# Defining and Constraining the Electrical Cardinality
of Multiport Converter Mission Profiles

Matthew Deakin This work was supported by the the Royal Academy of Engineering
under the Research Fellowship programme, and the EPSRC Multi-energy Control of
Cyber-Physical Urban Energy Systems (MC2) project (EP/T021969/1). M. Deakin is
with Newcastle University, Newcastle-upon-Tyne, UK. Contact:
<EMAIL_ADDRESS>
###### Abstract

Mission profiles describe a representative set of conditions that a power
converter is designed to operate under, and are known to be more complicated
for multiport converter applications due to a wider range of combinations of
powers that can be transferred between ports. This paper studies the
properties of mission profiles derived from operational optimization of
multiport converters in distribution system applications (e.g., soft open
points). The electrical cardinality of the mission profile is introduced as a
useful, naturally varying property of multiport mission profiles derived from
optimal operation within distribution system, with the cardinality equal to
the number of non-zero power transfers at a given time. Furthermore, it is
shown that the cardinality can be conveniently constrained within the
framework of conventional mixed-integer conic optimization problems, yielding
a family of mission profiles that can enable converter designers to simplify
multiport designs via converter reconfiguration. Results demonstrate the
potential to reduce the cardinality in a four-terminal multiport converter
from four to two whilst still effectively supporting congestion management and
achieving 91.7% of the loss reduction capabilities of a conventional design.

###### Index Terms:

Mission Profile, Multiplexed Soft Open Point, Multiport Converter, Power
Converter Reconfiguration, Hybrid AC/DC distribution systems.

##  I Introduction

Applications of power electronics-based power converters in power systems
today are rapidly growing, with industry requiring new methods of integrating
electric transportation and renewable generation into the ac grid. Multiport
converters are an important class of grid-connected power converters, allowing
power to be efficiently transferred between different feeders in a power
distribution system. For example, a three terminal multiport converter can act
as a soft open point (SOP), connecting two ac distribution feeders through a
dc link with an integrated energy storage system that can provide temporal
flexibility.

A mission profile is a representative time series collecting the parameters
that affect the key performance indices for a given power converter (e.g.,
reliability, lifetime or efficiency). This typically will consist of at least
the thermal or electrical stresses that the converter will be under, but can
also include other relevant environmental parameters such as humidity or
vibration. Accurate mission profile simulation can be complex and time
consuming, and so there has been significant research on this topic. For
example, recent works focus on fast emulation of thermal and electrical
aspects of mission profiles including realistic switching profiles [1], or
considering comprehensive sets of parameters, such as switching frequency or
the direction of real power transfer for increasing accuracy [2].

The electric power distribution community is also concerned with the
development of mission profiles, but typically from the point of deriving
optimal power transfers for a given converter design and optimization
approach, rather than simulating the impacts on long-term converter
reliability or state of health. For example, a three-terminal ac/dc/ac
multiport, operated as an integrated SOP-energy storage device, is scheduled
to provide loss reduction, price arbitrage and robust congestion management in
[3], with the main focus of the work exploring impacts of uncertainty on the
feeder voltages. Similarly, a robust strategy is developed for a five-terminal
SOP in [4], with the main contribution a two-stage optimization combining both
semidefinite programming and droop control, again considered over a
representative day.

Of those works that do consider properties of mission profiles that result
from distribution system optimization, the metrics considered typically do not
clearly link to potential impacts on converter design. For example, average
and time-varying utilization of power transferred through individual
converters are presented to summarise performance in [5] for a two-terminal
phase changing SOP. Similarly, in [6], the utilization of individual power
converter’s mission profiles are studied as a function of variable converter
sizes. The variability of the electrical parameters of the mission profile are
shown to influence efficiency in [7], proposing redeployment of modular
converters within a multiport converter. The dimension and volume of a
multiport’s capability chart is studied in [8], with it recognized that
multiport converters with degenerate capability charts (i.e., with a
capability chart hypervolume of zero) can still yield good performance within
a network. Indeed, this final observation clearly motivates the study of
cardinality of the power transfers within mission profiles to explore and
understand how prevalent these simpler designs might be in future [8, 9].

The contribution of this paper is to introduce the electrical cardinality of a
mission profile, providing an intuitive property that can be used to study
interactions between a distribution network’s requirements and the design of
reconfigurable multiport converters. The proposed definition is particularly
useful as it conveniently links to cardinality constraints that can be imposed
in mixed-integer conic optimization problems commonly used in distribution
optimal power flow problems. The cardinality is subsequently used to explore
the properties of optimal mission profiles of a given dimension, with case
studies demonstrating a halving of the cardinality whilst still achieving
91.7% of the potential system benefits.

The paper is structured as follows. In Section II, the electrical cardinality
is introduced, with a short discussion highlighting why this naturally varies
with the outputs of optimal distribution network scheduling, then showing how
reconfigurable power converters can exploit mission profiles with reduced
cardinality. Subsequently, cardinality constraints are proposed in Section III
to demonstrate how a family of mission profiles can be developed that could
allow a converter designer to explore a range of potential reconfigurable
designs with varying complexity. Two case studies are presented and solved in
Section IV to demonstrate qualitative and quantitative attributes of mission
profiles with varying cardinality constraints. Conclusions are drawn in
Section V.

##  II Mission Profile Electrical Cardinality

The cardinality of a vector is the number of non-zero elements of that vector.
In this section, the electrical cardinality of a mission profile is introduced
from this concept, with it demonstrated how it can be determined from the
powers that are transferred to and from the ac distribution grid.
Reconfigurable multiplexed soft open points (MOPs) are then introduced as a
specific multiport topology that can take advantage of reduced cardinality,
highlighting a direct application of the electrical cardinality.

###  II-A Defining Electrical Cardinality

In this work, the electrical cardinality of a mission profile is defined as
the cardinality of the powers transferred by a multiport converter into an ac
distribution grid at a given time. It is referred to as the electrical
cardinality so as to distinguish from the thermal aspects of the mission
profile which are also of significant interest (but are not considered further
in this work). For example, consider apparent powers transferred by an
$m$-terminal SOP over duration $\tau$ collected in a matrix
$S_{\mathrm{MP}}\in\mathbb{R}^{\tau\times m}$. In this case, the electrical
cardinality ($\mathrm{EC}$) at a given time instant $\tau$ is defined as

$\mathrm{EC}[\tau]=\mathrm{nnz}(S_{\mathrm{MP}}[\tau,:])\,,$ (1)

where $\mathrm{nnz}$ returns the number of non-zero elements of a vector, with
matrix $S_{\mathrm{MP}}$ indexed using ‘Matlab’ notation. The EC can then be
used to also define the maximum electrical cardinality ($\mathrm{MEC}$) of the
mission profile as

$\mathrm{MEC}=\max_{\tau}\,\mathrm{EC}[\tau]\,.$ (2)

When the EC needs to be calculated in practise (e.g., from a mission profile
$S_{\mathrm{MP}}$ returned by a numerical optimization routine), it can be
calculated based on a tolerance $\epsilon$, e.g.,

$\mathrm{EC}[\tau]=\sum_{i=1}^{m}I_{\epsilon}(S_{\mathrm{MP}}[\tau,i])\,,$ (3)

where the indicator function $I_{\epsilon}$ returns 1 if its argument is
greater than $\epsilon$, and zero otherwise. A relative value of $\epsilon$ of
$10^{-5}$ times the total converter capacity is used in this work.

There are several reasons that the value of the power transferred through a
given terminal of a multiport converter might have a value of zero and
therefore yield time-varying electrical cardinality $\mathrm{EC}$. If a
terminal is connected to a variable generator (such as solar) then the
generator might have zero output for significant time periods. A distribution
feeder might also be out of service. If a multiport is used most of the time
for system loss reduction and arbitrage, the losses within the converter might
be sufficiently high that it is not cost-effective to run the converter.

In fact, converter losses can tend to produce optimal solutions that are
sparse in the apparent power transfers $S_{c}$. For example, for converter
losses of an $m$-terminal multiport
$P_{\mathrm{Loss}}^{\mathrm{Conv.}}\in\mathbb{R}^{m}$, loss coefficient $k$
and linear model [10, 8]

$P_{\mathrm{Loss}}^{\mathrm{Conv.}}[i]=kS_{c}[i]\;\forall\;i\in[1,\,m]\,,$ (4)

the solution of optimization problems that involve minimization of total
converter losses will tend to result in a solution which is sparse in $S_{c}$
(i.e., has few non-zero entries). This is because (4) acts as a ‘shrinkage
operator’ on $S_{c}$ [11]. Increasing values of $k$ will yield increasingly
sparse solutions in $S_{c}$ [12, Ch. 6.2].

###  II-B Exploiting Partial Electrical Cardinality

The electrical cardinality $\mathrm{EC}$ can take integer values between 0 and
$m$ (for a multiport connected to $m$ ac feeders), with cases that are of most
interest in this work being when the $\mathrm{MEC}$ is less than $m$. This is
because an $\mathrm{MEC}$ less than $m$ implies that a multiport converter
with $m$ legs has a degree of redundancy, insofar as there is potential for
the exploitation of reconfigurable multiplexed soft open point (MOP) designs
with a reduced number of ac/dc legs [8].

(a) Conventional SOP with integrated DG

(b) Dual converter MOP (variable $\alpha$, with $0<\alpha<1$)

(c) Single converter MOP

Figure 1: Single line diagram of a conventional SOP (with integrated DG
connected to the DC link), (a), as compared to reconfigurable multiplexed SOP
(MOP) multiport designs with two converters (b), or a single reconfigurable
ac/dc converter (c). All designs have the same pu ac/dc converter capacity,
but different capability charts and varying complexity of construction in
terms of the number and type of components.

For example, three multiport designs are plotted in Fig. 1, both with and
without MOP reconfiguration stages. If the maximum cardinality $\mathrm{MEC}$
has value two (i.e., it is required that at some time periods power must be
transferred to or from point of common coupling (PCC) 1 to PCC 2
simultaneously), then the conventional design (Fig. 1a) or dual converter MOP
design (Fig. 1b) can be considered. In contrast, if the $\mathrm{MEC}$ has
value unity, such that the solar power is exported only through PCC 1 or PCC 2
(or reactive power is never required for voltage control concurrently in both
feeders), then the simpler MOP design Fig. 1c can also be considered. Such a
design only requires a single ac/dc converter and can inject a full 1 pu into
either PCC 1 and PCC 2, and so has advantages over both the conventional SOP
and dual converter MOP in at least one sense.

##  III System Modelling

In the previous section, the electrical cardinality and its maximum value
$\mathrm{MEC}$ were introduced as metrics that can be used to explore the
potential degree of redundancy of non-reconfigurable multiport converters. In
this section, we first consider an additional benefit of defining the
electrical cardinality: this cardinality can be varied in a computationally
efficient way, enabling a multiport converter designer to explore the
performance of a family of mission profiles. Cardinality constraints are
introduced and then formulated using a big-$M$ procedure, so that these
constraints can be appended to existing convex optimization problems.
Subsequently, a mixed-integer second order cone program is presented that
integrates these cardinality constraints into an examplar optimal power flow
problem for a distribution system, utilizing idealised MOP designs to bound
the performance of all profiles of a given cardinality.

###  III-A Cardinality Constraints Formulated Using the Big-M Method

For an optimization decision variable $y\in\mathbb{R}^{p}$ for some integer
$p$, a cardinality constraint [13]

$\mathrm{nnz}(y)\leq q\,,$ (5)

for some $q<p$ can be recast using a big-$M$ formulation (this reformulation
is required as the constraint as-written in (5) cannot be incorporated into
integer optimization packages). Specifically, if $y\geq 0$ and $M$ is an upper
bound for each element of $y$, then $p$ auxiliary binary variables
$z_{i}\in\mathbb{B}$ can be introduced alongside the constraint

$y[i]\leq Mz_{i}\;\forall\;i\,,$ (6)

such that if the $i$th element of $y$ is zero then so will the value of
$z_{i}$; if the $i$th element of $y$ is non-zero, then $z_{i}$ will have value
one. These auxiliary variables allow for (5) to then be rewritten

$\sum_{i=1}^{p}z_{i}\leq q\,.$ (7)

Therefore, a optimization problem with constraints (6), (7) is equivalent to
having constraint (5). These equations are linear in binary variables $z_{i}$,
and so can be included directly in conventional mixed-integer conic
optimization packages.

###  III-B System model

For a reconfigurable MOP connected to an $m$-feeder node within a distribution
network, the goal is to schedule the real and reactive power flows transferred
$P_{c}\in\mathbb{R}^{m},\,Q_{c}\in\mathbb{R}^{m}$ to minimize network and
converter losses, subject to congestion constraints. The optimization
developed to solve this problem builds on the approach outlined in [8], with a
number of substantive changes as follows.

  * •

The primary topic of this work is the exploration of the solution as the
cardinality varies, and so a cardinality constraint of the form (5) is added
and subsequently solved for a range of cardinality values $q$ up to the number
of terminals $m$.

  * •

Instead of considering large numbers of variable MOP or SOP designs for a
given converter capacity, instead idealised, fully reconfigurable designs are
considered within the modelling [8]. This provides a convenient upper bound
for the performance of a design with a given cardinality, although it is
possible the mission profiles may not be realisable as designs (this is
discussed further in Section IV-C).

  * •

To model network congestion, a linearized model is used that maps power
injections to changes in voltage magnitudes. The ‘First Order Taylor’ method
is used for linearization from [14], linearized at the no-load solution.

  * •

Finally, the model of losses in power injections is based only on a quadratic
model developed on the linearization of complex voltages [14] around the no-
load point (rather than calculating a linearization at all loading conditions
separately).

With these changes, the optimization problem can be written as follows.

$\displaystyle\min_{P_{c},\,Q_{c}}$ $\displaystyle
P_{\mathrm{Loss}}^{\mathrm{Ntwk.}}+\sum_{i}P_{\mathrm{Loss}}^{\mathrm{Conv.}}[i]$
(8) $\displaystyle\mathrm{s.t.}\>S_{c}[i]$
$\displaystyle\geq\sqrt{P_{c}^{2}[i]+Q_{c}^{2}[i]}\>\forall\>i\in[1,\,n]\,,$
(9) $\displaystyle\sum P_{\mathrm{dc}}$ $\displaystyle=0\,,$ (10)
$\displaystyle P_{\mathrm{dc}}[i]+P_{\mathrm{Loss}}^{\mathrm{Conv.}}[i]$
$\displaystyle=P_{c}[i]\>\forall\>i\in[1,\,m]\,,$ (11) $\displaystyle
P_{\mathrm{dc}}[m+1]$ $\displaystyle=P_{\mathrm{DER}}\,,$ (12) $\displaystyle
P_{\mathrm{Loss}}^{\mathrm{Conv.}}[i]$
$\displaystyle=kS_{c}[i]\>\forall\>i\in[1,\,m]\,,$ (13) $\displaystyle
P_{\mathrm{Loss}}^{\mathrm{Ntwk.}}$ $\displaystyle=x^{\mathrm{T}}\Lambda
x+\lambda x+\sigma\,,$ (14) $\displaystyle x$
$\displaystyle=[P_{c}^{\mathrm{T}},\,Q_{c}^{\mathrm{T}}]^{\mathrm{T}}\,,$ (15)
$\displaystyle V$ $\displaystyle=Kx+b\,,$ (16) $\displaystyle V_{-}$
$\displaystyle\leq V\leq V_{+}\,,$ (17) $\displaystyle\sum_{i}S_{c}[i]$
$\displaystyle\leq S_{c}^{\mathrm{Total}}\,,$ (18) $\displaystyle\mathrm{EC}$
$\displaystyle\leq n\,.$ (19)

The objective (8) is to schedule the converter real $P_{c}$ and reactive
$Q_{c}$ power flows to minimize converter losses
$P_{\mathrm{Loss}}^{\mathrm{Conv.}}$ and network losses
$P_{\mathrm{Loss}}^{\mathrm{Ntwk.}}$. The apparent power of each converter
$S_{c}[i]$ must be bounded by the capacity connected to that converter leg
(9). The powers injected into the dc node $P_{\mathrm{dc}}$ must balance (10),
and the power injected by a distributed energy resource $P_{\mathrm{DER}}$ is
the final element of the vector of powers at the dc node (12). The power must
balance across the lossy converters (11), with converter losses modelled as
being linear in apparent power (13) for loss coefficient $k$. Network losses
are quadratic in power injections (14), with $x$ the stacked vector of real
and reactive power injections (15) and $\Lambda,\,\lambda,\,\sigma$ the
parameters of the quadratic model. Voltage magnitudes $V$ are linear in
injection (16), with $K,\,b$ the sensitivity and offset components of the
power-voltage affine model, and upper and lower bounds on voltages
$V_{+},\,V_{-}$ imposed via (17). The total ac/dc converter capacity
$S_{c}^{\mathrm{Total}}$ limits the sum of power transfers into the ac feeders
(18), based on the idealised MOP of [8]. Finally, the number of non-zero power
injections (i.e., the electrical cardinality) is less than or equal to $n$
(19).

Note that the full optimization formulation includes auxiliary variables to
include the cardinality constraint (19) as described in Section III-A (these
are not written explicitly for brevity). Additionally, the quadratic network
losses (14) are converted to a relaxed second order cone constraint, as in
[8].

##  IV Case Studies

In the previous section, the operational optimization method was developed to
optimally dispatch an idealised reconfigurable power converter, subject to a
cardinality constraint that ensures the electrical cardinality is no greater
than a given value. In this section, the approach is demonstrated on a pair of
distribution network models, simulated over the course of a full calendar
year.

Fig. 2 shows the topology of the interconnected feeders of the two networks
studied, the 75 Bus UK Generic Distribution System (GDS) HV UG network [15]
and the IEEE 33 Bus Network [16]. Both of these models have a significant wind
and solar generator connected within the network. Additionally, to illustrate
how injections into the dc link of a multiport can influence the mission
profile, the GDS network also has an 0.8 MW solar generator connected to the
dc link. Each load is allocated an individual demand profile from real
measured annual profiles from a utility in the UK [17], and embedded wind and
solar profiles are taken from [18] to represent the renewable generation
profiles. A converter loss coefficient $k$ of 1% is used (i.e., ac-dc-ac
efficiency is 98%), as in [8].

(a) Case study 1: the 75 bus UKGDS HV UG network.

(b) Case study 2: the IEEE 33 Bus distribution system.

Figure 2: The topology and generators installed for the two case studies, with
the UKGDS having a three-port MOP with two ac feeders and a generator
connected to the dc link (similar to Fig. 1); and, the 33 Bus system with a
4-terminal MOP.

The solution approach has been developed using the Mosek Fusion API [19]. The
conic relaxation of (14) is accurate across all cases and time periods to
within a relative error of $3\times 10^{-5}$. Relative and absolute integer
optimization gaps of $10^{-4},\,10^{-5}$ are used, with all problems then
solved to optimality. The quadratic loss model (14) has good performance as
compared to the true non-linear solution obtained from OpenDSS. For example,
for the 33 Bus system and unconstrained cardinality (Section IV-B), the
correlation between modelled and true loss changes has a correlation
coefficient of 94.4% and slope (as determined via linear regression) of 0.914.
Linearization errors for voltage magnitudes are even more accurate–for
example, relative errors across the whole year are less than 3.5% for Case
Study 1 (in terms of the 2-norm of the changes in voltages from the linear
model and OpenDSS).

###  IV-A Case Study 1: UKGDS Distribution System

The first case considered is the UKGDS system (Fig. 2a), with total idealised
converter capacity $S_{c}^{\mathrm{Total}}$ of 3200 kVA. Fig. 3 plots the
operation of the device for five representative days in the summer, showing
the real and reactive flows of the two multiport converters with unrestricted
cardinality (Fig. 3a), the power flows for the converter with a cardinality
constraint (Fig. 3b), the additional losses in the system as compared to a
system with no converter or the 0.8 MW additional generator (Fig. 3c), and
finally normalised values of the range and quartiles of the demand, the solar
profile and wind profile (Fig. 3d). It can be observed in this figure that, as
expected, when the cardinality is unrestricted that power can be transferred
between feeders in any combination. In contrast, when the cardinality EC is
constrained, real and reactive powers can only be transferred into one feeder
at a time (Fig. 4a). This has a substantial effect on power flows and losses,
as there is voltage congestion at Bus 1132 (Fig. 4a) due to the large wind
generator on Feeder 6 (for the purposes of this work, a limit of 1.045 pu is
considered). For the restricted case with $n=1$, when there are high winds and
solar generation (as on June 25th and 26th), the converter must connect to
Feeder 6 to lower the voltage by drawing reactive power; however, this means
that real power is also injected from the solar plant, and so further reactive
power must be drawn to reduce the voltage even further. In contrast, the
mission profile with unrestricted cardinality can simply transfer the surplus
real power from the wind generator from Feeder 6 to Feeder 3.

(a) Power transfers, unrestricted cardinality ($\mathrm{EC}$ unconstrained).

(b) Complex power transfer, cardinality constraint $\mathrm{EC}\leq 1$.

(c) Additional system losses.

(d) Generation profiles and demand quartiles, Case 1.

Figure 3: Power transfers (a, b), changes to losses (c) and generation
profiles alongside the range, interquartile range and median demand (d) for
Case 1 (UKGDS HVUG network, Fig. 2a). These subfigures show that power flow is
feasible for the case with cardinality restricted to $n=1$, but the additional
reactive power that must be drawn to manage voltages increases losses
substantially.

Despite these increased system losses, congestion is still effectively managed
by the power converter and so there are no infeasible points (i.e., time
periods which the voltage is outside of the limits defined by the utility).
Across the year, the increase in losses is 19.1 MWh for the converter with
constrained cardinality as compared to the converter with unconstrained
cardinality. However, this may be offset by the reduced complexity of the
mission profile, as shown in Fig. 4. A monolithic ac/dc converter with
capacity 3200 kVA could be designed with a reconfiguration output stage to
meet the mission profile profile, rather than a more complex ac/dc/ac
converter.

(a) Unrestricted cardinality.

(b) Cardinality constraint $\mathrm{EC}\leq 1$.

Figure 4: Scatter plot of the mission profile for the unrestricted and
constrained mission profiles, demonstrating cardinality EC of the mission
profile has been reduced from 2 (a) to 1 (b). The thin red dashed line
indicates the apparent power constraint for the 3200 kVA converter considered.

###  IV-B Case Study 2: IEEE 33 Bus Network

The results with the IEEE 33 Bus system are plotted in Fig. 5 for
representative days of August 1st-3rd, with a total ac/dc converter capacity
$S_{c}^{\mathrm{Total}}$ of 750 kVA. Real power transfers are plotted the
unrestricted design in Fig. 5a and the design with cardinality $n$ restricted
to 2 in Fig. 5b; the reduction in system losses are plotted in Fig. 5c and the
relative fraction of the loss reduction as compared to the unrestricted
cardinality are plotted in Fig. 5d (to highlight more clearly the difference
between the three lines in Fig. 5c); the out-turn cardinality of the hourly
mission profile are plotted in Fig. 5e, and finally the normalised range and
quartiles of the demand profiles, and the solar and wind profiles, are plotted
in Fig. 5f. As expected, from this figure, it can again be seen that power can
be transferred between any combination of powers for a converter with
unrestricted cardinality, where the case with cardinality $n$ restricted to 2
or less can only transfer power between any two feeders. For example, on
August 1st, the high solar generation leads the MOP to draw from bus 17
$P_{c}[2]$ and export to bus 32 $P_{c}[1]$ due to high demand and low wind.

(a) Real power transfers, unrestricted cardinality ($\mathrm{EC}$
unconstrained).

(b) Real power transfers, cardinality constraint $\mathrm{EC}\leq 2$.

(c) Reduction in system losses.

(d) Fraction of reduction of system losses against cardinality constraint.

(e) Number of non-zero apparent power transfers.

(f) Generation profiles and demand quartiles, Case 2.

Figure 5: Results for Case 2. (a, b) plot real power transfers for the
unrestricted case ($n=4$) and for $n=2$; (c) plots the loss reduction, and (d)
plots the equivalent fraction of loss reduction against the restricted cases.
Finally, (e) plots the number of non-zero power transfers, whilst (f) plots
the generation profiles and demand quartiles for this case.

The results of Fig. 5 show clearly the fact that the cardinality EC can vary
significantly as a function of time (calculated according to (3)). As a
result, during the time periods with an EC of 2, all three of the converter
models perform the identically. Over the course of the full year (17520 half-
hours), the losses for systems with 2, 3 or no cardinality constraint values
are shown in Fig. 6, as is the distribution of EC values. In this system, more
than 4% of time periods have no power injections (i.e., an EC value of zero),
even with the low ac/dc conversion loss coefficient of $1$%.

(a) Total annual losses

(b) Distribution of EC values

Figure 6: Summary statistics for case study 2, in terms of (a) annual loss
reduction and (b) the distribution of electrical cardinality (EC) values.

###  IV-C Discussion

This work has considered the construction of mission profiles with a
constrained cardinality to explore the properties of these solutions. However,
in general the idealised MOP model [8], as used in the optimization (18), may
or may not be realisable with fixed converter sizes (as modelled in, e.g.,
[8]). Nevertheless, the proposed approach can show if there are significant
potential benefits of reduced cardinality, thereby motivating a future method
to consider an optimal sizing strategy for individual ac/dc legs for a given
cardinality.

Furthermore, this work has focussed on the electrical cardinality, as this
drives electrothermal stresses and is therefore the primary component of a
mission profile. However, the thermal aspect of a mission profile has not been
modelled explicitly, and the impact of changing cardinality may be
considerable if a reconfigurable model is designed which substantially changes
converter utilization. Conversely, distribution optimization problems could
also be developed that aim to directly avoid unnecessary thermal cycling of
power converter modules to improve reliability and lifetime of these devices.

##  V Conclusion

Multiport mission profiles have more complex mathematical structure than two-
terminal power converters, and so require more sophisticated techniques to
model, design and simulate. The electrical cardinality of a multiport’s
mission profile has been introduced, based on the outputs of distribution
system optimal power flow, and has been shown to have temporal variability
within systems composed of power converters embedded within electrical
distribution networks. Furthermore, a method to include cardinality
constraints within optimal power flow problems has been proposed that enables
a family of mission profiles to be provided to a power converter designer for
further analysis. Results show that converters with restricted cardinality can
still provide effective congestion management and provide grid services such
as loss reduction. A case study demonstrating a halving of the cardinality
only reducing the potential losses reduction by 91.7% as compared to an
unrestricted case.

Power flows in distribution systems are increasingly becoming controllable and
variable, whilst developments in power electronics are resulting in increasing
power density and reliability of converters. To maximise the potential
benefits of these devices within systems, much higher levels of dialogue will
be required between power converter designers and network operators. The
electrical cardinality of the mission profile provides one useful piece of
information that can be clearly communicated that affects both the power
converter and its operation within a network. It is concluded that new
integrated power distribution–power electronics design and modelling
approaches will be necessary to most effectively realise the potential of
power converters to support the integration of low carbon technologies in
power distribution.

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# On the natal kick of the black hole X-ray binary H 1705–250

Cordelia Dashwood Brown, Poshak Gandhi, and Yue Zhao
School of Physics and Astronomy, University of Southampton, Southampton SO17
1BJ, UK

(Accepted 2023 October 5. Received 2023 October 5; in original form 2023 Jun
23)

###### Abstract

When a compact object is formed, an impulse (kick) will be imparted to the
system by the mass lost during the core-collapse supernova (SN). A number of
other mechanisms may impart an additional kick on the system, although
evidence for these natal kicks in black hole systems remains limited. Updated
Gaia astrometry has recently identified a number of high peculiar velocity (in
excess of Galactic motion) compact objects. Here, we focus on the black hole
low-mass X-ray binary H 1705–250, which has a peculiar velocity
$\upsilon_{\mathrm{pec}}\,=\,221^{+101}_{-108}\,\mathrm{km}\,\mathrm{s}^{-1}$.
Using population synthesis to reconstruct its evolutionary history (assuming
formation via isolated binary evolution within the Galactic plane), we
constrain the properties of the progenitor and pre-SN orbit. The magnitude of
a kick solely due to mass loss is found to be $\sim$ 30 km s-1, which cannot
account for the high present-day peculiar motion. We therefore deduce that the
black hole received an additional natal kick at formation, and place limits on
its magnitude, finding it to be $\sim$ 295 km s-1 (minimum 90 km s-1). This
furthers the argument that these kicks are not limited to neutron stars.

###### keywords:

X-rays: binaries – black hole physics – supernovae: general – stars:
kinematics and dynamics

††pubyear: 2015††pagerange: On the natal kick of the black hole X-ray binary H
1705–250–References

##  1 Introduction

The majority of stellar mass black holes (BHs) detected in the electromagnetic
spectrum exist in binaries, where they are actively accreting from a companion
star, primarily emitting in the X-ray band. Despite a growing catalogue of
these BH X-ray binaries (XRBs) with confirmed masses and well-constrained
astrometry, it remains unclear how these objects form. The consensus is that
BHs form from the gravitational collapse of a massive star, with a core mass
$>\,8\,\mathrm{M}_{\odot}$. The BH may form directly after collapse;
alternatively, if the supernova (SN) is not energetic enough to successfully
eject all it’s envelope, the BH may form through fallback, after existing
briefly as a neutron star (NS) (see Repetto & Nelemans, 2015; Mirabel, 2017;
Mapelli, 2021, and references therein).

The deaths of massive stars can impart additional acceleration to compact
objects at the instant of SN, an impulse referred to as a natal kick. There
remains significant uncertainty in the role of natal kicks in the evolution of
BH systems. Previous literature has made a distinction between kicks
associated with NS (e.g., Gunn & Ostriker, 1970; Trimble, 1971; Lyne et al.,
1982; Lyne & Lorimer, 1994; Brandt & Podsiadlowski, 1995; Hobbs et al., 2005;
Verbunt et al., 2017) versus BH (e.g., Jonker & Nelemans, 2004; Gualandris et
al., 2005; Repetto et al., 2012; Janka, 2013a; Mandel, 2015), with the former
being more extensively studied. Whilst there exists evidence for NS kicks with
both high and low velocities, evidence of BHs receiving natal kicks is less
clear.

Understanding these natal kicks is of great importance in shaping SN
mechanisms and energetics, as well as providing constraints on the number of
compact objects that may be retained in globular or young stellar clusters
(e.g., Pfahl et al., 2002; Strader et al., 2012; Giesler et al., 2018). The
exact mechanisms leading to these kicks is unknown, but they may be the result
of the recoil due to baryonic ejecta (Blaauw, 1961; Brandt & Podsiadlowski,
1995; Nelemans et al., 1999) or anisotropy in gravitational attraction due to
asymetrically ejected mass (Janka, 2013b, 2017). Alternatively, the recoil
momentum could be due to asymmetric neutrino emission, related to the
hydrodynamic processes within the SN (Chugai, 1984; Dorofeev et al., 1985;
Arras & Lai, 1999; Lai, 2004).

These kicks leave imprints on the kinematics of XRBs, and the high peculiar
velocity ($\upsilon_{\mathrm{pec}}$) of many XRBs supports the hypothesis that
non-negligible natal kicks occurred at the instant of SN (Mirabel et al.,
2001; Gandhi et al., 2019; Atri et al., 2020; Fortin et al., 2023; O’Doherty
et al., 2023), although this is only a proxy for kick velocity.

Previous studies have considered whether BHs receive substantial natal kicks.
Willems et al. (2005) considered the case of GRO J1655–40, re-constructing its
evolutionary history to determine that the natal kick velocity
($\upsilon_{\mathrm{NK}}$) was small (a few 10s km s-1), and in contrast to
the high velocities associated in NSs. The high velocity of XTE J1118+480 was
discussed by Mirabel et al. (2001), and follow-up work suggested a lower limit
of $\upsilon_{\mathrm{NK}}$ as either 80 km s-1 (Fragos et al., 2009) or 93 –
106 km s-1 (Repetto & Nelemans, 2015). A similar analysis of Cyg X-1 by Wong
et al. (2012) constrained $\upsilon_{\mathrm{NK}}\,<$ 77 km s-1, and Kimball
et al. (2023) found MAXI J1305–704 to be consistent with a high
$\upsilon_{\mathrm{NK}}\,>$ 70 km s-1.

One source of post-SN velocity is symmetric mass-loss (Nelemans et al., 1999),
where mass is symmetrically and instantly ejected from the compact progenitor
by the SN, resulting in a recoil kick to the system. Considering this in
tandem with Kepler’s laws, one can derive an expression for the velocity
imparted to the binary system:

$\upsilon_{\mathrm{MLK}}=213\times\frac{m}{M_{\odot}}\frac{\Delta
M}{M_{\odot}}\left(\frac{P_{\mathrm{circ}}}{\mathrm{day}}\right)^{-\frac{1}{3}}\left(\frac{M_{\rm
tot}}{M_{\odot}}\right)^{-\frac{5}{3}}\mathrm{km\,s}^{-1},$ (1)

where $m$ and $M_{\rm tot}$ are the luminous companion and post-SN total
system masses, $\Delta M$ is the mass lost from the compact object progenitor
due to SN, and $P_{\mathrm{circ}}$ is the post-SN circularised orbital period.

Eq. 1 yields an estimate of the post-SN velocity of a system purely perturbed
by symmetric mass loss, subsequently referred to as the mass-loss kick
velocity ($\upsilon_{\mathrm{MLK}}$). These are conceptually identical to
Blaauw 1961 kicks We can examine the peculiar velocities of each system
compared to the expected $\upsilon_{\mathrm{MLK}}$ to determine if this
provides sufficient momentum, or if an additional kick (i.e. a natal kick
related to the mechanisms outlined above) is necessary.

Here we present evidence for another BH-XRB with a large
$\upsilon_{\mathrm{pec}}$, and test the argument that large kicks can be
imparted to BHs at formation by tracing its evolutionary history. Sec 2
describes the observations used in this work, Sec 3 presents the results of
simulations, and Sec 4 provides relevant discussions.

##  2 Observations & Analysis

Our target of interest was selected from a catalogue of 89 compact object
binaries that includes updated and improved calculated peculiar velocities
presented by Zhao et al. (2023) (hereafter Zhao23). The dataset is a
comprehensive collation of Galactic binaries hosting compact objects with
reliable astrometry from either Gaia or radio interferometric or timing
observations (Zhao23 and references therein). The updated parallax ($\varpi$),
and proper motion values provided by Gaia are used to derive 3D
$\upsilon_{\mathrm{pec}}$, following the formulation described by Reid et al.
(2009). Orbits are integrated back in time, and the values of
$\upsilon_{\mathrm{pec}}$ as the binary passes the Galactic plane are
recorded, as a proxy for the peculiar velocity at birth
($\upsilon_{\mathrm{pec}}^{z=0}$) — assuming that the binaries are born in the
Galactic plane (for details, see Zhao23). The primary findings of Zhao23 are
an anti-correlation between $\upsilon_{\mathrm{pec}}$ and $M_{{\rm tot}}$;
here we focus on the origins of high-$\upsilon_{\mathrm{pec}}$ systems and the
implications for natal kicks.

High natal kick candidates can be identified as follows. For each binary the
maximum velocity associated solely with mass-loss ($\upsilon_{\mathrm{MLK}}$)
can be estimated and compared to the calculated $\upsilon_{\mathrm{pec}}$.
From the virial theorem, it follows that a system that loses more than half
its total mass due to the SN will be disrupted; therefore the present-day
$M_{\rm tot}$ of the system provides an upper limit of $\Delta M$. We also
assume that the present-day circular orbital period ($P_{\mathrm{orb}}$) is
consistent with the $P_{\mathrm{circ}}$ (i.e. there has been no significant
change in $P_{\mathrm{orb}}$ due to mass transfer, magnetic braking, etc.).
This assumption is later improved upon with simulations. The
$\upsilon_{\mathrm{MLK}}$ can be estimated from Eq. 1.

Where systems have a $\upsilon_{\mathrm{pec}}$ higher than
$\upsilon_{\mathrm{MLK}}$, it can be inferred that an additional source of
momentum may have been imparted — an additional kick related to mechanisms
other than instantaneous mass ejection. Fig 1 shows the estimated
$\upsilon_{\mathrm{MLK}}$ compared to the measured $\upsilon_{\mathrm{pec}}$
for the Zhao23 sample. We highlight systems which have been the subject of
previous targeted natal kick studies and the focus of this paper (H 1705–250).

Figure 1: Present-day peculiar velocity ($\upsilon_{\mathrm{pec}}$) compared
with the estimated mass-loss kick velocity ($\upsilon_{\mathrm{MLK}}$) as a
function of total system mass for the Zhao23 sample. H 1705–250 (the subject
of this paper) is highlighted by the black cross. Previous studies of black
holes are distinguished by colour and shape: GRO J1655–40 (Willems et al.,
2005), XTE J1118+480 (Fragos et al., 2009), Cyg X-1 (Wong et al., 2012), and
MAXI J1305–704 (Kimball et al., 2023).

There are a number of systems (both NS & BH) which appear to require
additional kicks to explain their $\upsilon_{\mathrm{pec}}$. That a large
proportion of BH systems seem to require additional natal kicks is evidence
that, whilst there may be differences in the kick origins and mechanisms, both
types of systems can experience additional natal kicks at birth.

Whilst the estimated $\upsilon_{\mathrm{MLK}}$ is a good litmus test for
systems that may require additional kicks, it cannot be used in isolation, and
detailed evolutionary histories need to be examined to infer more robust
conclusions. We focus on one candidate that appears to have been kicked: H
1705–250 (or Nova Oph 1997, hereafter H1705), a short-period BH low mass X-ray
binary (LMXB), with $P_{\mathrm{orb}}=12.51\pm 0.03$ hrs, and masses
$M_{\mathrm{BH}}=6.4\pm 1.5\,\mathrm{M_{\sun}}$ and $m=0.34\pm
0.08\,\mathrm{M_{\sun}}$ for the BH and stellar companion, respectively
(Harlaftis et al., 1997; Remillard et al., 1996). This system has an observed
space velocity
$\upsilon_{\mathrm{pec}}=221.2^{+100.8}_{-108.0}\,\mathrm{km\,s^{-1}}$
(Zhao23). We emphasise that these are not sigma errors, but rather the result
of Monte-Carlo calculations of peculiar velocity using the astrometric and
radial velocity errors. We chose to focus on this system as Zhao23 suggests
higher mass systems are less likely to receive high natal kicks, and whilst
the velocity of this source has been commented on in previous literature
(Mandel, 2015; Repetto & Nelemans, 2015), there are inconsistencies in the
peculiar velocities quoted in these papers. Based on the assumptions outlined
above, $\upsilon_{\mathrm{MLK}}$ is estimated to be $25\pm
7\,\mathrm{km\,s^{-1}}$ (incorporating 1$\sigma$ uncertainties in the
component masses and $P_{\mathrm{orb}}$). We carry out detailed simulations of
this system to identify its evolutionary history, and the conditions required
to account for the present-day characteristics.

##  3 Simulations

In order to understand the history of H1705 and the impact of the SN, we
generate a population of 6 million binaries and identify those that evolve to
match the current system characteristics (primarily the masses and orbital
period). We do this with Compact Object Synthesis and Monte Carlo
Investigation Code (cosmic), a binary population synthesis code adapted from
Binary Stellar Evolution code (BSE) to include modified and updated evolution
prescriptions and parameters (Breivik et al., 2020).

We generate 6 million binaries under a range of different initial conditions
and track their evolution, searching for systems that will evolve to resemble
H1705 following the first SN. The majority of evolutionary parameters and
prescriptions are kept constant, with only the natal kick inputs changing, to
ensure we generate systems with a wide distribution of natal kick velocities
($\upsilon_{\mathrm{NK}}$) to determine which of these is most favoured. Our
pre-SN masses spanned 2 – 20 $M_{\odot}$ and 0.1 – 2 $M_{\odot}$ for the
compact progenitor and stellar companion, respectively, and $P_{\mathrm{orb}}$
= 2 – 48 hrs. Natal kick magnitudes were drawn from a log-uniform distribution
where $10^{0.1}\leq\upsilon_{\mathrm{NK}}\leq 10^{3}\,\mathrm{km\,s^{-1}}$,
such that none of the current natal kick models are favoured. Our aim is to
identify the magnitude of $\upsilon_{\mathrm{NK}}$ needed to explain the
present-day $\upsilon_{\mathrm{pec}}$, and therefore we chose this input
distribution such that $\upsilon_{\mathrm{NK}}$ encompass a broad range of
magnitudes. Initial eccentricities for all systems were set at zero; this is a
fair assumption given that tidal forces will act to circularise binaries
quickly, particularly closely separated binaries such as these. Note that
whilst evolutionary uncertainties such as common envelope efficiency, rates of
mass-loss from stellar wind, and changes in $P_{\mathrm{orb}}$ associated with
magnetic braking may all impact the evolution of simulated binaries, the focus
is on the effect of kicks. The common envelope efficiency parameter $\alpha$
was drawn from a uniform distribution between 0.1 and 1, and stellar
metallicity for both the compact progenitor and stellar companion is set to
0.02. Mass transfer is treated according to the standard BSE choice, outlined
in Hurley et al. (2002), and we use the magnetic braking prescription from
Ivanova & Taam (2003). We are not proposing any one of these choices to be
‘correct’ over alternative theories, but rather ensuring that the effects of
kicks imparted on the compact objects are isolated from models involved in
binary evolution.

The evolution of H1705 appears to follow ‘standard’ binary evolution. The more
massive component of the initial binary evolves, eventually breaching its
Roche-lobe and resulting in mass transfer; it then undergoes a core-collapse
supernova, leaving behind a BH. The results of these simulations are shown in
Fig. 2. There is a well-defined region of parameter space prior to the SN that
is conducive to forming systems resembling H1705 (referred to hereon as H1705
analogues). At SN, the BH progenitor (a stripped Helium star) likely had a
mass of 9.25 – 11.5 $\mathrm{M_{\sun}}$, and the companion had a mass of 0.25
– 0.45 $\mathrm{M_{\sun}}$. The expected $P_{\mathrm{orb}}\approx$ 0.2 – 1.5
days, with an orbital separation of 0.02 – 0.06 AU.

Figure 2: Properties of H1705 progenitor systems, at the time of SN. Blue
markers represent systems that evolve to form H1705 analogues after the SN,
whilst grey systems are unsuccessful (either disrupted or merged during SN, or
evolving to form different systems). Upper panel shows the masses of the
different components (with the primary being the BH progenitor). Lower panel
shows primary mass and orbital period

Within our sample of 6 million binaries, we focus on 2 million where the pre-
SN characteristics resemble those of the H1705 analogues. Of these, 0.5
million go on to form H1705 analogues, and the remainder act as a control
sample to identify the effects of SN and natal kicks.

##  4 Discussions

The components of our simulated systems have mean post-SN masses
$M_{\mathrm{BH}}=6.49\pm 0.89\,\mathrm{M_{\sun}}$ and $m=0.34\pm
0.05\,\mathrm{M_{\sun}}$, and $P_{\mathrm{orb}}=12.4\pm 1.8$ hrs, matching the
observed system parameters well. Systems where the BH progenitor mass
$\gtrapprox 11.5\,\mathrm{M_{\sun}}$ at the time of SN results in a post-SN
$M_{\mathrm{BH}}>9\,\mathrm{M_{\sun}}$. Conversely, if the primary mass
$\lessapprox 8.5\,\mathrm{M_{\sun}}$, then systems are far more likely
($>80\%$) to be disrupted, and the surviving systems have
$M_{\mathrm{BH}}\lessapprox 3.5\,\mathrm{M_{\sun}}$. Systems with pre-SN
$P_{\mathrm{orb}}>2\,$days are typically disrupted or forced into wider orbits
by the SN. The BH progenitor in the H1705 analogues loses about 4
$\mathrm{M_{\sun}}$: 30 – 45 % of the $M_{\rm tot}$ at the time of SN.

###  4.1 The effect of natal kicks on system survival

Of the sample of binaries with pre-SN conditions matching those of H1705
analogues, 42% will be disrupted by the SN, and a further 5% will merge. Only
$\approx$ 25% form H1705 analogues. Systems with orbital periods $<$0.5 days
are more likely to merge than longer period systems, and all systems with
$P_{\mathrm{orb}}<0.24$ days will merge. However, there is no obvious
dependence on the masses. Figure 3 shows $\upsilon_{\mathrm{NK}}$ &
$\upsilon_{\mathrm{MLK}}$ for our simulated systems, distinguishing those
disrupted by SN, those which do not evolve to match the characteristics of
H1705, and H1705 analogues. The majority ($\sim$ 70%) of systems with
$\upsilon_{\mathrm{NK}}$ $<$ 50 km s-1 merge. Systems that are disrupted by
the SN have, on average, higher $\upsilon_{\mathrm{NK}}$ than their surviving
counterparts, and all systems where $\upsilon_{\mathrm{NK}}$ > 1000 km s-1 are
disrupted.

###  4.2 The effect of natal kicks on system velocity

Simulations indicate that the compact progenitor loses $\sim$ 4 M⊙ as a
consequence of the supernova - 40% of its initial mass. Calculating the
magnitude $\upsilon_{\mathrm{MLK}}$ for each system shows an average
$\upsilon_{\mathrm{MLK}}\,\approx\,$16 km s-1 for H1705 analogues, with 95% of
systems having $\upsilon_{\mathrm{MLK}}\,<$ 25 km s-1. The distribution of
$\upsilon_{\mathrm{MLK}}$ is shown in Fig. 3. For the H1705 analogues,
$\upsilon_{\mathrm{MLK}}$ does not exceed 35 km s-1; we can therefore infer
that the momentum imparted to the surviving system from the mass ejection in
the H1705 system cannot explain the high system velocity observed. An
additional source of momentum, due to natal kick related to mechanisms other
than mass ejection, is required.

Figure 3: Histograms showing the natal kick velocity
($\upsilon_{\mathrm{NK}}$) and mass-loss kick velocity
($\upsilon_{\mathrm{MLK}}$) of simulated systems. Blue (dashed line) systems
mark H1705 analogues, grey (dotted line) marks those binaries which go on to
evolve into systems distinct from H1705 (including systems that merge), and
orange (solid line) marks systems which are disrupted during the SN.

In Fig. 4 we show the range of $\upsilon_{\mathrm{pec}}$ of our H1705
analogues compared to the $\upsilon_{\mathrm{MLK}}$, colour-coded by the
magnitude of $\upsilon_{\mathrm{NK}}$. Around 53% of systems have
$\upsilon_{\mathrm{pec}}$ consistent with $\upsilon_{\mathrm{pec}}^{z=0}$
(Zhao23). Assuming this is an accurate estimate of the birth
$\upsilon_{\mathrm{pec}}$, H1705 must have received an additional
$\large\upsilon_{\mathrm{NK}}\,\gtrapprox\,97$ km s-1. For systems where
$100<\upsilon_{\mathrm{pec}}<200\,\mathrm{km\,s^{-1}}$ (i.e., the lower end of
estimates for $\upsilon_{\mathrm{pec}}$ within errors),
$\upsilon_{\mathrm{NK}}$ must be in excess of $95\,\mathrm{km\,s^{-1}}$. These
values are lower than those found by (Repetto & Nelemans (2015), where
$\upsilon_{\mathrm{NK}}\approx$ 361– 441 $\mathrm{km\,s^{-1}}$), which is due
to the different methods in inferring the $\upsilon_{\mathrm{pec}}$ at
formation — however, the primary findings remain the same: the BH system H1705
must have received an additional kick in line with those expected for NS
systems.

Figure 4: Post-SN peculiar velocity ($\upsilon_{\mathrm{pec}}$) compared to
the mass-loss kick velocity ($\upsilon_{\mathrm{MLK}}$) for all H1705
analogues. The colour bar indicates the velocity of the natal kick imparted on
the system ($\log_{10}(\upsilon_{\mathrm{NK}})$). The observed present-day
$\upsilon_{\mathrm{pec}}$ of H1705 is indicated by the black marker, with
errors.

###  4.3 Caveats

####  4.3.1 H1705 distance uncertainties

The distance to H1705 is still highly uncertain. The Gaia parallax measurement
is only weakly significant, at $\varpi\,\approx 2.15\,\pm\,1.67\,mas$.
Combined with exponential distance priors (Zhao23), the Gaia predicted
distance peaks at $\sim$ 3 kpc, but with a significant tail to >10 kpc. Future
data releases should help narrow this prediction. Jonker & Nelemans (2004)
determine the distance to H1705 to be 8.6 $\pm$ 2.1 kpc, and this is the value
used by Zhao23 to calculate $\upsilon_{\mathrm{pec}}$. In Figure 5 we show the
effect of varying distances in calculating the peculiar velocity. It is
evident that for distances $>$ 2.5 kpc, $\upsilon_{\mathrm{pec}}$ is
substantial, at $>100$ km s-1. For this reason, we do not believe that the
uncertainty in the distance to H1705 has a significant impact on our analysis.

Figure 5: Peculiar velocity ($\upsilon_{\mathrm{pec}}$) as a function of
distance to H1705. The errors (dark grey) are derived from the 1-$\sigma$
uncertainties in parallax, proper motions, and radial velocity. The light-
shaded region indicates the quoted $\upsilon_{\mathrm{pec}}$ range
($221.2^{+100.8}_{-108.0}\,\mathrm{km}\,\mathrm{s}^{-1}$).

####  4.3.2 Calculations of peculiar velocity

One method of estimating peculiar velocity at birth (assuming birth in the
Galactic plane) is by the velocity needed to reach its current location above
the Galactic plane:

$\upsilon_{\perp,\mathrm{min}}=\sqrt{2\left[\Phi(R_{0},z)-\Phi(R_{0},0)\right]}$
(2)

where $\Phi(R,z)$ is the Galactic potential, and $R_{0}$ is the projected
distance from the Galactic center (Repetto et al., 2012).

The uncertainty in distance means estimates of initial velocity from the above
may be unreliable. This was highlighted by Mandel (2015), who suggests that
the uncertainties in distance led to an overestimation of the initial
velocity, by up to an order of magnitude, and consequently, this source is
therefore not suggestive of a significant natal kick. If one assumes the
lowest literature distance, then the inferred velocity at birth (within the
Galactic plane) would be less than 50 km s-1 and therefore explainable with a
mass-loss kick alone (though higher kicks are not ruled out in general).

In this analysis, we consider the peculiar velocities calculated from the
current positions and proper motions. Gaia astrometry has placed solid
constraints on the system’s proper motion, despite $\varpi$ remaining poorly
measured ($\frac{\varpi}{\sigma_{\varpi}}\,\sim$ 1.29). We comment that the
uncertainties in distance and proper motions
($\mu_{\mathrm{RA}}\approx-6.97\pm 2.83$ mas yr-1,
$\mu_{\mathrm{DEC}}\approx-8.51\pm 1.38$ mas yr-1) result in the high
uncertainty of present-day $\upsilon_{\mathrm{pec}}$
($221.2^{+100.8}_{-108.0}\,\mathrm{km}\,\mathrm{s}^{-1}$). However, even the
lower limit on $\upsilon_{\mathrm{pec}}$ is significantly higher than both the
estimated $\upsilon_{\mathrm{MLK}}$ and the mean peculiar velocities of all
systems within the Zhao23 sample (we again note that the peculiar velocity
distribution of H1705 is not Gaussian).

####  4.3.3 System origin

H1705 lies 1.35 kpc above the Galactic plane, just above the Galactic bulge.
Evolving the system back in time suggests that the system may have crossed the
plane at some point in its history, and therefore could have formed here.
However, given the height above the Galactic plane, we cannot exclude the
possibility that H1705 instead formed within a globular cluster via dynamical
interaction, and was subsequently ejected. In this scenario, some of our
analysis would not be applicable. Where the velocity dispersion of the
Galactic disc is $\sim$ 40 km s-1 (Carlberg et al., 1985), the dispersion in
the Galactic halo is larger: 50–120 km s-1 (Battaglia et al., 2005; Brown et
al., 2010). However, the present-day $\upsilon_{\mathrm{pec}}$ of H1705 is
still excessive compared to these estimates. The magnitude of the present-day
$\upsilon_{\mathrm{pec}}$, neglecting any inference on the peculiar velocity
at birth, is large enough that a high NK cannot immediately be ruled out if
the system were to form in a GC. More detailed analysis and kinematic study of
this system is needed to definitively determine the system’s origin.

####  4.3.4 Simulation restrictions

As discussed in Sec 3, the simulations carried out by cosmic do not explore
the full range of parameters involved in binary evolution. For instance, our
simulations are restricted to low eccentricities. Preliminary tests suggest
that higher initial eccentricities do not preclude systems from evolving to
resemble H1705, and $\upsilon_{\mathrm{NK}}$ must still be large in order to
explain the present-day system kinematics. We also ran smaller-scale
simulations whilst varying stellar metallicity by a factor of two each way, to
determine if this has a significant effect on our results. Whilst the
parameter space for systems that form H1705 analogues is slightly altered
(e.g., doubling stellar metallicity results in, on average, slightly higher
compact progenitor masses), our core findings still hold. Any variations in
mass are small enough such that $\upsilon_{\mathrm{MLK}}$ remains largely
unchanged, and the system still requires a substantial additional natal kick
in order to reproduce the observed $\upsilon{\mathrm{pec}}$. Factors such as
differing descriptions of mass-transfer, winds, common envelopes, magnetic
braking, etc. were kept constant in all simulations. These factors are well-
known uncertainties for binary population synthesis. We have assumed
‘canonical’ estimates of these factors here, which can all be considered as
physically plausible. Pathological or more complex scenarios cannot be ruled
out, but are beyond the scope of this work.

####  4.3.5 Kick mechanisms

As discussed in Section 1, there are a range of theories regarding the origin
of natal kicks. In this work, we do not constrain the nature of the mechanisms
behind the kick imparted to H1705, nor suggest whether the kick is applied to
the BH directly, or rather associated with a proto-NS before fallback. These
questions can only be addressed by future detailed analysis of large samples
and may allow us to infer likely sources of natal kicks and compare different
populations.

It is also not fully understood how the momentum is imparted to the system and
its constituents. It is believed that $\upsilon_{\mathrm{MLK}}$ is imparted to
the center of mass of the system, whereas $\upsilon_{\mathrm{NK}}$, associated
with mechanisms other than symmetric mass ejection, is imparted to the compact
object. At this time there is no clear expression for how the system velocity
will be affected by the combination of these two sources of momentum (and any
others). In cosmic, the system velocity is derived such that it does not
explicitly include the effects of mass-loss kicks as described by Nelemans et
al. (1999). For this system, we have demonstrated the
$\upsilon_{\mathrm{MLK}}$ is low, and therefore calculations carried out by
cosmic are unlikely to significantly underestimate the
$\upsilon_{\mathrm{pec}}$. However, it is important to recognise the
limitations in our understanding of how the kinematics of a system post-SN
evolve — this will be the subject of future work.

##  5 Conclusions

We have highlighted the high peculiar velocity of the BH-LMXB H 1705–250, and
presented evidence that the system requires a high natal kick (in addition to
the imparted velocity associated with mass-loss due to supernova) to explain
this. This marks another BH for which there is evidence of high natal kicks
(i.e. comparable to the magnitude of NS kicks), implying that the discussion
regarding natal kicks cannot be limited to NS. We do not favour specific
mechanisms or prescriptions for natal kicks. Applying the same analysis used
in this paper to a larger sample of binaries may help in constraining the
nature of these kicks — this will be the subject of future work. The birth
site of H1705 is still degenerate; here we assume formation within the
Galactic plane, however, we cannot rule out formation within the halo or a
globular cluster. In this instance, the case for a high natal kick is less
robust; though still appears to be required.

## Acknowledgements

We thank a referee for feedback and insight, and STFC for support. We also
thank Christian Knigge for helpful comments and discussion. We thank Katie
Breivik and all those involved in cosmic for their hard work, and for making
the code publicly available.

## Data Availability

No new data are presented in this work. For details of compact object binaries
(including H1705) refer to Zhao23. cosmic is publicly available on GitHub –
refer to Breivik et al. (2020). Future work will include the code used to
track binary evolution with COSMIC.

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