Dr João Pedro Ferreira
email: jpferreira@med.up.pt

Faiez Zannad

Michael S Kiernan

Marvin A Konstam

High versus low dose losartan and uric acid: an analysis from HEAAL

Keywords: uric acid, hyperuricemia, losartan dose

Background: Serum uric acid (SUA) is activated in catabolic, hypoxic, and inflammatory conditions characteristic of heart failure (HF) and is a source of reactive oxygen species. Losartan is unique among other angiotensin receptor blockers in reducing SUA.

Objectives: To study the patient characteristics and outcome associations by SUA levels, as well as the effect of high vs. low dose losartan on SUA levels in HF.

Methods: HEAAL was a double-blind trial, comparing the effect of two doses of losartan 150 (high dose) vs. 50 (low dose) mg/day among 3834 patients with symptomatic heart failure, a LVEF≤40%, and known intolerance to angiotensin converting enzyme inhibitors. In the present study we studied the associations of SUA with outcomes and the effect of high vs. low dose losartan on SUA levels, incident hyperuricemia and gout.

Results: Patients with higher SUA had more comorbidities, worse renal function, were more symptomatic, used diuretics more frequently, and were 1.5 to 2-fold more likely to experience HF hospitalizations and cardiovascular death. The benefit of high dose losartan to improve HF outcomes was not influenced by baseline SUA levels (interaction P>0.1). Compared with low dose, high dose losartan reduced SUA by -0.27 (-0.34 to -0.21) mg/dL, P<0.001. The incidence of hyperuricemia was reduced with high dose losartan, but the incidence of gout was not.

Conclusions: In HEAAL, hyperuricemia was associated with worse outcomes. High dose losartan reduced SUA and hyperuricemia more than low dose and the cardiovascular benefits of high dose losartan were not modified by SUA levels.

Introduction

Serum uric acid (SUA) is generated by the enzyme xanthine oxidase that is activated in catabolic, hypoxic, or inflammatory conditions characteristic of heart failure (HF) and is a source of reactive oxygen species. [START_REF] Doehner | Xanthine oxidase and uric acid in cardiovascular disease: clinical impact and therapeutic options[END_REF][START_REF] Maiuolo | Regulation of uric acid metabolism and excretion[END_REF][START_REF] Packer | Uric Acid Is a Biomarker of Oxidative Stress in the Failing Heart: Lessons Learned from Trials With Allopurinol and SGLT2 Inhibitors[END_REF] Comorbidities and medications such as chronic kidney disease (CKD) and diuretics are common in HF and may contribute to elevated SUA levels. Elevated SUA has been associated with higher risk of hospitalization and mortality, but causality has not been established. [START_REF] Vaduganathan | Relation of serum uric acid levels and outcomes among patients hospitalized for worsening heart failure with reduced ejection fraction (from the efficacy of vasopressin antagonism in heart failure outcome study with tolvaptan trial)[END_REF][START_REF] Anker | Uric acid and survival in chronic heart failure: validation and application in metabolic, functional, and hemodynamic staging[END_REF] Controlled trials targeting the reduction of SUA either by inhibition of the synthesizing enzyme xanthine oxidase or by uricosuric therapy failed to improve outcomes in HF. [START_REF] Doehner | Effects of xanthine oxidase inhibition with allopurinol on endothelial function and peripheral blood flow in hyperuricemic patients with chronic heart failure: results from 2 placebo-controlled studies[END_REF][START_REF] Givertz | Effects of Xanthine Oxidase Inhibition in Hyperuricemic Heart Failure Patients: The Xanthine Oxidase Inhibition for Hyperuricemic Heart Failure Patients (EXACT-HF) Study[END_REF][START_REF] Ogino | Uric acid-lowering treatment with benzbromarone in patients with heart failure: a double-blind placebocontrolled crossover preliminary study[END_REF] However, some therapies that improve HF outcomes also reduce SUA levels.

For example, sacubitril/valsartan is thought to lower SUA via uricosuric effects combined with urate synthesis inhibition. [START_REF] Selvaraj | Serum uric acid, influence of sacubitril-valsartan, and cardiovascular outcomes in heart failure with preserved ejection fraction: PARAGON-HF[END_REF] Sodium-glucose cotransporter 2 inhibitors (SGLT2i) markedly reduce SUA levels through a uricosuric effect, reduced oxidative stress and promotion of cellular autophagy. [START_REF] Doehner | Uric acid and sodiumglucose cotransporter-2 inhibition with empagliflozin in heart failure with reduced ejection fraction: the EMPEROR-reduced trial[END_REF][START_REF] Mcdowell | Dapagliflozin reduces uric acid concentration, an independent predictor of adverse outcomes in DAPA-HF[END_REF][START_REF] Ferreira | Empagliflozin and uric acid metabolism in diabetes: A post hoc analysis of the EMPA-REG OUTCOME trial[END_REF] Losartan has been shown to reduce SUA more than other angiotensin receptor blockers (ARBs), without dose-response relationship in patients with hypertension. [START_REF] Würzner | Comparative effects of losartan and irbesartan on serum uric acid in hypertensive patients with hyperuricaemia and gout[END_REF][START_REF] Matsumura | Effect of losartan on serum uric acid in hypertension treated with a diuretic: the COMFORT study[END_REF] Losartan is thought to exert an uricosuric effect through the inhibition of the renal organic anion transporter URAT1 (SLC22A12). [START_REF] Nakanishi | Functional cooperation of URAT1 (SLC22A12) and URATv1 (SLC2A9) in renal reabsorption of urate[END_REF][START_REF] Hamada | Uricosuric action of losartan via the inhibition of urate transporter 1 (URAT 1) in hypertensive patients[END_REF] In the present work we aim to study the patient characteristics and outcome associations by SUA levels, as well as the effect of high vs. low dose losartan on SUA levels in HF.

Methods

Study design and population

HEAAL was an international, multicentre, double-blind, event-driven trial, comparing the effect of two doses of losartan: 150 (high dose) vs. 50 (low dose) mg/day, among 3834 patients with symptomatic heart failure, a LVEF of 40% or less, stable cardiovascular medical therapy for at least 2 weeks, and known intolerance to angiotensin converting enzyme inhibitor (ACEi) defined as discontinuation of ACEi treatment because of one or more of the following adverse effects: cough, symptomatic hypotension, azotaemia, hyperkalemia, taste disturbance, gastrointestinal upset, or rash. Patients assigned to the high dose group received a mean of 129 ± 39 mg of losartan daily and the patients assigned to the low dose group received a mean of 46 ± 11 mg/daily. The median follow-up time was 4.7 (pct 25-75 , 3.5-5.5) years. Compared to low dose, high dose losartan reduced the composite of HF hospitalization or cardiovascular death: HR 0.87, 95%CI 0.79-0.97, P =0.009. [START_REF] Konstam | Effects of high-dose versus low-dose losartan on clinical outcomes in patients with heart failure (HEAAL study): a randomised, double-blind trial[END_REF] The study conformed to the principles outlined in the Declaration of Helsinki and was approved by the ethics committee or institutional review board of every site. All patients provided written informed consent before randomization. HEAAL is registered with ClinicalTrials.gov, number NCT00090259.

SUA was measured at months 1, 4, 9, and 12 and at every 6-month visit thereafter.

In the present analysis, we studied the associations of SUA with the outcomes of cardiovascular death or HF hospitalization, cardiovascular death, all-cause death or all-cause hospitalization, and all-cause death. The cause of death and the primary cause of admission were determined by an Endpoint Classification Committee, whose members were masked to treatment assignment. Adverse events were reported while patients were taking treatment and for 14 days after the end of treatment. Events were coded with the Medical Dictionary for Regulatory Activities (version 12).

Statistical analysis

The baseline characteristics were compared between the baseline SUA tertiles, using ANOVA or Kruskall-Wallis tests, as appropriate. Continuous variables are expressed as median (percentile 25-75 ) and categorical variables are presented as absolute numbers and percentages. The association between SUA and the study outcomes was studied across SUA tertiles with Cox models adjusted for age, sex, race, atrial fibrillation, ischemic HF etiology, hypertension, diabetes mellitus, prior myocardial infarction, NYHA class, left ventricular ejection fraction (LVEF), systolic blood pressure (SBP), heart rate, body mass index, estimated glomerular filtration rate (eGFR), sodium, hemoglobin, beta-blocker, mineralocorticoid receptor antagonists (MRA), and loop diuretic use, and randomized losartan dose. The effect of high vs. low dose losartan across the baseline SUA levels was assessed with a treatment-by-SUA interaction term in the Cox model. The effect of high vs. low dose losartan on SUA levels throughout the follow-up was studied using mixed effects models including baseline SUA levels and treatment-by-time interaction terms as dependent variables, and random intercept at the patient level. The effect of high vs. low dose losartan on incident gout was studied by means of a Cox model with treatment as independent variable and first occurrence of gout (i.e., first occurrence in a study visit recorded as free text) as outcome variable, in the subset of patients without gout at baseline. Landmark analysis at 1 month were used to study the associated between change in SUA levels (i.e., decrease vs. increase from baseline) and outcomes. A P-value <0.05 was accepted as the threshold for statistical significance without correction for multiplicity of tests given the exploratory nature of this work. Statistical analyses were conducted in Stata® version 17 (StataCorp LP).

Results

Patient's baseline characteristics by uric acid levels

A total of 3816 (99.5% of HEAAL population) patients with available baseline SUA levels were included in the analysis. The baseline SUA tertiles were: tertile 1 ≤5.70 mg/dL (n =1265), tertile 2 ≥5.70 to ≤7.30 mg/dL (n =1212), tertile 3 >7.30 mg/dL (n =1339). Compared to patients with lower SUA levels (tertile 1), those with higher SUA (tertile 3) were more frequently men, had more frequent atrial fibrillation and diabetes mellitus, were more symptomatic, had lower LVEF, SBP, and eGFR, and were more frequently treated with beta blockers, MRAs, and diuretics. Table 1.

Circulating SUA levels had a normal distribution. Supplementary Figure 1. Patient's baseline characteristics by sex-specific SUA tertiles are presented in the Supplementary Table 1.

Outcome associations by uric acid levels

Compared to patients with lower SUA levels (tertile 1), those with higher SUA (tertile 3) were more likely to experience the composite outcome of cardiovascular death or HF hospitalization: adjusted HR 1.53, 95%CI 1.33-1.77, cardiovascular death: adjusted HR 1.95, 95%CI 1.66-2.29, composite of all-cause death or all-cause hospitalization: adjusted HR 1.42, 95%CI 1.19-1.70, and all-cause death: adjusted HR 1.24, 95%CI 1.11-1.38. The association between SUA levels and the composite outcome of cardiovascular death or HF hospitalization was not modified by sex (interaction P =0.21). The benefit of high dose losartan to reduce the composite of cardiovascular death or HF hospitalization was not influenced by baseline SUA levels (interaction P =0.82). Table 2 & Figure 1. Outcome associations of sexspecific SUA tertiles are presented in the Supplementary Table 2, with similar results to those above described and presented in Table 2.

A decrease in SUA levels (compared to an increase) at 1 month was not associated with better outcomes: cardiovascular death or HF hospitalization HR 1.02, 95%CI 0.91-1.14, P =0.74.

Effect of high dose losartan on uric acid levels and gout episodes

Compared with low dose, high dose losartan reduced SUA levels by -0.27 (-0.34 to -0.21) mg/dL on average (P <0.001), a reduction that was observed early after 1 month of follow-up and corresponded to a -4% (-5% to -3%) average relative change. Figure 2 & Supplementary Figure 2. The SUA reduction was more pronounced in patients with hyperuricemia (SUA >7 mg/dL) at baseline: -0.37 (-0.49 to -0.26) mg/dL, than in patients without hyperuricemia (SUA ≤7 mg/dL) at baseline: -0.20 (-0.29 to -0.12) mg/dL, interaction P =0.011. The incidence of hyperuricemia (SUA levels > 7 mg/dL) were also reduced with high dose losartan throughout the follow-up. Figure 3.

During the follow-up 146 events of gout were reported without differences between low and high dose losartan groups: HR 0.95, 95%CI 0.69-1.31, P =0.76. Only 19 nephrolithiasis episodes were recorded without between group differences (P =0.26). Supplementary Figure 3.

Discussion

This study shows that higher SUA levels were associated worse HF symptoms, poorer renal function and more frequent use of diuretics, and with a higher risk of hospitalizations and mortality. High dose losartan reduced SUA levels and hyperuricemia more than low dose but did not reduce incident gout episodes. The benefit of high dose losartan to reduce the composite of HF hospitalization or cardiovascular death was not modified by baseline SUA levels.

Higher SUA levels in HF may reflect advanced stages of the disease, along with renal impairment and higher need of diuretics to mitigate congestion. Thus, hyperuricemia may be a marker of increased oxidative stress, inflammation, endothelial dysfunction and catabolic processes, all of which portend worse prognosis in HF. [START_REF] Leyva | Uric acid in chronic heart failure: a marker of chronic inflammation[END_REF] Compared to low dose, high dose losartan reduced SUA levels by -0.3 mg/dL (-4%) on average, a reduction that was even more pronounced among patients with hyperuricemia at baseline. Importantly, the incidence of hyperuricemia throughout the follow-up was reduced significantly with high dose losartan. Previous studies in hypertension suggested that the effect of losartan to reduce SUA was not dose dependent, but treatment adherence issues might have occurred in the high dose group. [START_REF] Würzner | Comparative effects of losartan and irbesartan on serum uric acid in hypertensive patients with hyperuricaemia and gout[END_REF] The present study shows that higher losartan doses produce greater SUA reductions than lower doses which are clinically important findings because high-dose losartan may be used to lower SUA levels beyond the proven clinical benefits of this drug. [START_REF] Konstam | Effects of high-dose versus low-dose losartan on clinical outcomes in patients with heart failure (HEAAL study): a randomised, double-blind trial[END_REF] Furthermore, the benefits of high dose losartan on morbidity and mortality were not modified by baseline SUA levels, supporting the up-titration of losartan to the maximum tolerated dose or 150 mg/day regardless of SUA levels.

Despite the hyperuricemia reductions, high dose losartan did not reduce incident gout. With only 146 events recorded (corresponding to a 4% cumulative incidence at 4 years), this study might have been underpowered to study the effect of high vs. low dose losartan on gout, because low dose losartan might have already mitigated the occurrence of gout. Still, despite the uricosuric mechanisms of losartan, high dose was not associated with an increased risk of nephrolithiasis, with only 19 events recorded throughout the follow-up. [START_REF] Nakanishi | Functional cooperation of URAT1 (SLC22A12) and URATv1 (SLC2A9) in renal reabsorption of urate[END_REF][START_REF] Hamada | Uricosuric action of losartan via the inhibition of urate transporter 1 (URAT 1) in hypertensive patients[END_REF] Xanthine oxidase inhibitors such as allopurinol seem more effective than losartan in lowering SUA levels. [START_REF] Givertz | Effects of Xanthine Oxidase Inhibition in Hyperuricemic Heart Failure Patients: The Xanthine Oxidase Inhibition for Hyperuricemic Heart Failure Patients (EXACT-HF) Study[END_REF][START_REF] Mackenzie | Allopurinol versus usual care in UK patients with ischaemic heart disease (ALL-HEART): a multicentre, prospective, randomised, open-label, blinded-endpoint trial[END_REF] The rapid effect of high dose losartan to reduce SUA (observed as early as 1 month i.e., the first post-randomization visit) may be of interest in the setting of concomitant use of other HF medications, SGLT2i in particular. This drug class, produces a rapid and pronounced SUA decrease (around -1 mg/dL on average). [START_REF] Doehner | Uric acid and sodiumglucose cotransporter-2 inhibition with empagliflozin in heart failure with reduced ejection fraction: the EMPEROR-reduced trial[END_REF][START_REF] Mcdowell | Dapagliflozin reduces uric acid concentration, an independent predictor of adverse outcomes in DAPA-HF[END_REF][START_REF] Ferreira | Empagliflozin and uric acid metabolism in diabetes: A post hoc analysis of the EMPA-REG OUTCOME trial[END_REF] Therefore, the combination of high dose losartan and a SGLT2i may be useful to reduce SUA levels in HF patients, and such combination may be worth investigating in future studies.

Limitations

Some limitations should be acknowledged in our study. First, this is a post-hoc analysis of a randomized controlled trial which is affected by the biases inherent to observational studies, particularly the impossibility to infer causality. Second, postrandomization medications including the use of SUA-lowering and anti-gout medications were not registered in the dataset, and we could not assess medication changes after randomization. Third, diuretic doses or diuretic initiation throughout the follow-up was not available in the dataset; thus, we could not ascertain if diuretic use (or diuretic doses) throughout the follow-up could be associated with incident hyperuricemia and gout.

Conclusions

In HEAAL, hyperuricemia was associated with worse outcomes. High dose losartan reduced SUA and hyperuricemia more than low dose, and the benefits of high dose losartan were not modified by baseline SUA levels. No significant interactions were found between randomized treatment (high vs. low losartan dose) and baseline uric acid levels: cardiovascular death or HF hospitalization interaction P =0.82; cardiovascular death interaction P =0.53; allcause death or all-cause hospitalization interaction P =0.13, all-cause death interaction P =0.49. Caption: Compared to low dose, high dose losartan reduced uric acid levels by -0.27 (-0.34 to -0.21) mg/dL on average, P <0.001. The serum uric acid reduction was more pronounced in patients with hyperuricemia (uric acid >7 mg/dL) at baseline -0.37 (-0.49 to -0.26) mg/dL than in patients without hyperuricemia (uric acid ≤7 mg/dL) at baseline -0.20 (-0.29 to -0.12) mg/dL, interaction P =0.011. Figure 3. Proportion of patients with hyperuricemia (uric acid levels >7 mg/dL) throughout the follow-up Caption: High dose losartan decreased the proportion of patients with hyperuricemia throughout the follow-up. The overall odds ratio for new onset hyperuricemia was 0.71, 95%CI 0.58-0.88, P =0.002 in favour of high dose losartan.

Figure 1 .

 1 Figure 1. Cumulative incidence of cardiovascular death or HF hospitalization events by tertiles of uric acid at baseline

Figure 2 .

 2 Figure 2. Effect of high vs. low losartan dose on uric acid levels over time

Table 1 .

 1 Baseline patient's characteristics by uric acid tertiles

	Characteristic	Uric acid level in tertiles (mg/dL) ≤5.70 5.71-7.30 >7.30	P-value
	N.	1265	1212	1339	-
	Age, years	66 (58, 73)	65 (56, 73)	65 (56, 72)	0.010
	Female	558 (44.1%)	315 (26.0%)	264 (19.7%)	<0.001
	White race	742 (58.7%)	774 (63.9%)	793 (59.2%)	0.015
	Atrial fibrillation	310 (24.5%)	356 (29.4%)	397 (29.6%)	0.005
	Ischemic heart disease	829 (65.5%)	767 (63.3%)	845 (63.1%)	0.36
	Hypertension	788 (62.3%)	727 (60.0%)	768 (57.4%)	0.037
	Diabetes mellitus	384 (30.4%)	339 (28.0%)	472 (35.3%)	<0.001
	Myocardial infarction	106 (8.4%)	144 (11.9%)	98 (7.3%)	<0.001
	Stroke	94 (7.4%)	106 (8.7%)	106 (7.9%)	0.48
	NYHA III/IV	347 (27.4%)	360 (29.7%)	460 (34.4%)	<0.001
	LVEF, %	34 (29, 38)	33 (27, 37)	31 (25, 36)	<0.001
	SBP, mmHg	130 (115, 140)	125 (112, 140)	120 (110, 135)	<0.001
	Heart rate, bpm	72 (64, 80)	72 (64, 80)	72 (65, 82)	0.008
	BMI, Kg/m 2	27 (24, 30)	27 (24, 30)	27 (25, 31)	<0.001
	eGFR, ml/min/1.73m 2	73 (60, 89)	69 (57, 84)	59 (47, 72)	<0.001
	eGFR ≤60 ml/min/1.73m 2	318 (25.1%)	363 (30.0%)	707 (52.8%)	<0.001
	Potassium, mmol/L	4.4 (4.2, 4.7)	4.5 (4.2, 4.8)	4.4 (4.1, 4.8)	0.61
	Sodium, mmol/L	140 (138, 142)	140 (138, 142)	140 (138, 142)	0.25
	Hemoglobin, g/dL	13.7 (12.7, 14.7) 14.2 (13.0, 15.1) 14.0 (12.9, 15.0) <0.001
	Uric acid, mg/dL	4.9 (4.3, 5.3)	6.5 (6.1, 6.9)	8.6 (7.9, 9.7)	<0.001
	Beta blocker	855 (67.6%)	889 (73.3%)	1001 (74.8%)	<0.001
	MRA	404 (31.9%)	452 (37.3%)	641 (47.9%)	<0.001
	Diuretics	937 (74.1%)	972 (80.2%)	1215 (90.7%)	<0.001
	Statin	489 (38.7%)	501 (41.3%)	514 (38.4%)	0.25
	Legend: NYHA, New York Heart Association: LVEF, left ventricular ejection fraction;
	SBP, systolic blood pressure; BMI, body mass index; eGFR, estimated glomerular
	filtration rate; MRA, mineralocorticoid receptor antagonist.

Table 2 .

 2 Outcome associations by tertiles of uric acid at baseline UA, uric acid; HF, heart failure. Adj., models adjusted on age, sex, race, atrial fibrillation, ischemic HF etiology, hypertension, diabetes, myocardial infarction, NYHA class, ejection fraction, systolic blood pressure, heart rate, body mass index, estimated glomerular filtration rate, sodium, hemoglobin, beta blocker use, MRA use, diuretic use, and randomized losartan dose.

	Outcome/UA tertile (mg/dL)	n/N events (%)	Adj. HR (95%CI)	P-value
	Cardiovascular death or HF hospitalization
	≤5.70	368/1265 (29.1)	Referent	-
	5.71-7.30	418/1212 (34.5)	1.11 (0.96-1.28)	0.15
	>7.30	673 /1339 (50.3)	1.53 (1.33-1.77)	<0.001
	Cardiovascular death
	≤5.70	232/1265 (18.3)	Referent	-
	5.71-7.30	251/1212 (20.7)	1.14 (0.96-1.37)	0.14
	>7.30	434/1339 (32.4)	1.95 (1.66-2.29)	<0.001
	All-cause death or all-cause hospitalization
	≤5.70	740/1265 (58.5)	Referent	-
	5.71-7.30	755/1212 (62.3)	1.05 (0.87-1.26)	0.63
	>7.30	997/1339 (74.5)	1.42 (1.19-1.70)	<0.001
	All-cause death
	≤5.70	355/1265 (28.1)	Referent	-
	5.71-7.30	355/1212 (29.3)	1.05 (0.94-1.16)	0.38
	>7.30	581/1339 (43.4)	1.24 (1.11-1.38)	<0.001
	Legend:

Funding

None.

Abbreviations

SUA, serum uric acid HF, heart failure CKD, chronic kidney disease ARB, angiotensin receptor blocker

Conflicts of interest

The authors report not having relevant conflicts of interest regarding the content of this study.

Ethics approval and consent to participate

The study was approved by all relevant ethics committees and regulatory bodies. All participants provided written informed consent prior to study specific procedures.

Consent for publication

There is no data of individual persons included in the manuscript.


H Ouadi

F Giri
email: giri@greyc.ensicaen.fr

L Dugard

ACCOUNTING FOR MAGNETIC SATURATION IN INDUCTION MACHINES MODELLING

A new model is developed for (uniform air-gap) induction motor that accounts for the saturation feature of the machine magnetic circuit. It is built-up starting from the basic electrical/mechanical laws and turns out to be quite different from the standard model that is widely used in control oriented literature. Specifically, the new model involves state-dependent parameters. An experimental validation using a real 7.5 KW machine proved the good accuracy of the proposed model. A simpler control-oriented version of the model is also developed and shown to be quite representative of the machine.

I. Introduction

It is widely recognized that induction motor is going to become the main actuator for industrial purposes. Indeed, as compared to the DC machine, it provides a better power/mass ratio, allows a simpler maintenance (as it includes no mechanical commutators) and costs relatively cheaper. However, the problem of controlling induction motor is more complex because of its multivariable and nonlinear nature. Besides, some of its state variables are not accessible to measurement. The problem of induction motor control and observation has been given a great deal of interest over the last decade see e.g. [START_REF] Ortega | On speed control of induction motors[END_REF]- [START_REF] Hu | Robust tracking control of an induction motor[END_REF]. However, most of previous works have been based on the standard model where the magnetic characteristic is described by a linear relation. As a matter of fact, such a characteristic is nonlinear in physical machines: it exhibits saturation and hysteresis features. To avoid drastic control performance deterioration, due to magnetic circuit saturation, special precautions should be taken when it comes to practical implementation of the obtained controllers. Specifically, two implementation strategies are generally followed to avoid control performance deterioration due to magnetic saturation. The first strategy consists in constraining the rotor flux to take small values so that the machine operates in the linear zone of its magnetic characteristic i.e. the domain where the standard model is representative of the machine. The drawback of such a solution is that the machine does not operate in optimal efficiency conditions especially in presence of high loads. Then, the consumed stator current is unnecessary large. The second control strategy consists in regulating the rotor flux norm around its nominal value which is generally located at the nonlinear part of the magnetic characteristic. To meet this objective, the parameters of the standard model (based upon when designing the controller) should be tuned so that they correspond to the above nominal operation point. Then, the machine efficiency is maximal when the machine load torque is close to its nominal value. But, the load is usually subject to large variations in practical applications.

If the load torque is small (with respect to the nominal load), there is a useless energy stored in stator inductances reducing the machine efficiency.

To achieve high-performance varying-speed operation mode for induction machines, it is necessary to use controllers that allow large flux variations. Indeed, allowing large flux variations makes possible the achievement of a suitable power factor and a high efficiency (by limiting current absorption). But, in order to allow large flux variations the controller must be developed by using a model that takes into account the nonlinear nature of the machine magnetic characteristic. The question is: how such model can be obtained?

The present work precisely focuses on such a modelling issue. This has been coped with, in [START_REF] Sullivan | Modelling the effect of magnetic saturation on electrical machine control systems[END_REF]- [START_REF] Novotnak | High-performance motion control of an induction motor with magnetic saturation[END_REF], [START_REF] Pedra | Saturation Model for squirrel cage induction motors[END_REF]- [START_REF] Sullivan | Modelling the effects of magnetic saturation on electrical machine control systems[END_REF] by just letting the mutual inductance coefficient associated with a given rotor flux direction (d or q) be a nonlinear function of the stator current along the same direction. This assumption is not realistic because it amounts to neglect the crosssaturation feature. Furthermore, neither the nonlinear functions approximating the mutual inductances nor the resulting (saturated) model are explicitly described in the mentioned works. A quite different approach to account for the saturation feature has been proposed for synchronous machines using energetic considerations [START_REF] Levi | Impact of Cross-Saturation on Accuracy of Saturated Induction Machine Models[END_REF]- [START_REF] Vagati | Impact of cross saturation in synchronous reluctance motors of the tranverse-laminated type[END_REF]. The obtained model is expressed in the complex notation which makes it only valid in steady-state harmonic regimes.

The present paper develops a new approach that appropriately accounts for the magnetic saturation phenomenon in induction machine modelling. It thoroughly contrasts with the previous approaches which heavily relied upon the standard unsaturated model. Presently, the starting point is the machine electric equivalent scheme where the stator and the rotor are represented by triphase coils [START_REF] Leonhard | Control of Electrical Drives[END_REF]. Additional physical laws will be applied to account for the (well known) coupling that exists (even in the case of a uniform air-gap machine) between both axes of an AC-machine, [START_REF] Levi | Impact of Cross-Saturation on Accuracy of Saturated Induction Machine Models[END_REF]. Such a coupling (usually called cross-saturation), is due to the nonlinear properties of magnetic materials. As suggested in [START_REF] Garrido | The matrix analysis of saturated electrical machines[END_REF], the magnetic characteristic can be approximated by a nonlinear function that could be polynomial, exponential, arctangent, etc. Such a function links the air-gap flux   to the magnetizing current  I (this includes the contribution of both stator and rotor currents). The obtained model is experimentally validated using a 7.5 KW AC-machine. The input signals are chosen so that the machine operates both in the linear and nonlinear parts of the corresponding magnetic characteristic. The resulting responses of the new model turn out to be sufficiently close to those of the true machine. On the contrary, the standard model responses are not so close, especially when the machine operates in the saturation part. The new model is quite suitable for the machine simulation but can hardly be used in control design, because it is highly nonlinear and involves many parameters depending on the machine state variables. Therefore, a simpler version is developed by gathering all magnetic leakage inductances at the stator side. The simplified model proves to be quite accurate and, due to its simplicity, convenient for designing controllers and observers. The paper is organized as follows: Section II is devoted to modelling the magnetic characteristic of the studied machine; the remaining electromechanical equations are established in Section III; in Section IV, the static magnetisation parameter is introduced and, in Section V, machine modelling is completed by establishing its state-space representation; an experimental validation of the obtained model as well as a comparison with the standard model are performed in Section VI. In section VII, a simplified version of the above model is derived.

 r

Rotor flux (through one phase) i s , i r

Stator and rotor currents (through one phase)

' r i Current in a rotor phase brought back to the stator,

 j r r e i i  ' i  Magnetizing current i   i s + ' r i ; I  is its norm k s, k
q d     (d, q) components of the magnetizing flux ) ( k P Park transformation ( is its angle) s s r r n k n k k 
Transformation report (ratio?) of the AC machine

II. Characterisation of magnetic saturation in AC machines

The nonlinear feature of the magnetic circuit in induction motors has been accounted for in many ways. Early solutions suggested capturing this through nonlinear approximations of the (B, H) characteristic, where B denotes the magnetic field and H the magnetic induction ( [START_REF] Leonhard | Control of Electrical Drives[END_REF], [START_REF] Seguier | Industrial Electronics[END_REF]). In more recent works, the magnetic saturation is accounted for through a flux-current relation called magnetic characteristic. This links the magnetizing flux norm (i.e. the useful flux at a stator phase) to the magnetizing current ( [START_REF] Levi | Impact of Cross-Saturation on Accuracy of Saturated Induction Machine Models[END_REF]). Several laws have been suggested to describe the magnetic characteristic, e.g.:

      I I  or      2 2 1 1 ) ( s b s s I n n n       .
Other functions such as arctangent or hyperbolic-tangent are also resorted to approximate the magnetic characteristic. In the present work, a spline polynomial approximation is developed to approach the nonlinearity of the magnetic circuit. This has several advantages such as: (i) simplicity of experimental determination, (ii) the larger the polynomial degree the better the approximation accuracy, (iii) simplicity of algebraic and differential transformations.

Let the (unknown) real magnetic characteristic be denoted

) (      I
. Our purpose is to build up a polynomial approximation for I  ). The first step consists in obtaining experimentally a set of points of the real machine characteristic. A spline interpolation of the experimental points is then performed, using suitable tools, to get a polynomial approximation, denoted P(.) of the unknown function  .). The larger the degree of P(.), the more smooth and accurate the approximation. Fig. 1 shows the polynomial P(.) obtained with 8 n  . In the sequel, we will also need an approximation of the inverse characteristic

) ( 1       I
. A polynomial approximation P inv (.) is obtained directly from the available experimental set of points, using the same tools as previously (Fig 2). Remark 1. Fig. 1 shows that the largest linear zone of the magnetic characteristic corresponds to small values of the flux (  < 0.7 wb) whereas the machine nominal point, presently equal to   = 0.9 Wb, is located at the saturation elbow. The proposed modelling approach is based on standard assumptions, namely: the machine is symmetrical, the air gap is smooth, the ferromagnetic losses are negligible, the induction distribution through the air gap is sinusoidal and all electromagnetic variables

        ,... , , 3 , 2 , 1 3 , 2 , 1 3 , 2 , 1 s r s i  
define a well balanced triphase system, i.e.

0 3 2 1    s s s   
. Applying the Park transformation to the triphase equations yields the following electrical equations, completed by the mechanical equation (see e.g. [START_REF] Leonhard | Control of Electrical Drives[END_REF], [START_REF] Seguier | Industrial Electronics[END_REF]):

sq s sd sd s sd dt d i R V       . ( 1
)
sd s sq sq s sq dt d i R V       . ( 2
)
rq s rd rd r dt d i R     ) ( . 0     (3) rd s rq rq r dt d i R     ) ( . 0     (4)
  J f J T J T dt d L e    (5)
III.2 Saturated flux equations in the (d, q) coordinates system

.The magnetic fluxes for the stator and rotor phases are given by (see [START_REF] Leonhard | Control of Electrical Drives[END_REF], [START_REF] Seguier | Industrial Electronics[END_REF]):

          s s st / leakage s i l (6) r r r r rt / leakage r i l           (7)
where   and r   respectively denote the magnetizing air-gap fluxes along one phase of the stator and rotor;



are respectively proportional to the stator and rotor currents. This is valid in real machines because the leakage flux, circulating in air, not in iron, is not large. Let k denote the machine transformation ratio. The flux equations ( 6)-( 7) become, in the (d,q) coordinates:

d sd s sd i l      (8) q sq s sq i l      ( 9
)
d rd r rd k i l      ( 10
)
q rq r rq k i l      (11)
The contribution of the stator and rotor in the air-gap flux generation is expressed in term of the magnetizing current, denoted i μ ( [START_REF] Leonhard | Control of Electrical Drives[END_REF], [START_REF] Levi | Impact of Cross-Saturation on Accuracy of Saturated Induction Machine Models[END_REF]). Therefore, the (d-q) components of the 123 ] [  i system satisfy:

rd sd µd i k i i   and rq sq µq i k i i   (12)
The nonlinear feature of the machine magnetic characteristic is responsible for the crosssaturation effect, see ( [START_REF] Levi | Impact of Cross-Saturation on Accuracy of Saturated Induction Machine Models[END_REF]- [START_REF] Vagati | Impact of cross saturation in synchronous reluctance motors of the tranverse-laminated type[END_REF]). Accordingly, the component of a given (stator, rotor or magnetizing) flux, along a given axis (d or q), turns out to be dependent on both the dand the q-components of both stator and rotor currents. Inspired from [START_REF] Garrido | The matrix analysis of saturated electrical machines[END_REF], the crosssaturation effect is accounted for letting the magnetizing flux be expressed as follows:

0 d q dq d d d i M i M         ( 13
)
0 q d dq q q q i M i M         (14)
where

q d M , M
and dq M are new inductive parameters and:

. d i  , q i  denote the (d,q) components of the magnetizing current . d d i M  is the magnetizing flux generated, along the d-axis, by the components (along the same axis) of the stator and the rotor currents . q q i M  is the magnetizing flux generated, along the q-axis, by the components (along the same axis) of the stator and the rotor currents . d dq i M  is the coupling magnetizing flux generated, along the q-axis, by the components (along the same axis) of the stator and the rotor currents . are not uniquely defined. A judicious definition, suggested in [START_REF] Garrido | The matrix analysis of saturated electrical machines[END_REF], consists in choosing them so that, when differentiating ( 13) and ( 14) with respect to d i  and q i  , one gets:

d q q d def dq q q def q d d def d i i M ; i M ; i M                         (15)
It follows from ( 13) and ( 14) that the functions

0 q 0 d   and
must undergo the following differential equations:

q d dq d d d d d i i M i i M i                0 ; d q dq q q q d q i i M i i M i                0 (16a) q q dq d q d q d i i M i i M i                0 and d d dq q d q d q i i M i i M i                0 (16b)
On the other hand, substituting ( 13)-( 14) in ( 8)-( 11) leads to the following expressions for the (d, q) flux coordinates:

    0 0 0 0 q d dq q q rq r rq d q dq d d rd r rd q d dq q q sq s sq d q dq d d sd s sd i M i M k i l i M i M k i l i M i M i l i M i M i l                                 (17) Remark 2. i)



) depend on all components of both the stator and rotor currents. Contrarily, the standard modelling (proposed or used in [START_REF] Sullivan | Modelling the effect of magnetic saturation on electrical machine control systems[END_REF], [START_REF] Novotnak | High-performance motion control of an induction motor with magnetic saturation[END_REF], [START_REF] Pedra | Saturation Model for squirrel cage induction motors[END_REF], [START_REF] Sullivan | Modelling the effects of magnetic saturation on electrical machine control systems[END_REF]) leads to the fact that a given flux component only depends on the current component along the same direction (e.g. rd  only depends on rd i and sd i ). This is a simplification of a more complex reality. ii) It also follows from the cross-saturation that the parameters

IV. Relation between the induction coefficients and the static magnetization parameter IV.1 Definition of static magnetization parameter

The nonlinear feature of the machine magnetic circuit is entirely accounted for through the magnetic characteristic

      i I I ) (  (18)
The forthcoming development involves the static magnetizing parameter m, see [START_REF] Garrido | The matrix analysis of saturated electrical machines[END_REF]:

) I h( I I I m def           ) ( (19)
In view of (18), the (d,q)-components of the magnetizing flux can be expressed as follows:

q q d d i m i m         , (20)
Then, the expressions ( 8)- [START_REF] Seguier | Industrial Electronics[END_REF] of the stator and rotor flux become:

q rq r rq d rd r rd q sq s sq d sd s sd i m k i l i m k i l i m i l i m i l                 , , (21)
More generally, it is shown in subsequent sections, that all machine parameters, including induction coefficients, can be expressed as a function of m. But first, let us rewrite m as a function of the machine state variables. Indeed from ( 19) and ( 17), one gets:

) ) ( ) ( ( ) ( ) ( ) ( 2 2 1 2 2 2 2 1 2 2 sq s sq sd s sd sq s sq sd s sd d d q d i l i l i l i l m                           (22)

IV.2 Analytical expressions of induction coefficients

The nonlinearity of the magnetic characteristic implies that the induction parameters M d , M q and M dq are varying with the state variables. As the model one develops involves such induction parameters, they need to be computable in real-time. This makes it necessary to explicitly express such parameters in terms of the machine state variables. To this end, one gets from ( 15) and (20) :

     I i dI dm m i i m m M d d d d 2       (23)    I i dI dm m M q q 2   ,     I i i dI dm M q d dq  (24)
using the fact that

2 2 2 q d i i I     
. Now, let us build up mathematical expressions that explicitly link the induction coefficients M d , M q and M dq to the state variables. Equations ( 23)-(24) show that this objective can be reached by expressing m and  dI dm / in terms of the machine state variables (measured or observed). First, recall that a set of experimental points

)) ( , ( ) , (       I I I  is available. A set of experimental couples )) ( , (   I h I
can thus be readily obtained, where (.) h is as in (19). Using ad-hoc interpolation tools, the experimental couples can then be used to get a smooth polynomial approximation of the function (.) h . The obtained polynomial approximation for the considered machine, denoted Q(.), is represented by , it also has to be approximated by a polynomial, denoted R(I μ ). To this end, a set of points 2. These approximations will now be based upon to get expressions that explicitly link the induction coefficients to the state variables. To this end, it follows from ( 19) and ( 22) that:

2 sq s sq 2 sd s sd ) i l ( ) i l (         (25)           2 sq s sq 2 sd s sd inv ) i l ( ) i l ( P I    (26)
Therefore, combining equations ( 23)-( 24) with (21) leads to the following expressions:

) ( ) ( ) ( ) ( 2 2      I Q I i l I R I Q M sd s sd d    ( 27
) ) ( ) ( ) ( ) ( 2 2      I Q I i l I R I Q M sq s sq q    ( 28
) ) ( ) )( ( ) ( 2      I Q I i l i l I R M sq s sq sd s sd dq    (29)
Remark 3. Equations ( 27)-(29) show that, unlike the standard unsaturated case, all the induction coefficients vary (with the state variables). Nevertheless, these coefficients can be computed on-line if the state variables (

s s i ,  ) are available.
) ( P I inv     ) I ( Q m   ) (   I R dI dm 

V. Induction machine model development accounting for magnetic saturation feature

Using the AC machine equations obtained in the previous section, the global model will now be progressively developed. Its final version must be described using only the usual state variables

T rq rd sq sd i i X ] , , , , [     .

V.1 Rotor flux equations

From (3) one readily gets that:

rq r rd r rd i R dt d       (30)
Since i rd is not a state variable, it should be removed from the above equation. To this end, one obtains from ( 21) and ( 12) that:

m k l i m k i r sd rd rd 2     (31)
which, together with (30), yields:

rq r sd rd rd i g g dt d         2 1 (32) with m k l R g r r 2 1   and m k l km R g r r 2 2  
. This is the first state equation. To obtain the second one, let (4) be rewritten as follows:

rd r rq r rq i R dt d       . ( 33
)
Since i rq is not a state variable it should be removed from the above equation. To this end, one gets from (21) and ( 12) that:

m k l kmi i r sq rq rq 2     (34)
which, together with (33), yields:

rd r sd rq rq i g g dt d         2 1 (35)
This is the second state equation.

V.2 Stator current equations

These are derived from equations (1)-(2) which are rewritten here for convenience:

sq s sd sd s sd dt d i R V       . ( 36
) sd s sq sq s sq dt d i R V       . ( 37
)
As sd  and sq  are not state variables, they should be removed from the above equations. To this end, one gets from equations (21) that:

) i l ( i l rd r rd k 1 sd s sd      ( 38
) ) i l ( i l rq r rq k 1 sq s sq      (39)
Substituting ( 38)-( 39) in ( 36)-(37), and using (31) and (34), one gets:

rq r s sq r r s s rd r rd sd s sd s sd m k l k m i m k l m l l dt di k l dt d k dt di l i R V     ) ( 1 2 2                  ( 40
) rd r s sd r r s s rq r rq sq s sq s sq m k l k m i m k l m l l dt di k l dt d k dt di l i R V     ) ( 1 . 2 2                  (41)
In this expression, dt di rd / and dt di rq / must be expressed in terms of the state variables. To this end, time-derivation of the rotor flux equations (17) gives, using (16):

dt di M k dt di kM dt di kM dt d dt di M k l dt di M k dt di kM dt di kM dt d dt di M k l rd dq sd dq sq q rq rq q r rq dq sq dq sd d rd rd d r 2 2 2 2 ) ( ) (             (
i m k l M k l km R a i m k l M mk R a M k l m k l M k R a M k m k l M k l R a dt di                                               2 2 0 2 3 0 2 2 2 0 2 2 2 0 ) ( ) ( ) (         dt di kM l a dt di M k M k l kM a sq dq r sd dq q r d 0 2 2 2 0 ) (     (43) with   1 2 4 2 2 0 ) )( (      dq q r q r M k M k l M k l a
. Substituting (43) and ( 32) in (40), the expression of stator voltage along the d axis becomes:

rq s r rd r sd sd a a a a a a i a a v      ) ( ) ( ) ( 3 3 3 2 2 2 1 1                dt di a a dt di a i a a sq sd sq s 6 0 5 4 4 ) (       (44)
where the meaning of the different parameters is given in Table 3. Table 3. Parameters newly introduced in equations ( 43)-( 44)

  1 2 dq 4 d 2 r q 2 r 0 M k ) M k l )( M k l ( a        m k l M k l m l R a a 2 r q 2 r r r 0 1      2 dq 2 q 2 r d r 0 s 5 M k ) M k l ( M l a l a     m k l m R R a 2 r r s 1       ) m k l ( k M k l l R a a 2 r q 2 r r r 0 2     , m k l m l l a 2 r r s 4     ) m k l ( mM k R l a a 2 r dq 2 r r 0 4   m k l km a 2 r 3     , 2 r dq 0 6 l M a a  ) m k l ( k R a 2 r r 2     dq r 0 2 kM l a a   , ) m k l ( kM R l a a 2 r dq r r 0 3     ) M k l ( l a 1 a q 2 r r 0 k 1 3    
Operating similar transformations on the stator voltage along the q axis, one gets:

rq r sd s sq sq a a b i a a i b a v    ) ( ) ( ) ( 2 2 2 4 4 1 1             dt di a dt di b a b a sd sq rd s r 6 5 3 3 3 ) (            (45)
where the newly introduced coefficients are defined in Table 4.

  m k l M k l m l R a b 2 r d 2 r r r 0 1    ) m k l ( k ) M k l ( R l a b 2 r d 2 r r r 0 2      2 dq 2 d 2 r q r 0 s 5 M k ) M k l ( M l a l b       ) M k l ( l a 1 k b d 2 r r 0 1 3    
Now the state equations of the stator currents can readily be obtained by solving equations ( 44) and (45) with respect to dt di sd / and dt di sq / . Doing so, one gets:

rq s r sd s sq s sd q q q i q q i q q dt di      ) ( ) ( ) ( 3 3 3 2 2 1 1              sq sd rd s r v q v q q q q 0 5 4 4 4 ) (            (46) rd s r sq s sd s sq d d d i d d i d d dt di      ) ( ) ( ) ( 3 3 3 2 2 1 1              sq sd qd s r v d v d d d d 6 5 4 4 4 ) (            (47)
These equations introduce new coefficients whose meaning is described in Table 5.

Table 5. Parameters newly introduced in ( 46)-(47)

2 6 5 5 5 0 a b a b q   , 6 4 5 0 1 a a b q q    , 4 0 2 a q q    ) ( 6 4 5 1 1 0 1 a a b b a q q      ) ( 1 1 5 1 6 4 0 2 a a b a a q q      , 3 0 4 a q q      ) ( 3 5 1 6 2 2 0 3 a b a b a q q       ) ( ) ( 2 0 1 5 1 0 6 3 0 0 4 a a a b a a a a q q        3 5 1 0 6 0 3 ) ( a b a a q q     ) ( 2 5 3 6 1 6 0 4 a b b a a q q      , 5 1 6 0 5 b a q q   ) ( 1 5 4 1 6 2 q a a a d    ) ( 2 5 1 1 1 6 1 q a a a a d      , 3 5 0 6 0 3 ) ( a b a a q q      2 5 1 6 1 q a a d        1 5 4 1 6 2 q a a a d        , ) ( 4 5 2 1 6 3 q a a a d         4 5 2 2 1 6 3 q a a a a d       4 5 1 6 3 q a a d        , ) ( 3 5 3 1 6 4 q a a a d          3 5 3 1 6 4 q a a a d    ) ( 3 5 3 1 6 4 q a a a d           , 1 6 5 0 6   a a q d   5 5 1 6 5 1 q a a d   

V.3 Mechanical equation

The mechanical power P m that induces the electromagnetic torque generation is given by (see e.g [START_REF] Leonhard | Control of Electrical Drives[END_REF]):

r rd rq rq rd s sd sq sq sd m dt d i i dt d i i p       ) ( ) (    
Using the rotor currents expressions (31) and (34), in the flux equations (21) yields:

) ( ) ( 2 r s sd rq sq rd r m dt d i i m k l m k p        
which implies that the torque is given by:

) i i ( m k l pkm T sd rq sq rd 2 r e     
. The rotor motion equation turns out to be:

    J f J T i i m k l J m k p dt d L sd rq sq rd r      ) ( ) ( 2 (48)
This is the fifth (and last) state equation. The model of a real (saturating) AC machine consists of the equations (32), ( 35), ( 46), ( 47) and (48), which are, for convenience, rewritten in a more condensed form:

u ) X ( g ) X ( f X    (49a)   T rq rd X h y 2 2 , ) (       (49b) with :   T rq rd sq sd i i X    , , , ,  ,   T sq sd v v u ,  ,           0 0 0 d d 0 0 0 q q ) X ( g 6 5 0 5 (49c)                                                                                           J f J T i i m k l J m k p i g g i g g d d d d d d i d d i d d q q q q q q i q q i q q X f L sd
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 1 2 1 4 4 4 3 3 3 2 2 1 1 4 4 4 3 3 3 2 2 1 1 (49d)
It is worth noting that, due to flux saturation, the model parameters ) , , , , ( q , q q d d d

i i i i i i      
are dependent on the static magnetisation parameter m. If the magnetic saturation is ignored, all previous parameters become constant. The proposed model then reduces to the standard model which is widely used in the control literature, e.g. [START_REF] Ortega | On speed control of induction motors[END_REF]- [START_REF] Hu | Robust tracking control of an induction motor[END_REF].

VI. Experimental validation

The experimental part of the study was performed in the Automatic Control Dept of GIPSA-Lab, using a real induction motor whose features are summarised in Table 5. The motor control inputs are the stator voltage amplitude (V s ) and frequency ) ( s  . The values of these control signals are imposed through a DC/AC converter. The measured variables are the stator currents, electric power and the rotor speed. The whole system is controlled by a PC through a DSP Card. The first experimental task consists in obtaining a number of experimental points of the motor magnetic characteristic. These are used to construct the spline approximation

) (      I (Fig 1)
. Presently, one proceeds with the second task that consists in performing a practical validation of the newly developed model (49a-d) using real measurements. First experiment. The model is compared with the real motor operating in the linear part of the magnetic characteristic (i.e. for small values of   and  I ). To this end, V s is a square signal that varies between 82 V and 140 V, s rd s / 300   and the load torque T L = 0 Nm. It is worth recalling that the inputs V s and s  of the real machine are generated using a converter which, in addition, generates high frequency harmonics that are observed on V s (see Fig 6). To make the model validation more convincing, a disturbing high frequency noise is added to the input signal (Fig 6). The resulting flux norm ( r  ) and current norm (I s ) for the true machine and the model are respectively compared in Figures 7 and8. These clearly show that the model behavior is very close to that of the real system. Fig 9 shows that the corresponding static induction parameter m is almost time-invariant (this shows a relative variation of just 2%). This is a direct consequence of the fact that the experimental conditions are such that the machine operates in the linear part of the machine magnetic characteristic. Note also that, in these conditions, the proposed model reduces to the usual standard one. It is clearly seen that, again, the model is sufficiently accurate in representing the real machine. The corresponding static induction parameter m is represented by Figure 13. Unlike the previous experiment, here the initial and final values of m are very different (relative variation 60%). Consequently, the proposed model cannot reduce, in these conditions, to the usual standard model.

VII. Simplified version of the saturating model

The new model (49a-d) was shown to be accurate through experimental validation and so it will be referred to experimentally validated model (EVM). Its accuracy makes it suitable for simulation purposes. But, it may not be quite convenient for control purpose because it involves many state dependent parameters. Therefore, it is of interest to find out if there are simplified versions that preserve the model reliability. In this section, one develops such a simplified model.

VII. 1. Simplified model development

A simplified version of the EVM can be obtained by gathering at the same side both stator and rotor leakage inductances, i.e. replacing them by a unique equivalent inductance that may be placed either at the stator side or at the rotor side, ( [START_REF] Leonhard | Control of Electrical Drives[END_REF], [START_REF] Seguier | Industrial Electronics[END_REF]). To fix idea, let seq L denote the equivalent leakage inductance at the stator. To this end, recall that the stator and rotor fluxes are given by equations (21). Letting 0 l r  and seq s L l  in (21), one gets:

d sd seq sd i m i L     , q sq seq sq i m i L     ( 50
)
d rd i m k    , q rq i m k    (51)
The

v L 1 i a i a dt di          (52d)   J T i i J p dt d L sd rq sq rd       (52e)
where:

m L k R seq r 3   k R a r  1 , seq r s L k R R a 2 2   , seq 3 kL 1 a  ( 53
)
Remark 4. The model (52a-e) obtained by simplification is still nonlinear but it involves only one parameter (namely ) depending on the machine magnetic state through the magnetization coefficient. It proved to be quite convenient for designing controllers [START_REF] Ouadi | Modelling, observation and control of saturated induction machines[END_REF].

VII. 2. Simplified model validation

The

VII.3. Supremacy of the (simplified) new model over existing models

We have shown that the simplified model (52a-e) is a quite accurate approximation of the experimentally validated model (49a-d). It will be referred to "simplified experimentally validated model" or simply "SEV model" to alleviate the text. The question is whether the existing simple modelling approaches can yield equally accurate approximations. The focus is presently made on two approaches: (i) the standard modelling approach that totally ignores the magnetic saturation feature; (ii) the Heinemann-Leonhard modelling approach that accounts for such feature but it does it in an ad hoc way. Let us first present the models obtained by both approaches.

VII.3.a The standard model

As mentioned earlier (Remark 3, part iii), the standard model can be derived from the general model (49a-b) by simply letting the static induction parameter m be constant. The latter is given (the quasi-constant) value obtained in the first experiment, namely 1 . 0  m (see Fig 9). The obtained standard model is the following:

sd s rq rd r sq s sd sd v l K T K i i dt di      1       (54a) sq s rd rq r sd s sq sq v l K T K i i dt di      1       (54b) rq s rd r sd r rd T i T M dt d      ) ( 1     (54c) rd s rq r sq r rq T i T M dt d      ) ( 1     (54d)       J f J T i i JL pM dt d L sd rq sq rd r ) (   (54e)
The involved parameters are defined in Table 7.

Table 7. Parameters of the standard model

M l L s s   , m M  M l L r r   L L M K s   r s L L M 2 1   2 2 r s r s s L L M R L R      r r r R L T 

VII.3.b Heinemann-Leonhard saturated model

Instead of going back to first-principles derivation (modifying it to include a saturation model of the iron) as we did in this paper, former authors chose to just add on, in a somewhat ad hoc manner, a model of magnetic saturation to standard models like (54). Following Heinemann and Leonhard [START_REF] Heinemann | Self-tuning field-oriented control of an induction motor drive[END_REF] (see also [START_REF] Novotnak | High-performance motion control of an induction motor with magnetic saturation[END_REF]), the model (54a-e) is first subject to flux-orientation along d-axis. Accordingly, the q-axis flux equation (54d) is cancelled and the d-axis equation (54c) becomes:

) ) ( ( 1 sd rd r rd i P T M dt d       (55)
where M is set to its nominal value that presently corresponds to the linearity region on the magnetic characteristic (.) P (Fig 1). Equation ( 55) is completed by equations (54a), (54b) and (54e) which remain unchanged. The resulting model is rewritten for convenience:

sd s rd r sq s sd sd v l T K i i dt di     1      (56a) sq s rd sd s sq sq v l K i i dt di     1      (56b) ) ) ( ( 1 sd rd r rd i P T M dt d       (56c)      J f J T i JL pM dt d L sq rd r  (56d) rd sq r s i T M p w     (56e)

VII.3.c. Limitations of the standard and Heinemann-Leonhard models

The standard model (described by (54a-e)) and Heinemann-Leonhard (HL) model (described by (56a-e)) will now be compared using numerical simulations with the proposed SEV model (52a-e). To be meaningful, the comparison is performed in different operational conditions. Accordingly, the machine operates in both linear and nonlinear zones of its magnetic characteristic. This is achieved by modulating the stator voltage amplitude with a square signal switching between 80 and 200V (Fig 15) The situation is quite different when the machine operates in the nonlinear region of its magnetic characteristic. This is illustrated in the time interval [6 s, 15s] (that characterizes the standard model). Then, it is not surprising that the standard model responses deviate significantly from those of the SEV model (Figs. 17-18). The responses show that the standard model underestimates the absorbed stator current but overestimates the rotor flux. The deviation of the standard model responses (with respect to those of the SEV model) is almost 65% for the current and nearly 25% for the rotor flux. Figs 17-18 also show that the responses of the HL model (56a-e) are not so different from those of the SEV model (52a-e). This is normal since the magnetic saturation is not entirely ignored in the model (as this is the case in the standard model (54a-e)). Nevertheless, the HL model is itself unable to perfectly imitate the SEV model as the difference between both models is nearly 7% for the stator currents and 5% for the rotor flux (Figs. 17-18). Such a deviation is simply explained by the fact that the magnetic saturation is accounted for in an ad-hoc way in the HL model. Obviously, such a deviation can even be made larger by enforcing the machine to operate in a more saturating regime. The larger the stator voltage and rotor flux are the larger the more important the deviation of the HL model from the SEV model. Finally, it is observed in Fig. 19 that the differences between the three models are minor as long as the speed response is concerned. Therefore, any of these three models can be based upon, to achieve a good speed regulation. However, the energetic cost will not be the same especially in presence of high load torque. As a matter of fact, the regulators obtained from the standard and HL models will absorb higher stator currents (than the new model) because they both underestimate the current responses. Obviously, the energetic cost will be higher using the standard model than with the HL model. It is worth pointing out that the energetic issue was never raised in the previous works where speed regulation was dealt with based on the standard model.

VIII. Conclusion

In this paper, a new model that accounts for magnetic saturation has been developed for (uniform air-gap) induction motors. As a consequence of the nonlinear feature of the magnetic characteristic, a coupling arises between the motor (d,q)-axes. The new model (49a-d) turns out to be nonlinear and involves state-dependent parameters. Its practical validation has been performed on a 7.5 KW induction motor. Different experiments have proved that the new model is well representative of the true machine. The supremacy of the new model over the standard (54a-e) and the Heinemann-Leonhard model (56a-e) is illustrated through simulations. On the other hand, the simplified version (52a-e) proved to be quite accurate and, consequently, can be based upon in control design.

Fig 1 .

 1 Fig 1. Magnetic characteristic   =  I  ). Crosses: experimental points ) , I (    Solid: interpolation

d  , 0 q

 0  is the coupling magnetizing flux generated, along the d-axis, by the components (along the same axis) of the stator and the rotor currents . 0 are additional terms whose values depend on the way the inductive parameters are definedIt is worth noticing that the coefficients

  As a consequence of the cross-saturation phenomenon, both rotor flux components ( rq rd ,

  time-varying as they are functions of the machine state variables. Formulas that link these parameters to the machine state variables are established in the next subsection. iii) In the case of unsaturated machine, one has 0 M dq  and q d M , M become constant. Then, the resulting machine model coincides with the widely used standard model.

  one has the relation:

Fig 3 .

 3 Notice also that the function Q(.) is time-derivable and bounded away from zero. On the other hand, as expressions (23)-(24) involve the derivative  dI dm /

  shown in Fig 4. The polynomial approximations, defined up to now are summarized in Table

  Fig 3. Experimental couples ) , ( m I  (crosses) and spline interpolation ) (  I Q (solid). Unities:  I (A), m (Wb/A)

Fig 6 .

 6 Fig 6. First experiment: the stator voltage amplitude Vs (V).

Fig 7 :

 7 Fig 7: First experiment: flux amplitude Φr (Wb). Real machine (solid); model (49a-d) (dashed).

Fig 8 .

 8 Fig 8. First experiment: the stator current amplitude I s (A). Real machine (solid); model (49a-d) (dashed).

Fig 9 :

 9 Fig 9: First experiment: static induction parameter m (Wb/A)

Fig 10 .Fig 11 .

 1011 Fig 10. Second experiment: the stator voltage amplitude Vs (V)

Fig 12 .

 12 Fig 12. Second experiment: stator current norm I s (A). Real machine (solid); model (dashed).

Fig 13 :

 13 Fig 13: Second experiment: static induction parameter m (in Wb/A)

  responses of the simplified model defined by (52a-e) are compared with those of EVM defined by (49a-d). The applied stator voltage input (V s ) is an amplitude-modulated sinusoid with a square modulating signal. The experimental conditions are such that the motor operates in the nonlinear part of the machine magnetic characteristic (large values of   and  I ). To this end, the stator voltage amplitude s V is a square signal switching between 160 and 192.5 V (Fig 17a), the stator current pulsation is set to 300  s  rd/s. Figs 17b-d show that the responses generated by the two models are very close, except at the switching instants. That is, the simplified model is actually representative of the AC motor. a) Stator voltage norm (in V) b ) Rotor speed (in rd/s): EVM (solid), simplified model (dashed) c) Rotor flux norm (in Wb): EVM (solid), simplified model (dashed) d) Stator current norm (in A) : EVM (solid), simplified (dashed).

Fig 14 .

 14 Fig 14. Responses of the EVM and simplified model (52a-e)

.

  The load couple T L switches between 0 Nm and 15Nm (Fig16).

Figs 17 -

 17 Figs 17-19 show that, as long as Vs = 80V (time-interval [0 s, 6s]), the responses (flux and stator current norms) of the standard and HL models are very close to those of the SEV model. This is normal because the machine operates in the linearity region of its magnetic characteristic. In fact, the rotor flux level is relatively low ) 6 . 0 ( Wb r   and the static magnetization parameter m (for the SEV model (52a-e) or the original model (49a-d)) is very close to the value used to obtain the standard model (i.e. m = 0.09). In these conditions, the standard and HL models are equally representative of the real machine.

  by setting the stator voltage to the high value Vs = 200V and the rotor flux to Wb r at the elbow region in the magnetic characteristic. The parameter m turns out to be varying, deviating far from the value 09 . 0

Fig. 15 .Fig 17 .Fig 18 .Fig 19 .

 15171819 Fig. 15. Stator voltage amplitude V s (V)

Table 1 .

 1 NotationsR s , R r Stator and rotor resistances l s , l

r Stator and rotor leakage inductances (constant parameters)   Magnetizing flux at a stator phase (flux within the air-gap)   Norm of the flux    s Stator flux (through one phase)

Table 2 :

 2 Spline approximations

Table 4 .

 4 Parameters newly introduced in equation (45)

Table 5 :

 5

	Induction Motor Characteristics
	Power	P N	7.5	KW
	Nominal speed	Ω N	1450	Tr / mn
	Nominal stator voltage	U sn	380	V
	Nominal stator current	I sn	16	A
	Nominal flux	Φ rn	1	Wb
	Nominal frequency	f s	50	HZ
	Poles pair number	p	2