, in accordance with the specifications. In the frame of electronic ballast and specifications considered by [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF], the values are presented in Table III. The IBBA will be performed with a selection of 13 types of piezoelectric ceramic. This selection is composed of soft and hard ceramics produced by two manufacturers (Ferroperm Piezoceramics A/S, Kristgaard, Denmark); and APC International, Ltd., Mackeyville, PA.

Added to these bounded values of the variables, a set of constraints are also defined in Table IV in order to obtain an efficient behavior of the transformer.

In comparison with the classical analytical method which is deduced with an imposed resonance frequency, we suggest here a range of convenient frequency f r in accordance with the specifications of the applications (zero voltage switching conditions). This newly defined bounded function allows the addition of one degree of freedom in the optimal design problem. The electric quality factor q i is bounded, too, and the values are chosen to satisfy an optimal efficiency for the rated resistive load.

We also fix a maximal height of the transformer to avoid the delamination. Moreover, this maximal height is con- strained by a ratio with the diameter of the transformer to promote the radial vibratory mode. The temperature rise at the functioning point must be under 200 • C to avoid the Curie point. This critical thermal point is considered equal for each type of ceramic. Finally, the thickness layers of the primary and secondary sectors are equal for the simplification of manufacturing and computations.

B. Single Criterion and Multicriteria Optimizations

This optimization is performed in the frame of electronic ballast application and the results are compared with the design obtained analytically by [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF]. The optimal resistive load, bounded values of frequency, and voltage gain are in accordance with the requirements presented in Table IV. The design optimization performed in this paper is about three essential criteria. First, the algorithm method is applied to each single criterion, i.e., the maximal efficiency, the maximal electromechanical coupling factor, and the minimal input capacitance. Then, an optimal design is carried out in order to satisfy the best compromise between these criteria. Table V groups the obtained results. These designs are in accordance with the imposed constraints (optimal load, satisfactory voltage gain, etc.).

Several comments are noteworthy from the results of single criterion optimizations:

• The optimization algorithm has given some slightly more efficient results than those in [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF], depending on the chosen criterion. • Since the secondary capacitance must be in accordance with the rated load (500 Ω), it is noticeable that stacked ceramics are more suitable than multilayer ceramics for this type of application. • For each optimal design, the efficiency is excellent and, therefore, it does not represent a sufficient criterion of optimization. Consequently, a real upgrading of transformer concerns the other parameters such as frequency, volume, dielectric capacitance, coupling factor, or temperature. • The expected voltage ratio requires few layers. This implies very high dielectric impedances and good efficiency. Consequently, primary and secondary dielectric impedances may be neglected in the present study so as to simplify the expression of efficiency. This is not an appropriate simplification in the case of an architecture with many layers. • Among the ceramic types considered (13 types), only hard ceramics present satisfying performances (PZ26, PZ24, and ACP-841). • Among the widest possible combinations among the parameters of Table III, the dimensions are slightly similar for the different optimized designs. This means that the system of equations is strongly constrained. In the present case, resonance frequency, electrical quality factor, and thermal limitations impose strong constraints on the dimensions. • About the temperature rise, the maximal value is strongly reduced for the optimal solutions compared to the results of [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF]. This value is obtained for the rated resistive load at the resonance frequency. This particular operating point will not be reached since the voltage ratio is higher than expected at this point. However, it underlines a destructive functioning point that must be avoided. • The compactness is almost similar to the reference solution in [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF]; however, the temperature rise is less. Furthermore, the reduction of total thickness permits reducing the risk of delamination.

The performances of each design are satisfactory regarding efficiency, coupling factor, and capacitance. A multicriteria optimization is investigated in order to verify if another design gives a better compromise.

The multi-objective optimization is obtained with the formulation of a new function deduced from the singlecriterion results. This "weight factor method" permits considering differently the weight of the criteria. multi

(X) = - η max (X) max(η max ) - k eff (X) max(k eff ) + C d1 (X) min(C d1 ) , ( 14
)
where max(η max ), max(k eff ), and min(C d1 ) are the three results obtained during the single-criterion optimization (refer to Table V). The minimization of this function [START_REF] Messine | Deterministic global optimization using interval constraint propagation techniques[END_REF] gives a compromise between the criteria, depending on the weights attributed to each criterion. Consequently, several optimal results can also be obtained. However, in this study the dimensions obtained are closer regardless of the criterion considered. So, the multi-objective optimization does not bring an important upgrade of performances. For this reason, the dimensioning obtained for the maximization of coupling factor k eff is selected to illustrate the results of the proposed method compared with the reference solution. Thus, the theoretical performances (according to the electrical equivalent circuit) are shown in the following figures and discussed. Fig. 6 shows that the optimal solution gives a maximal efficiency slightly better than the reference in [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF]. This maximal efficiency is obtained for an optimal R load near the rated value (500 Ω), thanks to the electrical quality factor q i ≈ 1. The evolution of efficiency at constant resistive load function of the operating frequency is illustrated in Fig. 7. This shows again a greater maximal efficiency, but reached at a lower frequency for the optimal PT. This characteristic still depends on the schedule of conditions and permits limiting the losses in an eventual upstream inverter. Furthermore, the efficiency is high on a widest range of frequency compared with the PT designed in [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF].

Verification that the voltage gain for the rated load value is superior to the expected minimal value is shown in Figs. 8 and9; this voltage gain can be reached by an appropriate operating frequency.

From these results, we can conclude that better control of performances is attained by the proposed algorithmic approach. The association of technical constraints makes it possible to determine the best design, avoiding the expensive phase of prototype making. Obviously, the results obtained by this approach depend on the accuracy of the analytical model and on the reliability of the piezoelectric characterization data.

As for all manufactured products obtained by sintering, this conception of ceramics implies an inevitable tolerance of the dimensions. This technical requirement can also be considered in the algorithm design problem: indeed, an epsilon value can be added to each inaccurate variable. Thus, the final result of the algorithm method will be an interval solution for each criterion, depending on the maximal bounded tolerances.

V. Conclusion

All piezoelectric transformers require a precise design so as to obtain the optimal performances for specific applications. The considered resonant structures are strongly affected by the operating frequency and the load. There-fore, this paper deals with the use of IBBA in order to solve the design of PT by the formulation of an inverse problem.

Contrary to a classical analytical optimization which takes into account a single criterion and generally requires a preselected starting point, the present method gives an exact solution (if it exists) for design problems. The IBBA includes a wide set of constraints such as the schedule of conditions, the technical limits, and the realistic properties of the available materials. The variables defined by interval offer more degrees of freedom to satisfy the optimal design without arbitrary constraints.

A typical application of electronic ballast has been chosen in order to illustrate the proposed design method. First, simulations provided from the electrical equivalent circuit have given an overview of the behavior and have also underlined the most important criteria of an optimal functioning. Then, the IBBA has been briefly detailed, including the formulation of the optimal design problem supplemented with some extensions. As a first conclusion, it was noticed that global performances of the chosen test transformer are quite satisfactory compared to the design results of [START_REF] Huang | Design of radial mode piezoelectric transformer for a charge pump electronic ballast with high power factor and zero voltage switching[END_REF]. However, by adding a set of bound parameters in accordance with the conditions, the performances can be significantly improved by a rational choice of both material and dimensions. The present method has shown good accuracy for the chosen step-up transformer application, and it can be adequately applied for specific applications (low-voltage, high frequency, PT matrix, etc.) and other architectures of PT. This study is concentrated on a particular vibratory mode but can be extended to several architectures of transformers for a broad whole of applications. Moreover, a great additional advantage of the proposed design algorithm lies in the possibility of taking into account the technological aspect by defining a precision for each geometrical parameter.

Fig. 1 .

 1 Fig. 1. Architecture and parameters of radial mode transformers.

Fig. 2 .

 2 Fig. 2. Electrical equivalent circuit around one vibratory mode.

Fig. 3 .

 3 Fig. 3. Input impedance and phase as functions of the operating frequency.

Fig. 4 .

 4 Fig. 4. Typical characteristics as functions of the operating frequency.

Fig. 5 .

 5 Fig. 5. Typical characteristics as functions of the resistive load.

Fig. 6 .

 6 Fig. 6. Efficiency as a function of the resistive load.

Fig. 7 .

 7 Fig. 7. Efficiency as a function of the operating frequency.

Fig. 8 .

 8 Fig. 8. Voltage gain as a function of the resistive load.

Fig. 9 .

 9 Fig. 9. Voltage gain as a function of the operating frequency.

TABLE I

 I Equations of Equivalent Circuit.

	Primary capacitance	C d1 =		d 2 31 (1-σ) T 33 .S E 11
	Secondary capacitance	C d2 =		d 2 31 (1-σ) T 33 .S E 11
	quivalent inductance	Lm =
	Equivalent capacitance	Cm =	π.S E	32.r 2 (d 31 .n 1 ) 2
	Equivalent impedance	Rm =	2ρ.S E 11 32.Qm.d 2 3 .(1-σ) 3 31

r 2 .n 1 .π. T 33 t 1 1r 2 .n 2 .π. T 33 t 2 1ρ.S E 11 2 .(1-σ) 2 .(n 1 t 1 +n 2 t 2 ) 16.π(n 1 d 31 ) 2 11 .(n 1 t 1 +n 2 t 2 )(1-σ)

TABLE III Bounded

 III Values of the Variables.

	Variables	Values
	Radius r	[4.5e -3 ; 30e -3 ]
	Layer thickness t 1	[20e -6 ; 2e -3 ]
	Layer thickness t 2	[20e -6 ; 2e -3 ]
	Layer number n 1	[1; 50]
	Layer number n 2	[1; 50]
	Resistive load R load	[500; 500]
	Ceramics type cat	[1; 13]

TABLE V

 V Results for Single-Criterion Design.

				Max	Min
		Unit Max η	k eff	C d1	Reference
	r	mm	13.07	11.9	11.8	10.5
	t 1 , t 2	mm	0.615	1.116	0.558	1.5
	n 1		3	3	3	4
	n 2		1	1	1	1
	Ceramic type		PZ24	PZ26	PZ24	ACP-841
	fr	kHz	83.4	81.37	92.42	98.3
	N		3	3	3	4
	R d1	kΩ	103.6	50.46	104.3	45.4
	R d2	kΩ	310.8	151.4	312.8	181.6
	C d1	nF	9.2	12.9	8.25	10.2
	C d2	nF	3.07	4.3	2.75	2.54
	ηmax	%	98.8	98.7	98.7	96.5
	k eff		0.4	0.448	0.4	0.4
	q i		0.804	1.1	0.8	0.786
	∆Θ	• C	163	183	199	900

Ph.D. degree from the Institut National Polytechnique de Toulouse in 1990. He is now a professor with the Electrical Engineering Department of INPT/ENSEEIHT. He teaches an electrodynamics and electrical machines course and is also the head of the Electrodynamics GREM3 research group of INPT-ENSEEIHT-LEEI Toulouse. The group consists of nine permanent academics working in the fields of electromechanical energy conversion, with particular interest in novel techniques and design methodologies such as electroactive materials, composite magnetic materials, electroactive fluids, analytical field calculation, optimal design, etc.


Philippe Evon
email: philippe.evon@ensiacet.fr

Guyonne De Langalerie

Laurent Labonne

Othmane Merah

Thierry Talou

Stéphane Ballas

Thierry Véronèse

Innovative low-density blocks from amaranth pith for the thermal insulation of buildings

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible Any correspondence concerning this service should be sent to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr

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Innovative low-density blocks from amaranth pith for the thermal insulation of buildings Philippe Evon 1 , G. de Langalerie 1,2 , L. Labonne 1 , O. Merah 1 , T. Talou 1 , S. Ballas 2 , T. Véronèse 2 1 Laboratoire de Chimie Agro-industrielle (LCA), Université de Toulouse, ENSIACET, INRA, INPT, Toulouse, France 2 Ovalie Innovation, Auch, France Philippe.Evon@ensiacet.fr Amaranth is an annual herb native to temperate and tropical regions. Cultivated by Native Americans for the nutritional properties of its seeds, the latter are very digestible and also an interesting source of starch and proteins. For the future, amaranth appears as a promising raw material for the biorefinery of whole plants, all parts of the plant being potentially usable for different food and non-food applications. This study especially aims to investigate the possible uses of pith from stems for material applications. For that, plants from the Amaranthus cruentus variety were cultivated in 2018 near Auch (Gers, France). Stems were manually harvested at plant maturity. Representing up to 90% w/w of the aerial part of amaranth plant, stems were then dried in a ventilated oven to facilitate their conservation. They are composed of a bark on their periphery and a pith fraction in their middle. The structure of stems was studied from ten samples, and the pith fraction was estimated manually to 27% w/w. Due to the difference in density between bark and pith fractions, a fractionation process associating grinding and blowing steps made possible the continuous separation between bark and pith. As for sunflower and corn, amaranth pith particles have an alveolar (i.e. a microporous) structure similar to that of expanded polystyrene, and they reveal a very low bulk density (e.g. 48-52 kg/m 3 for 4-16 mm particle size, and 58-61 kg/m 3 for 1.25-2.50 mm particle size), making them a promising raw material for the thermal insulation of buildings. Cohesive and low-density insulation blocks were successfully obtained from amaranth pith, primarily mixed with a starch-based binder, through compression moulding at ambient temperature. Different operating conditions were tested, especially including (i) the size distribution of amaranth pith particles, (ii) the binder content, and (iii) the filling level of the mould. All produced samples were then characterized in terms of (i) density, (ii) bending and compression properties, and (iii) thermal insulation properties (measured through the hot wire method). All insulation blocks revealed low density, ranging from 90 to 140 kg/m 3 . Such innovative materials could be sustainable and viable options for the thermal insulation of buildings.

Introduction Conclusion

Materials and methods

Results and discussion

Aims of the study :

1 To optimize a fractionation process for the separation of bark and pith.

2 To develop a process for producing thermal insulation blocks from pith.

Name

  This is an author's version published in: http://oatao.univ-toulouse.fr/2 5895 To cite this version: Evon, Philippe and de Langalerie, Guyonne and Labonne, Laurent and Merah, Othmane and Talou, Thierry and Ballas, Stéphane and Véronèse, Thierry Innovative low-density blocks from amaranth pith for the thermal insulation of buildings. (2019) In: 15th International Conference on Renewable Resources & Biorefineries (RRB15), 3 June 2019 -5 June 2019 (Toulouse, France). (Unpublished) .

►

  Amaranthus cruentus, an annual herb from temperate and tropical regions. ► Seeds are cultivated by Native Americans for their nutritional properties.► Stems are composed of 2 parts : bark in periphery and pith in the middle. ► Alveolar structure of the pith similar to expanded polystyrene (EPS).

  ) ; Thermal conductivity (λ) Bending properties : Bending stress (σ f ) and elastic modulus (E f ) 2 ρ (kg/m 3 ) λ (mW/m.K) σ f (kPa) E f (kPa) Amaranth pith 102.4 ± 5.9 54.8 ± 1.5 21.0 ± 6.7 299.0 ± 88.5 Sunflower pith [1] 32.0 ±1.92 35.3 ± 1.6 49.6 ± 6.0 422.0 ± 93.M. Abdellahi, Contribution to the elaboration and characterization of a 100% bio-based thermal insulating panel, using sunflower pith for building applications. (2016) [2] A. Ben Rhouma, Development and characterization of bio-based composite materials for building industry with low environment impact and corn stem based. and pith from the amaranth stem by a grinding, sieving and blowing (for pith purification) process.


domain: sdv info shs

text:
D Andueza
email: donato.andueza@clermont.inra.fr

A M Rodrigues

F Picard

N Rossignol

R Baumont

U Cecato

A Farruggia

Relationships between botanical composition, yield and forage quality of permanent grasslands over the first growth cycle

Keywords: floristic composition, chemical composition, organic matter digestibility, voluntary intake, permanent grasslands

This study examined the relationships between botanical composition and forage parameters (yield and forage quality variables) in 153 permanent grasslands located in the Massif Central of France. Grasslands were sampled at two vegetation stages in the first growth cycle. Botanical composition, yield, ash, crude protein, neutral detergent fibre, acid detergent fibre, acid detergent lignin, organic matter digestibility (OMD) and voluntary intake (VI) were estimated for each sample. Temporal variability in species-forage parameter relationships were accounted for using innovative multivariate analyses applied mainly in ecological science. Crude protein and OMD were weakly correlated when each harvest time was analysed separately. Species-forage parameter relationships remained stable during the first growth cycle. The stability of these relationships indicates that permanent grasslands dominated by competitive species were associated with high yield and forage quality values whereas permanent grasslands composed of conservative species and/or high proportions of senescent material were associated with high structural carbohydrate values and low yield, OMD and VI values. Based on these relationships, we propose a typology of permanent grasslands along with a set of indicator species enabling non-specialist botanists to easily classify grasslands for grassland management purposes.

Introduction

The EC definition of permanent grassland is a plot 'that has not been included in the crop rotation of the holding for five years or longer', yet European permanent grassland is usually kept for longer than five years and is not sown, although, in some cases, it may be oversown or resown depending on management intensity. Permanent pastures, as part of permanent grasslands are used to feed livestock and are typically composed of a mixture of native self-seeded grasses and forbs. These botanical communities are maintained by agricultural practices (livestock grazing and/ or mowing) but are not substantially modified by intensive inorganic fertilization, drainage or herbicide use (Reheul et al., 2007). They are also called unimproved permanent grasslands, but if one or more of these practices is used on them, they are reclassed as semi-improved or improved permanent grasslands according to degree of use.

Permanent grasslands are a major feature of land use in Europe, covering more than a third of the European agricultural area (FAO, 2008). Forage from permanent grasslands is a key component of ruminant diets and also plays a key role in differentiating products derived from livestock fed with this type of forage (Martin et al., 2005).

The feed value of the main forage grass species found in temporary grasslands is well established (Baumont et al., 2007), but there is a lack of adequate estimates of the feed value of forage obtained from permanent grasslands (Bruinenberg et al., 2002;Pelve et al., 2012). The large diversity of indigenous grasses and dicotyledons in permanent grasslands and their broad variability in phenology and competitiveness (Rossignol et al., 2014) may explain the difficulty in obtaining good estimates of nutritive value. Seasonal changes in plant abundances further complicate efforts to predict forage quality. Indeed, Todorova and Kirilov (2002) reported huge variation (19-69% of total yield) in the proportion of Agrostis capillaris in different periods of the first growth cycle of an Ag. capillaris-Festuca fallax grassland. Likewise, Andueza et al. (2013) reported important variations in the floristic composition of two botanically different extensively managed permanent grasslands studied at three dates during the first growth cycle in the Massif Central of France. This complexity is the reason why the variability in the feed value of permanent grasslands, and its evolution over the first growth cycle, are still poorly understood.

This study set out to (i) describe the relationship between the botanical composition of permanent grasslands (seminatural unimproved and semiimproved pastures and also grasslands improved in the past but unploughed for at least 5 years) and forage parameters:, i.e., yield, chemical composition and feed value, (ii) determine the stability of this relationship over the first growth cycle and (iii) characterize permanent grasslands on the basis of these variables. These objectives were achieved using innovative multivariate analyses that take into account how the relationship changes over time.

Materials and methods

Grasslands and sampling

A total of 153 permanent grasslands over 5 years old with no history of ploughing were randomly selected from locations in the Auvergne region of central France. The sampled grasslands were representative of different types of soil [granitic (n = 78) and volcanic (n = 75)], altitude [ranging from 300 to 1350 m a.s.l. (n = 83 for altitude > 900 m a.s.l., n = 70 for an altitude < 900 m a.s.l.)] and annual rainfall [ranging from 480 to 1440 mm (n = 23 for an annual rainfall > 1000 mm, n = 130 for an annual rainfall < 1000 mm)]. Plots included grasslands dominated by Lolium perenne or Dactylis glomerata but also Holcus lanatus, Festuca rubra or Ag. capillaris. once when farmers made silage (Silage stage, SS), that is, around the heading stage of dominant grass species, and once when they made hay (Hay stage, HS), that is, around the flowering stage of dominant grass species.

In each grassland, we used different plots for SS and HS samples. Sampling was performed on surfaces left uncut by farmers when they decided to harvest the grasslands between the two sampling periods. On average, HS sampling came three weeks after the SS sampling. For each sampling, five randomized 0Á25 m² subplots (0Á5 9 0Á5 m) were cut within each grassland at 5 cm above ground level using a small lawnmower. Forage obtained in the five subplots was pooled to give a single representative sample per grassland, then weighed and divided into two subsamples. The first subsample (125 g fresh material) was stored at À20°C and used to characterize botanical composition. The second subsample was dried at 60°C for 72 h to assess dry-matter (DM) content, then ground in a hammer mill through a 0Á8-mm screen.

Determination of botanical composition

First, each sample was sorted into fresh forage and senescent material (Sm). Botanical composition of fresh forage was then determined by hand-sorting the samples into plant species. Species identification and nomenclature followed Flora Europea (Tutin et al., 1964(Tutin et al., -1980)). Individual species and Sm were then dried at 60°C for 72 h and weighed. Proportion of each species and Sm was expressed as species and Sm contribution to total yield.

Determination of chemical composition and feed value

Ash and crude protein (CP) contents, in vivo organic matter digestibility (OMD) and voluntary intake (VI) by sheep were estimated by near-infrared reflectance spectroscopy (NIRS) according to the global model proposed by Andueza et al. (2011). Specific NIRS models were also built for the estimations of neutral detergent fibre (NDF), acid detergent fibre (ADF) and acid detergent lignin (ADL) contents. The reference methods used were proposed by Van Soest et al. (1991) for NDF and Van Soest and Robertson (1980) for ADF and ADL and performed on an Ankom analyzer (Ankom â Tech. Co., Fairport, NY, USA). For all determinations, before running the NIRS predictions, we first calculated Mahalanobis distance (H) between the spectra of each sample and the average calibration spectra to check the calibration models were compatible with the samples to be predicted. No samples were discarded (H > 3). Predicted values for chemical composition, OMD and VI of samples were within the range of values of the databases used to develop the calibration equations. Standard errors of cross-validation were 14Á0, 12Á1 and 6Á0 g kg DM À1 , and coefficients of determination of cross-validation were 0Á96, 0Á94 and 0Á90 for NDF, ADF and ADL respectively. Ash content, CP, VI, OMD, NDF, ADF and ADL were included in the analysis together with yield as variables representing forage quality.

Data analysis

Differences between sampling periods for forage parameter determinations were tested by ANOVA. The species proportions and forage parameters for each grassland and period of sampling generated two sequences of tables: one sequence of two tables for the species proportions at both SS and HS sampling periods and another sequence of two tables for the eight forage parameters, again at both the SS and HS sampling periods (Figure 1). The common relationship between botanical composition and forage parameters of permanent grassland and the stability of this relationship over both sampling periods were investigated using STATICO, a special kind of multivariate analysis (Simier et al., 1999;Thioulouse et al., 2004).

STATICO is a three-step method in which each table is first analysed separately by principal component analysis. The relationship between species and forage parameter is assessed for each period using a co-inertia analysis on each species-forage parameter pair of tables (Doledec and Chessel, 1994). Finally, a partial triadic analysis (Thioulouse and Chessel, 1987) is run on the sequence of cross-tables produced by the co-inertia analyses to assess the relationships that are stable along the sequence of cross-tables, that is over the two sampling periods here. The main outputs of this analysis are the interstructure, compromise and intrastructure. The compromise gives a picture of the common relationship between floristic composition and forage parameters over the two periods. The interstructure gives the importance of each period in the construction of the compromise, while the typological value gives an idea of how much the compromise expresses the information from each period. The intrastructure (trajectories) summarizes the variability of the species-forage parameters relationships around the common structure defined by the compromise between the two sampling periods (Thioulouse et al., 2004;Thioulouse, 2011).

Before implementing the STATICO analysis, botanical composition data were Hellinger-transformed to factor out double absence of species being considered a resemblance between sites (Legendre and Gallagher, 2001), and forage parameter data were standardized (divided by the variance). For each sampling period, the significance of the relationship between botanical composition and forage parameter tables revealed by the coinertia analyses was tested using a Monte Carlo test. Then, a grassland classification scheme was constructed using a hierarchical cluster analysis based on the row coordinates in the forage parameters space of a co-inertia analysis (Dray et al., 2003) performed between botanical composition and forage parameters of the SSperiod sampling. This method classifies swards by maximizing intragroup homogeneity and intergroup diversity. The number of groups was proposed by the scree test (Cattell, 1966). Multivariate analyses were carried out with GNU-project R software version R 2.13.2 (R Foundation for Statistical Computing, http://www.Rproject.org) using the ADE4 package (Thioulouse et al., 1997). Finally, the effect of these groups on the variables related to botanical composition and forage parameters was tested by ANOVA. The model included group and sampling period as fixed effects and grassland as random effect. Period of sampling was considered as a repeated measure. Arcsine square root transformation was performed on percentage species before statistical analysis to improve normality and homoscedasticity (Sokal and Rohlf, 1981). ANOVAs were performed using SAS statistical software (SAS, 2000).

Results

Yield, forage quality and species diversity

Yield and forage quality characteristics for both harvesting periods of the sampled grasslands are reported in Table 1. Estimated values for ash content, CP content, OMD and VI showed higher average values at SS than at HS, whereas average values for yield, NDF, ADF and ADL were higher at HS than at SS.

PCA PCA Z l (p X q) Z l (p X q) X l (n l X p) Y l (n l X q)
Data structure

Basic analyses

Co-inertia analysis

Cross-table

Partial triadic analysis et al., 2004).

In total, 125 different plant species (Table 2) were identified in the 306 samples (153 plots * 2 sampling periods), including thirty-six species of grasses (between 3 and 15 species per sample), seventy-eight species of forbs (0-23 species per sample) and eleven species of legumes (0-5 species per sample). Proportion of grasses ranged from 5 to 99% at SS and from 12 to 97% at HS. Proportion of forbs ranged from 0 to 66% at both SS and HS. Proportion of legumes ranged from 0 to 55% at SS and from 0 to 47% at HS. Proportion of Sm ranged from 0 to 81% at SS and from 0 to 47% at HS.

Relationship between botanical composition and forage parameters

Co-inertia analyses showed a significant correlation (P < 0Á001) between botanical composition and forage parameter for each sampling period, allowing us to implement the STATICO analysis.

Interstructure and typological values

The interstructure factor map (Figure 2a) allows an examination of how the relationships between species and forage parameters change according to the two different periods. The eigenvalue of the first axis (81% of overall inertia) clearly prevailed over the eigenvalue of the second axis (19% of overall inertia). The arrows on the interstructure factor map (Figure 2a) pointed in the same direction on the first axis which showed that the costructure between forage parameters and botanical composition of permanent grasslands was similar throughout the SS and HS periods. The data from both sampling periods thus had the same influence on the compromise. However, the typological value showed that the compromises better expressed the information contained in tables derived from the SS sampling period (cos 2 = 0Á95) than from the HS sampling period (cos 2 = 0Á82).

Compromise

The compromise expressed the relationship between species and forage parameters over the two periods. The projection of variables (species and forage parameters) on the first two axes of the compromise enabled us to spot associations among species (Figure 2c) and among forage parameters (Figure 2b). Furthermore, comparing the coordinates of the species and forage parameters on the first two axes of the compromise enabled us to highlight which of the species were related to forage-quality variables and yield. In this sense, the first axis was dominant, accounting for 72% of overall inertia, whereas the second axis accounted for just 11% of inertia. Looking at the factor map of forage parameters (Figure 2b), the first axis displayed an opposition between estimated NDF, ADF and ADL values on one side and estimated OMD, VI and yield of permanent grassland on the other, thus expressing a gradient of forage quantity and quality. The second axis was characterized by an estimated CP value gradient. Ash content was closely correlated with CP content. In Figure 2c, species were ranked according to their proportion in permanent grasslands. Running a comparison of both compromise figures allowed us to characterize permanent grasslands according to the relationship between forage parameters and characteristic species. Thus, proportion of Sm was positively associated with permanent grasslands characterized by high NDF, ADF and ADL values, low OMD and yield values and a high proportion of spe-Table 1 Mean, standard deviation (s.d.), minimum (min) and maximum (max) values obtained for yield and NIRS-estimated values for ash, crude protein (CP), neutral detergent fibre (NDF), acid detergent fibre (ADF), acid detergent lignin (ADL), organic matter digestibility (OMD) and voluntary intake (VI) of permanent grasslands sampled at two sampling periods (sp): silage stage (SS) and hay stage (HS).

Intrastructure

The intrastructure indicated the reproducibility of the compromise for each of the two sampling periods, making it possible to analyse how the species-forage parameters relationship differed between the two sampling periods. The projection of the forage parameter variables and the projection of the species on the compromise axes are shown in the trajectories factor map (Figure 3). The two periods were plotted on separate graphs to show (3a) projection of forage parameters in SS, (3b) projection of species proportions in SS, (3c) projection of forage parameters in HS and

6Á25 12Á16 80Á98 7Á02 11Á89 59Á39 Lolium perenne 6Á53 10Á72 57Á72 5Á76 10Á72 69Á00 Phleum pratense 0Á89 2Á61 15Á62 1Á61 4Á50 23Á46 Poa pratensis 4Á23 5Á68 41Á28 4Á46 5Á73 32Á60 Poa trivialis 2Á98 5Á89 41Á08 1Á85 3Á32 20Á03 Trisetum flavecens 3Á54 0Á41 5Á01 4Á61 6Á68 37Á26 Forbs Cerastium spp. 1Á60 2Á34 10Á83 1Á20 1Á82 8Á48 Gallium mollugo 0Á63 3Á30 29Á63 0Á59 2Á52 24Á62 Heracleum sphondylium 1Á00 3Á82 23Á77 1Á05 6Á02 52Á49 Plantago lanceolata 1Á22 3Á14 22Á24 0Á56 1Á44 8Á00 Ranunculus acris 1Á34 3Á04 22Á73 0Á88 2Á07 11Á00 Rumex acetosa 0Á54 2Á50 22Á16 0Á33 1Á37 10Á00 Taraxacum officinale 4Á42 7Á39 61Á08 1Á54 2Á81 16Á36 Urtica dioica 0Á46 4Á32 52Á80 0Á81 5Á59 49Á35 Legumes Lathyrus pratensis 0Á45 1Á43 10Á98 0Á37 1Á24 8Á49 Trifolium pratense 0Á68 4Á68 55Á80 0Á61 4Á04 47Á00 Trifolium repens 1Á27 2Á45 15Á77 1Á04 2Á71 21Á00 Vicia sativa 0Á42 1Á80 17Á06 0Á61 2Á29 17Á00 Other Senescent material 6Á12 10Á80 80Á91 6Á11 4Á97 26Á11
Minimum values are 0Á00 for all species, with the exception of senescent material at HS which presented a minimum value of 0Á37%. Twenty grasses, sixty-eight forbs, seven legumes and two monocotyledons not grasses are not listed.

(3d) projection of species proportions in HS. The stable part of the species-forage parameter dynamics revealed by the compromise analysis was better expressed for SS. Both species and forage parameters showed lower variability in HS than in SS, although there was strong similarity in the structure of the forage parameters and botanical composition. Consequently, for both sampling periods, yield and estimated VI and OMD values were mainly opposed to ADL but also to ADF and NDF in the first axis, whereas the second axis was explained by a gradient of CP values. There were no differences in botanical composition between the two sampling periods. However, Sm was less correlated with forage parameters in HS than in SS. Dactylis glomerata was more correlated with OMD in SS but it was more correlated with yield in HS.

A grassland typology according to botanical composition and forage parameters

The relationship between floristic composition and forage parameters as described prompted us to base the grassland typology on co-inertia analysis at the SS sampling period. The co-inertia and hierarchical cluster analyses led to a typology of permanent grasslands grouped into four classes according to the aggregate of proportion of different species and feed-quality values. Significant differences (P < 0Á01) between classes were found for all forage parameters (Table 3) and for the more representative species of each class (Table 4), except for proportion of F. pratensis. Significant differences (P < 0Á01) between periods were found for all forage parameters and for proportions of Ag. capillaris, B. mollis, T. officinale and Ranunculus acris. Class 9 period interaction was significant (P < 0Á01) for all forage parameters with the exception of yield and CP content, and for proportion of T. officinale and Sm. On the factor map of the co-inertia analysis (Figure 4), classes 1 and 4 were opposed to classes 2 and 3 according to the first axis (i.e., gradient of forage quantity and quality), while classes 1 and 3 were opposed to classes 2 and 4 on the second axis (i.e., gradient of CP).

Class 1 (n = 28) was characterized by low yield and low feed value. Average values were 1Á7 and 2Á6 t DM per ha yield, 117 and 88 g DM CP, 0Á66 and 0Á61 OMD and 65Á9 and 61Á3 g per kg BW 0Á75 VI for SS and HS respectively (Table 3). This class was chiefly characterized by abundant senescent material in SS (Sm 20%) decreasing to 7% in HS and high proportions of Ar. elatius, Ho. lanatus, E. repens, Ag. capillaris, Al. pratensis, U. dioica and Gallium mollugo in both sampling periods (Table 4).

Class 2 (n = 49) was characterized by high yield and high feed-value parameters (except CP). Average values were 4Á51 and 5Á98 t DM per ha yield, 108 and 81 g kg DM À1 CP, 0Á72 and 0Á62 OMD and 71Á4 and 65Á7 g per kg BW 0Á75 VI for SS and HS respectively (Table 3). The characteristic species of Class 2 were Ag. capillaris, L. perenne, D. glomerata, Fe. rubra, Ho. lanatus, Al. pratensis, T. officinale and Cerastium spp. (Table 4).

Class 3 (n = 48) was characterized by high yield and high feed-value parameters. Average values were 5Á83 and 6Á98 t DM per ha yield, 121 and 86 g kg DM À1 CP, 0Á75 and 0Á64 OMD, 73Á3 and 66Á3 g per kg BW 0Á75 VI, and 535 and 600 g per kg NDF for SS and HS respectively (Table 3). The characteristic species of Class 3 were D. glomerata, T. officinale and L. perenne. Taraxacum officinale decreased significantly between SS and HS (Table 4).

Class 4 (n = 28) was differentiated from other classes by low feed-value parameters, including low CP. Average values were 3Á43 and 4Á52 t DM per ha yield, 107 and 79 g per kg DM CP, 0Á68 and 0Á60 OMD, 61Á3 and 60Á3 g per kg BW 0Á75 VI, and 583 and 608 g per kg NDF for SS and HS respectively (Table 3). The characteristic species of Class 4 were Fe. rubra, Ag. capillaris, Ho. lanatus, G. mollugo and Sm (Table 4).

Discussion

The correlation between forage parameters and botanical composition

We found very broad variability in the chemical composition and feed value of the grassland-derived forages used in this study. The range of values in our dataset was similar to the range reported by Baumont et al. (2007) for green forages from the first growth cycle in French permanent grasslands. This large range in terms of both species identity and relative abundance reflects the diversity of plant communities sampled in this study.

An interesting finding of this study was the lack of correlation between CP and other feed-value parameters. Several studies have reported that CP and OMD decrease in a similar way with time (Andrieu et al., 1981;Andres et al., 2005). Our data did not confirm this pattern, even though both variables were well correlated (r = 0Á86) when the two sampling periods were combined, but this correlation almost disappeared when the CP-OMD linkage was investigated separately for each sampling period. Accordingly, for a given point in time, grassland forage can be characterized as either low OMD and high CP or low OMD and low-CP.

A surprising finding of this study was the positive correlation between Sm and CP content. The possible presence of fungal diseases of grasses on Sm could explain this result. According to Queiroz et al. (2012), plants characterized by moderate rust diseases showed lower CP content than disease-free plants whereas plants characterized by high rust infection showed higher CP content than disease-free plants. Potter (1987) also reported that the presence of crown rust increased the CP content of perennial and Italian ryegrasses. According to Dimmock and Gooding (2002), rust causes physical damage to plants thereby causing leaves to senesce prematurely through desiccation. This reduces translocation but greatly increases retention of amino acids which could increase the CP content of forage. Table 3 Mean values and statistical effects for yield and NIRS-estimated values for ash, crude protein (CP), neutral detergent fibre (NDF), acid detergent fibre (ADF), acid detergent lignin (ADL), organic matter digestibility (OMD) and voluntary intake (VI) in four classes of permanent grassland sampled at two periods during the first growth cycle:

silage stage (SS) and hay stage (HS). Our results further confirmed the well-known negative correlation between structural carbohydratelignin contents and OMD, VI and yield. Several papers have reported similar relationships between ADF and ADL content and OMD (Demarquilly et al., 1995) and between cell-wall content and VI (Mertens, 1994) in forages.

Looking at the relationship between botanical composition and forage parameters, the first factor map of the compromise revealed a species gradient based on ability to compete using natural resources (nutrients, light etc.; Grime, 1977) associated with a feed-value gradient. Permanent grasslands rich in competitive grasses such as L. perenne and D. glomerata associated with high forage parameter values (Schubiger et al., 2001;Bruinenberg et al., 2002) were opposed to more conservative grasses such as Fe. rubra and Ag. capillaris associated with low forage parameter values (Frame, 1991;Pavlu et al., 2006;Pontes et al., 2007). The presence of these functional groups of species (Cruz et al., 2002) is generally associated with a gradient of several environmental variables (precipitation, relative irradiation, soil type, air temperature and nitrogen) (Ejrnaes and Bruun, 2000;Jeangros et al., 2000) and/or with different management history [fertilization, drainage (Hopkins, 1986) and intensity of management (Pavlu et al., 2006)].

Other major species playing an important role in this study were forbs. The feed value of some of these species is not yet sufficiently described in the literature. Forbs may contain secondary compounds which could modifypositively or negativelythe feed value of permanent grasslands. The few papers citing feed values for this functional group do not always (2012) found that the nutritive value of herbage from diverse grasslands was in some cases higher than grass-dominated swards, but Hofmann and Isselstein (2005) concluded that the introduction of forbs in a permanent grassland decreased forage digestibility. In our study, the presence of T. officinale was positively correlated with abundance of L. perenne and was associated with high OMD and VI values, as suggested by Tilley and Terry (1963). Other species associated with high-nutritive forage were Cerastium spp, Ra. acris and Heracleum sphondylium. However, the forbs that negatively correlated to high-nutritive forage included G. mollugo and U. dioica as reported by Hofmann and Isselstein (2005).

Another interesting finding was that the relationships between botanical composition and forage parameters remained stable during the period of the first growth cycle despite the fact that differences in nutritive value and yield of permanent grasslands were smaller in HS than in SS. This means that forage parameters for permanent grassland composed mainly of competitive species were higher than forage parameters for permanent grassland composed of conservative species during both SS and HS, despite the contrasting patterns of change in feed value of these groups. Duru et al. (2008) andAndueza et al. (2010) have already shown that permanent grasslands composed of species that employ resource-capture strategies (competitive species), such as L. perenne and D. glomerata, had higher digestibility values at the beginning of the cycle but also showed a stronger decline than permanent grasslands composed of species that employ resource-conservation strategies, such as Fe. rubra and Ag. capillaris. Similarly, Andueza et al. (2013) found the same pattern for VI on two different permanent grasslands. In this study, we showed similar patterns, but the novelties are as follows: first, that the grasslands dominated by competitive species preserve their higher forage quality (OMD and VI) compared to grasslands dominated by conservative species despite the sharp decline observed between SS and HS for OMD; secondly, that the trends identified using our database can be generalized to a large range of permanent grasslands.

Characterization of permanent grasslands

The permanent grasslands selected and analysed in this study can be categorized into four plant community classes that provide different forage qualities. The co-analysis of botanical composition and forage parameters underlined the association between permanent grasslands with high proportions of competitive species such as D. glomerata, L. perenne, T. officinale and Cerastium spp. with high yield and feed values (classes 2 and 3), especially when they are cut or grazed early (as shown by the comparison at SS vs. HS). On the other hand, permanent grasslands composed mainly of Fe. rubra, Ag. capillaris, Ar. elatius and Ho. lanatus with high proportions of Sm were associated with high fibre contents and low feed values (classes 1 and 4), particularly in the early stages of the growth cycle (SS). Our analysis also showed that grassland types can be further discriminated according to their forage CP and ash contents, allowing us to separate Class 1 from 4 and Class 2 from Class 3. This study thus demonstrates that each class can be characterized by a given species assemblage. However, several species can be found in different classes and in significant proportions (e.g. Ag. capillaris in Class 2 and Class 4 or L. perenne in Class 2 and Class 3). This result underlines that grassland class cannot be determined by a single characteristic species but rather by the co-occurrence of a set of characteristic species. Permanent grasslands dominated by Ag. capillaris and Fe. rubra (Class 4) are thus characterized by low feed value, whereas permanent grasslands dominated by L. perenne and Ag. capillaris (Class 2) showed high feed value. Forbs can also be indicative species despite their lower proportions in Figure 4 The four classes of permanent grassland charted in the two-dimensional space of the co-inertia analysis defined by principal component axes 1 and 2. values give grid size. PC1 = principal component 1; PC2 = principal 2; d value gives grid size.

the classes. T. araxacum officinale, Cerastium spp. and He. sphondylium were found in high feed-value classes whereas G. mollugo and U. dioica belonged to low feedvalue classes. Consequently, an important result of this study was that we were able to identify several indicator species or species associations characterizing the yield and feed value of permanent grasslands: D. glomerata and T. officinale for grasslands with high yield potential and high nutritive value, Ag. capillaris and L. perenne for grasslands with similar characteristics but low CP content, Fe. rubra and Ag. capillaris for permanent grasslands with low yield and low feedvalue parameters, and the association of high proportions of Sm with species such as E. repens, Ho. lanatus or Ar. elatius indicated permanent grasslands with low yield, OMD and intake values but high CP content. This typology differs from that proposed by Daccord et al. (2006), who also linked nutritive value to botanical composition of permanent grasslands. However, the typology proposed by Daccord et al. (2006) was mainly based on the proportions of different families in the grasslands, although they did take into account the proportion of certain individual species like ryegrass.

Senescent material was found to be associated with permanent grasslands of Class 1. This can be explained by the fact that Class-1 pastures were associated with more extensive management (data not shown) than the other more intensively managed grassland classes, which would equate to a higher quantity of rejections at the end of autumn especially in the SS. The difference of Sm between SS and HS in this class is a result that can be explained by the increase in green forage between SS and HS periods in relation to Sm.

An outcome of this study is the proposed permanent grasslands typology scheme using forage quality and yield based on a set of common grassland species including grasses and forbs. This typology, based on assemblages of species easily identified in permanent grasslands, can be used without deep knowledge of botany. We anticipate these results to be complementary to other tools available, such as the functional classification of permanent grasslands according to grass growth strategies (Cruz et al., 2002). Practical use of this permanent grasslands classification by farmers and land managers (earlier use of grasslands composed of competitive species and then later use of permanent grasslands composed of conservative species) would allow more sustainable and efficient farm-scale management of permanent grasslands.

Conclusions

Applying STATICOan original statistical method mainly used in ecological scienceenabled us to iden-tify the relationships between botanical composition and forage parameters (yield and forage quality) of permanent grasslands at two important maturity stages of the first growth cycle. We established relationships between the dominant species and forage quality parameters of permanent grasslands in a large range of samples and proved that these correlations remain stable over the first growth cycle. This study highlights links between the abundance of certain species in permanent grasslands and their forage quality. We identified a set of indicator species or species associations that characterized the yield and forage quality of permanent grasslands. These findings offer a useful tool for more sustainable and efficient management of permanent grasslands that can also be used in addition to other functional typology tools that farmers already use in practice.

Figure 1

 1 Figure 1 STATICO flow chart (adapted from Thioulouse et al., 2004).

Figure 2

 2 Figure 2 Interstructure (a) and compromise factor maps of the STATICO analysis of forage parameters (b) and botanical composition (c). The interstructure map shows the importance of each sampling period in the compromise. The compromise factor maps show the projection of forage parameter variables (b) and species (c) on the first and second axes, and the stable part of the species-forage parameter variables. d values give grid size. Ag.ca, Agrostis capillaris; Br.mo, Bromus mollis; Ce.sp, Cerastium spp; El.re, Elytrigia repens; Da.gl, Dactylis glomerata; Fe.pr, Festuca pratensis; Fe.ar, Festuca arundinacea; Fe.ru, Festuca rubra; Ph.pr, Phleum pratense; Po.pr, Poa pratensis; Ar.el, Arrhenatherum elatius; Ga.mo, Gallium mollugo; Tr.fl, Trisetum flavescens; Ac.mi, Achillea millefolium; Ho.la, Holcus lanatus; Sm, Senescent material; Ur.di, Urtica dioica; Po.tr, Poa trivialis; Ta.of, Taraxacum officinale; L.pe, Lolium perenne; Tr.re, Trifolium repens; Al.pr, Alopecurus pratensis; Biom, yield; ADL, acid detergent lignin; ADF, acid detergent fibre; NDF, neutral detergent fibre; CP, crude protein; OMD, organic matter digestibility; VI, voluntary intake; SS, silage stage; HS, hay stage; PC1, principal component 1; PC2, principal component 2; d value gives grid size.

Figure 3

 3 Figure 3 Trajectories factor map projection of the coordinates of the eight forage parameters and 125 species proportions on the principal component factor map of the compromise. PC1 = first principal component. PC2 = second principal component. The two periods [Silage stage (SS) and Hay stage (HS)] have been plotted on separate graphs to show (a) projection of forage parameters in SS, (b) projection of species proportions in SS, (c) projection of forage parameters in HS, (d) projection of species proportions in HS. d values give grid size. Ag.ca, Agrostis capillaris; Br.mo, Bromus mollis; Ce.sp, Cerastium spp; El.re, Elytrigia repens; Da.gl, Dactylis glomerata; Fe.pr, Festuca pratensis; Fe.ru, Festuca rubra; Ar.el, Arrhenatherum elatius; Ac.mi, Achilea millefolium; Sm, Senescent material; Po.tr, Poa trivialis; Ta.of, Taraxacum officinale; L.pe, Lolium perenne; Al.pr, Alopecurus pratensis; Biom, yield; ADL, acid detergent lignin; ADF, acid detergent fibre; NDF, neutral detergent fibre; CP, crude protein; OMD, organic matter digestibility; VI, voluntary intake.

  ., Standard error; ns, P > 0Á05; *P < 0Á05; **P < 0Á01; ***P < 0Á001. © 2015 John Wiley & Sons Ltd. Grass and Forage Science, 71, 366-378

  The mean AE s.d. soil chemical and physical properties of the studied area according to Hulin et al. (2011) were as follows: 45 AE 14% sand, 34 AE 10% loam, 21 AE 8% clay,

	7Á2 AE 5Á6%	organic	matter,	5Á9 AE 0Á34	pH,
	280 AE 73 mg per kg Olsen-P status, 350 AE 200 mg per
	kg K status and 22Á7 AE 10Á39 CEC. Each grassland was
	sampled twice in 2005 during the first cycle of growth:

Table 2

 2 Mean, standard deviation (s.d.) and maximum (Max) values obtained for the most representative species (proportion of dry matter, as %) of permanent grasslands sampled at two sampling periods: ensilage stage (SS) and hay stage (HS).

			SS			HS
	Species	Mean	s.d.	Max	Mean	s.d.	Max
	Grasses
	Agrostis capillaris	8Á27	9Á21	39Á82	12Á43	12Á99	60Á81
	Alopecurus pratensis	3Á30	8Á91	49Á78	2Á89	7Á28	35Á00
	Anthoxanthum odoratum	1Á23	3Á00	21Á24	0Á68	1Á81	13Á35
	Arrhenatherum elatius	3Á88	8Á46	47Á14	4Á98	11Á52	60Á82
	Bromus mollis	3Á76	8Á45	47Á32	1Á48	4Á00	31Á83
	Dactylis glomerata	10Á49	14Á41	79Á52	11Á84	17Á63	87Á75
	Elytrigia repens	1Á50	4Á70	32Á81	2Á14	6Á38	44Á98
	Festuca arundinacea	0Á87	4Á49	35Á94	1Á11	5Á84	41Á89
	Festuca pratensis	1Á61	6Á21	48Á75	2Á22	8Á11	56Á95
	Festuca rubra	7Á31	12Á50	61Á02	6Á15	10Á85	59Á90
	Holcus lanatus

Table 4

 4 Mean values and statistical effects for proportion of different species in four classes of permanent grasslands sampled at two periods during the first growth cycle: ensilage stage (SS) and hay stage (HS).

			SS					HS
		Class	Class	Class	Class	Class	Class	Class	Class			Class
		1	2	3	4	1	2	3	4	Class	Period	3 period
	Agrostis capillaris	5Á47	12Á78	2Á99	12Á23	7Á54	16Á71	8Á43	16Á66	***	**	ns
	Bromus mollis	1Á07	5Á14	5Á90	0Á39	0Á74	1Á39	2Á73	0Á21	***	**	ns
	Cerastium spp.	0Á71	2Á03	2Á14	0Á79	1Á19	1Á54	1Á34	0Á35	***	ns	ns
	Elytrigia repens	7Á18	0Á05	0Á54	0Á02	9Á82	0Á04	1Á04	0Á01	***	ns	ns
	Dactylis glomerata	4Á61	6Á31	23Á55	1Á30	5Á93	8Á32	22Á56	5Á50	***	ns	ns
	Festuca pratensis	0Á21	1Á76	2Á20	1Á76	2Á05	2Á20	2Á41	2Á09	ns	ns	ns
	Festuca	3Á02	0Á22	0Á14	1Á08	4Á35	0Á08	0Á00	1Á56	***	ns	ns
	arundinacea
	Festuca rubra	1Á61	6Á58	0Á16	26Á56	0Á90	6Á89	0Á61	19Á61	***	ns	ns
	Phleum pratense	0Á56	0Á70	1Á46	0Á57	0Á55	1Á30	2Á63	1Á47	**	ns	ns
	Anthoxanthum	0Á00	2Á04	1Á40	0Á77	0Á04	1Á10	0Á66	0Á59	**	ns	ns
	odoratum
	Arrhenatherum	13Á55	0Á93	3Á15	0Á60	15Á94	2Á27	3Á89	0Á65	**	ns	ns
	elatius
	Gallium mollugo	1Á58	0Á19	0Á03	1Á51	0Á17	0Á37	0Á00	2Á38	***	ns	ns
	Lathyrus pratensis	0Á00	0Á58	0Á21	1Á06	0Á05	0Á062	0Á02	0Á86	***	ns	ns
	Heracleum	0Á00	0Á95	2Á21	0Á00	0Á00	1Á24	2Á07	0Á00	**	ns	ns
	sphondylium
	Holcus lanatus	13Á85	5Á75	3Á45	4Á34	14Á04	4Á44	7Á19	4Á19	***	ns	ns
	Senescent material	19Á94	2Á12	2Á00	6Á36	7Á39	4Á92	5Á49	7Á99	***	ns	***
	Urtica dioica	2Á13	0Á08	0Á15	0Á00	4Á31	0Á06	0Á00	0Á00	***	ns	ns
	Poa trivialis	0Á92	3Á04	5Á75	0Á17	0Á82	2Á04	3Á17	0Á31	***	ns	ns
	Taraxacum officinale	1Á19	3Á41	9Á84	0Á13	1Á00	1Á45	2Á76	0Á18	***	***	***
	Plantago lanceolata	0Á32	2Á47	0Á54	1Á09	0Á04	0Á98	0Á17	1Á00	***	ns	ns
	Lolium perenne	0Á69	11Á63	8Á36	0Á32	1Á82	9Á45	7Á33	0Á50	***	ns	ns
	Ranuculus acris	0Á10	1Á64	1Á91	1Á09	0Á29	1Á53	0Á96	0Á21	***	**	ns
	Rumex acetosa	0Á00	0Á84	0Á82	0Á06	0Á00	0Á51	0Á52	0Á04	**	ns	ns
	Trifolium repens	0Á00	1Á83	1Á91	0Á48	0Á07	1Á37	1Á71	0Á29	***	ns	ns
	Alopecurus pratensis	5Á75	4Á44	2Á52	0Á19	4Á01	5Á23	1Á33	0Á055	***	ns	ns
	ns, P > 0Á05; **P < 0Á01; ***P < 0Á001.

give consistent results. Schubiger et al. (2001), using the in vitro method of Tilley and Terry (1963), reported OMD values for T. officinale that were similar to ryegrass, whereas Wilman and Riley (1993) and Derrick et al. (1993), using in vivo digestibility, reported lower values than for ryegrass. Derrick et al. (1993) however, reported similar intake values between T. officinale and leafy ryegrass. In addition, Seither et al.

© 2015 John Wiley & Sons Ltd. Grass and Forage Science, 71, 366-378 Botanical composition, feed value and yield of permanent grasslands 369

© 2015 John Wiley & Sons Ltd. Grass and Forage Science, 71, 366-378

© 2015 John Wiley & Sons Ltd. Grass and Forage Science, 71, 366-378 Botanical composition, feed value and yield of permanent grasslands 375

Acknowledgements

The authors thank the staff at experimental stations 1354 UE RT and 1296 UEMA, the agricultural extension services (EDE63 and CA63) and the farmers that participated in this study. A.M. Rodrigues received a 'CAPES' Fellowship from the Brazilian Government.

Fourrages, 172, 335-354. (eds) Forage Quality, Evaluation and Utilization, pp. 450-532. Madison, WI, USA: American Society of Agronomy, Inc., Crop Science Society of America, Inc., Soil Science Society of America, Inc.