Joao Pedro Ferreira
email: j.ferreira@chru-nancy.fr

MD John G F Cleland

MD, PhD Carolyn S P Lam

MD, PhD Stefan D Anker

MD, PhD Mandeep R Mehra

MD, PhD Dirk J Van Veldhuisen

MD William M Byra

David La Police

MD Bertram Pitt

MD Barry Greenberg

MD, PhD João Pedro Ferreira

BS David A La Police

MD, PhD Faiez Zannad
email: f.zannad@chru-nancy.fr

Professor Faiez Zannad or Dr Pedro João

Ferreira

Heart failure re-hospitalizations and subsequent fatal events in coronary artery disease: insights from COMMANDER-HF, EPHESUS, and EXAMINE

Keywords: failure, myocardial infarction, stroke

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

Introduction

Strategies to prevent and treat coronary artery disease (CAD) have led to reductions in the incidence of myocardial infarction (MI) and subsequent mortality, driven by early restoration of myocardial blood flow and better secondary prevention by controlling cardiovascular risk factors and pharmacological interventions [START_REF] Smolina | Determinants of the decline in mortality from acute myocardial infarction in England between 2002 and 2010: linked national database study[END_REF][START_REF] Bandosz | Decline in mortality from coronary heart disease in Poland after socioeconomic transformation: modelling study[END_REF] . In contrast, the incidence and prevalence of heart failure has increased, especially amongst older people, due (at least partially) to better survival after MI [START_REF] Conrad | Temporal trends and patterns in heart failure incidence: a population-based study of 4 million individuals[END_REF] . The development of heart failure often leads to substantial disability and impaired quality of life and is associated with high rates of hospitalizations and a poor prognosis [START_REF] Conrad | Temporal Trends and Patterns in Mortality After Incident Heart Failure: A Longitudinal Analysis of 86000 Individuals[END_REF] . Hospitalisations are often for worsening heart failure (HFH) that may be triggered by ischemia or many other factors [START_REF] Kobayashi | Heart failure etiologies and clinical factors precipitating for worsening heart failure: Findings from BIOSTAT-CHF[END_REF][START_REF] Arrigo | Precipitating factors and 90-day outcome of acute heart failure: a report from the intercontinental GREAT registry[END_REF] .

In the COMMANDER-HF trial, which included patients with a left ventricular ejection fraction (LVEF) <40%, heart failure with reduced ejection fraction (HFrEF) and coronary artery disease (CAD) in sinus rhythm plus a recent hospitalisation for worsening congestion, rivaroxaban 2.5mg bd added to background anti-platelet therapy did not reduce the rate of hospitalisation or death due to worsening heart failure but may have reduced vascular occlusive events [START_REF] Zannad | Rivaroxaban in Patients with Heart Failure, Sinus Rhythm, and Coronary Disease[END_REF][START_REF] Mehra | A comprehensive analysis of the effects of rivaroxaban on stroke or transient ischaemic attack in patients with heart failure, coronary artery disease, and sinus rhythm: the COMMANDER HF trial[END_REF][START_REF] Greenberg | Association of Rivaroxaban With Thromboembolic Events in Patients With Heart Failure, Coronary Disease, and Sinus Rhythm: A Post Hoc Analysis of the COMMANDER HF Trial[END_REF] . These findings suggest that myocardial dysfunction rather than coronary events drive the progression of HFrEF with CAD [START_REF] Cleland | Myocardial Dysfunction and Coronary Artery Disease as Therapeutic Targets in Heart Failure[END_REF] . Accordingly, we compared event-rates for HFH, MI and stroke as well as the risk of subsequent death after a non-fatal event (HFH, MI, and stroke) in COMMANDER-HF compared to EPHESUS (Eplerenone, a Selective Aldosterone Blocker, in Patients with Left Ventricular Dysfunction after Myocardial Infarction) [START_REF] Pitt | The EPHESUS trial: eplerenone in patients with heart failure due to systolic dysfunction complicating acute myocardial infarction. Eplerenone Post-AMI Heart Failure Efficacy and Survival Study[END_REF] and EXAMINE (Alogliptin after Acute Coronary Syndrome in Patients with Type 2 Diabetes) [START_REF] White | Alogliptin after acute coronary syndrome in patients with type 2 diabetes[END_REF] .

Methods

Study populations

COMMANDER-HF was an international, double-blind, randomized trial comparing the factor-Xa inhibitor rivaroxaban (2.5 mg twice daily) vs. placebo. Key inclusion criteria included history of chronic HF for 3 or more months, left ventricular ejection fraction (LVEF) of 40% or less, history of coronary artery disease, absence of atrial fibrillation or other indication for chronic anticoagulation, and treatment for an episode of decompensated heart failure (i.e., the index event) within the previous 21 days. Decompensated HF was defined by symptoms of worsening dyspnea or fatigue, objective signs of congestion, and/or adjustment of HF medications requiring hospital admission or unscheduled parenteral diuretic [START_REF] Zannad | Rivaroxaban in Patients with Heart Failure, Sinus Rhythm, and Coronary Disease[END_REF][START_REF] Zannad | Rationale and design of a randomized, doubleblind, event-driven, multicentre study comparing the efficacy and safety of oral rivaroxaban with placebo for reducing the risk of death, myocardial infarction or stroke in subjects with heart failure and significant coronary artery disease following an exacerbation of heart failure: the COMMANDER HF trial[END_REF] .

EPHESUS was an international, double-blind, randomized trial comparing eplerenone vs. placebo. Patients were eligible for enrolment if they had an acute MI, and a LVEF ≤40% plus HF or diabetes within 3 to 14 days before randomization [START_REF] Pitt | The EPHESUS trial: eplerenone in patients with heart failure due to systolic dysfunction complicating acute myocardial infarction. Eplerenone Post-AMI Heart Failure Efficacy and Survival Study[END_REF] .

EXAMINE was an international, double-blind, randomized trial comparing the DPP-IV inhibitor alogliptin vs. placebo. Patients were eligible for enrolment if they had type-2 diabetes mellitus and had experienced an acute coronary syndrome within 15 to 90 days before randomization [START_REF] White | Alogliptin after acute coronary syndrome in patients with type 2 diabetes[END_REF][START_REF] White | EXamination of cArdiovascular outcoMes with alogliptIN versus standard of carE in patients with type 2 diabetes mellitus and acute coronary syndrome (EXAMINE): a cardiovascular safety study of the dipeptidyl peptidase 4 inhibitor alogliptin in patients with type 2 diabetes with acute coronary syndrome[END_REF] .

The three trials were conducted in accordance with the Declaration of Helsinki and approved by the site ethics committees. All participants gave written informed consent to participate in the studies.

Study outcomes

The primary efficacy outcome in COMMANDER-HF was the composite of death from any cause, MI, or stroke. In EPHESUS the primary efficacy outcomes were a composite of death from cardiovascular causes or hospitalization for cardiovascular causes (including heart failure, acute myocardial infarction, stroke, or ventricular arrhythmia), and death from any cause. In EXAMINE the primary endpoint was a composite of death from cardiovascular causes, nonfatal MI, or nonfatal stroke. HFH was a main secondary end-point of both the COMMANDER-HF and EXAMINE trials.

Statistical analysis

Similar methods were applied across the studied trials. Number and proportion of events, person-time exposures and time-to-first event-rates (per 100 person-years) were computed.

A "smoothed" hazard function was plotted for each time-to-first event to assess the event pattern throughout the follow-up. To make the comparisons easier to interpret we plotted all-cause death, HFH, MI, and stroke over time using the same x-scale and y-scale for all trials. Cumulative incidence rates were computed using the Nelson-Aalen method in the placebo arm of the respective trials to ease the between-trail comparison without the effect of treatments. To assess the risk of death after the first non-fatal events; time-updated Cox models were performed in the placebo groups of the respective trials adjusting for potential baseline confounders, as described in previous analysis [START_REF] Ferreira | Income level and inequality as complement to geographical differences in cardiovascular trials[END_REF] . In COMMANDER-HF the adjustment was performed on age, sex, race, diabetes, prior MI, prior stroke, history of percutaneous coronary intervention or coronary artery bypass grafting (PCI/CABG), NYHA class, body mass index, systolic blood pressure, anemia, estimated glomerular filtration rate (eGFR), dual anti-platelet therapy, rivaroxaban or placebo allocation, and geographical region stratification. The treatment effect of rivaroxaban vs. placebo was tested by the intention-to-treat principle at different follow-up times with stratification by region and "right censored" times (6 months, 12 months, and/or last censor). The proportional hazards assumption was tested by plotting the Schoenfeld residuals over time; the residuals were independent of time and no evidence of proportional hazards violation was found.

Results

Baseline characteristics

In COMMANDER-HF a total of 5022 patients were included with a median follow-up duration of 21.1 months (percentile 25-75 , 12.9-32.8), the mean age was 66±10 years, 77% were male, 82% white, and 41% had diabetes. In EPHESUS a total of 6632 patients were included with a median follow-up duration of 15.6 months (percentile 25-75 , 11.9-20.6), the mean age was 64±11 years, 71% were male, 90% white, and 32% had diabetes. In EXAMINE a total of 5380 patients were included with a median follow-up duration of 19.2 months (percentile 25-75 , 12.0-26.4), the mean age was 61±10 years, 68% were male, 73% white, and all patients had diabetes (by inclusion criteria). The characteristics of the patients enrolled in the respective trials are described in the Supplemental Table 1.

Event rates

In COMMANDER-HF the primary outcome (a composite of MI, stroke or all-cause death) overall event rate was 13.5 (12.8-14.3) events per 100 person-years; the event rate was higher in the first 6 months and decreased thereafter: 16.8 (15.2-18.5) in the first 6 months vs. 11.9 (11.0-12.9) events per 100 person-years beyond the 12th month of follow-up. Similar findings were observed for the composite of MI, stroke or cardiovascular death. The major contributor for the observed event rate was all-cause death with an overall event rate of 11.2 (10.6-11.9) with similar findings observed for cardiovascular death. Table 1. The overall rate of MI was low: 2.2 (2.0-2.6) events per 100 person-years, but slightly higher within the first 6 months of follow-up: 3.2 (2.6-4.0) in the first 6 months vs. 1.6 (1.3-2.0) events per 100 person-years beyond the 12th month of follow-up. The rate of stroke was even lower and more homogeneous over time with an overall event rate of 1.3 (1.1-1.6) events per 100 person-years. Table 1. The rate of HFH was the highest among all outcomes with an overall rate of 16.9 (16.1-17.9) events per 100 person-years that was much higher in the first 6 months: 29.3 (27.1-31.6) vs. 9.3 (8.4-10.3) after 12 months of follow-up. Table 1, Figure 1 & Figure 2. Bleeding rates were overall low: ≤2 events per 100 person-years. 2. Details on multiple other events (e.g., sudden death, "pump failure" death, non-HFH) are also presented in the Supplemental Table 2. No major variations of the treatment effect (rivaroxaban vs. placebo) over time were found. Supplemental Table 3.

Supplemental Table

In EPHESUS the overall event rate of the composite outcome of MI, stroke or cardiovascular death was 16.2 (15.4-17.1) events per 100 person-years; the event rate was higher in the first 6 months and decreased thereafter: 28.6 (26.7-30.6) in the first 6 months vs. 7.8 (6.8-9) events per 100 person-years beyond the 12th month of follow-up. In EPHESUS, the MI rates were more than 3-fold higher than in COMMANDER-HF, with an overall event rate of 7.2 (6.7-7.8) events per 100 person-years, also highest in the first 6 months. The rate of stroke was similar to that found in COMMANDER-HF with an overall value of 1.9 (1.7-2.3) events per 100 person-years. Table 1. The HFH rates were lower than those found in COMMANDER-HF. The overall rate was 10.6 (9.9-11.3) events per 100 person-years, but also highest early in the follow-up. Table 1, Figure 1 & Figure 2.

In EXAMINE the overall event rate of the composite outcome of MI, stroke or cardiovascular death was 7.7 (7.1-8.4) events per 100 person-years; the event rate was higher in the first 6 months and decreased thereafter. In EXAMINE, the MI rates were 2-fold higher than in COMMANDER-HF, with an overall event rate of 4.4 (4.0-4.9) events per 100 person-years, also highest in the first 6 months. The rate of stroke was lower than in COMMANDER-HF with an overall value of 0.7 (0.6-0.9) events per 100 person-years. Table 1.

The HFH rates were much lower than those found in COMMANDER-HF and EPHESUS. The overall rate was 2.4 (2.0-2.7) events per 100 person-years, but also highest early in the follow-up. Table 1, Figure 1 & Figure 2.

Deaths after a non-fatal event

In COMMANDER-HF, having a non-fatal event during the follow-up increased the risk of subsequent death by ≈2-fold. The risk was similar for all non-fatal events, but slightly higher after an MI: MI adjusted HR (95%CI) =2.31 (1.73-2.61), p<0.001; HFH adjusted HR (95%CI) =1.81 (1.59-2.05), p<0.001; stroke adjusted HR (95%CI) =1.75 (1.31-2.33), p<0.001. The absolute numbers and proportion of HFH events preceding death was much higher than the proportion of MI or stroke: HFH 517 (46.9%), MI 106 (9.6%), Stroke 53 (4.8%). Table 2, Figure 3 & Figure 4.

In EPHESUS, having a non-fatal event during the follow-up increased the risk of subsequent death by 2 to 3-fold. The risk was higher after an MI or stroke, than after a HFH: MI adjusted HR (95%CI) =3. 16 (2.72-3.67), p<0.001; stroke adjusted HR (95%CI) =3.01 (2.38-3.81), p<0.001; HFH adjusted HR (95%CI) =2.23 (1.94-2.58), p<0.001. The absolute numbers and proportion of HFH events preceding death was also higher than the proportion of MI or stroke: HFH 316 (30.6%), MI 261 (25.3%), Stroke 82 (8.0%). Table 2, Figure 3 &Figure 4.

In EXAMINE, having a non-fatal event during the follow-up increased the risk of subsequent death by 1.5 to 2.5-fold. In EXAMINE the risk was higher after a HFH than after a MI or stroke: HFH adjusted HR (95%CI) =2.50 (1.80-3.48), p<0.001; MI adjusted HR (95%CI) =1.59 (1.15-2.19), p=0.005); stroke adjusted HR (95%CI) =1.49 (0.70-3.18), p=0.30. The absolute numbers and proportion of HFH events preceding death were similar to MI and higher than stroke: HFH 48 (14.7%), MI 46 (14.1%), Stroke 7 (2.2%). Table 2, Figure 3 & Figure 4.

Discussion

Our study shows that HFHs are frequent and associated with poor outcomes in patients with CAD. However, depending on the population studied the type of events and their rate may vary widely. Patients with decompensated HF but without an acute MI (as in COMMANDER-HF), present a high rate of HF re-hospitalizations frontloaded early within the follow-up; the HFH rates are much superior (8-fold or even higher) to those of MI or stroke (despite the presence of underlying CAD) and greatly increase the risk for subsequent death (≈2-fold higher). In EPHESUS, with high-risk acute MI patients, HF re-hospitalizations were more frequent than recurrent MI and also independently associated with a poor prognosis. In EXAMINE the recurrent MI rates were higher than HFH, but HFH was also frequent and associated with a poorer prognosis than MI. Together, these data show that a large window of opportunity to reduce events still exists and that targeting HF is key in this endeavour.

COMMANDER-HF was designed with the hypothesis that thrombin would play a central role in the HF progression of patients with CAD [START_REF] Zannad | Rationale and design of a randomized, doubleblind, event-driven, multicentre study comparing the efficacy and safety of oral rivaroxaban with placebo for reducing the risk of death, myocardial infarction or stroke in subjects with heart failure and significant coronary artery disease following an exacerbation of heart failure: the COMMANDER HF trial[END_REF] . Inhibiting the factor Xa with rivaroxaban could lead to the reduction of thrombin and related pathways associated with myocyte injury, inflammation, endothelial dysfunction and microvascular thrombosis, which could then result in a reduction of ischemic/thrombotic events and also HF hospitalizations and mortality via thrombin-related pathways [START_REF] Zannad | Is thrombosis a contributor to heart failure pathophysiology? Possible mechanisms, therapeutic opportunities, and clinical investigation challenges[END_REF][START_REF] Ferreira | Antithrombotic therapy in heart failure patients with and without atrial fibrillation: update and future challenges[END_REF] . The inclusion of patients early (within 21 days) of a decompensated HF episode would enrich the trial for outcome events. This was indeed what happened, with the vast majority of these events being HFH on which rivaroxaban had no effect [START_REF] Mehra | A comprehensive analysis of the effects of rivaroxaban on stroke or transient ischaemic attack in patients with heart failure, coronary artery disease, and sinus rhythm: the COMMANDER HF trial[END_REF][START_REF] Ferreira | Stroke Risk in Patients With Reduced Ejection Fraction After Myocardial Infarction Without Atrial Fibrillation[END_REF] . By computing the risk of death after the non-fatal events, one may observe that the risk of dying after a HFH was similar to the risk of dying after a stroke or MI. However, HFH were a much more frequent event than MI or stroke, making HF rehospitalizations as a main target for improvement. Even in patients who had an acute MI (as in EPHESUS and EXAMINE), HF re-hospitalizations represent a major burden with deleterious consequences. Similarly, in a post-hoc analysis of the ALTITUDE trial (Aliskiren Trial in Type 2 Diabetes Using Cardio-Renal Endpoints) that included patients with type 2 diabetes plus chronic kidney disease, cardiovascular disease or both, the proportion of patients experiencing a HFH was higher than that of MI or stroke, and the greatest number of deaths occurred in the group who had a HFH first [START_REF] Jhund | Mortality following a cardiovascular or renal event in patients with type 2 diabetes in the ALTITUDE trial[END_REF] .

Up-to-date HFrEF treatment including an angiotensin-receptor neprilysin inhibitor (ARNI), a sodium glucose co-transporter inhibitor (SGLT2i), and a mineralocorticoid receptor antagonist (MRA) may lead to a 50% event reduction compared with treatment with angiotensin converting enzyme/angiotensin receptor blocker (ACEi/ARB) and a betablocker [START_REF] Vaduganathan | Estimating lifetime benefits of comprehensive disease-modifying pharmacological therapies in patients with heart failure with reduced ejection fraction: a comparative analysis of three randomised controlled trials[END_REF] ; however, it should be noted that the implementation of HF-modifying therapies remains an issue even in developed countries [START_REF] Tromp | Heart failure around the world[END_REF][START_REF] Greene | Titration of Medical Therapy for Heart Failure With Reduced Ejection Fraction[END_REF][START_REF] Armstrong | Vericiguat in Patients with Heart Failure and Reduced Ejection Fraction[END_REF][START_REF] Mcmurray | Dapagliflozin in Patients with Heart Failure and Reduced Ejection Fraction[END_REF][START_REF] Mcmurray | Angiotensin-neprilysin inhibition versus enalapril in heart failure[END_REF] . Notwithstanding, while the therapeutic armamentarium for chronic HFrEF may sharply reduce events rates, if implemented; post worsening HF remains an unmet need with high recurrent HFH and mortality, and offers an opportunity to test and implement novel strategies. Such opportunity was tested in the PIONEER-HF (Angiotensin-Neprilysin Inhibition in Acute Decompensated Heart Failure) trial, where initiation of sacubitril/valsartan upon stabilization of a decompensated HF episode led to a greater reduction in the NT-pro BNP than enalapril without increasing adverse events [START_REF] Velazquez | Angiotensin-Neprilysin Inhibition in Acute Decompensated Heart Failure[END_REF] . In VICTORIA-HF (Vericiguat in Patients with Heart Failure and Reduced Ejection Fraction) patients had to have evidence of worsening HF and most patients (67%) had been hospitalized for HF in the previous 3 months which resulted in a high rate of HFH and death in this trial, slightly reduced with vericiguat treatment [START_REF] Armstrong | Vericiguat in Patients with Heart Failure and Reduced Ejection Fraction[END_REF] . In COMPASS (Rivaroxaban with or without Aspirin in Stable Cardiovascular Disease) patients with a HF history also had a higher rate of all the studied events, including HFH [START_REF] Branch | Rivaroxaban With or Without Aspirin in Patients With Heart Failure and Chronic Coronary or Peripheral Artery Disease[END_REF][START_REF] Eikelboom | Rivaroxaban with or without Aspirin in Stable Cardiovascular Disease[END_REF] .

These examples show that the opportunity for improvement in populations that had a recent HF hospitalization, MI or both is considerable and optimizing the use of therapies that have been shown to improving outcomes is key to achieve this goal [START_REF] Zannad | Rising incidence of heart failure demands action[END_REF] .

Limitations

The findings here depicted may only be generalized to patients with similar characteristics of those included in these trials, due to the relatively young, white and male populations included, our findings may not be applicable to patients with different characteristics.

Furthermore, the inclusion/exclusion criteria and level of monitoring performed in a trial context may further limit the generalization of these results to patients in the community, but allows a detailed assessment of adjudicated cause-specific events. The timing of enrollment after the acute event was similar in COMMANDER-HF and EPHESUS (less than 21 and 3 to 14 days, respectively), but more widespread in EXAMINE (15 to 90 days) which might have contributed to the lower overall event rate in EXAMINE. The follow-up times were slightly different between trials (median in months of 21 in COMMANDER-HF, 16 in EPHESUS, and 19 in EXAMINE), which might have influenced the event rates and the assessment of fatal events after non-fatal ones. EPHESUS was published almost 20 years ago; the MI therapeutics have improved much since and the observed high death risk after MI could be lower in a contemporary cohort.

Conclusions

The burden of HF re-hospitalizations is enormous in patients with CAD and is associated with a high risk of subsequent death. Reducing HF in CAD represents an unmet need that should be addressed in future studies.

Disclosures

Dr Ferreira is a consultant for Boehringer-Ingelheim. Dr Zannad reports steering committee personal fees from Applied Therapeutics, Amgen, Bayer, Boehringer, Novartis, Janssen, Caption: the figure represents the proportion of all deaths that occurred after a non-fatal event in each trial. For example, in COMMANDER-HF 47% of all deaths occurred after a HFH.

  A 2-sided p value of <0.05 was considered significant. Stata® version 16 (StataCorp. 2019. College Station, TX: StataCorp LLC) was used for the analyses.

  Abbott Diagnostics, Amgen, Applied Therapeutics, AstraZeneca, Bayer, Biofourmis,

Figure 1 .

 1 Figure 1. Smoothed hazard function over time

Figure 2 .

 2 Figure 2. Cumulative incidence of events in the placebo groups of the respective trials

Figure 3 .

 3 Figure 3. Proportion of deaths following a non-fatal event in each trial

Figure 4 .

 4 Figure 4. Cumulative incidence of death after a non-fatal event

Table 1 .

 1 Event rates per 100 person-years by intervals of follow-up time MI, myocardial infarction; CVD, cardiovascular death; ACM, all-cause mortality; HFH, heart failure re-hospitalization.

	Time (months)	Person-years	Events	Rate (per100py)
	COMMANDER-HF (n =5022)
	MI, stroke, ACM
	0 to 6	2403	404	16.8 (15.2-18.5)
	6 to 12	2151	290	13.5 (12-15.1)
	Beyond 12	4946	590	11.9 (11-12.9)
	Total	9500	1284	13.5 (12.8-14.3)
	MI, stroke, CVD
	0 to 6	2403	367	15.3 (13.8-16.9)
	6 to 12	2151	263	12.2 (10.8-13.8)
	Beyond 12	4946	491	9.9 (9.1-10.8)
	Total	9500	1121	11.8 (11.1-12.5)
	MI
	0 to 6	2412	78	3.2 (2.6-4)
	6 to 12	2169	59	2.7 (2.1-3.5)
	Beyond 12	5048	79	1.6 (1.3-2)
	Total	9629	216	2.2 (2-2.6)
	Stroke
	0 to 6	2420	41	1.7 (1.2-2.3)
	6 to 12	2183	30	1.4 (1-2)
	Beyond 12	5075	56	1.1 (0.8-1.4)
	Total	9679	127	1.3 (1.1-1.6)
	ACM
	0 to 6	2429	320	13.2 (11.8-14.7)
	6 to 12	2202	239	10.9 (9.6-12.3)
	Beyond 12	5180	543	10.5 (9.6-11.4)
	Total	9810	1102	11.2 (10.6-11.9)
	CVD
	0 to 6	2429	282	11.6 (10.3-13)
	6 to 12	2202	212	9.6 (8.4-11)
	Beyond 12	5180	435	8.4 (7.6-9.2)
	Total	9810	929	9.5 (8.9-10.1)
	HFH
	0 to 6	2265	663	29.3 (27.1-31.6)
	6 to 12	1872	344	18.4 (16.5-20.4)
	Beyond 12	4009	373	9.3 (8.4-10.3)
	Total	8146	1380	16.9 (16.1-17.9)
	EPHESUS (n =6632)
	MI, stroke, CVD
	0 to 6	2998	857	28.6 (26.7-30.6)
	6 to 12	2676	280	10.5 (9.3-11.8)

Table 2 .

 2 Death after a non-fatal event

	Events	Event and death	Event and survival	Adj. HR (95%CI)	P
	COMMANDER-HF
	HFH
	N=1380	517 (46.9%)	863 (22.0%)	1.81 (1.59-2.05)	<0.001
	MI
	N=216	106 (9.6%)	110 (2.8%)	2.13 (1.73-2.64)	<0.001
	Stroke
	N=127	53 (4.8%)	74 (1.9%)	1.75 (1.31-2.33)	<0.001
	Total	1102	3920
	EPHESUS
	HFH
	N=855	316 (30.6%)	539 (9.6%)	2.23 (1.94-2.58)	<0.001
	MI
	N=604	261 (25.3%)	343 (6.1%)	3.16 (2.72-3.67)	<0.001
	Stroke
	N=168	82 (8.0%)	86 (1.5%)	3.01 (2.38-3.81)	<0.001
	Total	5600	1032
	EXAMINE
	HFH
	N=195	48 (14.7%)	147 (2.9%)	2.50 (1.80-3.48)	<0.001
	MI
	N=360	46 (14.1%)	314 (6.2%)	1.59 (1.15-2.19)	0.005
	Stroke
	N=61	7 (2.2%)	54 (1.1%)	1.49 (0.70-3.18)	0.30
	Total	5054	326
	Legend: MI, myocardial infarction; HFH, heart failure re-hospitalization.

Funding

The COMMANDER-HF trial was supported by Janssen Research and Development.


Vectorial Modeling and Control of Multiphase Machines with Non-salient Poles Supplied by an Inverter

Introduction and presentation of the electrical machines we will study

This chapter is devoted to the modeling and control of electrical machines with at least two independent statoric currents. The star-coupled three-phase machine without a neutral terminal and the triangle-coupled three-phase machine are the most basic ones. More precisely, this chapter aims to emphasize the particularities created by a number of independent currents greater than two with respect to the classic three-phase machine.

To reach this goal, we will restrict ourselves to machines fulfilling certain hypotheses:

-constant magnetic air-gap (without variable reluctance effect); -without magnetic saturation effect; -built regularly, i.e. it is impossible to discriminate against phases as all of the phases are characterized by the same technological realisation.

In practice, these hypotheses allow us to deal with at least two large families of machines:

-synchronous machines with surface-mounted magnets;

Chapter written Xavier KESTELYN and Eric SEMAIL.

-squirrel-cage induction machines.

Obviously real machines do not perfectly fulfill the hypotheses, but we will assume that the phenomena induced by the non-fulfilling of these hypotheses generate second-order phenomena that can be implicitly compensated for by robust control.

For control, squirrel-cage induction machines can be distinguished from synchronous machines with magnets, mainly because the magnetization of the machine is entirely controlled by the inverter supplying the stator. In fact, for synchronous machines with magnets, part of the magnetic field within the machine is not controlled by the supply. Hence, the study of the control of synchronous machines has more constraints than that of induction machines. For this reason, we will restrict the study in this chapter to that of synchronous machines with permanent magnets. The particularity of these machines lies in the number of phases and taking both space and time harmonics into account. This problem of the impact of harmonics on control has already been tackled in the case of three-phase synchronous machines with trapezoidal electromotive forces in [GRE 94] and [LOU 10]. In this chapter, we emphasize the originality induced by increasing the number of phases.

Finally, from among the machines studied we can also distinguish between several subfamilies by focusing on the connections that can be observed between the different coils of the statoric phases:

-machine without coupling between phases: each phase that has two connection terminals is generally supplied by a monophase H inverter; -simple star machine: n coils constituting the n phases are connected by a common point, the neutral terminal of the machine; -multi-star machine: k stars we will be connected to n/k coils by a common point. We will therefore have k neutral points; -machine with polygonal coupling: n coils constituting the n phases are connected in series. For the three-phase machines, we refer to this as triangle coupling.

This chapter will only deal with the control of independent phase machines and simple star machines. The five-phase machine will be the reference example allowing easy generalization to n-phase machines from the vectorial formulation choosen.

Control model of synchronous machines with permanent magnets and supplied by an inverter

The following hypotheses and notations will be used to model the machine:

-the n phases are identical and shifted by an angle n / 2π α = , and p is the number of pole pairs of the machine;

-the machine has smooth poles and is not saturated.

Figure 5.1 represents a two-pole n-phase machine in which variable g (a voltage, current, flux, etc.) is written g k with respect to phase k.

Characteristic spaces and generalization of the notion of an equivalent twophase machine

Equations in the natural basis of the stator and general vectorial expression

A Euclidean vectorial space E n of dimension n as well as an orthonormal basis

{ } n n n n n x x x B ,..., , 2 1 =
is combined with the n-phase machine. This is referred to as natural, since the coordinates of vector g in this basis are measurable variables g k of stator phases:

∑ = + + + = = n k n k k n n n n n x g x g x g x g g 1 2 2 1 1 ... [5.1]
Thus, the following vectors can be defined:

-voltage:

n n n n n x v x v x v v + + + =
... . By considering the resistance R s of a stator phase, we can determine a single voltage vectorial equation that gathers the scalar voltage equations of each phase:

n B s dt d i R v / ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = φ [5.2]
From the hypothesis of non-saturation, equation [5.2] can be written as: is due to the contribution of the rotor (permanent magnets or inductor coil for a synchronous machine, and rotor coils or bars for an induction machine).

In the case of a synchronous machine with smooth poles, equation [5.3] is more classically written as:

e dt i d i R v n B s + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = / λ [5.4] in which:
-λ is a linear application (or morphism) so that ( ) ss i λ φ = . This application is commonly written in natural basis B n in the form of a symmetrical matrix ( [ ] ( )

⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = = nn n n n n n n s L L L L L L L L L B mat L 2 1 2 22 21 1 12 11 , λ [5.5] - / n sr B d e dt φ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
is the electromotive forces (emfs) vector, which is written in the form of a function ε (speed-normalized emf) depending only on θ, the relative position of the rotor with respect to the stator coils, and on speed Ω of the rotor, see [5.6]:

( )

e ε θ = Ω [5.6]
More than allowing synthetic writing, the vectorial relationships ease the calculations of powers and torque. The instantaneous power flowing through the machine is obtained by the simple scalar product of the voltage and current vectors:

1 . n k k k p v i v i = = = ∑ [5.7]
By replacing expression [5.4] of the voltage vector in equation [5.7], we get:

( ) i e i dt i d i R p n B s . . / 2 + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = λ [5.8]
Here, we recognize the following within the framework of the hypotheses:

- ( ) 2 i R p s j =
, the stator Joule losses;

- i dt i d p n B mag . / ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = λ
, the power linked to the magnetic energy stored;

- i e p em . =
, the electromechanical power developed by the machine that lies in the origin of the creation of an electromagnetic torque c, expressed by:

∑ = = = Ω = Ω = n k n n em i i i e p c 1 . . ε ε [5.9] 5.2.1.2

. Determination of a decoupling basis

If vectorial relationship ) (i ss λ φ =
between the stator flux and stator current vectors remains true whichever the basis of space E n choosen, it is not true when it comes to matricial relationships between the coordinates of these vectors. The vector coordinates are obtained by projecting the vectors on the generating vectors of the basis. We can then understand that if we project the vectors in another basis, the relationships between the coordinates will change. The matrix that characterizes morphism λ in the natural basis is generally a full matrix. Therefore, couplings between the different phases that appear are not appreciated within the control framework.

If there is a basis in which the matrix of morphism λ is diagonal, we prefer to work in this basis when considering control. The coordinates in this basis of voltage, current and flux vectors are no longer measurable, but fictitious.

In the case of morphisms characterized by a symmetrical matrix (which is the case of λ ), we are assured of the existence of such bases that ensure decoupling of the different coordinates. Furthermore, these bases are orthogonal and the eigenvalues are real. )

A basis can be determined by analyzing the inductance matrix

0 det = - Λ n s n L I [5.10]
The detailed calculation of these eigenvalues and the associated eigenvectors can be found in [SEM 00], and in an analoguous form in [WHI 59].

In the new decoupling basis , presented in [5.14], is also defined by coordinates in the natural basis and the eigenvectors that make up the new decoupling basis. We clarify that in equation [5.14]:

{ } d n d d d x x x B ,..., , 2 1 = , characteristic matrix [ ] d s L of morphism λ is expressed: [ ] ( ) ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ Λ Λ Λ = = n d d s B mat L 0 0 0 0 0 0 , 2 1 λ [5.11] Matrix [ ]
2 n α π = ; c k = cos (kα); s k = sin (kα); if n is even, K = (n -2) α/2; if n is odd, K = (n -1) α/2 and we will omit the last line of the transformation matrix.

The analysis that has just been carried out aims to emphasize the specificity of [ ]

nn T or [ ] 1 - nn T
, whose synthesis does not come from the aim to simplify calculation but directly from the analysis of the inductance matrix.

[ ] [ ]

] [ ] [ 1 nn n s nn d s T L T L - = [5.12] ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - n n n n nn d n d d x x x T x x x 2 1 1 2 1 ] [ [5.13] 1 2 3 3 2 1 1 2 3 3 2 1 2 4 6 6 4 2 2 4 6 6 4 2 3 6 9 9 6 3 3 6 9 9 6 3 1 2 3 3 2 2 3 3 2 1 0 1 0 1 0 2 [ ] 1 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 nn K K K K K K K K K K K K c c c c c c s s s s s s c c c c c c s s s s s s c c c c c c s s s s s s T n c c c c c c s s s s s s - ⎛ ⎜ - - - ⎜ ⎜ ⎜ - - - ⎜ ⎜ ⎜ - - - ⎜ ⎜ = ⎜ ⎜ ⎜ - - - ⎜ - - - - ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎠ [5.14]
Besides relationships [5.11] to [5.14], some properties of morphism λ are very useful. Given the circularity of inductance matrix [ ] n s L , the eigenvalues k Λ of λ are generally "double", i.e. we can associate two independent eigenvectors with them. Only one of the eigenvalues is "single" (i.e. associated with a single eigenvector) in the case of an odd number of phases n; two are "single" in the case of an even number of phases n.

Thus, the latter properties allow us to justify the use of star couplings for machines with an odd number of phases and multi-star couplings for machines with an even number of phases when considering control. In fact, these couplings allow us to ensure that the currents associated with the single eigenvalues (also called homopolar currents) are rigorously kept at 0. Finally, it is the presence of a double eigenvalue, better known as cyclical inductance, that has allowed us to introduce the notion of the equivalent two-phase machine, which well accepted in the three-phase case.

Equations in the decoupling basis and independent energy fluxes

In section 5.2.1.2 we have shown that it is possible to determine an orthonormal decoupling basis in which inductance matrix [ ] d s L is diagonal. Furthermore, we have assumed that the eigenvalues of this matrix were double, except for one single in the case odd n and two singles in the case even n.

The initial vectorial space can therefore be broken down into a sum of vectorial subspaces of one or two dimensions, each combined with an eigenvalue k Λ of [ ] d s L , double or single. These subspaces, also referred to as eigenspaces of morphism λ are orthogonal because all the eigenvectors of morphism λ are orthogonal. A vector g belonging to vectorial space E n can then be broken down into a unique sum of vectors of one or two dimensions, each belonging to a vectorial Eigen subspace E se of one or two dimensions. The E se vectors are obtained by orthogonal projection of vector g on each of subspaces E se .

Applied to the voltage equation [5.4], this break down leads to the following equation, in which N is the number of subspaces, E se :

1 1 / . . d d N N d d d m k sm m m m m B di v v R i e dt = = ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = = +Λ + ∑ ∑ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ [5.15]
If we now look for the electromagnetic torque using a power assessment, by recalling that Eigen subspaces E se are orthogonal, we can find:

( ) ( ) 1 1 1 2 1 / . . . . . . d N N N d d d d k k k k m m m d N d dd d m s m k m m m m B p v i v i v i di R i i ei dt = = = = ⎛ ⎞⎛ ⎞ = = = ∑ ∑ ∑ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = + Λ + ∑ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ [5.16]
In [5.16] a term that associates with Joule losses appears, then another term with the storage of magnetic energy and finally a term with the electromechanical conversion. By expanding this last term according to equation [5.17], it appears that the total mechanical energy of the machine is the sum of N mechanical energies, each associated with a subspace, E se . It is therefore possible to consider that the torque supplied by the real machine is the sum of torques supplied by N fictitious machines. Each of these machines is characterized by its resistance, R s , its inductance, k Λ , and its emf vector, d m e . According to the dimension of the eigenspace with which it is associated, the fictitious machine will be monophase or two-phase.These machines all run at the same speed. They can therefore be considered mechanically coupled. . .

d d N N N d d d m m m m m m m m e i c c i ε = = = = = = ∑ ∑ ∑ Ω [5.17]
Figure 5.2 illustrates the equivalence between an n-phase machine and a set of fictitious monophase and rtwo-phase machines.

It has to be noted that if we can find an infinity of nn T type transformations, there one and only one break down into fictitious machines (mathematically, breaking down a vector on the eigenspaces of morphism λ ). This uniqueness is a key point of the vectorial approach with respect to matricial approaches using transformations of infinite number. Naturally this break down can also be applied to the three-phase case: the three-phase machine is a priori equivalent to two machines -one monophase and the other two-phase. We will see why only the two-phase machine is kept when the machine is star-coupled.

Fundamental harmonic properties of fictitious machines

The concept of fictitious machines allows us to transform the real machine, whose phases are magnetically coupled, into a sum of magnetically decoupled oneand two-phase fictitious machines. In this section, we show that the variables Real n-phase machine ( )

n v v … , 1 Ω , c
Decoupling transformation of dimension n ( )

d d v v 2 1 , ( ) n v v … , 1 Ω , 1 d c Ω , 2 d c Ω , d N c ( ) d d v v 4 3 , d n v
Fictitious 1-and 2-phase machines associated with a fictitious machine consist of a harmonic group of the real machine's variables. This aspect is fundamental when working out the control of a machine or during its design phase, particularly in the case of fault-tolerant machines. Each fictitious machine possesses its own features that need to be known for proper design and control.

Characteristic harmonics groups

Let g be a vector variable associated with the real machine. We assume that each of the vector variable coordinates has a period p / 2π , i.e. it can be broken down into a Fourier series. We define θ as the mechanical angle between the rotor and the stator.

∑ = = n k n k k x g g 1 [5.18] with: ( ) max 1 2 sin 1 k h h g g h p k n π θ ∞ = ⎛ ⎞ ⎛ ⎞ = -- ∑ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ [5.19]
We calculate the coordinates of the vector variables associated with fictitious machines by projecting the vector variable on the different subspaces E se associated with fictitious machines. We will find the details of the calculations in [KES 03]. We only discuss the basic idea here.

Assuming that a subspace associated with fictitious machine number m

( { } 1, m N ∈
) is generated by vectors { } ( )

2 1 2 1 2 1 2 2 2 1 2 1 2 2 . . d d d d d d d d d m m m m m m m m m g g x x g x x g x g x - - - - - = + = + [5.20]
In [KES 03] and [SEM 04a] it is shown that the coordinates of the vector variable written in the basis generating a subspace (or fictitious machine) consist of a harmonics group of the variables of the real machine. With the presence of harmonics families, we find a result expressed by Klingshirn [KLI 83] within the framework of the supply of a multiphase asynchronous machine with inverter in full square wave mode in steady state. σ allows us to take into account whether a vector is homopolar ( )

0 σ = , rotating in clockwise (direct) direction ( ) 1 σ = + or anticlockwise (retrograde) direction ( ) 1 σ = -.
( )

m nl h with hp g n g l h d m σ θ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∑ ∞ = - 0 max 1 2 sin 2 [5.21] ( ) m nl h with hp g n g l h d m σ θ σ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ∑ ∞ =0 max 2 cos 2 [5.22]
Table 5.1 summarizes the harmonics groups associated with three-phase, fivephase and seven-phase machines. [SEM 03a] contains the case of the six-phase double-star machine, referred to as a double-star three-phase machine. The fictitious machine is the principal machine associated with the first harmonic. That machine associated with the second harmonic is secondary. The machine associated with the third harmonic, is the tertiary machine and that associated with harmonic n is homopolar. In the case of a monophase fictitious machine ( )

0 σ = ,

.1. Summary table of harmonic groups associated with three-phase, five-phase and seven-phase machines

For instance, harmonics 1 and 2 associated with a five-phase machine generate rotating vectors in the direct direction, 3 and 4 in the retrograde direction, 6 and 7 in the direct direction, etc.

We again find the well-known results for the three-phase machine. The odd harmonics of electromotive force of rank 5 (retrograde) and rank 7 (direct), by interaction with the first current harmonic, induce (direct) torque pulsations of rank 6. Similarly, the harmonics of rank 11 (retrograde) and 13 (direct) induce torque pulsations of rank 12.

For a five-phase machine, the harmonics of rank 9 (retrograde) and 11 (direct), by interaction with the first current harmonics, induce torque pulsations of rank 10. The harmonics of rank 7 (direct) and 13 (retrograde) interact with the harmonics of rank 3 (retrograde) to generate torque pulsations of rank 10.

It must be noticed that in general the harmonics of even ranks have a value of 0 except in the case of pole asymmetry. Therefore their case has not been developed.

Relationship between emf harmonics and torque generated by a fictitious machine

If we apply the "real variables to decoupled variables" transformation to the emfs of a machine possessing n phases, we can formulate different remarks:

-If the emfs of the real machine are sinusoidal; only the principal fictitious machine has an emf. In this case, according to equation [5.17], only the fictitious machine can generate a torque.

-If the number of phases of the machine is odd, the emf harmonics of the real machine that has a rank multiple of the number of phases (h=an, a integer) cannot generate a constant torque. In fact, these harmonics are assigned to the monophase fictitious machine referred to as being homopolar. This remark partially justifies the quasi-systematic use of star coupling between the phases, which ensures 0 current in the homopolar machine.

-If the emfs of the real machine have odd harmonics whose rank h is less than or equal to the number of phases n, the fictitious machines have a sinusoidal emf (or some are 0 if h<n).

-If the emfs of the real machine have more harmonics than the number of phases, there is at least one fictitious machine possessing non-sinusoidal emf.

We will notice that these remarks increase in importance when designing the control of the machine. 5.2.1.4.3. Inductances and electrical time constants of fictitious machines: the impact of harmonics If we can restrict ourselves to estimating (with suitable precision) the value of inductances of a three-phase machine with smooth poles, that is not saturated and is star coupled by considering that the magnetomotive forces are sinusoidal, it is imperative to take account of the magnetomotive force harmonics when estimating the inductances of a multiphase machine. In the opposite case, there will be considerable error in the estimation of electrical time constants associated with fictitious machines, the parameters necessary for proper design of the machine's supply system and in tuning the associated servo-systems. Thus, we can conclude that a multiphase machine with sinusoidal magnetomotive forces possessing few leaks is not a "good" machine in the way we mean when discussing a three-phase star-coupled machine. In fact, some fictitious machines have very weak inductances (because they are only equal to the leakages inductances) and will require the machine to be supplied with inverters with a very high chopping frequency. If this condition is not fulfilled, we observe large amplitude interference currents induced by the pulse width modulation (PWM).

Similarly, modeling a multiphase machine by assuming the systematic hypothesis that magnetomotive forces are sinusoidal can lead us to make quite significant errors in the evaluation of the inductances associated with fictitious machines [LOC 06, SEM 03b]. This error leads us to considerably overdimension the supply system and to mistune the correctors of associated currents.

If we take the magnetomotive force harmonics into account, we need to add the contribution due to the harmonics to the fundamental inductance expressed by equation [5.23]. The magnetic system being considered linear, we can apply a superposition theorem. The inductances associated with the real machine are then expressed by the following equation in which q are the magnetomotive force harmonic ranks that are kept:

( ) cos q jk jk leaks q L L q l δ = + ∑ [5.24]
In the particular case of concentrated coils with diametral pitch (located in only two notches and separated by a pole pitch), the harmonic inductance q L is expressed by:

odd q e DL N q L s q , 2 1 2 0 2 π μ = [5.25]
If we consider the magnetomotive force harmonics, the expression of inductance associated with m fictitious machines is given by:

∑ + = Λ h leakage h m l L n 2 [5.26]
According to the fictitious machine, the harmonic ranks h to consider keeping are recorded in Table 5.1. In equation [5.26], we are reminded that that h nl m σ = +

with l being the integer and m the number of fictitious machines related.

According to equation [5.26], we conclude that the inductances associated with fictitious machines other than the principal fictitious machine, are not only equal to the leakage inductance but also to a particular group of harmonic inductances. Thus, the existence of magnetomotive force harmonics allows us to increase the electrical time constant of a fictitious machine and allows the use of a smaller chopping frequency when the machine is supplied by an inverter.

It is this multiharmonics approach that fundamentally distinguishes the results presented in this chapter from those presented in Chapter IX of [WHI 59], for which only a first harmonic approach was used to model multiphase machines. This first harmonic approach had as a corollary in that only the two-phase machine associated with the first harmonic was susceptible to torque generation and that the other twophase and monophase machines were reduced to circuits characterized by the leakage inductance of the machine and the stator resistance.

We remember with the proposed approach several fictitious machines can contribute to the generation of torque (average but also pulsating), but there are design constraints so that a multiphase machine can be supplied with inverters whose PWM frequency is not too high. Here, we must take into account the harmonics whose impact is no longer a second-order phenomenon with respect to supply by an inverter [SCU 09]. 5.2.1.4. Examples

Three-phase machine

In this section, we consider a classic example of a three-phase synchronous machine with sinusoidal emf. It is equipped with distributed stator coils that generate a magnetomotive force with sinusoidal spatial distribution.

We associate a vectorial three-dimensional space with the machine that is equipped with an orthonormal natural basis

{ } n n n n x x x B 3 2 1 , , =
. The vectorial voltage equation of this machine is given by equation [5.4], where the inductance matrix, characteristic of linear application λ , and the emf vectors are detailed by:

2 2 1 c o s c o s 3 3 2 2 cos 1 cos 3 3 2 2 cos cos 1 3 3 leaks n leaks s leaks l L l L L L l L π π π π π π ⎛ ⎞ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ = + ⎜ ⎟ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎝ ⎠ [5.27]
and:

( )

max 1 2 3 2 4 sin sin sin 3 3 n n n e p x p x p x π π ε θ θ θ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = Ω + - + - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ [5.28]
If we apply the Concordia transformation expressed by [5.29], which is determined by equation [5.14] in the case where n = 3, the real machine of referenced variables (1,2,3) is broken down into two fictitious machines: a twophase machine, referred to as the principal machine, referenced (α,β); and a homopolar monophase machine, referenced z.

[ ] [ ] { } 1 1 1 33 33 2 3 2 2 1 cos cos 3 3 2 2 2 0 sin sin , , , , 3 3 3 1 1 1 2 2 2 z g g T g T g g v i e g g α β π π π π - - ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = - = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ [5.29]
The expression of the vectors of the new basis is given as a function of the vectors of the natural basis by: ( )

1 2 3 2 3 1 2 3 2 2 2 cos cos 3 3 3 2 2 2 sin sin 3 3 3 1 3 z x x x x x x x x x x x α β π π π π ⎧ ⎛ ⎞ = + + ⎪ ⎜ ⎟ ⎝ ⎠ ⎪ ⎪ ⎛ ⎞ ⎪ = - ⎨ ⎜ ⎟ ⎝ ⎠ ⎪ ⎪ ⎪ = + + ⎪ ⎩ [5.30]
The vectorial voltage equations associated with the fictitious machines are given by:

/ / n n s B z z sz z z B di v Ri e dt di v R i e dt αβ αβ αβ αβ αβ ⎧ ⎛ ⎞ = +Λ + ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎨ ⎛ ⎞ ⎪ = +Λ + ⎜ ⎟ ⎪ ⎜ ⎟ ⎝ ⎠ ⎪ ⎩ [5.31]
Finally, their characteristic variables are given by:

3 0 0 2 0 0 0 0 leaks d s z leaks L l L l αβ αβ Λ Λ Λ ⎡ ⎤ = + ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ = ⎢ ⎥ ⎣ ⎦ [5.32]
and:

( ) ( ) ( ) max 3 sin cos 2 0 z z e p x p x e x αβ α β ε θ θ ⎧ = Ω - ⎪ ⎨ ⎪ = ⎩ [5.33]
Analysis of the characteristic variables of fictitious machines gives us information about their specificities. Only the principal fictitious machine can generate a torque, its emf being sinusoidal and non-zero. Its electrical time constant,

s leakage R l L / 2 3 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +
, allows us to determine the chopping frequency of the associated inverter. The homopolar fictitious machine does not possess an emf and thus cannot generate a torque. Its supply can only generate losses. If it possessed a non-zero emf linked to the presence of harmonics of rank 3, it could admittedly generate a torque of non-zero mean value [GRE 94], but due to its monophase feature it would have a significant pulsating component that would have to be compensated for by the two-phase machine. Finally, its electrical time constant, which would have a small value, / leaks s l R , that would demand supply from an inverter with a very high chopping frequency.

These conclusions bring an additional argument for a quasi-systematic coupling of three-phase machines. Such coupling will ensure the non-supply of the homopolar machine. The main economical argument is that we use three legs instead of six.

Five-phase machine

In this section, we illustrate the notions previously introduced, by means of a five-phase synchronous machine with surface-mounted permanent magnets at the surface and concentrated coils with a diametral pitch. 5.2 gives the relative harmonic content of the emf of the fivephase machine considered.

1 2 2 1 1 1 2 2 2 1 1 2 2 2 1 1 1 2 2 1 n s L M M M M M L M M M L M M L M M M M M L M M M M M L ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ = ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ [5.34] ( ) max 1 1 2 . sin 1 n h n k k h e hp k x n π ε θ ∞ = = ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = Ω -- ∑ ⎜ ∑ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ [5.35] Harmonics h = 1 h = 3 h = 5 h = 7 h = 9
Relative rate 100% 28.5% 12.4% 5.1% 1.7% By applying the Concordia transformation of the fifth dimension to the fivephase machine, we show that the real machine is equivalent to the association of three fictitious machines: a two-phase fictitious machine referred to as the "principal", a two-phase fictitious machine referred to as "secondary" and a monophase fictitious machine referred to as "homopolar". Each of these fictitious machines is characterized by a harmonic group of variables associated with the real machine. We will find these groups with the help of Table 5.1. We are initially interested in the partition property of emf harmonics among fictitious machines. Figure 5.4 shows a recording of one of the emfs of the real machine and emfs of the associated fictitious machines at 1,500 rpm. We check that the emf of fictitious machines are mainly composed of: -fundamental and harmonic of rank 9 for the principal machine, -harmonics of ranks 3 and 7 for the secondary machine; and -harmonic of rank 5 for the homopolar machine.

We then conclude that each of these machines can supply a torque. Obviously, for a given amplitude of current in the fictitious machines, the torque generated is greater for the principal machine than for the secondary and homopolar machine.

We are now interested in the inductances associated with the fictitious machines.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p p d s s s z L Λ ⎛ ⎞ ⎜ ⎟ Λ ⎜ ⎟ ⎜ ⎟ ⎡ ⎤ = Λ ⎣ ⎦ ⎜ ⎟ Λ ⎜ ⎟ ⎜ ⎟ Λ ⎝ ⎠ [5.
Λ = ⎧ ⎪ Λ = ⎨ ⎪ Λ = ⎩ [5.38]
As shown in section 5.1.4.3, the hypothesis that of only considers the fundamental component of the magnetomotive force leads to large errors in the evaluation of inductances associated with the secondary and homopolar fictitious machines. When dimensioning the chopping frequency of the inverter, these errors lead to unnecessary oversizing of the machine's supply system, in particular the apparent power of the inverter. We will, however, notice that even by considering the harmonics, the ratio between the inductance of the principal machine and that of the secondary machine, / 4.3

p s Λ Λ =
, is still high in the case of this five-phase machine.

From another point of view, if we imagine that the machine had been designed to have low magnetic leakages and a single harmonic of magnetomotive force (and thus emf), the determination of the frequency of the PWM from the single time constant of the principal machine would lead to the observation of parasitic currents of very high amplitude in practice.

Finally, as long as the inductance linked to the homopolar machine is the smallest, the star coupling will ensure that there is no current, at least in the homopolar machine. It only remains to control the currents in the secondary machine.

The inverter seen from the machine

For the model of an n-leg inverter, this section will take up, the main results of Chapter 8 [KES 09a].

Let us remind ourselves that the function of the inverter is to apply the voltages calculated by the control to the electrical machine. More precisely, knowing that we are positioned within the framework of a PWM control, we will try to apply voltages whose mean values correspond to those calculated by the control. The real voltages will in fact consist of a "rolling mean" component and a "noise" component that will be filtered by the inductive circuit of the machine. We will then obtain the mean currents desired, which cause the mean electromagnetic torque developed by the machine.

According to the control strategies close to the inverter, noise voltages can be significantly different, even if the mean value of the voltage is the same. These noise voltages will induce parasitic currents whose amplitudes will be added to the "effective" mean current. When the amplitude of these noise currents becomes important, these currents become dimensioning for the choice of inverter transistor rating. For machines with more than three phases, it has been seen in section 5.2.1.4 that, contrary to the star-coupled three-phase machine without a neutral point, there are several inductive circuits (one per fictitious machine). The analysis of fictitious voltages imposed by the inverter in each fictitious machine, both for its mean component and its "noise" component, is therefore fundamental in controlling the mean currents and the parasitic currents of the real machine.

At this level, two cases have to be distinguished, which have already been seen for the three-phase machine. In other words, depending on the way the phases of the machine are coupled, a fictitious machine can be dependent or not dependent on the inverter. For this reason we will show the case of the three-leg inverter using an approach that is easy to generalize to the case of a n-leg inverter supplying a n-phase machine.

Three-leg inverter seen from the star-coupled three-phase machine

Here we consider a three-phase machine assumed to be star-coupled and to fulfill the general hypotheses expressed at the beginning of this chapter, and particularly the hypothesis in section 2.1.4.1 (magnetomotive and sinusoidal emfs).

In order to emphasize the problem of parasitic currents, we will assume that the neutral N' of the machine is physically linked to neutral point N of the inverter by an impedance of capacitor type. If we consider the capacitance to be 0, we will again find the ideal case and point N will not exist in reality. The capacitance will allow us to roughly model the impedance of the machine's earthing circuits and the inverter. We use this type of modeling when we are interested, for instance, in parasitic currents going through the bearings of a machine [DAH 08] where one of the origins is linked to the homopolar components of voltages imposed by the inverter [LEE 01].

Figure 5.5 shows the classical supply topology of the three-phase machine: starcoupled phases and supplied with a three-leg inverter.

k 32 k C 1 v 2 v 3 v N ' N AN v 1 i 2 i 3 i A B C
The eight possible combinations of switches k ij allow us to generate the eight inverter voltage vectors,

1 2 3 k ond n n n AN BN CN v v x v x v x = + + : 0 1 2 3 1 1 2 3 2 1 2 3 3 1 2 3 4 1 2 3 5 1 2 3 6 1 2 3 7 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 n n n n n n N N n n n n n n N N n n n n n n N N n n n n N N E E E E E E v x x x v x x x E E E E E E v x x x v x x x E E E E E E v x x x v x x x E E E E E v x x x v x x =- - - =- - + =- + + =- + - =+ - - =+ - + = + + - = + + 3 2 n n E x + [5.39]
Any one of these eight vectors can be broken down by applying the Concordia transformation in a unique sum of a voltage vector belonging to a principal vectorial subspace and another homopolar vectorial subspace:

( ) ( ) z Nk Nk Nk N N Nz z v v v v x v x v x αβ α α β β = + = + + [5.40]
The detail of these vectors is given by: . 6 2

N N N N N N N N E E v x x v x x E E E v x x v x x E E E v x x v x x E E v x x v x x
z z N z N z z z N z N z z z N z N z z z N z N z E E v x v x E E v x v x E E v x v x E E v x v x = - = - = - = + = - = + = + = + [5.42]
Voltages v αβ and z v applied to the principal and secondary fictitious machines are respectively expressed by:

s N di v Ri v dt αβ αβ αβ αβ αβ = +Λ = [5.43] ' 3 z N N z sz z N z di V v Ri v dt = +Λ = - [5.44]
The capacitor being run through by a current equal to the sum of currents in the phases of the machine, we have:

' 3 z N N i V d t C = ∫ [5.45]
An ideal star coupling (zero capacitance, C) implies a current 1 2 3 0 3

z i i i i + + = =
and hence a voltage applied to the homopolar machine of 0

z v = ,

whichever voltage

Nz v is imposed by the inverter. The homopolar machine is therefore never supplied when the real machine is star coupled and supplied by a three-leg inverter without a neutral point.

When capacitance C is not null, a non-zero current z i circulates if voltage Nz v is not zero. This is always the case when we consider the eight vectors z Nk v that can be imposed by the three-leg inverter. We will understand why some inverter control strategies aim to choose those voltage vectors presenting the weakest homopolar components (excluding inverter voltage vectors 0 REMARK 5.1.-these results still remain true if the emfs and magnetomotive forces have harmonics of rank 3 because this modifies the homopolar components of emf as well as the homopolar inductance. Only the homopolar current will be modified if the capacitance is non-zero. REMARK 5.2.-if we consider a triangle coupling, the impact of an emf harmonic of rank 3 is fundamental. In fact, in the case of triangle coupling it is the homopolar voltage z v of the machine that we make 0. Current z i is only 0 when the emf z e is also 0.

By considering the ideal case of the star-coupling with no capacitance and the homopolar machine not being supplied, the real machine supplied by a real inverter is strictly equivalent to the principal fictitious machine supplied by a fictitious inverter. Instead of studying the supply of the real machine using one of the eight real In Figure 5.6 the grey circle of radius R 1 gives the limit of mean voltage vector ( )

t v αβ
that can be imposed if we generate three mean sinusoidal voltages of reference at the level of the inverter (voltages v AN , v BN and v CN in Figure 5.5). In this

α x β x αβ 4 v αβ 5 v αβ 1 v αβ 3 v αβ 6 v αβ 2 v 1 R 2 R 2 1 2 1,15 3 R R = =1.15
case, by control we ensure that the mean value on a PWM period of the homopolar voltage of the inverter is 0:

( )

1 0 MLI Nz Nz T MLI v t v dt T = = ∫ [5.46]
We then have a mean value of:

( ) ' 0 N N v t = .
The black concentric circle, of radius R 2 > R 1 , indicates the limit of the mean voltage vector if we inject a homopolar component (often a harmonic of rank 3) in the reference voltages of the inverter. We are thus reminded that this overmodulation allows us to use at best, from the point of view of the DC voltage of the bus, the degree of freedom left by the star coupling (unsupplied homopolar with a harmonic multiple of 3 present in the inverter voltages).

In summary, the star coupling without a neutral point allows us to ensure that a fictitious machine will have 0 current and frees a degree of freedom for control, allowing us to better use the DC voltage of the bus (+15.5% of excursion).

W The reader who would like to deepen his of her knowledge of the elements relative to the vectorial modeling of inverters is invited to read Chapter 8 "Multiphase voltage source inverters" of [MON 11].

Generalization to n-leg inverters: fictitious two-phase inverters. Example of machine-five-leg inverter association

The study of machines with more than three phases, supplied with inverters with a number of legs equal to the number of phases follows the same approach as that used in section 5.2.2.1.

The star coupling without a neutral point always allows the the current in the homopolar fictitious machine to be 0 and frees a degree of freedom for control. In Table 5.3 [LEV 08], we will find that the benefit introduced by the homopolar weakens as the number of phases increases. We could conclude that an increase in the number of phases is unfavorable for good use of the DC bus. We do, however, need to note that the calculations leading to Table 5.3 assume the machine is supplied in sinusoidal regime with an injection of homopolar voltage. In the "fictitious machine" approach, this means that we assume that we are only supplying a single fictitious machine. If this approach is enough in three-phase it is no longer the case if the number of phases n is greater than three because the optimal use of the DC bus depends on all the fictitious machines being supplied. This in [RYU 05], we find a modulation rate of 1.23 in the case of a five-phase machine that has been properly designed and whose two fictitious machines are supplied.

Numberof phases n

Level of harmonics injection n

Index of maximum modulation Mi

Percentage of increase in the fundamental component

3

- If we omit this aspect of homopolar injection, which only intervenes in the cases where we work at the limits of the voltage possibilities of the inverter, supplying a multiphase machine has as a sole difference with respect to the three-phase case. It has to consider several projections of the 2 n characteristic vectors of the inverter (instead of 3

2 ) in several planes instead of a single plane. Thus, for a five-leg inverter we will find two planes with 30 non-zero vectors (2 5 -2), see Figure 5.7. At the level of the inverter control, the different steps for the determination of duty cycles will be the following: a) obtaining the voltage vectors for each fictitious machine (in three-phase we has only a single vector) as a function of torques required for each fictitious machine; b) vectorial summation of the different voltage vectors of these machines; c) possible addition of a homopolar voltage vector in order to optimize the use of the bus voltage; d) calculation of the conduction durations by assuming we are in linear regime (non-saturation of the inverter) according to the expressions given in [KES 09a]; e) taking into account, if necessary, the saturation of the inverter by generating adapted references.

The analysis of the different steps emphasizes the fact that we apply a superposition theorem as long as there is no saturation of the inverter. This means that everything is as if each fictitious machine was supplied with a fictitious inverter. Thus, as long as we are not interested in the nonlinear aspects of the inverter control, we can use very classic intersective control techniques that implicitly fulfill step d, knowing that step c is less important as the number of phases increases.

Torque control of multiphase machines

Efficient position and speed servo-controls require a torque servo-control, i.e. stator currents. If several sets of n stator currents can generate the same torque, we will try and use the degrees of freedom given by the redundancy of phases to fulfill some criteria. Among the most frequently used is the criterion that minimizes Joule losses for a given torque. This will be used in the following sections.

Servo-control of currents in the natural basis

Statement of the method

We speak of servo-control of currents in the natural basis when the servocontrolled currents are the real currents measured in the machine. Figure 5.8 gives the synoptic diagram of the torque control of a multiphase machine in natural basis. We will recognize :

-c * : the reference torque;

-k : the criterion used to elaborate the reference currents; The first operation consists of elaborating the current references. Since we need to elaborate a vector (reference current vector) from a scalar (reference torque), it is necessary to introduce a criterion, and therefore a vectorial. Equation [5.47] clarifies the generation of current references from a reference torque:

* * i k c = [5.47]
We can, for instance, choose to work with a maximum torque for given statoric Joule losses. Given that these Joule losses are proportional to the square of the modulus of the current vector, this criterion is easily translated from equation [5.9] by the fact that the current vector has to be colinear to the emf vector. For a given current vector modulus, we then obtain the maximum torque. It therefore becomes: -If the emfs only have harmonics of rank that are smaller than the number of phases n of the machine, term 2 ε is constant and the currents possess the same harmonic ranks as the emfs. If we take the classic case of a machine with sinusoidal emf, we again find sinusoidal currents in phase with the emf.

-If the emfs possess a number of harmonics greater than the number of phases, n, term 2 ε is no longer constant and the currents possess a number of harmonics greater than the emfs. Finally, if it's possible that term 2 ε cancels out for a particular position θ of the rotor, there is no combination of currents that allows us to maintain a constant torque.

None the cases shown previously have constant curents in steady-state regime and their servo-control usually requires the use of correctors with large bandwidth (often with hysteresis). This is prohibited in the domain of high power because of the rich and poorly controlled spectral content. Let us note, however, that if the harmonic content of current is finite, the use of multifrequential resonating controllers can allow the perfect tracking of references in steady state [LIM 09].

Example: five-phase machine with trapezoidal emf

Here we show the results of controlling currents in the natural basis of the fivephase machine that were presented in section 5.2.1.4.1 for a machine that is star coupled and supplied with a five-leg inverter [KES 09b]. Figure 5.9 is a synoptic diagram of the experimental system put in place for these trials.

Torque reference *

c comes from the proportional integral (PI) controller ensuring the servo-control of speed Ω . This speed is estimated by filtered numerical differentiation from the position measured by a synchro-resolver. The "optimal" current references are calculated from equation [5.49], the emf (i.e. vector ε ) being estimated in real time from mechanical angle θ .

The set is controlled by real-time set DSpace ® 1005. The mechanical load is generated by a powder brake. The effective torque is measured by a torquemeter placed between the motor and the load. The currents, which are servo-controlled by hysteris controllers, allow us to obtain a constant torque with minimal Joule losses, as shown in Figure 5.11. The amplitude of torque oscillations results from the bandwidth of the hysteresis corrector, which is itself linked to the maximum authorized chopping frequency.

Servo-control of currents in a decoupling basis

Statement of the method

If we apply the decoupling transformation in section 5.2.1 to the characteristic equations of an n-phase machine, we are no longer studying the control of the real machine but a sum of N fictitious one-or two-phase machines. Figure 5.12 gives a synoptic diagram of the torque control of the machine in a decoupling basis. The variables carry indices d to indicate that it is about a decoupled variable and 1 to N to indicate the number of the fictitious machine of interest.

The torques reference distribution can be done in several ways according to the field of application. Either we use the principal machine (in which the fundamental component of the emf is projected) in normal running and the other fictitious machines are only used transiently in "transient overtorque" mode; or we make demands on all the fictitious machines. In this last mode, we can again take the case of control with minimum global Joule losses. By projecting the reference current vector expressed by [5.49] in each eigenspace associated with a fictitious machine, we obtain the reference current for each fictitious machine: According to the harmonic content of the real machine, the reference torque imposed on each of the fictitious machines can vary, although that of the real machine is constant. In the latter case, other strategies will be devised. given torque leads us to impose constant torque references in each fictitious machine if the torque reference imposed to the real machine is constant. We then end up controlling each two-phase fictitious machine as we would control the equivalent two-phase machine in the case of a star-coupled three-phase machine (where control is referred to as in the Park basis).

) ( ) ( ) _ m a x _ _ _ sin cos 2 m m m m m m n e h px h px αβ α β ε θ σ θ = Ω - [5.52]
Here, h m corresponds to the emf harmonics of the real machine, which is projected in fictitious machine number m. This harmonic, assumed to be unique, generally corresponds to the first odd harmonics of each machine (see Table 5.1).

We are reminded that σ allows us to take into account whether the vector rotates in the direct ( )

1 σ = + or the retrograde direction ( ) 1 σ = -.
If in each fictitious machine we have a new basis change, obtained by rotation of the basis ( ) αβ as expressed by:

( ) ( ) ( ) ( ) _ _ _ _ cos sin sin cos d m m m m q m m m m x x h p h p x x h p h p α β θ σ θ σ θ θ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ [5.53]
We obtain emf, referred to as axes (dq):

( )

_ m a x _ _ _ 0. . 2 dq i i d i q i n e x x ε σ = Ω - [5.54]
Insofar as some harmonics generate vectors rotating in the direct direction and others in the retrograde direction, the rotation can be in one direction or in the other.

The emfs of axes (dq), which are constant in steady-state regime, lead to "optimal" current references such as * _ 0

d i i = and * * _ . q i d i kc =
. These justify the use of current PI-type controllers and a control strategy in PWM with constant frequency of the inverter. The parameters of current controllers can be determined using classic methods (compensation of dominant compensation pole method, symmetrical optimum method, etc.) insofar as the fictitious machines are modeled by a first-order circuit made of the resistance of the stator phase, the inductance and the emf of the fictitious machine considered.

Figure 5.13 shows the synoptic diagram of current control of an n-phase machine in an extensive Park basis. If we wish to obtain better dynamic performances, we add the emf compensations (constant in a steady-state regime given the hypotheses taken in this section) to the PI controllers ensuring the control of currents.

Case of machines whose fictitious machines are not sinusoidal emf

In the case of non-sinusoidal fictitious machines, there are two alternatives.

The first imposes minimum global Joule losses for a given torque. In this case, the torque references of fictitious machines are not constant when the real machine is constant, and the choice of current correctors then arises. This approach, which has little advantage with respect to control in the natural basis detailed in section 5.3.1, will not be developed further.

The second alternative uses the approach developed in section 5.3.2.1. We then make a basis change in each fictitious machine by rotation of the basis associated with the being machine considered. We chose a rotation angle m h pθ so that the mean value of the emf of axis q is as high as possible. In this case, imposing constant currents dq in a steady-state regime does not allow us to generate a constant electromagnetic torque but does however possess a mean value for minimum Joule losses.

Besides generating torque undulations, the emf harmonics of ranks different from h m generate harmonic currents that are responsible for additional losses and torque ripples. The solution here lies compensating for these harmonics by injecting them in the reference voltages. The synoptic diagram in Figure 5.13 remains valid, although the terms corresponding to the compensation of emf are not only necessary for obtaining good dynamic performance, but for cancelling out harmonics current that are harmful in steady state. Here we give the results of current control in the extensive Park basis of the fivephase machine, described in section 5.2.1.4.1, that is star coupled and supplied with a five-leg inverter. Figure 5.14 shows a synoptic diagram of the current control system put in place for these trials.

We are reminded that the five-phase star-coupled machine is equivalent to the association of two two-phase fictitious machines: a principal machine, indexed p, in which we have harmonics 1 and 9 of the magnetomotive forces and emfs of the real machine; and a secondary machine, indexed s, in which we mainly find harmonics 3 and 7 of the same variables. The emf of fictitious machines indexed d or q in the new basis are obtained by rotation of variables of the principal machine of angle pθ and variables of the secondary machine of angle 3 pθ , see Figure 5.15. These emfs are not constant because they are not made of a single harmonic, the real machine possessing quasitrapezoidal emfs. In the emfs of fictitious machines we again find a constant value that is predominantly associated with a harmonic of rank 10, which comes from the rotation of harmonic 9 byθ in the principal machine and harmonic 7 by 3θ in the secondary machine.

For all the trials that have been carried out, the inverter is controlled in intersective PWM centered at a chopping frequency of 5 kHz. This frequency has The current references are constant and the currents are controlled via PI-type controllers. Inputs, allowing the emf to be compensated, are added to the output of current correctors. for the secondary machine.

The emfs are not compensated.

Figure 5.16 shows the current in one phase. We observe a strong harmonic 7 due to the corresponding component of the emf. The PI current controller as it is set is not capable of rejecting this perturbation at the speed considered. This current harmonic is responsible for additional losses as well as torque ripples. We realize these harmonic currents by checking in Figure 5.17 that, although the current references of the secondary machine are set to 0, strong harmonic currents (of rank 10) exist. . Finally, if we assume that the electromagnetic torque generated by the machine is proportional to current of axis q, the references of currents of axis q then fulfill the ratio: * * 0.3 sq p q i i = .

We keep the same current references for the principal machine ( * 0 The injection of current of harmonic 3 allows an increase in electromagnetic torque of almost 9% (30% of current of rank 3 and 30% of emf of rank 3) which leads to an increase in Joule losses of 9%. An increase of 9% in torque using only the principal machine would require an increase in Joule losses of almost 19% (1.09 times the current of the principal machine).

The use of fictitious machines to generate of the total torque thus allows optimal use of the multiphase machine. Control in the extensive Park basis allows the use of PI controllers associated with an inverter controlled in PWM. The modeling and torque control of multiphase machines in degraded supply mode is a broad topic that cannot be developed in detail in this chapter. However, the techniques shown previously are, with some modifications, applicable to torque control in degraded supply mode.

Modeling of a machine with a supply defect

A first method models a machine with n symmetrical phases whose m phases would no longer be supplied as a machine with n-m asymmetrical phases. By using a change of basis, such as the extensive Park transform, we control the system by controlling the constant current references [RYU 06]. This method has the major drawback of having to develop as many transforms (of models) as there are default cases. We can easily apply this method to machines with a small number of phases, but it is increasingly limited as the number of phases increase.

Another method models degradation and adds it to the initial model of the machine [CRE 10]. In this case the complete model is unique and therefore generally applicable to any number of phases. The use of particular controllers is necessary, however, to obtain satisfactory performance.

Torque control of a faulty machine

As for machines in normal functioning mode, control of the machine in degraded supply mode can be done in the natural basis and in a decoupling basis.

If we control the machine in the natural basis, we again have to calculate the current references allowing us to obtain a constant torque. Several authors propose calculations using optimization methods that cannot be used in real time (off-line method: [FU 94, PAR 07]). The method shown in section 5.3.1, however, can be applied in real time provided that we modify the emf vector according to the nonsupplied phases [KES 09b].

If we control the machine in a decoupling basis, two cases arise. The first consists of designing a new model of the machine for each default case. In this case, the current references remain constant [RYU 06]. If we keep the same model of the machine and add a degradation model to it, we will either have to calculate new current references and check that the controllers are able to track them, or keep the same current references as in normal mode (constant) but adapt the number of

Figure 5 . 1 .

 51 Figure 5.1. Symbolic representation of a synchronous two-pole n-phase machine

  equation[5.5]. This matrix, given the hypothesis of constructive regularity that leads to all phases being equivalent, is circulating, i.e. we get line n°2 of the matrix from line n°1 by simple shifting of one rank (L 21 = L 1n , L 22 = L 11 , L 23 = L 12 , etc.);

  characteristic matrix of morphism λ between the stator flux and stator current vectors in the natural basis. In this basis, eigenvectors are associated with the eigenvalues k Λ of morphism λ . We recall that eigenvalues k Λ are the solutions to characteristic equation [5.10], in which [I n ] is the identity matrix of dimension n: [ ] [ ] (

LT

  is obtained by classic basis change, as described by [5.12] with [ ] nn T the matrix of basis change. We are reminded that obtaining the coefficients of matrix [ ] nn T is easy since, by definition, each column of [ ] nn T consists of coordinates, in the natural basis, of one of the new eigenvectors that constitute the orthonormal decoupling basis. Things being as they are, in practice matrix [ ] nn Tis less useful than its inverse, fact, it is relationship [5.13] that allows us to obtain the coordinates of a vector in the new decoupled basis as a function of coordinates in the natural basis.Although obtaining [ ] is rarely, it is in our case, since the inverse matrix is identical to the transposed matrix: [ ] [ ]

Figure 5 . 2 .

 52 Figure 5.2. Equivalence between an n-phase machine and a set of fictitious oneand two-phase mechanically coupled machines

-

  of decoupling basis B d , defined by equation [5.13] and corresponding to the lines of matrix [ ] 1 nn T given by [5.14]. Equation [5.20] reminds s that projection 2 1 d mgof vector variable g is obtained by simple scalar products.

Equations [ 5 +

 5 .21] and[5.22] give the expression of the coordinates of 2 1 d m gin an Eigen subspace. In these equations, the harmonic ranks (assumed to be positive) that appear for m fictitious machines are h nl m and l ∈ .

  Equation [5.23] reminds us of the analytical expression of the inductance between phases j and k by considering sinusoidal magnetomotive forces [LOU 04d, WHI 59]: : permeability of air; -k s : coil coefficient; -N s : the number of turns of a coil (with p coils per phase); -D: the inner diameter of the stator; -L: the effective length of the stator; -e: the thickness of the magnetic air-gap (air + magnets); -separating phases j and k; and -l leaks : leakage inductances (generally considered zero-valued if k j ≠ ). In this case, the calculation of inductances associated with fictitious machines leads to the following conclusions: -the inductance associated with the principal fictitious machine (or cyclical inductance) is 1 2 leaks n L l Λ = + , with L being the self-inductance of a phase (j=k in equation [5.23]); Vectorial Modeling and Control of Multiphase Machines 177 -the inductances associated with the other fictitious machines are

Figure 5 . 3 .

 53 Figure 5.3. Views of the disassembled machine and of a cut

Figure 5 . 4 .

 54 Figure 5.4. The emf of the real machine (top) and of fictitious machines (bottom) at 1,500 rpm

  Equation [5.36] gives the inductance matrix d s L ⎡ ⎤ ⎣ ⎦ of the machine in the decoupling basis. p Λ , s Λ and z Λ represent the inductances associated with the principal, secondary and homopolar machines, respectively.

  36]If we only consider the fundamental component of the magnetomotive force, the inductances associated with the fictitious machines, whose calculations come from equation [5.23], are: we now take into account harmonics 1, 3 and 5 of the stator magnetomotive force, the inductances, stemming from the calculations from equations [5.24] and [5.25] become:

Figure 5 . 5 .

 55 Figure 5.5. Star-coupled three-phase machine supplied with a three-leg inverter

N v and 7 Nv

 7 ) from among the eight voltage vectors [LEE 01].

  vectors Nk v given by [5.39], we prefer to study the supply of the fictitious machine by one of the eight principal vectors, projection of the real vectors into the subspace associated with the principal machine (see equation[5.41]). These two-dimensional vectors are classically represented in a plane and form the centered hexagon given in Figure5.6. Vectors 0 v αβ and 7 v αβ appear in this figure only via the central point of the hexagon because they possess no projection in the principal plane.

Figure 5 . 6 .

 56 Figure 5.6. The two-dimensional representation of voltage vectors supplying the principal machine

Figure 5 . 7 .

 57 Figure 5.7. Graphical representation of a five-phase inverter in the principal (left) and secondary (right) subspaces

  -*i and i : the current references and the measured currents;-* v and v : the voltage references and the voltages applied to the machine; and finally -V bus : the voltage of the DC bus supplying the inverter.

Figure 5 . 8 .

 58 Figure 5.8. Synoptic diagram of the torque control of a multiphase machine in the natural basis

  49] leads to several remarks:-Vectors i and ε being colinear, the generated torque induces minimum global Joule losses. If there are effectively several strategies available, generating a given k Joule losses allows optimal use (by neglecting the ion losses) from a thermal point of view of the machine.

Figure 5 .

 5 Figure 5.10 shows the current references obtained for a control speed of 20 rad/s and a resistant torque of 2 N.m.

Figure 5 . 9 .

 59 Figure 5.9. Synoptic diagram of the experimental system of control of currents in the natural basis of a five-phase machine

Figure 5 . 11 .

 511 Figure 5.11. Experimental torque of a five-phase machine obtained by servo-control of currents in the natural basis

Figure 5 . 12 .d ε and 2 ε

 5122 Figure 5.12. Synoptic diagram of the torque control of a five-phase machine in a decoupling basis

  Each fictitious machine is assumed to possess an emf of the type:

(

Figure 5 .

 5 Figure 5.13. Synoptic diagram of the current control of a multiphase machine in the generalized Park basis

  2.3. Example: five-phase machine with trapezoidal emf

Figure 5 . 14 .

 514 Figure 5.14. Synoptic diagram of the current control of a five-phase machine in generalized Park basis

  the smallest time constant of the system corresponding to the electrical time constant of the secondary machine,

Figure 5 . 15 .

 515 Figure 5.15. The emfs of principal and secondary machines before (e αβ ) and after rotation (e dq ) compared to the emf of phase 1 (e 1 )

Figure 5 . 16 .Figure 5 . 17 .

 516517 Figure 5.16. Current in a phase and associated frequential spectrum when the emfs are not compensated for

  , the references associated with the secondary machine becoming * 0 sd i = and * 2.1 sq i A = . Figures 5.20 and 5.21 show the currents in one phase of the real machine and in the associated fictitious machines.

Figure 5 . 20 .

 520 Figure 5.20. Current in a phase and associated frequential spectrum when the two fictitious machines generate a torque

Figure 5 .

 5 Figure 5.21. The dq currents in the principal and secondary machines when emf harmonic 7 is compensated for

Figure 5 . 22 .

 522 Figure 5.22. The dq currents in the principal and secondary machines and speed response in the case of a servo-controlled system

Table 5 . 2 .

 52 Relative harmonics content of the emf of the five-phase machine considered

Comment [IRL1]:Ok ?

Current control with compensation of emfs

In the trials in Figures 5.18 and 5.19, the only compensation is for emf harmonic 7. The current in a phase is quasi-sinusoidal and the currents in the fictitious machines being properly controlled. The fictitious machines now being considered properly controlled, we decide to generate a torque with each of the machines.