# Abundance ratios in dwarf elliptical galaxies

Ş. Şen1,2, R. F. Peletier1, A. Boselli3, M. den Brok4, J. Falcón-Barroso5,6,
G. Hensler7, J. Janz8,9,16, E. Laurikainen9, T. Lisker10, J. J. Mentz1,11, S.
Paudel12, H. Salo9, A. Sybilska13, E. Toloba14, G. van de Ven15, A.
Vazdekis5,6 and C. Yesilyaprak2

1Kapteyn Astronomical Institute, University of Groningen, P. O. Box 800, 9700
AV Groningen, Netherlands
2Dept. of Astronomy and Astrophysics, Faculty of Science, Atatürk University,
25030, Erzurum, Turkey
3Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de
Marseille), UMR 7326, 13388 Marseille, France
4Institute for Astronomy, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich,
Switzerland
5Instituto de Astrofísica de Canarias,Calle Vía Láctea s/n, E-38200 La Laguna,
Tenerife, Spain
6Departamento de Astrofísica, Universidad de La Laguna, E-38205 La Laguna,
Tenerife, Spain
7Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17,
1180 Vienna, Austria
8Centre for Astrophysics and Supercomputing, Swinburne University, Hawthorn,
VIC 3122, Australia
9Astronomy Research Unit, University of Oulu, FI-90014, Finland
10Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg, Mönchhofstraße 12-14,69120 Heidelberg, Germany
11Center for Space Research, North-west University, Potchefstroom 2520, South
Africa
12Department of Astronomy & Center for Galaxy Evolution Research, Yonsei
University, Seoul 03722, Republic of Korea
13European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching
bei München, Germany
14University of the Pacific, Department of Physics, 3601 Pacific Avenue,
Stockton, CA 95211, USA
15Max-Planck-Institut für Astronomie, Königsstuhl 17, 69117 Heidelberg,
Germany
16Finnish Centre of Astronomy with ESO (FINCA), University of Turku,
Väisäläntie 20, 21500 Piikkiö, Finland E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We determine abundance ratios of 37 dwarf ellipticals (dEs) in the nearby
Virgo cluster. This sample is representative of the early-type population of
galaxies in the absolute magnitude range -19.0 < Mr < -16.0. We analyze their
absorption line-strength indices by means of index-index diagrams and scaling
relations and use the stellar population models to interpret them. We present
ages, metallicities and abundance ratios obtained from these dEs within an
aperture size of ${R}_{e}$/8. We calculate [Na/Fe] from NaD, [Ca/Fe] from
Ca4227 and [Mg/Fe] from Mgb. We find that [Na/Fe] is under-abundant with
respect to solar while [Mg/Fe] is around solar. This is exactly opposite to
what is found for giant ellipticals, but follows the trend with metallicity
found previously for the Fornax dwarf NGC 1396. We discuss possible formation
scenarios that can result in such elemental abundance patterns and we
speculate that dEs have disk-like SFH favouring them to originate from late-
type dwarfs or small spirals. Na-yields appear to be very metal-dependent, in
agreement with studies of giant ellipticals, probably due to the large
dependence on the neutron-excess in stars. We conclude that dEs have undergone
a considerable amount of chemical evolution, they are therefore not uniformly
old, but have extended SFH, similar to many of the Local Group galaxies.

###### keywords:

galaxies: dwarf elliptical - galaxies: evolution - galaxies: individual
(Virgo) - galaxies: abundances ratios - galaxies: stellar populations

††pubyear: 2017††pagerange: Abundance ratios in dwarf elliptical galaxies–5

##  1 Introduction

Early-type dwarf galaxies (dEs) play a key role in understanding galaxy
cluster evolution. dEs161616The term dE has traditionally been used to refer
to dwarf elliptical galaxies, whereas we loosely use the term here to include
dwarf ellipticals and dwarf lenticulars (dS0)., the low luminosity
(${M}_{\text{B}}$ > -18) and low surface brightness ($\mu_{\text{B}}$ > 22 mag
arcsec-2) population of the Early Type Galaxy (ETGs) class are found in high-
density environments and are very rare in isolation (Gavazzi et al., 2010;
Geha et al., 2012; Blanton et al., 2005). dEs are found abundantly in groups
and clusters of galaxies where they dominate in numbers (Binggeli et al.,
1988).

The Lambda cold dark matter (CDM) hierarchical merging scenario predicts that
CDM haloes are formed because of gravitational instabilities and evolve
hierarchically via mergers (White & Rees 1978; Frenk et al. 1988; White &
Frenk 1991; Lacey & Cole 1993; Cole et al. 2000). These models predict that
dwarf-size dark matter haloes form first and then merge forming more massive
haloes.

The class of dEs contains objects covering a wide range of internal
properties, with sometimes rather complicated structures. Taking advantage of
deep photometric studies, we know that several of them contain substructures
such as disks, sprial arms and irregular features (e.g. Jerjen et al. 2000;
Barazza et al. 2002; Geha et al. 2003; Graham & Guzmán 2003; De Rijcke et al.,
2003; Lisker et al., 2006a; Ferrarese et al., 2006; Janz et al., 2012, 2014).
Apart from this, dEs also show a complicated variety of internal kinematics
and dynamics. dEs with similar photometric properties can have different
stellar populations (Michielsen et al. 2008; Paudel et al. 2010; Koleva et al.
2009, 2011; Ryś et al. 2015) and different rotation speeds (Pedraz et al.
2002; Simien & Prugniel 2002; Geha et al. 2002, 2003; van Zee et al. 2004;
Chilingarian 2009; Toloba et al. 2009, 2011, 2014b, 2015; Koleva et al. 2009,
2011; Ryś et al. 2013, 2014). Kormendy (1985) suggested that they developed
their spheroidal non-star forming appearance, that is probably highly
flattened (Lisker et al. 2006a; Lisker et al. 2007), during a transformation
from a late-type galaxy that fell into a cluster; it is thought that this
transformation is induced by the environment because the morphology-density
relation largely depends on the environment (e.g. Boselli & Gavazzi 2014).

Processes for that transformation include ram-pressure stripping (Gunn & Gott
1972; Lin & Faber 1983) and galaxy harassment(Moore et al. 1998). Ram-pressure
stripping should be able to remove the galaxy’s remaining gas from the system
on short time scales, so that star formation stops quickly. The effect of ram
pressure stripping depends strongly on the density of the environment and it
is expected that their angular momentum and structure should be preserved (Ryś
et al. 2014) while galaxy harassment by tidal interactions between a galaxy
and the potential of the cluster can heat up the object, increasing the
velocity dispersion, slow its rotation down and remove stellar mass so that
disks are transformed into more spheroidal objects (Moore et al. 1998). In
this case, a galaxy can lose some of its intrinsic angular momentum. For a
more detailed review on these effects, see Boselli & Gavazzi (2006, 2014).

How exactly ram pressure stripping and harassment transform objects is still
rather unclear. Ryś et al. (2013) concluded that a transformation mechanism
should be able to not only lower the angular momentum but also increase the
stellar concentration of dEs compared to their presumed progenitors. Toloba et
al. (2015) show that even a combination of these two mechanisms can not easily
remove all of the angular momentum, something which is needed to explain some
observations. Since ram pressure stripping is happening on short timescales,
it might be a standard mechanism to transform late type star-forming galaxies
into dwarf early-type galaxies. After being in the cluster for a long time,
the galaxy goes through its center several times, during which it can heat up,
lose stellar rotation and also lose its disky structure. Since fast rotators
in the outer parts of the cluster are rotating faster than the fast rotators
found in the inner part of the cluster (Toloba et al. 2014a, here after T14),
Springel et al. (2005) suggested that clusters were formed by the accretion of
small groups of galaxies. According to this scenario, the properties of slow
and non-rotating dEs in the center of the cluster can be explained as well as
the existence of kinematically decoupled cores observed in some of the SMAKCED
(Stellar content, MAss and Kinematics of Cluster Early-type Dwarfs) dEs
(Toloba et al. 2014b) Therefore, galaxy clusters, as a place with many dwarf
ellipticals with a range of environmental properties, are excellent places to
study the evolution and formation of the dEs.

Not only can be used the morphology, or their kinematics, to study the
evolution of galaxies: more detailed information can be obtained by studying
the stellar populations, since the distribution of ages, metallicities and
abundance ratios provide important information that can be used to study the
evolutionary history of galaxies since chemical abundances of the gas are
locked into the stars when they form. This information can be obtained via two
general techniques. The first is by studying ages and abundances from
observations of individual stars, which can be done for nearby galaxies where
individual stars can be resolved. The second is by studying the integrated
light from more distant galaxies to derive star formation histories and
abundance distributions. This second technique is the only one currently
available for galaxies at the distance of the Virgo cluster (16.5 Mpc, Mei et
al. 2007).

A non-trivial problem when analyzing the spectra of galaxies is the degeneracy
between age and metallicity. One can break the age-metallicity degeneracy
using a wide wavelength baseline, a combination of line indices, and accurate
data. These then are compared to evolutionary stellar population models (e.g.,
Vazdekis 1999; Bruzual & Charlot 2003; Thomas et al. 2003; Maraston 2005;
Schiavon 2007; Maraston et al. 2009; Vazdekis et al. 2010; Conroy & van Dokkum
2012). By comparing model predictions with observational galaxy parameters age
and metallicity distributions of the stars in that galaxy can be obtained. One
can compare observations with models of a single age and metallicity,
obtaining SSP-equivalent parameters. More complicated approaches, (e.g.
STECKMAP (Ocvirk et al. 2006), STARLIGHT (Cid Fernandes et al. 2005)) provide
full star formation histories. A problem still remains the unicity of the
solutions. As a rule of thumb one can say that the larger the wavelength range
is, the more unique the solution.

Stellar population studies show that dEs have on average a lower metal content
than giant ellipticals, as expected from the metallicity-luminosity relation
(Michielsen et al. 2008; Skillman et al. 1989; Sybilska et al. 2017). Their
ages are somewhat younger on average. However, recent studies show that the
stellar populations of dEs show indications of both young and old ages and a
range in gradients (e.g. Koleva et al. 2009, 2011; den Brok et al. 2011; Ryś
et al. 2015). Studies about detailed abundance ratios in dEs are scarce.
Gorgas et al. (1997); Michielsen et al. (2008) and Sybilska et al. (2017) show
that [Mg/Fe] is similar to solar, lower than what is found in giant
ellipticals. [Mg/Fe] can, however provide important information about the
formation of a galaxy. Individual galaxy abundances are the result of chemical
evolution, involving element enrichment in stars, supernova explosions and
galactic winds from e.g. AGB stars. As a result, the measurement of abundances
of many elements can give us a very detailed picture of the formation and
evolution of a galaxy (see e.g. Tolstoy et al. 2009 for Local group galaxies).
We often call this way of studying galaxies galactic archaeology.

Measured abundances of various elements allow us in principle to understand
which enrichment processes have been dominant at different epochs of galaxy
formation because of their different nucleosynthetic origin. It is thought
that a group of lighter elements, the so called $\alpha$-elements, such as O
and Mg, are produced by type II supernovae, supernovae originated from massive
stars, which therefore occur on short timescales (Worthey, Faber & Gonzalez
1992). Most of the Fe, on the other hand, is predominantly produced by a
different group of supernovae, those of type Ia, which occur on a much longer
timescale. As a result, elemental ratios of [$\alpha$/Fe] give us information
about the relative contribution from the two types of supernovae at a given
time, i.e., about the timescale of star formation. The observed correlation of
[$\alpha$/Fe] abundance ratio and galaxy mass is an indication of the
downsizing (Vazdekis, Trujillo & Yamada 2004; Nelan et al. 2005; Thomas et al.
2005). For dEs, Gorgas et al. (1997), measured that Virgo dEs are consistent
with solar [$\alpha$/Fe] abundance ratio and showed that star formation must
have happened on longer time scales in these systems. Several works confirmed
these results and found that dEs have younger ages and lower metallicities
than normal Es (Geha et al. 2003; van Zee et al. 2004). Interpretation of
abundance ratios of other elements is more complicated, and has been limited
mostly to the Local Group. They are also used to obtain a more detailed
picture and information on the IMF and SFH (McWilliam 1997). In dEs, at the
moment, very little information is available on abundance ratios of elements
(apart from Mg and Fe), mainly because of the lack of high S/N spectra, but
also because of the lack of methods to analyse them. Since this has changed in
recent years, we have been able to start a program to obtain and analyze
abundance ratios in dwarf ellipticals. The results on the pilot galaxy NGC
1396 are presented in Mentz et al. (2016). In the current paper a sample of 37
galaxies from the SMAKCED sample is analyzed.

To determine the abundance ratios from integrated spectra, we use the hybrid
model calibration by Conroy, Graves & van Dokkum (2014)(CvD hereafter). They
calculate spectra using standard stellar population models with solar
abundance ratios, which are modified by using theoretical responses of spectra
to abundance ratio variations, following a method developed by Walcher et al.
(2009). Here, we will study the abundance ratios of a few elements in dwarf
ellipticals, obtaining data which allow us to compare the formation history of
dEs with those of giant ellipticals, the Milky Way, and other galaxies of the
Local Group. We will focus on the Na doublet absorption features in the
optical wavelength range at 5890 and 5896 Å (NaD hereafter) and the Ca4227
line-strength indices, to study the abundances of Na and Ca, as well as the
better studied Mg. Although interpreting the observational results is at
present extremely difficult, already many conclusions can be derived by
comparing them to other types of galaxies.

This work is part of the SMAKCED project, aimed at studying the nature of dEs
in the Virgo Cluster. More details about the galaxies discussed here, and
their properties, can be found in the other SMAKCED papers (Janz et al., 2012,
2014; Toloba et al., 2014b, a, 2015). In Toloba et al. (2014a) a description
is given of the spectroscopic part of the survey, while the H-band photometry
is described in Janz et al. (2014). In Toloba et al. (2014a), kinematics of
two dEs are presented that show kinematically decoupled cores. In Toloba et
al. (2015) the stellar kinematics of dEs is presented as a function of
projected distance to the center of the Virgo cluster.

The Virgo cluster is an ideal laboratory to study dEs because it contains
hundreds of them, is close enough to resolve their detailed structure, and is
a dynamically young cluster that is still evolving today (Binggeli et al.,
1993; Boselli et al., 2014). In this paper we focus on the abundance ratio
distribution of dEs and compare them with other types of galaxies.

This paper is organized as follows. In Section 2, we present the general
properties of our samples, the observations, and the main data reduction
steps. We describe the measurements of age-sensitive and metallicity-sensitive
Lick indices. In Section 3, we derive the ages and metallicity based on the
Lick indices and the abundance ratios. In Section 4, our results are
summarized and discussed. In Section 5, conclusions are given.

##  2 Observations and Data Reduction

Our sample consists of 37 galaxies, the full spectroscopic sample of the
SMAKCED project. For each of them, we obtained data suitable for a detailed
stellar population study, with relatively high spectral resolution and high
signal-to-noise (S/N) ratio. Two galaxies in the sample (VCC1684, VCC2083)
were not included, since no ages could be determined because of lack of
observed Balmer lines. The spectroscopic data were obtained at three different
telescopes. Twenty six dEs were observed at the 4.2m WHT telescope using the
double-arm ISIS spectrograph, of which the blue-arm covered the wavelength
range 4200 - 5000 Å and the red-arm covered the wavelength range 5500 - 6700 Å
. Ten dEs were observed at the 2.5m INT telescope using the IDS spectrograph
covering the wavelength range 4600 - 5600 Å , while the remaining three dEs
were observed at the 8m VLT telescope using the FORS2 spectrograph that covers
the wavelength range 4500 - 5600 Å .

Table 1 summarizes the main properties of these 37 dEs (see also T14).

The data were reduced following the standard procedure for long-slit spectra
using the package REDUCEME (Cardiel, 1999). Details on sample selection,
observations and data reduction are presented in T14.

Table 1: Properties of the SMAKCED dEs. Column 1: galaxy name. Columns 2 and 3: right ascension and declination in J2000. Columns 4 and 5: r-band magnitude (in the AB system) and half-light radius Janz & Lisker (2008, 2009). Column 6: velocity dispersion. Galaxy | RA | Dec | Mr | Re | $\sigma_{\text{e}}$
---|---|---|---|---|---
| (J2000) | (J2000) | (mag) | (arcsec) | (km/s)
VCC0009 | 12:09:22.25 | 13:59:32.74 | -18.2 | 37.2 | 26.0$\pm 3.9$
VCC0021 | 12:10:23.15 | 10:11:19.04 | -17.1 | 15.2 | 28.9$\pm 2.9$
VCC0033 | 12:11:07.79 | 14:16:29.19 | -16.9 | 09.8 | 20.8$\pm 4.9$
VCC0170 | 12:15:56.34 | 14:26:00.33 | -17.6 | 31.3 | 26.6$\pm 4.6$
VCC0308 | 12:18:50.90 | 07:51:43.38 | -18.0 | 18.6 | 24.1$\pm 2.4$
VCC0389 | 12:20:03.29 | 14:57:41.70 | -18.1 | 18.0 | 30.9$\pm 1.2$
VCC0397 | 12:20:12.18 | 06:37:23.51 | -16.8 | 13.6 | 35.7$\pm 1.9$
VCC0437 | 12:20:48.10 | 17:29:16.00 | -18.0 | 29.5 | 40.9$\pm 4.0$
VCC0523 | 12:22:04.14 | 12:47:14.60 | -18.7 | 26.1 | 42.2$\pm 1.0$
VCC0543 | 12:22:19.54 | 14:45:38.59 | -17.8 | 23.6 | 35.1$\pm 1.4$
VCC0634 | 12:23:20.01 | 15:49:13.25 | -18.5 | 37.2 | 31.3$\pm 1.6$
VCC0750 | 12:24:49.58 | 06:45:34.49 | -17.0 | 19.5 | 43.5$\pm 2.9$
VCC0751 | 12:24:48.30 | 18:11:47.00 | -17.5 | 12.3 | 32.1$\pm 2.4$
VCC0781 | 12:25:15.17 | 12:42:52.59 | -17.2 | 13.4 | 38.0$\pm 2.8$
VCC0794 | 12:25:22.10 | 16:25:47.00 | -17.3 | 37.0 | 29.0$\pm 3.9$
VCC0856 | 12:25:57.93 | 10:03:13.54 | -17.8 | 16.5 | 31.3$\pm 4.1$
VCC0917 | 12:26:32.39 | 13:34:43.54 | -16.6 | 09.9 | 28.4$\pm 1.4$
VCC0940 | 12:26:47.07 | 12:27:14.17 | -17.4 | 19.8 | 40.4$\pm 1.3$
VCC0990 | 12:27:16.94 | 16:01:27.92 | -17.5 | 10.2 | 38.7$\pm 1.3$
VCC1010 | 12:27:27.39 | 12:17:25.09 | -18.4 | 22.2 | 44.6$\pm 0.9$
VCC1087 | 12:28:14.90 | 11:47:23.58 | -18.6 | 35.4 | 42.0$\pm 1.5$
VCC1122 | 12:28:41.71 | 12:54:57.08 | -17.2 | 17.3 | 32.1$\pm 1.7$
VCC1183 | 12:29:22.51 | 11:26:01.73 | -17.9 | 21.1 | 44.3$\pm 2.4$
VCC1261 | 12:30:10.32 | 10:46:46.51 | -18.5 | 23.8 | 44.8$\pm 1.4$
VCC1304 | 12:30:39.90 | 15:07:46.68 | -16.9 | 16.5 | 25.9$\pm 2.7$
VCC1355 | 12:31:20.21 | 14:06:54.93 | -17.6 | 30.3 | 20.3$\pm 4.7$
VCC1407 | 12:32:02.73 | 11:53:24.46 | -17.0 | 12.1 | 31.9$\pm 2.1$
VCC1431 | 12:32:23.41 | 11:15:46.94 | -17.8 | 09.8 | 52.4$\pm 1.6$
VCC1453 | 12:32:44.22 | 14:11:46.17 | -17.9 | 18.9 | 35.6$\pm 1.4$
VCC1528 | 12:33:51.61 | 13:19:21.03 | -17.5 | 09.6 | 47.0$\pm 1.4$
VCC1549 | 12:34:14.83 | 11:04:17.51 | -17.3 | 12.1 | 36.7$\pm 2.3$
VCC1695 | 12:36:54.85 | 12:31:11.93 | -17.7 | 24.0 | 24.4$\pm 2.2$
VCC1861 | 12:40:58.57 | 11:11:04.34 | -17.9 | 19.0 | 31.3$\pm 1.5$
VCC1895 | 12:41:51.97 | 09:24:10.28 | -17.0 | 16.3 | 23.8$\pm 3.0$
VCC1910 | 12:42:08.67 | 11:45:15.19 | -17.9 | 13.4 | 37.0$\pm 1.2$
VCC1912 | 12:42:09.07 | 12:35:47.93 | -17.9 | 22.5 | 36.0$\pm 1.5$
VCC1947 | 12:42:56.34 | 03:40:35.78 | -17.6 | 09.3 | 48.3$\pm 1.3$

###  2.1 Line-strength measurements

Observed spectral data can be studied either by fitting the full spectrum or
by focusing on selected line indices. In this work we study selected line
indices. We measured Lick indices (Worthey et al., 1994) in the LIS-5 Å flux
calibrated system (Vazdekis et al., 2010). The new Line Index System (LIS) has
numerous advantages over the Lick system. It is defined at three different
resolutions, namely 5.0 Å, 8.4 Å, and 14.0 Å. As such, it is particularly
useful for analyzing small galaxies and globular clusters. As the resolution
of the Lick/IDS library (FWHM $\sim$ 8-11 Å) is much lower than what is
available with our high resolution spectra, we broadened the spectra to the
LIS-5 Å system. The LIS-5 Å system choice expresses a factor of 2 improvement
in resolution over Lick/IDS system. Also, the fact that the LIS system is flux
calibrated, makes it easier to reproduce data from other authors.

The spectra were broadened to the LIS-5 Å system taking into account the
velocity dispersion of the spectra. This ensures that every galaxy spectrum
has the same spectral resolution, which is important and necessary to compare
them with each other.

We measured a total of 23 Lick indices (Faber et al., 1985; Gorgas et al.,
1993; Worthey et al., 1994, Worthey & Ottaviani, 1997). In this paper we use
${H}\gamma_{\text{F}}$, ${H}\beta$, Fe4383, Fe4531, Fe5270, Fe5335, Fe5406,
Fe5709, Mgb, Ca4227 and NaD. The summary of the indices used are tabulated
Table 2. However, because of the different wavelength range covered by our
spectra, not all lines are used for all galaxies. Galaxies observed at the VLT
do not cover the ${H}\gamma_{\text{F}}$, Fe4383 and NaD lines. Also, in some
spectra which are obtained by WHT, ${H}\beta$ is located at the end of the
spectrum, and since the spectra suffer from vignetting there, we could not
determine this index here. In Table 3 and Table 4, we show the line-strength
index measurements for each galaxy used.

Table 2: A summary of the indices used Telescope | WHT | INT and VLT
---|---|---
Age indicator | ${H}\gamma_{\text{F}}$ | ${H}\beta$
Metal indicator | Fe4383, Fe4531 | Fe4531, Fe5270, Fe5335
| Fe5709 | Fe 5406, Fe5709, Mgb
Na abundances | NaD | . . .
Ca abundances | Ca4227 | . . .
Mg abundance | . . . | Mgb

##  3 Results

###  3.1 Derived ages and metallicities

Luminosity-weighted ages and metallicities are estimated using age-sensitive
(${H}\beta$ and ${H}\gamma_{\text{F}}$) and metallicity-sensitive (Fe4383,
Fe4531, Fe5270, Fe5335, Fe5406, Fe5709 and Mgb) Lick spectral indices
(Worthey, 1994) measured in the LIS-5 Å system (Vazdekis et al., 2010). For
this we used an MCMC (Markov Chain Monte Carlo) code to derive the age and
metallicity of the best fitting MILES single stellar population model.

We estimate the best luminosity weighted age and metallicity from all
available index-index combinations by effectively computing the "distance"
from our measured indices to all predicted values on the model grids, and
finding the age and metallicity combination with the minimum total distance.
The age and metallicity values for each index-index diagram and associated
uncertainties were derived using 1000 MCMC iteration of the fit. To reduce the
effects of the grid discretization, the two relevant parameters (e.g. age and
metallicity) were interpolated. Uncertainties were calculated by performing
Monte Carlo simulations, making use of the observational error in each index.
In Table 5 we list the best fitting parameters for ages and metallicities that
are determined using combinations of all age sensitive lines and all metal
indicators.

Figure 1 shows index-index plots where we have restricted the age to the
interval 1.0 - 14.0 Gyr, and the metallicity range from -1.76 to 0.26, which
includes the range covered by the galaxies in our sample. We use the solar-
scaled theoretical isochrones in the model grids from Vazdekis et al. (2010)
in Figure 1.

Table 3: Lick spectral indices measured at LIS-5 Å resolution for WHT objects. Galaxy | Ca4227 | ${H}\gamma_{\text{F}}$ | Fe4383 | Fe4531 | Fe5709 | NaD
---|---|---|---|---|---|---
| (Å) | (Å) | (Å) | (Å) | (Å) | (Å)
VCC0033 | 1.25$\pm 0.38$ | 1.55 $\pm 0.39$ | 2.62$\pm 0.85$ | 3.09$\pm 0.65$ | 0.60$\pm 0.22$ | 0.83$\pm 0.373$
VCC0170 | 0.75$\pm 0.21$ | 2.43 $\pm 0.22$ | 1.48$\pm 0.51$ | 1.70$\pm 0.40$ | 0.48$\pm 0.09$ | 1.21$\pm 0.173$
VCC0308 | 0.99$\pm 0.12$ | 1.69 $\pm 0.14$ | 2.53$\pm 0.32$ | 3.02$\pm 0.24$ | 0.85$\pm 0.09$ | 1.57$\pm 0.125$
VCC0389 | 1.17$\pm 0.14$ | 0.33 $\pm 0.14$ | 3.90$\pm 0.29$ | 2.87$\pm 0.21$ | 0.99$\pm 0.07$ | 1.68$\pm 0.095$
VCC0397 | 1.11$\pm 0.12$ | 1.33 $\pm 0.14$ | 3.92$\pm 0.31$ | 3.50$\pm 0.24$ | 0.94$\pm 0.11$ | 1.92$\pm 0.136$
VCC0437 | 0.87$\pm 0.25$ | -0.76$\pm 0.27$ | 2.49$\pm 0.57$ | 2.01$\pm 0.41$ | 0.80$\pm 0.10$ | 2.08$\pm 0.165$
VCC0523 | 1.13$\pm 0.07$ | 0.93 $\pm 0.09$ | 3.25$\pm 0.19$ | 3.14$\pm 0.15$ | 0.91$\pm 0.10$ | 1.50$\pm 0.177$
VCC0543 | 1.41$\pm 0.13$ | -0.17$\pm 0.16$ | 3.96$\pm 0.32$ | 3.23$\pm 0.23$ | 0.77$\pm 0.06$ | 1.95$\pm 0.093$
VCC0634 | 1.21$\pm 0.15$ | 0.55$\pm 0.16$ | 3.70$\pm 0.33$ | 3.05$\pm 0.25$ | 0.75$\pm 0.13$ | . . .
VCC0750 | 1.09$\pm 0.20$ | 0.75 $\pm 0.21$ | 3.36$\pm 0.44$ | 3.00$\pm 0.33$ | 0.80$\pm 0.08$ | 1.61$\pm 0.122$
VCC0751 | 1.62$\pm 0.25$ | -0.23$\pm 0.27$ | 4.80$\pm 0.53$ | 3.93$\pm 0.40$ | 1.16$\pm 0.16$ | . . .
VCC0781 | 0.66$\pm 0.22$ | 2.74 $\pm 0.21$ | 1.31$\pm 0.49$ | 2.19$\pm 0.37$ | . . . | . . .
VCC0794 | 1.06$\pm 0.24$ | 0.43 $\pm 0.26$ | 2.84$\pm 0.55$ | 2.18$\pm 0.41$ | 0.62$\pm 0.08$ | 1.45$\pm 0.117$
VCC0917 | 0.99$\pm 0.10$ | 0.41 $\pm 0.13$ | 3.28$\pm 0.27$ | 2.96$\pm 0.21$ | 0.65$\pm 0.09$ | 1.26$\pm 0.129$
VCC1010 | 1.42$\pm 0.07$ | -0.83$\pm 0.09$ | 4.49$\pm 0.17$ | 3.25$\pm 0.13$ | 0.85$\pm 0.04$ | 2.33$\pm 0.054$
VCC1087 | 1.33$\pm 0.10$ | -0.51$\pm 0.12$ | 4.79$\pm 0.23$ | 2.98$\pm 0.18$ | 0.79$\pm 0.08$ | . . .
VCC1122 | 1.11$\pm 0.10$ | 0.86 $\pm 0.12$ | 3.20$\pm 0.26$ | 2.52$\pm 0.21$ | 0.71$\pm 0.20$ | . . .
VCC1304 | 0.81$\pm 0.13$ | 1.72 $\pm 0.15$ | 2.31$\pm 0.34$ | 2.60$\pm 0.26$ | 0.52$\pm 0.07$ | 1.95$\pm 0.097$
VCC1355 | 1.39$\pm 0.33$ | 0.59 $\pm 0.35$ | 2.77$\pm 0.74$ | 3.06$\pm 0.55$ | 0.78$\pm 0.10$ | 1.32$\pm 0.188$
VCC1407 | 0.83$\pm 0.18$ | 0.12 $\pm 0.20$ | 3.35$\pm 0.40$ | 2.81$\pm 0.30$ | 0.53$\pm 0.08$ | 1.55$\pm 0.129$
VCC1453 | 1.52$\pm 0.13$ | -0.43$\pm 0.14$ | 4.38$\pm 0.29$ | 3.40$\pm 0.21$ | 0.99$\pm 0.06$ | 2.08$\pm 0.065$
VCC1528 | 1.27$\pm 0.18$ | -0.40$\pm 0.19$ | 4.49$\pm 0.38$ | 3.28$\pm 0.28$ | 0.96$\pm 0.07$ | 2.33$\pm 0.093$
VCC1695 | 1.03$\pm 0.09$ | 1.47 $\pm 0.10$ | 2.82$\pm 0.23$ | 2.75$\pm 0.18$ | 0.80$\pm 0.08$ | 1.63$\pm 0.123$
VCC1861 | 1.43$\pm 0.11$ | -0.66$\pm 0.14$ | 3.71$\pm 0.29$ | 2.87$\pm 0.22$ | 0.70$\pm 0.10$ | . . .
VCC1895 | 1.09$\pm 0.20$ | 0.83 $\pm 0.23$ | 2.73$\pm 0.49$ | 2.75$\pm 0.37$ | 0.70$\pm 0.10$ | 1.26$\pm 0.155$
Table 4: Lick spectral indices measured at LIS-5 Å resolution for INT and VLT objects. Galaxy | Fe4531 | ${H}\beta$ | Mgb | Fe5270 | Fe5335 | Fe5406 | Fe5709
---|---|---|---|---|---|---|---
| (Å) | (Å) | (Å) | (Å) | (Å) | (Å) | (Å)
VCC0021 | 1.23$\pm 0.40$ | 3.98$\pm 0.19$ | 0.96$\pm 0.23$ | 1.25$\pm 0.25$ | 1.52$\pm 0.29$ | 0.51$\pm 0.23$ | 0.11$\pm 0.20$
VCC0856 | 3.35$\pm 0.62$ | 2.30$\pm 0.29$ | 2.44$\pm 0.33$ | 2.38$\pm 0.36$ | 1.87$\pm 0.41$ | 0.67$\pm 0.32$ | 1.39$\pm 0.27$
VCC0940 | 2.33$\pm 0.17$ | 2.22$\pm 0.09$ | 2.64$\pm 0.08$ | 2.45$\pm 0.09$ | 1.78$\pm 0.10$ | 1.30$\pm 0.07$ | …
VCC0990 | 3.20$\pm 0.30$ | 2.81$\pm 0.15$ | 2.49$\pm 0.17$ | 2.62$\pm 0.19$ | 2.29$\pm 0.21$ | 1.58$\pm 0.16$ | 0.83$\pm 0.15$
VCC1183 | 3.49$\pm 0.35$ | 2.61$\pm 0.16$ | 2.95$\pm 0.18$ | 2.82$\pm 0.20$ | 2.23$\pm 0.23$ | 1.59$\pm 0.17$ | 0.95$\pm 0.15$
VCC1261 | 2.33$\pm 0.21$ | 2.47$\pm 0.11$ | 2.19$\pm 0.12$ | 2.52$\pm 0.13$ | 2.13$\pm 0.15$ | 1.47$\pm 0.12$ | 0.94$\pm 0.11$
VCC1431 | 3.53$\pm 0.33$ | 1.95$\pm 0.16$ | 3.17$\pm 0.18$ | 2.46$\pm 0.19$ | 1.93$\pm 0.22$ | 1.41$\pm 0.17$ | 0.46$\pm 0.15$
VCC1549 | 2.72$\pm 0.46$ | 1.73$\pm 0.22$ | 3.02$\pm 0.25$ | 2.85$\pm 0.26$ | 2.48$\pm 0.30$ | 1.74$\pm 0.22$ | 0.99$\pm 0.20$
VCC1910 | 3.24$\pm 0.33$ | 1.75$\pm 0.15$ | 2.82$\pm 0.16$ | 2.50$\pm 0.18$ | 2.84$\pm 0.19$ | 2.06$\pm 0.15$ | 0.30$\pm 0.13$
VCC1912 | 2.31$\pm 0.24$ | 3.66$\pm 0.11$ | 1.32$\pm 0.13$ | 2.21$\pm 0.14$ | 2.30$\pm 0.16$ | 1.15$\pm 0.13$ | 0.16$\pm 0.11$
VCC1947 | 3.26$\pm 0.32$ | 1.82$\pm 0.15$ | 3.18$\pm 0.17$ | 2.99$\pm 0.18$ | 2.69$\pm 0.20$ | 1.85$\pm 0.16$ | 0.94$\pm 0.14$
Figure 1: Spectral index-index diagrams used to estimate the stellar
populations using solar-scaled theoretical isochrone grids with IMF slope of
1.3 from Vazdekis et al. (2010) in the system LIS-5 Å, solid lines indicate
constant age 1.0, 2.0, 3.5, 5.5, 10.0 and 14.0 Gyr, respectively while dotted
lines indicate constant [M/H] -1.76, -1.26, -0.65, -0.35, +0.06 and +0.26,
respectively.

###  3.2 Elemental abundance ratios

To calculate the abundance ratios [E/Fe] from the indices we first calculate
age and metallicity as described above, assuming that the galaxies can be
represented by an SSP model. We then measure the difference between the
observed index related to the element E and the index value expected from
stellar population models for the age and metallicity that we measure for the
galaxy, and divide this difference by the sensitivity of the index to
variations in [E/Fe] (equation 1). For Na (NaD) and Mg (Mgb) we use the Na-
MILES model of (La Barbera et al. 2017) based on Teramo isochrones, with
variable Mg and Na abundance ratios, and with a (standard) bimodal IMF with
$\Gamma_{b}$=1.3. Unfortunately, Mgb was measured only in a few galaxies,
those observed with the VLT and the INT. For Ca, we do the same using the
Ca4227 index. The problem here is that for this element no stellar population
models are available. The only available model, by CvD, has solar metallicity,
and a fixed, old, age of 13.5 Gyr, which is different from the ages and
metallicities of the objects we discuss here. In this paper we will use this
latter model, but move the analysis and discussion to Appendix A, since we do
not know how appropriate this is.

So, for element E based on index i the elemental abundance ratios were
calculated using

$[E_{i}/Fe]=\frac{i_{observed}-i_{model}}{\frac{\Delta{i_{model}}}{\Delta{[E_{i}/Fe]_{model}}}}.$
(1)

where, $\it{i_{observed}}$ is the value of the index measured from the
observations, $\it{i_{model}}$ is the index expected from the model and
$\it{E_{i}}$ is the elemental abundance.

In Table 5, we provide the elemental abundances for each galaxy determined
according to the procedure described above. The spectral range covered allowed
the measurement of [Ca/Fe] and [Na/Fe] for most galaxies that are observed by
WHT, and [Mg/Fe] for some that are observed by INT and VLT. We do not have
galaxies for which all three abundance ratios have been measured.

To establish the reliability of these abundance ratios, we made a comparison
of the sensitivity of $\Delta{NaD}$ to variations in [Na/Fe] and $\Delta{Mgb}$
to variations in [Mg/Fe] between the SSP models of CvD and MILES. For a
meaningful comparison, we have computed some additional $\alpha$ enhancement
SSP models for a total metallicity of [M/H]=0.3 (for [Fe/H] 0.0, from equation
4 of Vazdekis et al. 2012). For the Na comparison, a Na-MILES model (La
Barbera et al. 2017) is computed for the same age and metallicity as CvD.

We find that the Na-MILES model give a slightly larger sensitivity for Na than
the one of CvD and that we get similar results from both CvD and MILES for
Mgb. Note, however, that the sensitivities vary considerably when going to
lower ages and metallicities.

We report the values of the $\Delta{NaD}$ and $\Delta{Mgb}$ variations as a
function of these elemental abundance ratios for models with different ages
and metallicities in Table 6. When we compare the sensitivities at the same
age, we find that the NaD sensitivity generally increases with metallicity,
while the one for Mgb generally decreases. When one increases the age, both
sensitivities tend to increase.

Figure 2: Example of a spectral index-index diagram for ${H}\gamma_{\text{F}}$ versus NaD, showing that the NaD values are much lower than the predicted by models. Models from Vazdekis et al. 2010. Constant age and [M/H] as in Figure 1. Figure 3: Example of spectral index-index diagram for ${H}\gamma_{\text{F}}$ versus Ca4227, showing that the Ca4227 values are slightly higher than predicted by models. Models from Vazdekis et al. 2010. Constant age and [M/H] as in Figure 1. Figure 4: Mg and Na abundances as a function of metallicity [Fe/H]. Blue pluses are the Milky Way stars from Venn et al. (2004), red crosses are from the Fornax local dwarf from Letarte et al. (2010) and Shetrone et al. (2003), green asterisks are the LMC red giants from Pompéia et al. (2008), purple triangles are giant ellipticals from Conroy et al. (2014), orange circles are various radial bins of the NGC1396 in the Fornax cluster from Metz et al. (2016) and black circles are the dEs in Virgo cluster analyzed here. Table 5: Elemental abundances, metallicity and ages. Column 1: galaxy name. Column 2 and 3: measurement of [Ca/Fe] and [Na/Fe] for galaxies that are observed with the WHT. Column 4: measurement of [Mg/Fe] for galaxies that are observed using the INT and VLT. Column 5 and 6: metallicity [M/H] and logarithmic ages in Gyr with errors. Galaxy | [Ca/Fe] | [Na/Fe] | [Mg/Fe] | [Fe/H] | log (age)
---|---|---|---|---|---
|  |  |  |  | (Gyr)
VCC0009 | 0.05 | -0.52 | . . . | -1.06 $\pm 0.15$ | 0.94 $\pm 0.03$
VCC0021 | . . . | . . . | 0.03 | -1.17 $\pm 0.18$ | 0.20 $\pm 0.07$
VCC0033 | 0.32 | -0.57 | . . . | -1.00 $\pm 0.01$ | 0.83 $\pm 0.17$
VCC0170 | 0.17 | -0.32 | . . . | -1.05 $\pm 0.08$ | 0.47 $\pm 0.09$
VCC0308 | 0.14 | -0.24 | . . . | -0.40 $\pm 0.24$ | 0.34 $\pm 0.04$
VCC0389 | 0.15 | -0.23 | . . . | -0.37 $\pm 0.16$ | 0.70 $\pm 0.04$
VCC0397 | 0.05 | -0.20 | . . . | 0.06 $\pm 0.35$ | 0.23 $\pm 0.04$
VCC0437 | 0.12 | -0.06 | . . . | -0.71 $\pm 0.01$ | 0.98 $\pm 0.07$
VCC0523 | 0.17 | -0.33 | . . . | -0.19 $\pm 0.29$ | 0.41 $\pm 0.03$
VCC0543 | 0.21 | -0.17 | . . . | -0.44 $\pm 0.07$ | 0.83 $\pm 0.05$
VCC0634 | 0.30 | . . . | . . . | -0.93 $\pm 0.35$ | 0.84 $\pm 0.30$
VCC0750 | 0.13 | -0.24 | . . . | -0.47 $\pm 0.10$ | 0.62 $\pm 0.06$
VCC0751 | 0.45 | . . . | . . . | -0.93 $\pm 0.36$ | 0.85 $\pm 0.29$
VCC0781 | 0.11 | . . . | . . . | -0.90 $\pm 0.24$ | 0.84 $\pm 0.30$
VCC0794 | 0.25 | -0.31 | . . . | -0.81 $\pm 0.03$ | 0.90 $\pm 0.08$
VCC0856 | . . . | . . . | 0.05 | -0.64 $\pm 0.31$ | 0.92 $\pm 0.16$
VCC0917 | 0.08 | -0.43 | . . . | -0.47 $\pm 0.12$ | 0.69 $\pm 0.05$
VCC0940 | . . . | . . . | 0.40 | -0.97 $\pm 0.35$ | 0.84 $\pm 0.29$
VCC0990 | . . . | . . . | 0.07 | -0.23 $\pm 0.12$ | 0.43 $\pm 0.08$
VCC1010 | 0.18 | -0.07 | . . . | -0.31 $\pm 0.10$ | 0.94 $\pm 0.03$
VCC1087 | 0.35 | . . . | . . . | -0.94 $\pm 0.25$ | 0.84 $\pm 0.30$
VCC1122 | 0.27 | . . . | . . . | -0.93 $\pm 0.30$ | 0.85 $\pm 0.29$
VCC1183 | . . . | . . . | 0.15 | -0.26 $\pm 0.02$ | 0.67 $\pm 0.11$
VCC1261 | . . . | . . . | 0.04 | -0.33 $\pm 0.13$ | 0.52 $\pm 0.06$
VCC1304 | 0.15 | 0.06 | . . . | -0.68 $\pm 0.10$ | 0.55 $\pm 0.07$
VCC1355 | 0.34 | -0.32 | . . . | -0.64 $\pm 0.01$ | 0.75 $\pm 0.12$
VCC1407 | 0.11 | -0.27 | . . . | -0.73 $\pm 0.07$ | 0.95 $\pm 0.05$
VCC1431 | . . . | . . . | 0.11 | -0.49 $\pm 0.02$ | 1.07 $\pm 0.06$
VCC1453 | 0.23 | -0.15 | . . . | -0.27 $\pm 0.13$ | 0.76 $\pm 0.05$
VCC1528 | 0.14 | -0.05 | . . . | -0.25 $\pm 0.05$ | 0.79 $\pm 0.06$
VCC1549 | . . . | . . . | 0.08 | -0.41 $\pm 0.09$ | 1.04 $\pm 0.09$
VCC1695 | 0.15 | -0.21 | . . . | -0.40 $\pm 0.22$ | 0.39 $\pm 0.05$
VCC1861 | 0.38 | . . . | . . . | -0.90 $\pm 0.57$ | 0.85 $\pm 0.29$
VCC1895 | 0.22 | -0.37 | . . . | -0.77 $\pm 0.15$ | 0.82 $\pm 0.11$
VCC1910 | . . . | . . . | 0.05 | -0.37 $\pm 0.40$ | 0.93 $\pm 0.11$
VCC1912 | . . . | . . . | -0.07 | -0.07 $\pm 0.28$ | 0.14 $\pm 0.01$
VCC1947 | . . . | . . . | -0.09 | -0.93 $\pm 0.47$ | 0.88 $\pm 0.11$
Table 6: Comparison of MILES and CvD model predictions (column 1) with varying metallicity (column 2) and age (column 3). Columns 4 and 5: $\Delta{NaD}$ variations in [Na/Fe], $\Delta{Ca4227}$ variations in [Ca/Fe] and $\Delta{Mgb}$ variations in [Mg/Fe] based on the models shown in col.1 Model | [M/H] | Age | $\frac{\Delta{NaD_{*}}}{\Delta{[Na/Fe]_{*}}}$ | $\frac{\Delta{Mgb_{*}}}{\Delta{[Mg/Fe]_{*}}}$ | $\frac{\Delta{Ca4227_{*}}}{\Delta{[Ca/Fe]_{*}}}$
---|---|---|---|---|---
(*) | dex | (Gyr) |  |  |
CvD | 0.00 | 13.50 | 3.085 | 4.260 | 2.790
MILES | 0.06 | 14.00 | 3.547 | 4.350 | …
MILES | -0.35 | 1.00 | 0.973 | 3.133 | …
MILES | -0.35 | 7.00 | 2.317 | 4.830 | …
MILES | -0.35 | 14.00 | 2.683 | 5.609 | …
MILES | 0.06 | 1.00 | 1.453 | 3.009 | …
MILES | 0.06 | 7.00 | 3.167 | 3.298 | …
MILES | 0.26 | 1.00 | 1.583 | 3.463 | …
MILES | 0.26 | 7.00 | 3.380 | 3.336 | …
MILES | 0.26 | 14.00 | 3.830 | 3.768 | …

##  4 Discussion

We present ages, metallicities and abundance ratios obtained for 37 dEs within
an aperture size of Re/8. This aperture size is commonly used in conventional
long-slit studies. The radius has been chosen to provide a measure of the
stellar populations in the central regions of these galaxies, in a region with
a constant relative size. Following the classification of Lisker et al.
(2006a); Lisker et al. (2006b, 2007) using high-pass filtered Sloan Digital
Sky Survey (SDSS; Adelman-McCarthy et al. 2006), 36 of the galaxies of our
sample are classified as nucleated. VCC0021 and VCC1431 are galaxies with
large blue core. All the analysis comes from the central regions, in which the
nuclear cluster’s light is contributing significantly. The resulting abundance
ratios can thus provide insight about the luminosity-weighted stellar
population in this region. Although we could have tried to derive two-burst or
more complicated Star Formation Histories, it would have been very hard with
the current low-resolution data to be able to distinguish between a one- or
two-burst solution (see e.g. Ryś et al. 2015; Mentz et al. 2016). For that
reason, we postpone this to a future paper, in which we analyze spectra at a
spectral resolution of R=5000 using the SAMI instrument at the AAT.

One of the main results of this paper is the unusual behaviour of the Na
abundances in dEs, when compared with massive ellipticals and Local Group
dwarfs. For a sample of quiescent dwarfs with effective velocity dispersion in
the range 20 - 55 km s-1 and with absolute r-band magnitude ranging from -19
to -16, we find that our sample of dEs is underabundant in Na when compared to
the solar neighbourhood (Figure 4). At the same time, the Mg abundances are
around solar. In the following, we try to argue what this means for the
evolution of dwarf ellipticals, based on what we know from theory, and
observations of individual stars in the Milky Way and the Local Group, and
integrated abundance ratios in giant elliptical galaxies. For dwarf
ellipticals, such an analysis using integrated light has not been done before,
although several papers have tried to derive abundance ratios in giant
ellipticals (e.g. Worthey 1998; Thomas et al. 2010; Worthey et al. 2011;
Conroy et al. 2014; Spiniello et al. 2014; Smith et al. 2015; Yamada et al.
2006).

In this paper we are analyzing light elements. It is our current belief that
in the most-studied environments the relative abundance [X/Fe] of an element X
at low [Fe/ H] represents the abundance ratio from only those sources of
material processed through the nucleosynthesis channels that were active at
very early times, i.e. SNII from core collapse of massive stars, events
characteristic of the earliest epochs of star formation (Cohen & Huang, 2009).
After these early times, [X/Fe] is modified by other sources, such as SNIa,
SNII, AGB stars, novae, etc., and also by the accretion of primordial material
and of galactic winds. This gives a knee-like pattern in the relation between
[X/Fe] and [Fe/ H], of which the position of the bend is determined by the
delay time between the core collapse SNII and the other processes. This delay
depends on various parameters (IMF, star formation efficiency, the rate of
mass loss in stars, and the rate of accretion of primordial gas consisting
mostly of H), as well as on the element production yields (e.g. Greggio et al.
2008). These processes lead to the characteristic knee in the Milky Way, where
low metallicity halo stars generally have [Mg/Fe] values around 0.4, a value
reached by SNII enrichment only, going down to solar mass disk stars, with
[Mg/Fe] values around solar, an equilibrium values reached when all processes
have contributed. From these solar abundance ratios in the disk of our Milky
Way, one can then conclude that star formation has been taken place on long
timescales compared to the halo.

We will now have a more detailed look at the individual abundance ratios.

###  4.1 Na abundances

Na is believed to be made in the interiors of massive stars and to depend on
the neutron excess, which in turn depends on the initial heavy element
abundance in the star. Na thus has both a primary and a secondary
nucleosynthesis channel (Arnett 1971; Clayton 2003). Ni is assumed to
originate predominantly from SNe Ia. However, the production of Ni might also
be linked to the production of Na in SNe II (Thielemann et al. 1990; Timmes et
al. 1995). The amount of Na produced is controlled by the neutron excess,
where ${}^{23}Na$ is the only stable neutron rich isotope produced in
significant quantity during the C and O burning stage. The production of Ni
depends on the neutron excess and the neutron excess will depend primarily on
the amount of ${}^{23}Na$ previously produced. Hence, a Na-Ni correlation is
expected when the chemical enrichment is dominated by SNe II. The advent of
the SNe Ia explosions can break (or flatten) this relationship, as Ni is
produced without Na in the standard model of SN Ia (e.g., Iwamoto et al.
1999). Since the neutron excess is strongly metallicity-dependent, this could
explain the low [Na/Fe] that are being found for the Fornax dwarf (Letarte et
al. 2010). At the same time, it would explain the high values in giant
ellipticals (see later).

It is important to mention is that the abundance ratios found here are very
different from those in other stellar systems, most massive Galactic globular
clusters, and giant elliptical galaxies. In most massive globular clusters a
strong Na-O anticorrelation is observed in the RGB stars (Kraft 1994, and see
reviews of Carretta 2016; Gratton et al. 2001). For these stars, Oxygen is
depleted, and Na enhanced, just like N. Since this effect is not see in the
interior of the presently observed GC low mass stars, it is thought that this
is a second generation enrichment effect from massive stars, who have enhanced
their Na-abundance during C-burning through the NeNa cycle (Langer et al.
1993). It is thought that second generation (SG) stars are formed by nuclear
ejecta processed in the most massive first generation (FG) stars, diluted with
different amounts of unprocessed gas, generating a number of anticorrelations,
including the one of Na and O. This process, as far as we know, does not take
place in the halo of our Milky Way, and in field stars of local group
galaxies. In this paper we see that this is also not the case in field stars
of dwarf ellipticals in nearby galaxy clusters.

Four strong Na absorption features can be found in the optical and NIR
wavelength range: NaD (5890 and 5896 Å), NaI 0.82 (8183 and 8195 Å), NaI1.14
(11400 Å) and NaI2.21 (22100 Å). The equivalent widths(EWs) of NaD were
studied first by O’Connell (1976) and Peterson (1976), where they reported
that NaD was much stronger than expected from Ca and Fe indices in giant
early-type galaxies. Spiniello et al. (2015) reported that the NaD feature is
very sensitive to [Na/Fe] variations, while the NaI index seems to depend
mainly on the IMF (e.g. Vazdekis et al. 2012). La Barbera et al. (2017)
perform detailed fits to all 4 Na-lines in a number of giant ellipticals, and
find that both [Na/Fe] needs to be considerably larger than solar, and the
IMF-slope needs to be dwarf-enhanced (Smith et al. 2015). For dwarf
ellipticals, however, there are no indications that the IMF-slope is different
from our Galaxy (Mentz et al. 2016). Here, we find the behaviour that [Na/Fe]
is lower than solar, opposite to the behavior in giant ellipticals.

Remarkable is the strong trend between [Na/Fe] and [Fe/H], already mentioned
by Mentz et al. (2016) when joining the Fornax dwarf with our dwarf
ellipticals and the giant ellipticals. For [Fe/H] values of $\sim$ -0.8 very
low [Na/Fe] values are obtained of $\sim$ -0.6 - -0.8 in the Fornax dwarf.
This contrasts with the high, positive [Na/Fe] values that are found in giant
ellipticals. Such a strong correlation is what you could expect if the Na-
abundance depends strongly on the neutron excess, or equivalently, the
metallicity.

###  4.2 Abundance ratios and the formation of dEs

In this paper we find [Mg/Fe] values that are very close to solar (the mean
value of the [Mg/Fe] is 0.07), or slightly larger. [Na/Fe] is (the mean value
of the [Na/Fe] is -0.25), however, considerably lower than solar.

When comparing to stellar populations that show similar abundance patterns,
there are not many, but one could consider the younger, and more metal-rich
stellar populations in the center of the Fornax Dwarf galaxy, published by
Letarte et al. (2010). The majority of those stars are 1-4 Gyr old, and have
unusually low [Mg/Fe], [Ca/Fe] and [Na/Fe] compared to the Milky Way stellar
populations at the same [Fe/H], and are therefore at the end of their chemical
evolution. The difficulty is that, although [Mg/Fe] is approximately solar for
these stars, [Ca/Fe] lies considerably below this value. Letarte et al. (2010)
hypothesize that this means that a large fraction of Ca and Ti is produced in
processes that do not produce much Mg, such in SNe Ia. In this way, the low
[Ca/Fe] and [Ti/Fe] could be a consequence of the low-metallicity of their
progenitors compared to the Milky Way. At the same time, [Na/Fe] is found to
be rather low (between -0.9 and -0.4), but correlating strongly with [Ni/Fe]
(Nissen & Schuster, 1997, 2009).

For the dwarf ellipticals in Virgo analyzed in this paper, we conclude,
analogous to Letarte et al. (2010) that the low [Mg/Fe] values (w.r.t. the
thick disk of the Milky Way and giant ellipticals) show that the galaxies have
undergone a considerable amount of chemical evolution. This means that the
galaxies are not uniformly old, but have extended star formation histories,
similar to many of the Local Group galaxies. [Na/Fe] is lower than solar, but
still higher than in the Fornax dwarf. This is expectable, since the
metallicities of the dwarf ellipticals in Virgo are a bit larger, resulting in
a larger neutron excess and higher Na-abundances.

For dEs we find that (a) [Mg/Fe] $\sim$ 0 and (b) [Na/Fe] < 0\. Result (a)
implies that star formation is slow, like in the Milky Way disk. Result (b) is
consistent with the same formation mechanism. Just like the stars of Letarte
et al. (2010) in the center of the Fornax dwarf galaxy, the stars in the dEs
have undergone a considerable amount of enrichment and have an extended star
formation history. The dependence of [Na/Fe] on the neutron-excess causes
[Na/Fe] to be below 0, since for the Virgo dwarfs [Fe/H] is lower than solar
($\sim$ -0.5). The extended star formation history then could also cause
considerable Ca-enrichment, leading for the dwarf ellipticals to larger
[Ca/Fe] values, since they accrete material from the metal-rich cluster
environment, rather than the Fornax dwarf, which accretes more pristine gas in
the Local Group, which causes lower [Ca/Fe] ratios. An important clue might be
strong correlation between [Na/Fe] and [Fe/H], when one considers Local Group
dwarf galaxies like Fornax and the LMC, dwarf elliptical galaxies, the disk of
the Milky Way, and the centers of giant elliptical galaxies. All this could be
due to Na-yields that depend strongly on metallicity. The abundance of sodium
influences the electron pressure so that the strength of many other features
are affected. For example, Conroy & van Dokkum (2012) show that for massive
galaxies an increase in the sodium abundance causes a decrease in the
abundance of CaII, which causes a decrease in the EW of CaT. Increasing the
sodium abundance can mimic the effects of a more bottom-heavy IMF. For dwarfs,
however, we see a different behavior comparing Ca and Na, although there is no
evidence that the IMF is responsible here. And also, for the LMC both [Na/Fe]
and [Ca/Fe] have the same sign, implying that another parameter has to
responsible for the difference between LMC and the dEs, presumably the star
formation history. However, with this considerable ($\sim$ 0.2) difference in
[Ca/Fe] between the LMC and the dEs, it is not so obvious that objects like
the LMC should be the progenitors of dEs, unless Ca-enrichment by SN Ia in the
cluster environment was particularly effective.

What is clear is that the abundance ratio pattern in dwarf ellipticals is very
different from that in massive Galactic globular clusters, which show enhanced
[Na/Fe], a strong Na-O anticorrelation etc. in many stars. It is of course
still possible that a fraction of the stars displays these effects, but that
fraction is so small that it cannot be detected in integrated light. This
difference probably indicates that star formation timescales in dwarf
ellipticals are long, on the order of Gyrs, since these globular clusters must
have been formed on very small timescales, with their ages being so large
(Gratton et al. 2001).

##  5 Conclusions

  * •

In this paper, we determine abundance ratios of a sample 37 dEs in the Virgo
cluster as a part of the SMAKCED project. This sample is representative of the
analysis of the kinematic properties (Toloba et al., 2014a) and also all
morphological sub-classes found by Lisker et al. (2006a); Lisker et al.
(2006b, 2007).

  * •

We use optical spectroscopy to measure a total of 23 Lick indices in the LIS-5
Å flux calibrated system and apply the MILES models to interpret them. We
derive new age and metallicity estimates for these galaxies. Taking advantage
of high resolution spectral data we are able to calculate the abundance ratios
of Na and Mg using the models of MILES.

  * •

We find the unusual behaviour that [Na/Fe] is under-abundant w.r.t. solar.
This is opposite to what is found in massive giant elliptical galaxies. We
also find that dEs fall on a relatively tight relation between [Na/Fe] and
[Fe/H], which we recently presented in Mentz et al. (2016), including Local
Group dwarf galaxies, the Milky Way and giant elliptical galaxies. From our
results, we try to sketch a possible scenario for the evolution of dEs in the
Virgo cluster. We find that dEs show disk-like star formation histories
favouring them to originate from star forming spirals or dwarfs.

  * •

Na-yields appear to be very metal-dependent, in agreement with studies of
giant ellipticals, probably due to the large dependence on the neutron-excess
in stars.

  * •

We conclude that dEs have undergone a considerable amount of chemical
evolution, they are therefore not uniformly old, but have extended star
formation histories, similar to many of the Local Group galaxies.

## Acknowledgements

We thank an anonymous referee for his critical comments, helping to improve
the paper. We also thank to F. La Barbera for kindly providing the new models.
The research was supported by The Scientific and Technological Research
Council of Turkey (TUBITAK) under project number 1059B141401204 and
1649B031406124. RFP, AV and JF-B acknowledge support from grant
AYA2016-77237-C3-1-P from the Spanish Ministry of Economy and Competitiveness
(MINECO). Paudel acknowledges the support by Samsung Science & Technology
Foundation under Project Number SSTF-BA1501-0.

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##  Appendix A Ca abundances

Given the fact that the conversion between Ca4227 and [Ca/Fe] is more
uncertain than the ones beween NaD and [Na/Fe] and between Mgb and [Mg/Fe], we
have put the discussion of the Ca abundances in the appendix.

For our dEs sample, the measured Ca4227 values are slightly higher than the
model predictions for solar-abundance models (Fig. 3), implying that Ca is
slightly over-abundant in dEs. One might get a very rough idea about the
values of [Ca/Fe] when using the only available models, those by CvD for old
ages and solar metallicities, to convert Ca4227 to [Ca/Fe]. Although we know
that these models are inaccurate, since they must depend on age and
metallicity (see the MILES ones), they can serve here to give a crude idea.
Using this calibration, Figure 5 shows that [Ca/Fe] is a bit larger than in
the Milky Way disk, but lower than in the thick disk and halo. An
interpretation of these high values could that SN Ia’s are partially
responsible for the Ca-enrichment (see Travaglio et al., 2004 and above).

Previous abundance measurements of Ca (Michielsen et al. 2003) in dEs show two
groups: the ones with CaT* $\sim$ 7Å , higher than expected for solar
abundance ratio models, and a sample of 3 objects with CaT* $\sim$ 5Å . Such
lower values could be obtained by having a range in ages, while including some
very young objects. The higher values of 7Å , however, can only be explained
by [Ca/Fe] values that are somewhat larger than solar, consistent with this
paper. Our results are also consistent with the line strengths in the Fornax
cluster dwarf NGC 1396 (Mentz et al., 2016). Such high [Ca/Fe] values are not
seen in giant ellipticals, where the CaT* index is lower than expected from
stellar population models with solar abundance ratios (Saglia et al., 2002;
Cenarro et al., 2003). While the latter can be explained by a bottom-heavy
IMF, a much easier and more natural explanation would be in [Ca/Fe] < 0 for
such galaxies (Cenarro et al., 2003, see also Conroy & van Dokkum, 2012). For
dEs the CaT* values are so high that IMF-slopes steeper than Salpeter are
excluded (Mentz et al., 2016), so solar or super-solar [Ca/Fe] values are
unavoidable. Note, however, the difference between the dEs and the LMC, which
has much lower [Ca/Fe] values, but a similar metallicity. We argue that the
difference is due to the enrichment by SN type Ia and different star formation
histories.

Prochaska, Rose & Schiavon (2005) suggested that a CN band might be affecting
the Ca4227 index in giant ellipticals, but since dEs do not show strong CN
bands, the Ca abundance can be measured reliably from the Ca4227 index
(Vazdekis et al., 1997).

Figure 5: Ca abundances as a function of metallicity [Fe/H]. Description of
the points: as in Figure 4.


# Active Front-End Rectifier Modeling and Analysis

Aidar Zhetessov and Nathan Petersen

###### Abstract

Modeling and comprehensive analysis of the Active Front-End (AFE) rectifier is
addressed in this work. The particular emphasis is on the comparison of two
state feedback control frameworks: state-space and frequency domain. The
equivalence of both frameworks for the AFE case was shown. Moreover,
analytical relations between control gains and system parameters was derived,
allowing for explicit gain scheduling as a function of operating point over
the mission profile. The state observation of grid currents has also been
addressed. Finally, the work shows the example of robustness evaluation in
presence of system variation using the Lyapunov functions approach. The system
in nominal conditions was shown to be robust with respect to $50\%$ AC
hardware parameter variations.

##  I Introduction

The power electronic supply of high-power electrical systems from the three-
phase ac mains is usually carried out using three-phase rectifier systems.
Unlike passive rectifiers, Active Front-End (AFE) rectifiers utilize their
active components and controls to regulate the output dc voltage to the
desired values while keeping high quality of the power drawn from the grid
[1]. In addition to establishing a smooth grid-load power electronic
interface, the same AFE topology can be used to provide further grid services
like reactive power compensation and, more recently, even primary grid-forming
control [2].

Proper operation of the AFE in mentioned applications depends on the control
structure employed with AFE. Moreover, as it was shown in [3], not only the
operation, but even the AFE hardware design itself might depend on the
employed control structure for a given set of performance requirements.
Namely, [3] has found the regulator-induced constraints on electrical
parameters of the rectifier ($L,C,f_{sw}$). These constraints limit the design
search space of the rectifier, while guaranteeing the prescribed performance
requirements. Overall, AFE rectifier control influences hardware
design/optimization as well as proper operation in various classical and
emerging applications. Thus, it is interesting to perform a comprehensive
analysis of the AFE rectifier controls from different perspectives.

This project addresses the AFE rectifier modeling, control, and state
observation aspects in two frameworks: state-space and frequency domain. The
state-space approach to the control and observation problems, taught in
ECE717, is to be juxtaposed to the frequency domain approach, undertaken in
[3] and ME746. By doing this, one expects to gain a better understanding of
the two control frameworks and some insights that each of these perspectives
can provide using the example of the AFE rectifier system.

The rest of the project is organized as follows: Section II derives the AFE
rectifier small-signal model of interest. The control and observation problems
are analyzed and discussed from different perspectives in Sections III and IV,
respectively. The robust evaluation of the system under parameter variations
is addressed in Section V. Finally, the work is summarized in Section VI.

##  II AFE Rectifier Modeling

The proposed idea was to model, analyze, and control the grid-connected three-
phase AFE rectifier, realized as a standard six-switch two-level three-phase
topology. The rectifier can be modeled to feed the resistive, constant
current/power, or full ZIP-load. An example schematic of the rectifier with
resistive load is shown in Fig. 1.

Figure 1: Active Front End Rectifier Schematic Diagram

Here, the power comes from the grid, which is denoted by the balanced three-
phase voltage sources $V_{A},V_{B},V_{C}$. The three-phase grid current passes
through primary passive filter components ($L$ and $R$ same for three phases)
to enter the switching stage of the rectifier. The switching stage transfers
AC currents to DC side as prescribed by the control law of the rectifier (yet
to be defined). The converted DC current $i_{dc}$ feeds the DC-link capacitor
$C$ and resistive load $R_{load}$ as per Fig. 1.

The initial nonlinear model and electrical parameters are taken from [3].
Table I summarizes the respective specifications.

TABLE I: System Specifications Parameter | Symbol | Value | Unit
---|---|---|---
AC Input LL RMS Voltage | $V_{gLL}$ | 230 | Vrms
AC Input Ph Pk Voltage | $V_{g}$ | 187.8 | V
DC-bus Voltage | $V_{dc}$ | 400 | V
Rated Power | $P$ | 25 | kW
Rated Load Resistance | $R$ | 6.4 | $\Omega$
Switching Frequency | $f_{sw}$ | 10 | kHz
Grid Frequency | $f_{o}$ | 60 | Hz
Input Inductance | $L$ | 0.34 | mH
DC-bus Capacitance | $C$ | 1300 | uF
$L$ Series Resistance | $r$ | 5 | m$\Omega$
Operating Point d-Current | $I_{gd}$ | 88.96 | A
Operating Point q-Current | $I_{gq}$ | 0 | A
Operating Point d-Duty | $M_{d}$ | 0.4684 | -
Operating Point q-Duty | $M_{q}$ | -0.0285 | -
Common-Mode Duty | $M_{CM}$ | 0.5 | -
Current Control Bandwidth | $f_{i}$ | 1 | kHz
Voltage Control Bandwidth | $f_{v}$ | 100 | Hz

The nonlinear large-signal model in [3] comes from the averaged
voltage/current equations written in the dq-frame aligned with the grid
voltage ($v_{gd}$):

$\begin{split}&L\frac{i_{gd}}{dt}=v_{gd}-m_{d}v_{dc}-i_{gd}r+\omega_{0}Li_{gq}\\\
&L\frac{i_{gq}}{dt}=v_{gq}-m_{q}v_{dc}-i_{gq}r-\omega_{0}Li_{gd}\\\
&C\frac{v_{dc}}{dt}=\frac{3}{2}\big{(}m_{d}i_{gd}+m_{q}i_{gq}\big{)}-\frac{v_{dc}}{R}\\\
\end{split}$ (1)

After perturbing/linearizing the nonlinear plant equations (1) one can arrive
at the following steady-state operating point relations (2) and state-space
plant model (1):

$\begin{split}M_{d}V_{dc}&=V_{gd}-I_{gd}r\\\
M_{q}V_{dc}&=-\omega_{0}LI_{gd}\\\
P=\frac{V_{dc}^{2}}{R}&=\frac{3}{2}\big{(}M_{d}V_{dc}I_{gd}\big{)}\\\
\end{split}$ (2)
$\begin{split}&\underbrace{\begin{bmatrix}\dot{\tilde{i}}_{gd}\\\
\dot{\tilde{i}}_{gq}\\\
\dot{\tilde{v}}_{dc}\end{bmatrix}}_{\mathbf{\dot{x}}}=\underbrace{\begin{bmatrix}-\frac{r}{L}&\omega_{0}&-\frac{M_{d}}{L}\\\
-\omega_{0}&-\frac{r}{L}&-\frac{M_{q}}{L}\\\
\frac{3M_{d}}{2C}&\frac{3M_{q}}{2C}&-\frac{1}{CR}\end{bmatrix}}_{\mathbf{A}}\underbrace{\begin{bmatrix}\tilde{i}_{gd}\\\
\tilde{i}_{gq}\\\ \tilde{v}_{dc}\end{bmatrix}}_{\mathbf{x}}+\\\
&\underbrace{\begin{bmatrix}-\frac{V_{dc}}{L}&0\\\ 0&-\frac{V_{dc}}{L}\\\
\frac{3I_{gd}}{2C}&\frac{3I_{gq}}{2C}\end{bmatrix}}_{\mathbf{B_{1}}}\underbrace{\begin{bmatrix}\tilde{m}_{d}\\\
\tilde{m}_{q}\end{bmatrix}}_{\mathbf{u_{1}}}+\underbrace{\begin{bmatrix}\frac{1}{L}&0\\\
0&\frac{1}{L}\\\
0&0\end{bmatrix}}_{\mathbf{B_{2}}}\underbrace{\begin{bmatrix}\tilde{v}_{gd}\\\
\tilde{v}_{gq}\end{bmatrix}}_{\mathbf{u_{2}}}\\\
&\underbrace{\begin{bmatrix}\tilde{v}_{dc}\end{bmatrix}}_{\mathbf{y}}=\underbrace{\begin{bmatrix}0&0&1\\\
\end{bmatrix}}_{\mathbf{C}}\underbrace{\begin{bmatrix}\tilde{i}_{gd}\\\
\tilde{i}_{gq}\\\
\tilde{v}_{dc}\end{bmatrix}}_{\mathbf{x}},\mathbf{D}=\mathbf{0}\end{split}$
(3)

Here the state variables were selected to be small-signal grid dq-frame
currents and DC voltage. The input $B$ matrix was split in two parts to
differentiate between manipulated inputs (small-signal modulation indices) and
disturbances (small-signal grid voltage disturbances). Output is one of the
state variables to be controlled - small-signal DC voltage. The other state
variable of interest is the small-signal q-axis grid current. $M_{d}$,
$M_{q}$, $I_{gd}$, $I_{gq}=0$, $V_{dc}$, $V_{gd}$ and $V_{gq}=0$ are the
operating point large-signal parameters, around which the linearization was
performed. At rated conditions and in dq-frame aligned with grid $v_{gd}$
voltage, these steady state parameters are evaluated in Table I. Note that the
common-mode duty cycle $M_{CM}$ reflects the steady state voltage between the
grid star point and the lower rail of the rectifier. For a standard PWM
modulation assumed in this work, the mentioned voltage is kept constant at
$V_{CM}=M_{CM}V_{dc}=0.5V_{dc}$.

##  III AFE Control in Two Frameworks: State-Space and Frequency Domain

This section investigates the State Feedback (SFB) Control applied to the AFE
rectifier and looks at the control problem from two perspectives: State-Space
(SS, ECE717) and Frequency Domain (FD, ME746). Originally, full state feedback
control method pertains to the linear systems theory and state-space framework
taught in linear systems classes like ECE717. On the other hand, there is so-
called ”full state feedback” approach to the control in the frequency domain
framework taught in ME746 controls class, originating from the work of Prof.
Lorenz at UW-Madison. The objective of this section is to compare/contrast the
two perspectives and conclude whether “full state feedback” control in
frequency domain (FD) is actually implementing the same control law as the
original state feedback control in state-space (SS). AFE rectifier serves as
the plant to be controlled in both cases, with the modeling addressed in
Section II. Note that since this section addresses control problem only, it is
assumed that the full state of the system is known and the grid voltage
disturbances are known and negligible. Subsequently we relax some of these
assumptions in the context of observation problem. The other assumptions are:
perfect knowledge of the system matrices, perfect system modeling via these
matrices, and the constant linearized operating point.

This section comprises four subsections:

  1. 1.

Controllability Analysis: a prerequisite to the control design of the AFE
plant.

  2. 2.

State-Space SFB Controller Design: conventional approach to full state
feedback control.

  3. 3.

Frequency Domain SFB Controller Design: “physics-based” approach to full state
feedback control.

  4. 4.

Comparison of SS & FD SFBs: representation of FD SFB block diagram in
equivalent SS SFB matrix form, vice versa, and comparison.

###  III-A Controllability Analysis

Figure 2: Frequency domain state feedback controller design using the
techniques from ME746: Disturbance Input Decoupling (DID), State Feedback
Decoupling (SFBD), Virtual Zero Reference Decoupling (VZRD), Command
Feedforward (CFF).

From linear system theory, the controllability of the system can be checked by
controllability matrix:

$\displaystyle\mathcal{C}_{\text{CTR}}=\begin{bmatrix}\mathbf{B}&\mathbf{AB}&\cdots&\mathbf{A}^{n-1}\mathbf{B}\end{bmatrix}$
(4)

and if $\text{rank}(\mathcal{C}_{\text{CTR}})=n$, the system is called
controllable. This means that the state vector $\mathbf{x}$ can be steered
back to the origin from any initial state $\mathbf{x}_{0}\in\mathbb{R}^{n}$.

Plugging in $n=3$ and the matrices $\mathbf{A}$ and $\mathbf{B}_{1}$ from (3),
the controllability matrix is computed as:

$\displaystyle\mathbf{\mathcal{C}}_{\text{CTR}}=\begin{bmatrix}-\frac{V_{dc}}{L}&0&\frac{3I_{gd}}{2C}\\\
0&-\frac{V_{dc}}{L}&\frac{3I_{gq}}{2C}\\\
\mathcal{C}_{\text{CTR},1,3}&\mathcal{C}_{\text{CTR},2,3}&\mathcal{C}_{\text{CTR},3,3}\\\
\mathcal{C}_{\text{CTR},1,4}&\mathcal{C}_{\text{CTR},2,4}&\mathcal{C}_{\text{CTR},3,4}\\\
\mathcal{C}_{\text{CTR},1,5}&\mathcal{C}_{\text{CTR},2,5}&\mathcal{C}_{\text{CTR},3,5}\\\
\mathcal{C}_{\text{CTR},1,6}&\mathcal{C}_{\text{CTR},2,6}&\mathcal{C}_{\text{CTR},3,6}\\\
\end{bmatrix}^{\top}$ (5)

where the third column entries are derived as follows:

$\begin{split}\mathcal{C}_{\text{CTR},1,3}&=\frac{rV_{dc}}{L^{2}}-\frac{3I_{gd}M_{d}}{2CL}\\\
\mathcal{C}_{\text{CTR},2,3}&=\frac{\omega_{0}V_{dc}}{L}-\frac{3I_{gd}M_{q}}{2CL}\\\
\mathcal{C}_{\text{CTR},3,3}&=-\frac{3I_{gd}}{2C^{2}R}-\frac{3M_{d}V_{dc}}{2CL}\end{split}$
(6)

The rest of the entries are not included due to their size. Checking the rank
of $\mathcal{C}_{\text{CTR}}$ from (LABEL:eq:c_ctr) using MATLAB, it is indeed
3, meaning that the system is controllable.

###  III-B State-Space State Feedback Controller Design

Provided controllability of the system, the state feedback controller can
place the closed-loop eigenvalues to any desired locations. Generally, the
desired eigenvalue locations are selected as close to the real axis and as far
to the left-half-plane of the real-imaginary plane as possible. This is to
ensure faster convergence, less overshoots, less oscillations during
transients, and greater disturbance attenuation/rejection. On the other hand,
the eigenvalue locations are limited by the system saturation/slew rate
limits, noise attenuation/rejection specifications, and model inaccuracies,
which tend to become more pronounced at higher frequencies and set the
constraints on system robustness.

Within the guidelines above, state feedback allows placing eigenvalues near or
far from each other. This quite different from the conventional FD control
design methods, which often utilize time scale separation of the cascaded
control loops. It would be interesting to investigate how does the state
feedback eigenvalue placement impact the manipulated input requirements,
transients etc. This work, though, is limited only to the comparison of two
frameworks of state feedback control design. Therefore, for both SS SFB and FD
SFB controllers the conventional AFE eigenvalue locations are selected.

Given the switching frequency of $10$ kHz from Table I, within the
conventional setting, two inner current loop closed-loop poles are chosen to
be at $1$ kHz ($\lambda_{1,2}=-2\pi f_{i}=-2\pi\cdot 1000$ Hz). The outer
voltage loop pole is selected to be at $100$ Hz ($\lambda_{3}=-2\pi
f_{v}=-2\pi\cdot 100$ Hz) due to a decade frequency separation between the
loops. These poles are to be fixed for both controllers of this work.

Taking into account all the assumptions above, the state dynamics from (3)
with SS SFB controller take the following form:

$\mathbf{\dot{x}}=\mathbf{A}\mathbf{x}+\mathbf{B_{1}}\underbrace{(-\mathbf{K_{ss}}\mathbf{x})}_{\mathbf{u_{1}}}$
(7)

Here $\mathbf{K_{ss}}\in\mathbb{R}^{2\times 3}$ is a SS SFB controller matrix
that moves the closed-loop eigenvalues of the state dynamics to the desired
locations. Using the Place command in MATLAB one can find $\mathbf{K_{ss}}$
provided $\mathbf{A}$, $\mathbf{B_{1}}$, and the list of desired eigenvalues
$\mathbf{\Lambda}=[\lambda_{1},\lambda_{2},\lambda_{3}]$:

$\begin{split}\mathbf{K_{ss}}&=\textsc{Place}(\mathbf{A},\mathbf{B_{1}},[\lambda_{1},\lambda_{2},\lambda_{3}])=\\\
&=\begin{bmatrix}-5.9682&-0.3204&-2.3864\\\ 0.3204&-5.3283&-0.0713\\\
\end{bmatrix}\cdot 10^{-3}\end{split}$ (8)

Note that for multi-input systems the algorithm uses extra degrees of freedom
to find a solution $\mathbf{K}$ that minimizes the sensitivity of closed-loop
poles to perturbations in $\mathbf{A}$ and $\mathbf{B_{1}}$ [4]. Obtained SS
SFB controller is to be compared/contrasted to its FD SFB counterpart.

###  III-C Frequency Domain State Feedback Controller Design

The classical AFE frequency domain controller is a cascaded design with inner
current and outer voltage loops, as in [3]. For such cascaded designs,
additional ”physics-based” FD techniques were proposed in ME746 course, which
result in so-called ”state feedback” (SFB) controller, just like the
conventional SFB in state-space framework. Although these techniques do not
nominally change the cascaded control structure, they end-up eliminating
additional control states associated with integrators, making the control
structure depend only on the feedback states. The original SS SFB control also
does not introduce additional states, hence the similarity and the name is
deduced. Generally, there are practical reasons why one might want to keep the
cascaded control structure as opposed to explicit SS SFB configuration and
vice versa. Those will be touched upon in the next subsection. In this
subsection, starting from the control structure in [3] and applying the
aforementioned techniques from ME746, one derives the FD SFB controller design
for the AFE in Fig. 1 and (3).

The classical cascaded controller (in black) with ”physics-based”
modifications (in color) for the AFE plant is depicted in the closed-loop
block diagram of Fig. 2. To the right of the dashed vertical line one can find
the detailed block diagram of the AFE plant (3). To the left of the dashed
vertical line one can find the controller. The integrator gains were removed
from the conventional controller in black due to the modifications in color.
Classical structure uses d- and q-manipulated inputs to regulate d- and
q-current components respectively. The q-component current reference is zero,
while the d-component current reference is dictated by the outer cascaded DC
voltage controller.

Four frequency domain techniques from ME746 were used to bring the classical
controller to the ”SFB” form: Disturbance Input Decoupling (DID), State
Feedback Decoupling (SFBD), Virtual Zero Reference Decoupling (VZRD), and the
Command Feedforward (CFF). DID (red in Fig. 2) attempts to cancel the external
disturbances by measuring and applying the same and opposite signals through
manipulated inputs. In the AFE case, grid voltage small-signal disturbances
could be decoupled as shown. SFBD (blue in Fig. 2) attempts to cancel the
influence of all other states on the given state dynamics by measuring and
applying the same and opposite signals through manipulated inputs. For
example, from the plant in can be seen that the $\tilde{i}_{gd}$ dynamics are
influenced by the $\tilde{v}_{dc}$ through $M_{d}$ and the relevant SFBD
decouples this effect. VZRD (green in Fig. 2) can be regarded as the subset of
SFBD, which accounts for the influence of the state itself on its own
dynamics. Together, DID, SFBD, and VZRD break down the plant dynamics such
that it becomes the first order system, which can be easily controlled with
zero steady state error using only the proportional gain. For example,
focusing only on the $\tilde{i}_{gd}$ dynamics, DID, SFBD, and VZRD cancel all
the influences of the plant on $\tilde{i}_{gd}$, leaving simple $1/sL$ as the
plant to be controlled. By choosing the gain $K_{id}=\omega_{i}L$, one simply
places the closed-loop pole of the inner $\tilde{i}_{gd}$ loop to the desired
location $\omega_{i}$. Finally, CFF (gray in Fig. 2) accounts for potentially
non-constant commands into the respective loops and feeds those forward to the
manipulated inputs to ensure the best command tracking (no phase lag and
signal attenuation).

Once all the control modifications are made, one can tune the respective
cascaded loops to the desired bandwidths that match SS SFB controller and
Table I. For the q-axis current loop gain:

$T_{q}(s)=\frac{K_{iq}}{sL}\rightarrow K_{iq}=2\pi f_{i}L=\omega_{i}L$ (9)

For the d-axis cascaded current and voltage closed-loop transfer function:

$\begin{split}G_{d}(s)&=\frac{1}{\frac{s^{2}CL}{K_{id}K_{v}}+\frac{sC}{K_{v}}+1}=\frac{1}{\frac{s^{2}}{\omega_{i}\omega_{v}}+\frac{s(\omega_{v}+\omega_{i})}{\omega_{i}\omega_{v}}+1}\\\
\rightarrow K_{id}&=(\omega_{i}+\omega_{v})L,\quad
K_{v}=\frac{\omega_{i}\omega_{v}C}{(\omega_{i}+\omega_{v})}\\\ \end{split}$
(10)

Finally, by setting the $\tilde{v}^{*}_{dc}=0$ and multiplying out all the
gains for dc-conditions (since both references are at zero) one arrives at the
following equations for the manipulated inputs:

$\begin{split}\tilde{m}_{d}&=-k_{11}\tilde{i}_{gd}-k_{12}\tilde{i}_{gq}-k_{13}\tilde{v}_{dc}\\\
\tilde{m}_{q}&=-k_{21}\tilde{i}_{gd}-k_{22}\tilde{i}_{gq}-k_{23}\tilde{v}_{dc}\\\
\end{split}$ (11)

Where the gains $k_{ij}$ are as follows:

$\begin{split}k_{11}&=\frac{r-K_{id}}{V_{dc}}=-5.8623\cdot 10^{-3}\\\
k_{12}&=-\frac{\omega_{o}L}{V_{dc}}=-0.3204\cdot 10^{-3}\\\
k_{13}&=\frac{M_{d}}{V_{dc}}-\frac{2K_{id}V_{dc}[K_{v}-R^{-1}]-3K_{id}I_{gd}M_{d}}{3V_{dc}[M_{d}V_{dc}-I_{gd}r]}=\\\
&=-2.4338\cdot 10^{-3}\\\ k_{21}&=\frac{\omega_{o}L}{V_{dc}}=0.3204\cdot
10^{-3}\\\ k_{22}&=\frac{K_{iq}-r}{V_{dc}}=-5.3282\cdot 10^{-3}\\\
k_{23}&=\frac{M_{q}}{V_{dc}}=-0.0713\cdot 10^{-3}\\\ \end{split}$ (12)

Or, after collecting together in the matrix $\mathbf{K_{fd}}$:

$\mathbf{K_{fd}}=\begin{bmatrix}-5.8623&-0.3204&-2.4338\\\
0.3204&-5.3282&-0.0713\\\ \end{bmatrix}\cdot 10^{-3}$ (13)

Obtained FD SFB controller in matrix form is to be compared to its SS SFB
counterpart.

###  III-D Comparison of State-Space & Frequency Domain SFBs

Comparing the controllers in (8) and (13) one can see that they are within 2%
difference from each other in the worst case for the operating point at hand.
Other operating points were also evaluated, and the observed alignment holds
true for those points as well. Therefore, at least for the AFE case, one can
conclude that ”physics-based” full state feedback control in frequency domain
does implement the original state feedback control in state space and that the
”state feedback” term usage in this case is correct.

Moreover, one can claim that the designed FD SFB controller has the highest
robustness to parameter variations in $\mathbf{A}$ and $\mathbf{B_{1}}$, since
it is the practically the same as the SS SFB controller, for which the
robustness claim is true due to Place command and [4]. More on robustness
evaluation can be found in Section V.

Comparing the design effort of the controllers, state space approach is more
attractive. Indeed, the state space approach facilitates fast and precise
closed-loop pole placement, proven robustness guarantees [4], and easy command
feedforward realization without the need to consider cascaded loop
interactions.

Comparing the practical realization aspects of the controllers, frequency
domain is more appealing. Indeed, even without additional controller states,
it uses cascading, which is intuitively useful to separate the low-energy fast
state dynamics from high-energy slow state dynamics. Although in the case of
AFE this separation is not that obvious, at least from operating point
evaluation the capacitive energy storage $\frac{CV_{dc}^{2}}{2}=104J$ is far
greater than inductive energy storage $\frac{LI_{gd}^{2}}{2}=1.34J$. From the
safety perspective, it is useful to have an explicit current loop, on which
the current reference limiter can be applied. From the discrete control
perspective, faster state (current) can be sampled at much higher frequency
than slower state (voltage), implying less computational hardware allocation
requirements for the outer loops computations. Also, lower sampling frequency
for the outer states increases the resolution per sample. From the mission
profile operation perspective, the fact that FD SFB controller provides
explicit analytic relations (12) between controller gains and the operating
points is very useful, as one can implement gain scheduling with variable
gains for secure operation at different operating points over the mission
profile. On the other hand, the SS SFB controller $\mathbf{K_{ss}}$ relation
to the parameters in $\mathbf{A}$ and $\mathbf{B_{1}}$ is not that obvious.
Finally, time scale separating the inner and outer loops makes system more
robust. Qualitatively, when the eigenvalues are far from each other, it takes
a lot of parameter variation to significantly change the closed-loop dynamics,
while close eigenvalues can move/break off real axis with some parameter
variation, significantly changing the nature of the dynamics. More on
robustness is in Section V.

##  IV AFE State Observer

The system formulation (3) is such that only voltage sensors are used for
feedback. This means that the dc voltage $\tilde{v}_{dc}$ is measured, but
also the dq-frame grid voltages $\tilde{v}_{gd}$ and $\tilde{v}_{gq}$ are
measured. In (3), the grid voltages are contained in the input
$\mathbf{u}_{2}$; they are not a state variable. To implement control of the
AFE, the state vector is required including the currents $\tilde{i}_{gd}$ and
$\tilde{i}_{gq}$. Since the currents are not measured directly, they must be
estimated.

In this section, the observability of the AFE is investigated. Then, an
observer structure is proposed to estimate the unmeasured currents.

###  IV-A Observability Analysis

From linear system theory, controllability is related to observability through
the dual system formulation. In the end, the observability matrix is defined
as

$\displaystyle\mathcal{O}_{\text{OBS}}=\begin{bmatrix}\mathbf{C}\\\
\mathbf{CA}\\\ \vdots\\\ \mathbf{CA}^{n-1}\end{bmatrix}$ (14)

and if $\text{rank}(\mathcal{O}_{\text{OBS}})=n$, the system is called
observable. This means that the state vector $\mathbf{x}$ can be inferred from
knowledge of the output $y$ and input $u$, and the system matrices.

Plugging in $n=3$ and the matrices $\mathbf{A}$ and $\mathbf{C}$ from (3), the
observability matrix is computed as:

$\displaystyle\mathbf{\mathcal{O}}_{\text{OBS}}=\begin{bmatrix}0&0&1\\\
\frac{3M_{d}}{2C}&\frac{3M_{q}}{2C}&-\frac{1}{RC}\\\
\mathcal{O}_{\text{OBS},3,1}&\mathcal{O}_{\text{OBS},3,2}&\mathcal{O}_{\text{OBS},3,3}\\\
\end{bmatrix}$ (15)

where

$\displaystyle\mathcal{O}_{\text{OBS},3,1}$
$\displaystyle=\frac{-3R(LM_{q}\omega_{0}+M_{d}r)C-3M_{d}L}{2C^{2}LR}$
$\displaystyle\mathcal{O}_{\text{OBS},3,2}$
$\displaystyle=\frac{3R(LM_{d}\omega_{0}-M_{q}r)C-3M_{q}L}{2C^{2}LR}$
$\displaystyle\mathcal{O}_{\text{OBS},3,3}$
$\displaystyle=\frac{-3R^{2}(M_{d}^{2}+M_{q}^{2})C+2L}{2C^{2}LR^{2}}$

The rank of $\mathcal{O}_{\text{OBS}}$ from (LABEL:eq:o_obs) is indeed 3,
meaning that the system is observable.

###  IV-B Observer Design

Figure 3: Proposed observer structure to estimate the unmeasured currents,
$\tilde{i}_{gd}$ and $\tilde{i}_{gq}$, which are contained in
$\hat{\mathbf{x}}$.

Since the system is observable per Section IV-A, an observer can be
constructed which can accurately estimate the unmeasured states:
$\tilde{i}_{gd}$ and $\tilde{i}_{gq}$. The observer is formed using the system
matrices from (3): $\mathbf{A}$, $\mathbf{B}_{1}$, $\mathbf{B}_{2}$, and
$\mathbf{C}$. It is assumed that all matrices are known perfectly and that the
real system ideally follows the linear model. Furthermore, the operating point
at which the system is linearized must be constant.

The block diagram of the proposed observer is shown in Fig. 3 using the
notation from (3). The input to the observer is split into two parts: (i) the
control input: $\mathbf{u}_{1}$, and (ii) the measured voltages:
$\mathbf{u}_{2}$ and $\mathbf{y}$. The feedback matrix gain $\mathbf{L}$ is
used to tune the performance of the observer, and thus the estimation
convergence properties.

###  IV-C Observer Tuning

The observer feedback gain $\mathbf{L}$ is critical to the estimation
performance. The observer dynamics are governed by:

$\displaystyle\dot{\hat{\mathbf{x}}}$
$\displaystyle=\mathbf{A}\hat{\mathbf{x}}+\mathbf{B}_{1}\mathbf{u}_{1}+\mathbf{B}_{2}\mathbf{u}_{2}+\mathbf{L}(\mathbf{y}-\hat{\mathbf{y}})$
$\displaystyle\hat{\mathbf{y}}$ $\displaystyle=\mathbf{C}\hat{\mathbf{x}}$
(16)

Defining the state estimation error as
$\mathbf{e}=\mathbf{x}-\hat{\mathbf{x}}$, the observer dynamics can be written
in terms of the $\mathbf{e}$ as:

$\displaystyle\dot{\mathbf{e}}=(\mathbf{A}-\mathbf{L}\mathbf{C})\mathbf{e}$
(17)

The dynamics of the estimation error (17) can be arbitrarily set by placing
the eigenvalues of $\mathbf{A}-\mathbf{L}\mathbf{C}$ via selecting
$\mathbf{L}$. Recall that this is possible because the system is observable.
Typically, the state estimation dynamics are selected to be at least an order
of magnitude faster than the control dynamics.

##  V Robustness Evaluation through Lyapunov Functions

Throughout the work there were several mentions of system robustness with
respect to parameter variations. MATLAB Place command uses the eigenvalue
sensitivity to $\mathbf{A}$ and $\mathbf{B_{1}}$ parameter variations as the
robustness representation. Qualitative discussion on FD SFB controller
mentions closed-loop pole separation as the sign of a robust system. To
evaluate system robustness more precisely, one can use the Lyapunov stability
approach to the time-varying systems.

For a closed-loop time varying systems with certain bounds on parameter
variations in $\mathbf{A_{cl}}(t)$, one can conclude asymptotic stability of
the system irrespective of the temporal nature of parameter variations within
the defined bounds, if there exists a positive-definite Lyapunov function
$V(\mathbf{x})$ with a negative-definite time derivative for all possible
$\mathbf{A_{cl}}(t)$.

Consider $50\%$ parameter variations in $L$ and $r$ due to
thermal/saturation/proximity effects for example. Since $L,r$ enter the system
matrices $\mathbf{A}$ and $\mathbf{B_{1}}$, entries of those will change,
affecting $\mathbf{A_{cl}}(t)=\mathbf{A}(t)-\mathbf{B_{1}}(t)\mathbf{K}$. As
those variations are not modeled, the controller matrix will not change (both
SS and FD SFB). Robust asymptotic stability of the system in such conditions
can be concluded if there exists $\mathbf{P}=\mathbf{P}^{\top}>0$ such that:

$\begin{split}V(\mathbf{x})&=\mathbf{x}^{\top}\mathbf{P}\mathbf{x}>0\\\
\dot{V}(\mathbf{x})&=\mathbf{x}^{\top}\big{(}\mathbf{A_{cl}}^{\top}(t)\mathbf{P}+\mathbf{P}\mathbf{A_{cl}}(t)\big{)}\mathbf{x}<0,\forall\mathbf{x}\neq
0,\forall t\\\ \end{split}$ (18)

To solve for $\mathbf{P}$, one can define $\mathbf{Q}=\mathbf{I}$, equate
$\dot{V}(\mathbf{x})=-\mathbf{Q}$, and use the MATLAB Lyap command to find
$\mathbf{P}$. The resulting matrix form served as the inspiration to come up
with the following energy-like matrix $\mathbf{P}$:

$\mathbf{P}=\begin{bmatrix}\frac{L}{2}&0&\frac{\sqrt{LC}}{4}\\\
0&\frac{L}{2}&0\\\ \frac{\sqrt{LC}}{4}&0&\frac{C}{2}\\\
\end{bmatrix}=\mathbf{P}^{\top}>0$ (19)

The range of parameters of interest ($0.5L\rightarrow 1.5L,0.5r\rightarrow
1.5r$) was swept with high resolution to evaluate the $\mathbf{A_{cl}}(t)$ in
each case, assuming $\mathbf{K}$ is constant. For each case, negative-
definiteness of
$\mathbf{A_{cl}}^{\top}(t)\mathbf{P}+\mathbf{P}\mathbf{A_{cl}}(t)$ was
evaluated by checking the respective eigenvalues. Here, if all eigenvalues
throughout the sweep are strictly negative, one can conclude robust asymptotic
stability of the system in presence of parameter variation. The respective
eigenvalues are depicted in Fig. 4.

Figure 4: Eigenvalue sweep of the
$\mathbf{A_{cl}}^{\top}(t)\mathbf{P}+\mathbf{P}\mathbf{A_{cl}}(t)$ matrix.

Thus, it was shown that even with AC-side parameter variations in $\pm 50\%$
range, the closed-loop system is still robustly asymptotically stable. Note
that such conclusions cannot be drawn simply from evaluating the closed-loop
eigenvalues of the system over the parameter range. Lyapunov functions offer a
powerful tools for analysis and control design. Mastering those tools further
is definitely an interesting outlook for the future.

##  VI Conclusions

This work addressed modeling and comprehensive analysis of the Active Front-
End (AFE) rectifier system from two perspectives: state-space and frequency
domain. First, the tight hardware-control interconnection was mentioned in the
introduction, motivating a deeper understanding of the AFE rectifier control
for hardware design and overall system operation. Next, small-signal
linearized model has been derived in Section II.

Using the derived model, Section III compared two state feedback control
frameworks - state-space and frequency domain. It turned out that, at least
for the AFE case, the ”physics-based” frequency domain SFB controller is
actually the original state-space SFB controller, so the term ”state feedback”
for FD SFB controller was applied correctly. In the process, one could
identify the analytical relations between SFB controller gains and the system
parameters, including operating points, which can be useful for mission
profile regulation.

Next, section IV addressed the AFE state observer design, considering only
voltage measurements being available. This can be quite useful from the
practical perspective, as the current sensors require not only sensors
themselves, but all the accompanying peripherals like traces, a/d conversion,
layout space etc.

Finally, the Section V showed how the Lyapunov functions can be used for
robustness evaluation of the time varying system under certain parameter
constrained variation conditions. It turns out that there exists a Lyapunov
function that can prove system robust asymptotic stability in the presence of
$50\%$ variation in AC hardware parameters.

## Contributions

The AFE Modeling, Control in Two Frameworks, and Robustness Evaluation were
addressed by Aidar Zhetessov. The AFE State Observation was addressed by
Nathan Petersen. Other sections were addressed in collaboration.

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