GLQ | $t_{1,0}=0000$ | $t_{2,0}=0900$ | $t_{3,0}=1800$ | $t_{4,0}=2700$ | $t_{1,\star,0}=3600$
BBG | — | — | — | — | —
LNZ |  ($t_{1,2}=0500)$ | $(t_{2,2}=1501)$ | $(t_{3,2}=2318)$ |  $\bullet$ $(t_{4,2}=3265)$ | $(t_{1,\star,2}=4089)$
CRO | $t_{1,3}=0831$ | $(t_{2,3}=1721)$ | $t_{3,3}=2648$ | $(t_{4,3}=3522)$ | $t_{1,\star,3}=4449$
FKK | $t_{1,4}=1416$ | $t_{2,4}=2255$ |  $\bullet$ $t_{3,4}=3231$ | $t_{4,4}=4051$ | $t_{1,\star,4}=5023$
PMT | $(t_{1,5}=1665)$ | $t_{2,5}=2589$ | $(t_{3,5}=3453)$ | $t_{4,5}=4382$ | $(t_{1,\star,5}=5277)$
LIN | $(t_{1,6}=2110)$ | $t_{2,6}=3023$ | $(t_{3,6}=3907)$ | $t_{4,6}=4777$ | $(t_{1,\star,6}=5692)$
WGJ | — | — | — | — | —
HYM | $t_{1,8}=2955$ | $t_{2,8}=3860$ | $t_{3,8}=4717$ | $t_{4,8}=5645$ | $t_{1,\star,8}=6583$
EDB | $t_{1,9}=3175$ | $t_{2,9}=4148$ | $t_{3,9}=4841$ | $t_{4,9}=5826$ | $t_{1,\star,9}=6780$
Table 11: Case Study 6. Trial $1$. Typical timetable with no violations of the
separation conditions. The minimum separation is $34$ s. The bullets $\bullet$
show where the minimum separation occurs with $t_{4,2}=t_{3,4}+34$.

We will now show that the observed distribution of minimum separation times
can be supported by a theoretical argument. The minimum separation on each
trial is a random variable defined by $r=\min\\{r_{1},r_{2},r_{3}\\}$ where

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\min\\{t_{2,2}-t_{1,4},t_{3,2}-t_{2,4},t_{4,2}-t_{3,4},t_{1,\star,2}-t_{4,4}\\}$
$\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle\min\\{t_{2,3}-t_{1,5},t_{3,3}-t_{2,5},t_{4,3}-t_{3,5},t_{1,\star,3}-t_{4,5}\\}$
$\displaystyle r_{3}$ $\displaystyle=$
$\displaystyle\min\\{t_{2,6}-t_{1,8},t_{3,6}-t_{2,8},t_{4,6}-t_{3,8},t_{1,\star,6}-t_{4,8}\\}$

and where the individual time differences are normally distributed random
variables defined by

$\displaystyle t_{2,2}-t_{1,4}$ $\displaystyle=$ $\displaystyle
60+\theta\left[(h_{3}+60-900)+(h_{3}-h_{2})\right]^{1/2}z_{(2,2),(1,4)}\approx
60+s_{11}z_{(2,2),(1,4)}$ $\displaystyle t_{3,2}-t_{2,4}$ $\displaystyle=$
$\displaystyle
60+\theta\left[(h_{7}+60-1800)+(h_{7}-h_{4}-60)\right]^{1/2}z_{(3,2),(2,4)}\approx
60+s_{12}z_{(3,2),(2,4)}$ $\displaystyle\vdots$ $\displaystyle\vdots$
$\displaystyle t_{4,3}-t_{3,5}$ $\displaystyle=$ $\displaystyle
60+\theta\left[(h_{11}-h_{10})+(h_{11}-h_{10})\right]^{1/2}z_{(4,3),(3,5)}\approx
60+s_{23}z_{(4,3),(3,5)}$ $\displaystyle\vdots$ $\displaystyle\vdots$
$\displaystyle t_{1,\star,6}-t_{4,8}$ $\displaystyle=$ $\displaystyle
60+\theta\left[(h_{5}-h_{4})+(h_{5}+3540-h_{12}-60)\right]^{1/2}z_{(1,\star,6),(4,8)}\approx
60+s_{34}z_{(1,\star,6),(4,8)}$

in which $z_{(2,2),(1,4)},z_{(3,2),(2,4)},\ldots,z_{(4,3),(3,5)}\ldots
z_{(1,\star,6),(4,8)}\sim{\mathcal{N}}(0,1)$ are standard normal random
variables. The constants $S=[s_{i,j}]$ are given by

$S\approx\left[\begin{array}[]{cccc}18.1580&17.4201&18.1818&17.2538\\\
12.0576&13.6297&11.8889&13.9229\\\
19.4615&19.2753&18.3476&19.5650\end{array}\right].$

We can now find a theoretical distribution which is close to the observed
frequencies for the minimum separation. The cumulative distribution function
for the standard normal distribution is defined in terms of the standard
tabulated error function $\mbox{\rm erf}(x)$ by the formula
$\Phi(x)=(1/2)\left[1+\mbox{\rm erf}(x/2^{1/2})\right]$ for all
$x\in{\mathbb{R}}$. It follows that the cumulative distribution function for
the random variable $r_{i}$ is

$\displaystyle F_{i}(x)$ $\displaystyle=$
$\displaystyle\Phi\left((x-60)/s_{i1}\right)+\Phi\left((x-60)/s_{i2}\right)\cdot\left[1-\Phi\left((x-60)/s_{i1}\right)\right]$
$\displaystyle\hskip
14.22636pt+\Phi\left((x-60)/s_{i3}\right)\cdot\left[1-\Phi\left((x-60)/s_{i2}\right)\right]\cdot\left[1-\Phi\left((x-60)/s_{i1}\right)\right]$
$\displaystyle\hskip
14.22636pt+\Phi\left((x-60)/s_{i4}\right)\left[1-\Phi\left((x-60)/s_{i3}\right)\right]\cdot\left[1-\Phi\left((x-60)/s_{i2}\right)\right]\cdot\left[1-\Phi\left((x-60)/s_{i1}\right)\right]$

for each $i=1,2,3$. Now the cumulative distribution function for the random
variable $r$ is

$F(x)=F_{1}(x)+F_{2}(x)\cdot\left[1-F_{1}(x)\right]+F_{3}(x)\cdot\left[1-F_{2}(x)\right]\cdot\left[1-F_{1}(x)\right]$

for all $x\in{\mathbb{R}}$. We used Matlab to sketch the probability density
function $f(x)=F^{\,\prime}(x)$. The graph is shown on the right in Figure 9.
The theoretical mean and standard deviation are given approximately by
$\mu_{\mbox{\scriptsize\rm msep}}\approx 32.2897$ and
$\sigma_{\mbox{\scriptsize\rm msep}}\approx 10.3397$. The probability of a
violation is ${\mathbb{P}}[r<0]\approx 0.0051$. $\hfill\Box$

###### Remark 7

Case Study 6 shows that by choosing a buffer time approximately double the
observed standard deviation for the entire journey it is possible to eliminate
almost all potential safe-separation violations caused by normal stochastic
variation in segment traversal times. $\hfill\Box$

##  12 Conclusions

We found an analytic solution to the problem of minimizing tractive energy
consumption for a fleet of similar trains subject to active clearance-time
constraints that ensure safe separation and also compress the line-occupancy
timespan for the fleet. Compared to the schedule required for safe separation
when each train uses the classic single-train optimal strategy our method
provides a substantial reduction in line-occupancy timespan with only a
minimal increase in journey costs. We demonstrated our methods using a
sequence of case studies for an existing timetable on the Glasgow to Edinburgh
line using hypothetical separation constraints. Our calculations showed that
violations of the separation conditions which would occur if all trains use
individual optimal strategies can be avoided using our recommended strategies
with no change to the scheduled initial departure and final arrival times and
only a small increase in costs. We showed that an optimal timetable for
constant-speed strategies with safe separation constraints could be calculated
efficiently using a rapidly convergent Newton iteration. Our algorithm for
calculation of the optimal clearance times using the constant-speed strategies
is stable and efficient. Finally we showed that realistic strategies could be
implemented using an initial schedule obtained from a weighted optimal
constant-speed timetable and that this initial schedule could be improved
using a multi-dimensional method of steepest descent to find a near optimal
schedule for the realistic strategies.

The calculations for realistic strategies are much more demanding than the
calculations for the constant-speed strategies. However we reiterate that the
Energymiser® system is already used in practice on very fast trains to
continually calculate updated optimal driving strategies every few seconds for
journeys of more than one hundred km. It is envisaged that commercial
algorithms could be developed to implement rapid calculation of the schedules
devised here. Significant reductions in line-occupancy timespan for scheduled
services and cost reductions of between $5\%$ and $10\%$ obtained by
implementing optimal driving strategies can save many millions of dollars each
year for large rail organisations.

##  Appendix A Appendix: Mathematical background

###  A.1 Optimal strategies with intermediate time constraints

A train travels from $x=x_{0}$ to $x=x_{2}$ on level track. The time allowed
for the journey is $h_{2}=T$. We use the model described in Section 1.3 to
show that the optimal strategy is a long-haul strategy with optimal driving
speed $V$. Now suppose there is an intermediate time constraint and the train
must pass through $x_{1}$ at time $h_{1}$ without stopping. If we assume that

$\int_{0}^{V}dt_{a}(v)+(1/V)\left[x_{1}-\int_{0}^{V}dx_{a}(v)\right]<h_{1}.$

the long-haul strategy is no longer feasible. We will consider two plausible
alternatives. In each case we assume a strategy on $(x_{0},x_{1})$ of maximum
acceleration to speed $v=V_{1}<V$, speedhold with $v=V_{1}$, and maximum
acceleration to $v=U_{1}$ at $x_{1}$. Our first alternative on $(x_{1},x_{2})$
is a strategy of maximum acceleration starting from $v(x_{1})=U_{1}$ to
$v=V_{2}>V$, speedhold with $v=V_{2}$, coast to $v=U_{2}$ and maximum brake.
If this alternative is feasible then it is optimal. Our second alternative on
$(x_{1},x_{2})$ is a strategy of maximum acceleration starting from
$v(x_{1})=U_{1}$ to a maximum speed $v=V_{2}>V$, coast to $v=U_{2}$ and
maximum brake. The second alternative is more robust but is only optimal if
the first strategy is not feasible.

Case $1$: We assume the strategy is maximum acceleration, speedhold with
$v=V_{1}$, maximum acceleration to pass through $x_{1}$ with speed
$v(x_{1})=U_{1}$ and reach speed $V_{2}$, speedhold at speed $V_{2}$, coast to
speed $U_{2}$ and maximum brake. The cost of the strategy is

$J(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})=\int_{0}^{V_{2}}H(v)dx_{a}(v)+r(V_{1})\xi_{1}(U_{1})+r(V_{2})\xi_{2}(U_{1},U_{2},V_{2})$

where we have defined

$\xi_{1}(U_{1})=x_{1}-x_{0}-\int_{0}^{U_{1}}dx_{a}(v)$ (57)

and

$\xi_{2}(U_{1},U_{2},V_{2})=x_{2}-x_{1}-\int_{U_{1}}^{V_{2}}dx_{a}(v)-\int_{U_{2}}^{V_{2}}(-1)dx_{c}(v)-\int_{0}^{U_{2}}(-1)dx_{b}(v).$
(58)

This strategy is valid if $\xi_{1}(U_{1})\geq 0$ and
$\xi_{2}(U_{1},U_{2},V_{2})\geq 0$. The time taken to traverse $[x_{0},x_{1}]$
is

$\tau_{1}(U_{1},V_{1})=\int_{0}^{U_{1}}dt_{a}(v)+\xi_{1}(U_{1})/V_{1}$

and the time taken to traverse $[x_{1},x_{2}]$ is

$\tau_{2}(U_{1},U_{2},V_{2})=\int_{U_{1}}^{V_{2}}dt_{a}(v)+\int_{U_{2}}^{V_{2}}(-1)dt_{c}(v)+\int_{0}^{U_{2}}(-1)dt_{b}(v)+\xi_{2}(U_{1},U_{2},V_{2})/V_{2}.$

We wish to minimize
$J(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})$ subject to
$\tau_{1}(U_{1},V_{1})-h_{1}=0$ and
$\tau_{2}(U_{1},U_{2},V_{2})-h_{2}+h_{1}=0$. Define

${\mathcal{J}}(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})=J(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})+\mu_{1}[\tau_{1}(U_{1},V_{1})-h_{1}]+\mu_{2}[\tau_{2}(U_{1},U_{2},V_{2})-h_{2}+h_{1}]$

where $\mu_{1},\mu_{2}$ are Lagrange multipliers. Differentiation and some
elementary algebra gives

$\displaystyle\frac{\partial{\mathcal{J}}}{\partial U_{1}}$ $\displaystyle=$
$\displaystyle\frac{U_{1}[r(V_{2})-r(V_{1})]+\mu_{1}(1-U_{1}/V_{1})-\mu_{2}(1-U_{1}/V_{2})}{H(U_{1})-r(U_{1})}$
$\displaystyle\frac{\partial{\mathcal{J}}}{\partial U_{2}}$ $\displaystyle=$
$\displaystyle\left[\rule{0.0pt}{11.38092pt}U_{2}r(V_{2})-\mu_{2}(1-U_{2}/V_{2})\right]\left[\frac{1}{r(U_{2})}-\frac{1}{K(U_{2})+r(U_{2})}\right]$
$\displaystyle\frac{\partial{\mathcal{J}}}{\partial V_{1}}$ $\displaystyle=$
$\displaystyle\left[\rule{0.0pt}{11.38092pt}r^{\,\prime}(V_{1})-\mu_{1}/V_{1}^{2}\right]\xi_{1}(U_{1})$
$\displaystyle\frac{\partial{\mathcal{J}}}{\partial V_{2}}$ $\displaystyle=$
$\displaystyle\left[\rule{0.0pt}{11.38092pt}r^{\,\prime}(V_{2})-\mu_{2}/V_{2}^{2}\right]\xi_{2}(U_{1},U_{2},V_{2}).$

Setting the third and fourth partial derivatives equal to zero gives
$\mu_{j}=V_{j}^{2}r^{\,\prime}(V_{j})=\psi(V_{j})$ for each $j=1,2$. If we set
the second partial derivative to zero we get

$U_{2}=\psi(V_{2})/\varphi^{\,\prime}(V_{2})=U_{b}(V_{2}).$ (59)

Finally, if we set the first partial derivative equal to zero we get

$U_{1}=[\psi(V_{2})-\psi(V_{1})]/[\varphi^{\,\prime}(V_{2})-\varphi^{\,\prime}(V_{1})]=U_{s}(V_{1},V_{2}).$
(60)

This strategy of optimal type is defined by two independent variables $V_{1}$
and $V_{2}$, and two dependent variables $U_{1}=U_{s}(V_{1},V_{2})$ and
$U_{2}=U_{b}(V_{2})$. The values of $V_{1}$ and $V_{2}$ are fixed by the
signal times $h_{1}$ and $h_{2}$.

Case $2$: We assume the strategy is maximum acceleration, speedhold at speed
$V_{1}$, maximum acceleration to pass through $x_{1}$ with speed
$v(x_{1})=U_{1}$ and reach a maximum speed $V_{2}$, coast to speed $U_{2}$ and
maximum brake to stop at $x_{2}$. The cost of the strategy is

$J(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})=\int_{0}^{V_{2}}H(v)dx_{a}(v)+r(V_{1})\xi_{1}(U_{1}).$

The time taken to traverse $[x_{0},x_{1}]$ is

$\tau_{1}(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})=\int_{0}^{U_{1}}dt_{a}(v)+\xi_{1}(U_{1})/V_{1}$

and the time taken to traverse $[x_{1},x_{2}]$ is

$\tau_{2}(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})=\int_{U_{1}}^{V_{2}}dt_{a}(v)+\int_{U_{2}}^{V_{2}}(-1)dt_{c}(v)+\int_{0}^{U_{2}}(-1)dt_{b}(v).$

We wish the minimize
$J(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})$ subject to the
distance constraint $\xi_{2}(U_{1},U_{2},V_{2})=0$ and the time constraints
$\tau_{1}(U_{1},V_{1})-h_{1}=0$ and
$\tau_{2}(U_{1},U_{2},V_{2})-h_{2}+h_{1}=0$. Define

${\mathcal{J}}(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})=J(\mbox{$\mbox{\boldmath$U$}$},\mbox{$\mbox{\boldmath$V$}$})+\lambda\xi_{2}(U_{1},U_{2},V_{2})+\mu_{1}[\tau_{1}(U_{1},V_{1})-h_{1}]+\mu_{2}[\tau_{2}(U_{1},U_{2},V_{2})-h_{2}+h_{1}]$

where $\lambda,\mu_{1},\mu_{2}$ are Lagrange multipliers. Differentiation with
respect to the four independent variables and some elementary algebra gives

$\displaystyle\frac{\partial{\mathcal{J}}}{\partial U_{1}}$ $\displaystyle=$
$\displaystyle\frac{\mu_{1}-\mu_{2}-[r(V_{1})-\lambda+\mu_{1}/V_{1}]U_{1}}{H(U_{1})-r(U_{1})}$
$\displaystyle\frac{\partial{\mathcal{J}}}{\partial U_{2}}$ $\displaystyle=$
$\displaystyle\left[\rule{0.0pt}{11.38092pt}\lambda
U_{2}-\mu_{2}\right]\left[\frac{1}{r(U_{2})}-\frac{1}{K(U_{2})+r(U_{2})}\right]$
$\displaystyle\frac{\partial{\mathcal{J}}}{\partial V_{1}}$ $\displaystyle=$
$\displaystyle\left[\rule{0.0pt}{11.38092pt}r^{\,\prime}(V_{1})-\mu_{1}/V_{1}^{2}\right]\xi_{1}(U_{1})$
$\displaystyle\frac{\partial{\mathcal{J}}}{\partial V_{2}}$ $\displaystyle=$
$\displaystyle\left[\rule{0.0pt}{11.38092pt}\varphi(V_{2})-\lambda
V_{2}+\mu_{2}\right]\frac{H(V_{2})}{H(V_{2})-r(V_{2})}.$

Setting the partial derivatives equal to zero gives $\mu_{1}=\psi(V_{1})$,
$\lambda=\varphi(V_{2})/(V_{2}-U_{2})$,
$\mu_{2}=U_{2}\varphi(V_{2})/(V_{2}-U_{2})$ and

$U_{1}=[\psi(V_{1})-U_{2}\varphi(V_{2})/(V_{2}-U_{2})]/[\varphi^{\,\prime}(V_{1})-\varphi(V_{2})/(V_{2}-U_{2})]=U_{s}^{{\dagger}}(U_{2},V_{1},V_{2}).$
(61)

The formula for $U_{1}$ is valid provided
$U_{2}\in[\psi(V_{2})/\varphi^{\,\prime}(V_{2}),V_{2})$. For
$U_{2}=\psi(V_{2})/\varphi^{\,\prime}(V_{2})$ the formula reduces to (60). The
strategy is defined by the three independent variables $U_{2}$, $V_{1}$ and
$V_{2}$. The values are fixed by the times $h_{1}$ and $h_{2}$ and the
distance $x_{2}-x_{1}$.

###  A.2 Cost gradient with respect to journey time for journeys with
intermediate time constraints

We outline the derivation of the key formulæ for the partial rate of change of
journey cost with respect to the prescribed signal times. We consider Case $1$
and Case $2$ from Appendix A.1 and use the same notation.

Case $1$: We assume the optimal strategy takes the form described in Case $1$
of Appendix A.1. The cost of the strategy is

$J=\int_{0}^{V_{2}}H(v)dx_{a}(v)+r(V_{1})\xi_{1}(U_{1})+r(V_{2})\xi_{2}(U_{1},U_{2},V_{2})$
(62)

where $V_{1}$ and $V_{2}$ are independent variables,
$U_{1}=U_{s}(V_{1},V_{2})$ is the optimal speed at $x_{1}$ given by (60) and
$U_{2}=U_{b}(V_{2})$ is the optimal braking speed given by (59). The time
constraints are

$\int_{0}^{U_{1}}dt_{a}(v)+\xi_{1}(U_{1})/V_{1}=h_{1}$ (63)

and

$\int_{U_{1}}^{V_{2}}dt_{a}(v)+\int_{U_{2}}^{V_{2}}(-1)dt_{c}(v)+\int_{0}^{U_{2}}(-1)dt_{b}(v)+\xi_{2}(V_{1},V_{2})/V_{2}=h_{2}-h_{1}.$
(64)

Differentiation of (63) with respect to $h_{1}$ and $h_{2}$ and rearrangement
gives

$\xi_{1}\frac{\partial V_{1}}{\partial
h_{1}}=-V_{1}^{2}+V_{1}(V_{1}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}$ (65)

and

$\xi_{2}\frac{\partial V_{2}}{\partial
h_{1}}=V_{2}^{2}-V_{2}(V_{2}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}.$ (66)

Differentiation of (64) with respect to $h_{1}$ and $h_{2}$ and rearrangement
gives

$\xi_{2}\frac{\partial V_{2}}{\partial
h_{1}}=V_{2}^{2}-V_{2}(V_{2}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}$ (67)

and

$\xi_{2}\frac{\partial V_{2}}{\partial
h_{2}}=-V_{2}^{2}-V_{2}(V_{2}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial
h_{2}}+V_{2}(V_{2}-U_{2})[t_{c}^{\,\prime}(U_{2})-t_{b}^{\,\prime}(U_{2})]\frac{\partial
U_{2}}{\partial h_{2}}.$ (68)

Differentiation of (62) with respect to $h_{1}$ and $h_{2}$ gives

$\frac{\partial J}{\partial h_{1}}=r^{\,\prime}(V_{1})\xi_{1}\frac{\partial
V_{1}}{\partial h_{1}}-r(V_{1})x_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}+r^{\,\prime}(V_{2})\xi_{2}\frac{\partial
V_{2}}{\partial h_{1}}+r(V_{2})x_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}$ (69)

and

$\displaystyle\frac{\partial J}{\partial h_{2}}$ $\displaystyle=$
$\displaystyle r^{\,\prime}(V_{1})\xi_{1}\frac{\partial V_{1}}{\partial
h_{2}}-r(V_{1})x_{a}^{\,\prime}(U_{1})\frac{\partial U_{1}}{\partial
h_{2}}+r^{\,\prime}(V_{2})\xi_{2}\frac{\partial V_{2}}{\partial h_{2}}$ (70)
$\displaystyle\hskip 113.81102pt+r(V_{2})x_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial
h_{2}}-r(V_{2})[x_{c}^{\,\prime}(U_{2})-x_{b}^{\,\prime}(U_{2})]\frac{\partial
U_{2}}{\partial h_{2}}.$

It follows from (65) and (67) that (69) can be rewritten as

$\displaystyle\frac{\partial J}{\partial h_{1}}$ $\displaystyle=$
$\displaystyle\psi(V_{2})-\psi(V_{1})+\left[\rule{0.0pt}{11.38092pt}\psi(V_{1})-\varphi^{\,\prime}(V_{1})U_{1}-\psi(V_{2})+\varphi^{\,\prime}(V_{2})U_{1}\right]t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}$ (71) $\displaystyle=$
$\displaystyle\psi(V_{2})-\psi(V_{1}).$

It follows from (66) and (68) that (70) can be rewritten as

$\displaystyle\frac{\partial J}{\partial h_{2}}$ $\displaystyle=$
$\displaystyle-\psi(V_{2})+\left[\rule{0.0pt}{11.38092pt}\psi(V_{1})-\varphi^{\,\prime}(V_{1})U_{1}+\varphi^{\,\prime}(V_{2})U_{1}-\psi(V_{2})\right]t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{2}}$ (72) $\displaystyle\hskip
142.26378pt+\left[\rule{0.0pt}{11.38092pt}\psi(V_{2})-\varphi^{\,\prime}(V_{2})U_{2}\right][t_{c}^{\,\prime}(U_{2})-t_{b}^{\,\prime}(U_{2})]\frac{\partial
U_{2}}{\partial h_{2}}$ $\displaystyle=$ $\displaystyle-\psi(V_{2}).$

Case $2$: We assume the optimal strategy takes the form described in Case $2$
of Appendix A.1. The cost of the strategy is

$J=\int_{0}^{V_{2}}H(v)dx_{a}(v)+r(V_{1})\xi_{1}(U_{1})$ (73)

where $U_{2}$, $V_{1}$ and $V_{2}$ are independent variables and
$U_{1}=U_{s}^{{\dagger}}(U_{2},V_{1},V_{2})$ given by (61) is the optimal
signal location speed at $x_{1}$. There is one distance constraint

$x_{2}-x_{1}-\int_{U_{1}}^{V_{2}}dx_{a}(v)-\int_{U_{2}}^{V_{2}}(-1)dx_{c}(v)-\int_{0}^{U_{2}}(-1)dx_{b}(v)=0$
(74)

and two time constraints

$\int_{0}^{U_{1}}dt_{a}(v)+\xi_{1}(U_{1})/V_{1}=h_{1}$ (75)

and

$\int_{U_{1}}^{V_{2}}dt_{a}(v)+\int_{U_{2}}^{V_{2}}(-1)dt_{c}(v)+\int_{0}^{U_{2}}(-1)dt_{b}(v)=h_{2}-h_{1}.$
(76)

Differentiation of (74) with respect to $h_{1}$ and $h_{2}$ and using the
relationship $dx(v)=vdt(v)$ gives

$V_{2}[t_{a}^{\,\prime}(V_{2})-t_{c}^{\,\prime}(V_{2})]\frac{\partial
V_{2}}{\partial h_{1}}-U_{1}t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial
h_{1}}+U_{2}[t_{c}^{\,\prime}(U_{2})-t_{b}^{\,\prime}(U_{2})]\frac{\partial
U_{2}}{\partial h_{1}}=0,$ (77)

and

$V_{2}[t_{a}^{\,\prime}(V_{2})-t_{c}^{\,\prime}(V_{2})]\frac{\partial
V_{2}}{\partial h_{2}}-U_{1}t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial
h_{2}}+U_{2}[t_{c}^{\,\prime}(U_{2})-t_{b}^{\,\prime}(U_{2})]\frac{\partial
U_{2}}{\partial h_{2}}=0.$ (78)

Differentiation of (75) with respect to $h_{1}$ and $h_{2}$ and some
elementary algebra gives

$\xi_{1}\frac{\partial V_{1}}{\partial
h_{1}}=-V_{1}^{2}+V_{1}(V_{1}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}$ (79)

and

$\xi_{1}\frac{\partial V_{1}}{\partial
h_{2}}=V_{1}(V_{1}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial U_{1}}{\partial
h_{2}}.$ (80)

Differentiation of (76) with respect to $h_{1}$ and $h_{2}$ gives

$[t_{a}^{\,\prime}(V_{2})-t_{c}^{\,\prime}(V_{2})]\frac{\partial
V_{2}}{\partial h_{1}}-t_{a}^{\,\prime}(U_{1})\frac{\partial U_{1}}{\partial
h_{1}}+[t_{c}^{\,\prime}(U_{2})-t_{b}^{\,\prime}(U_{1})]\frac{\partial
U_{2}}{\partial h_{1}}=-1$ (81)

and

$[t_{a}^{\,\prime}(V_{2})-t_{c}^{\,\prime}(V_{2})]\frac{\partial
V_{2}}{\partial h_{2}}-t_{a}^{\,\prime}(U_{1})\frac{\partial U_{1}}{\partial
h_{2}}+[t_{c}^{\,\prime}(U_{2})-t_{b}^{\,\prime}(U_{1})]\frac{\partial
U_{2}}{\partial h_{2}}=1.$ (82)

Differentiation of (73) with respect to $h_{1}$ and $h_{2}$ gives

$\frac{\partial J}{\partial
h_{1}}=H(V_{2})x_{a}^{\,\prime}(V_{2})\frac{\partial V_{2}}{\partial
h_{1}}+r^{\,\prime}(V_{1})\xi_{1}\frac{\partial V_{1}}{\partial
h_{1}}-r(V_{1})x_{a}^{\,\prime}(U_{1})\frac{\partial U_{1}}{\partial h_{1}}$
(83)

and

$\frac{\partial J}{\partial
h_{2}}=H(V_{2})x_{a}^{\,\prime}(V_{2})\frac{\partial V_{2}}{\partial
h_{2}}+r^{\,\prime}(V_{1})\frac{\partial V_{1}}{\partial
h_{2}}-r(V_{1})x_{a}^{\,\prime}(U_{1})\frac{\partial U_{1}}{\partial h_{2}}.$
(84)

We can combine (77) and (81) to eliminate the terms in $\partial
U_{2}/\partial h_{1}$ and obtain

$(V_{2}-U_{2})[t_{a}^{\,\prime}(V_{2})-t_{c}^{\,\prime}(V_{2})]\frac{\partial
V_{2}}{\partial h_{1}}+(U_{2}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}=U_{2}.$

Substitution of the basic formulæ $t_{a}^{\,\prime}(v)=1/(H(v)-r(v))$,
$t_{c}^{\,\prime}(v)=-1/r(v)$ and $t_{b}^{\,\prime}(v)=(-1)/(K(v)+r(v))$ and
rearrangement now gives

$H(V_{2})x_{a}^{\,\prime}(V_{2})\frac{\partial V_{2}}{\partial
h_{1}}=\frac{\varphi(V_{2})U_{2}}{V_{2}-U_{2}}+\frac{\varphi(V_{2})(U_{1}-U_{2})}{V_{2}-U_{2}}t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}.$ (85)

Now (79) and (85) can be used to expand (83) and show that

$\displaystyle\frac{\partial J}{\partial h_{1}}$ $\displaystyle=$
$\displaystyle\frac{\varphi(V_{2})U_{2}}{V_{2}-U_{2}}-\psi(V_{1})+\left\\{\left[\frac{\varphi(V_{2})}{V_{2}-U_{2}}-\varphi^{\,\prime}(V_{1})\right]U_{1}+\psi(V_{1})-\frac{\varphi(V_{2})U_{2}}{V_{2}-U_{2}}\right\\}t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{1}}$ (86) $\displaystyle=$
$\displaystyle\frac{\varphi(V_{2})U_{2}}{V_{2}-U_{2}}-\psi(V_{1}).$

We can combine (78) and (82) to eliminate the terms in $\partial
U_{2}/\partial h_{2}$ and obtain

$(V_{2}-U_{2})[t_{a}^{\,\prime}(V_{2})-t_{c}^{\,\prime}(V_{2})]\frac{\partial
V_{2}}{\partial h_{2}}+(U_{2}-U_{1})t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{2}}=-U_{2}.$

Substitution of the basic formulæ $t_{a}^{\,\prime}(v)=1/(H(v)-r(v))$,
$t_{c}^{\,\prime}(v)=-1/r(v)$ and $t_{b}^{\,\prime}(v)=(-1)/(K(v)+r(v))$ and
rearrangement now gives

$H(V_{2})x_{a}^{\,\prime}(V_{2})\frac{\partial V_{2}}{\partial
h_{2}}=\frac{-\varphi(V_{2})U_{2}}{V_{2}-U_{2}}+\frac{\varphi(V_{2})(U_{1}-U_{2})}{V_{2}-U_{2}}t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{2}}.$ (87)

Now (80) and (87) can be used to expand (84) and show that

$\displaystyle\frac{\partial J}{\partial h_{2}}$ $\displaystyle=$
$\displaystyle\frac{-\varphi(V_{2})U_{2}}{V_{2}-U_{2}}+\left\\{\left[\frac{\varphi(V_{2})}{V_{2}-U_{2}}-\varphi^{\,\prime}(V_{1})\right]U_{1}+\psi(V_{1})-\frac{\varphi(V_{2})U_{2}}{V_{2}-U_{2}}\right\\}t_{a}^{\,\prime}(U_{1})\frac{\partial
U_{1}}{\partial h_{2}}$ (88) $\displaystyle=$
$\displaystyle\frac{-\varphi(V_{2})U_{2}}{V_{2}-U_{2}}.$

###  A.3 The Donsker invariance principle

The Donsker invariance principle can be seen as a specific generalization of
the central limit theorem. Let $(\Omega,\Sigma,\mu)$ be a probability space
and suppose that $\\{\xi_{k}\\}_{k\in{\mathbb{N}}}$ is a sequence of
independent and identically distributed random variables on
$(\Omega,\Sigma,\mu)$ with mean ${\mathbb{E}}[\xi_{k}]=0$ and variance
${\mathbb{E}}[\xi_{k}^{2}]=1$. Let $S_{m}=\sum_{k=1}^{m}\xi_{k}$ for all
$m\in{\mathbb{N}}$ and define a random continuous function
$X_{n}:C[0,1]\times\Omega\rightarrow{\mathbb{R}}$ by the formula

$X_{n}(t,\omega)=\left(1/\sqrt{n}\,\right)\mbox{$\sum_{1\leq k\leq\lfloor
nt\rfloor}$}\xi_{k}(\omega)+(nt-\lfloor
nt\rfloor)\left(1/\sqrt{n}\,\right)\xi_{\lfloor nt\rfloor+1}(\omega).$

The function $X_{n}(t,\omega)$ is a piecewise linear approximation to the
Wiener distribution. When $t=1$ we have

$X_{n}(1,\omega)=\left(1/\sqrt{n}\,\right)\mbox{$\sum_{1\leq k\leq
n}$}\xi_{k}(\omega)\rightarrow z(\omega)\sim{\mathcal{N}}(0,1)$

as $n\rightarrow\infty$ by the central limit theorem. When $t=p/q$ where
$p,q\in{\mathbb{N}}$ and $0\leq p<q$ we have

$X_{nq}(p/q,\omega)=\left(1/\sqrt{nq}\,\right)\mbox{$\sum_{1\leq k\leq
np}$}\xi_{k}(\omega)\rightarrow\sqrt{p/q}\cdot
z(\omega)\sim{\mathcal{N}}\left(0,p/q\,\right)$

as $n\rightarrow\infty$, once again by the central limit theorem. The Donsker
invariance principle [Donsker (1952), Dudley (1999)] extends this
convergence—in distribution—to all $t\in[0,1]$ with
$X_{n}(t,\omega)\rightarrow\sqrt{t}\cdot z(\omega)\sim{\mathcal{N}}(0,t)$ as
$n\rightarrow\infty$. The simpler step function $S_{n}(t,\omega)$ can be used
in place of the continuous function $X_{n}(t,\omega)$ in practice because
$\|X_{n}(t,\omega)-S_{n}(t,\omega)\|_{\infty}\rightarrow 0$ as
$n\rightarrow\infty$. The usual notational convention is to suppress the
dependence on $\omega$. Thus, for a scaled Wiener process $\epsilon(t)$ with
scale parameter $\theta$ we will normally write $\theta
X_{n}(t)\rightarrow\epsilon(t)=\theta W(t)=\theta\sqrt{t}\cdot
z\sim{\mathcal{N}}(0,\theta^{2}t)$.

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# Small-signal stability in inverter-dominated grids: exploring the role of
gains, line dynamics, and operating conditions

Ruth Kravis⋆, Gabriel E. Colón-Reyes⋆, Duncan S. Callaway⋆,† ⋆Department of
Electrical Engineering and Computer Sciences, University of California,
Berkeley, United States
†Energy and Resources Group, University of California, Berkeley, United States
{ruth.kravis, gecolonr<EMAIL_ADDRESS>
###### Abstract

In a power grid with growing penetrations of renewable energy sources,
inverters play a larger role in the dynamic interactions among network
components. However, much is yet to be studied regarding inverter-dominated
grid stability. This has initiated a re-examination of long-established
assumptions made when conducting power systems studies.

In this work we study the small-signal stability characteristics of a power
system comprised of inverter and synchronous machine sources, and consider the
effect of inverter control gains, transmission line dynamics, and network
operating conditions.

We utilize PowerSimulationsDynamics.jl to perform this study on the IEEE WSCC
9 Bus test system. We vary inverter parameters across reported ranges in the
literature, we consider three transmission line models of varying fidelity,
and compare results across different operating conditions.

We find that line dynamics are generally important for correctly assessing
small-signal stability in networks with large penetrations of grid-following
converters, and that gain tuning in these networks should utilize high
fidelity line models.

###### Index Terms:

Grid-forming inverters, grid-following inverters, small-signal analysis,
transmission line dynamics, low-inertia grids

##  I Introduction

As we continue to integrate converter interfaced generation (CIG) into the
grid, the potential for unstudied/under-studied dynamics to emerge increases.
This is due to the influence of inverter control gains, and in particular, due
to interactions of fast acting inverter control loops and fast electromagnetic
dynamics of power systems [1].

In this work, we study the small-signal stability of inverter-dominated grids,
while varying three elements: (1) the operating condition, which we
characterise as the combination of loading and share of load supplied by each
generator [2], (2) the gains/parameters of the generators, and (3) the
transmission line model. For (1), we choose representative scenarios for an
inverter-dominated grid, which includes a grid-forming (GFM), grid-following
(GFL), and synchronous machine (SM). For (2), we focus on the gains of the GFM
control loops, which include the virtual-synchronous machine (VSM) outer
control loop, and the cascaded voltage and current inner control loops [3]. We
use controller gains reported in the literature [4], and propose a method to
bring gains from different sources together based on time-scale separation
arguments. For (3), we consider three transmission line models of increasing
fidelity: $statpi$: an algebraic $\pi$ line model, $dynpi$: a dynamic $\pi$
line model, and multi-segment single-branch ($MSSB$): a dynamic, segmented
$\pi$ line model that captures higher frequency dynamics [5]. The higher
fidelity transmission line models will allow us to capture interactions
between states that are not captured by lower fidelity models.

We seek to answer the following research question: how do transmission line
models affect small-signal stability conclusions under different operation
conditions and gain choices?

We find that: 1) line models matter most in operating conditions with high GFL
share, and when they do matter, they have a stabilizing effect, 2) high GFL
share conditions tend to be less stable, which is consistent with results
reported in the literature, 3) under certain operating conditions, one set of
gains will result in a mostly unstable system using one line model, but an
always stable system using a different line model. This indicates that
inverter gain tuning requires using the highest fidelity line model possible,
particularly when working with GFL-dominated grids.

Notation: Dot notation indicates the time derivative of a variable, i.e.,
$\dot{x}=\frac{dx}{dt}$. Bold lower-case symbols are used to represent complex
variables in the $dq$ reference frame, e.g. $\boldsymbol{x}=x_{d}+jx_{q}$.

##  II Models

We use PowerSimulationsDynamics.jl (PSID) to model the system and perform
small-signal stability analysis.

(a)

(b)

Figure 1: Line model schematics: (a) $\pi$ model ($statpi$ and $dynpi$), and
(b) $MSSB$

###  II-A Power System Components

####  II-A1 Generator

Generators are modeled with five main components: a stator, a shaft, a turbine
governor, a power system stabilizer (PSS), and an automatic voltage regulator
(AVR). The models we choose, as well as parameters, can be found in Chapters
15 and 16 of [6]. We adopt the six-state Anderson-Fouad machine stator model.
We choose a shaft model given by the swing equations with damping. We opt for
a fixed input turbine governor, a Type 1 AVR, and we do not include a PSS.

####  II-A2 Inverter

Inverters are modeled with six main components: a DC voltage source, a model
for the switches, an output filter, an outer GFM or GFL control loop, an inner
control loop, and a frequency estimator. We choose models and parameters from
[7]. This includes a fixed DC voltage source model, an LCL filter, a VSM GFM
model for the outer loop, nested proportional-integral (PI) loops for the
inner control loops, and a phase locked loop (PLL) for damping of the virtual
frequency. We choose an averaged model for the switches [8]. For the GFL
control mode, we use active and reactive power PI controllers for the outer
loop, and a single inner current control PI loop [9].

###  II-B Transmission lines

To model TLs of arbitrary length $\ell$, we use per unit length parameters for
impedance, $z_{km}=r_{km}+j\omega l_{km}$, and shunt admittance,
$y_{km}=g_{km}+j\omega c_{km}$. At a particular operating frequency $\omega$,
we compute the lumped parameter equivalent $\pi$ model according to the
following equations, which include hyperbolic correction factors from the
steady state solution to Telegrapher’s equations:

$\displaystyle z_{\pi}$
$\displaystyle=z_{km}\ell\left(\frac{\sinh(\gamma\ell)}{\gamma\ell}\right)$
(1) $\displaystyle y_{\pi}$
$\displaystyle=y_{km}\ell\left(\frac{\tanh(\gamma\ell/2)}{\gamma\ell/2}\right)$
(2) $\displaystyle\gamma$ $\displaystyle=\sqrt{z_{km}y_{km}}$ (3)

where $\ell$ is line length (distinct from inductance $l$). We assume
$g_{km}=0$, which does not imply that $g_{\pi}=\mathrm{Re}(y_{\pi})=0$, but we
choose to overwrite $g_{\pi}=0$, since it is several orders of magnitude
smaller than the other line parameters. From Equations (1)–(3), we can compute
equivalent $r,l,c$ as
$r_{\pi}=\mathrm{Re}(z_{\pi}),l_{\pi}=x_{\pi}/\omega=\mathrm{Im}(z_{\pi})/\omega,c_{\pi}=b_{\pi}/\omega=\mathrm{Im}(y_{\pi})/\omega$.

####  II-B1 Algebraic $\pi$ model (statpi)

The algebraic $\pi$ model has the form shown in Fig. 1(a). It assumes that any
voltage/current dynamics are stable and settle quickly, compared to other
system dynamics. Therefore, the differential terms associated with the line
capacitance and inductance are set to zero, giving:

$\displaystyle\boldsymbol{i_{in}}$
$\displaystyle=\left(\frac{1}{z_{\pi}}+y_{\pi}\right)\boldsymbol{v_{1}}-\frac{1}{z_{\pi}}\boldsymbol{v_{2}}$
(4) $\displaystyle\boldsymbol{i_{out}}$
$\displaystyle=-\frac{1}{z_{\pi}}\boldsymbol{v_{1}}-\left(\frac{1}{z_{\pi}}+y_{\pi}\right)\boldsymbol{v_{2}}$
(5)

Since this model is purely algebraic, any dynamics on $\boldsymbol{i_{1}}$,
$\boldsymbol{i_{2}}$, $\boldsymbol{v_{1}}$, and $\boldsymbol{v_{2}}$ that
arise due to the interconnection of the line with dynamic devices will
instantaneously appear at the other end of the line.

####  II-B2 Dynamic $\pi$ model (dynpi)

The dynamic $\pi$ model has the same structure as shown in Fig. 1(a), but it
includes dynamics on line current and bus voltage states:

$\displaystyle\frac{l_{\pi}}{\omega_{b}}\frac{d\boldsymbol{i}}{dt}$
$\displaystyle=(\boldsymbol{v_{1}}-\boldsymbol{v_{2}})-z_{\pi}\boldsymbol{i}$
(6)
$\displaystyle\frac{c_{\pi}}{2}\frac{1}{\omega_{b}}\frac{d\boldsymbol{v_{1}}}{dt}$
$\displaystyle=(\boldsymbol{i_{in}}-\boldsymbol{i})-y_{\pi}\boldsymbol{v_{1}}$
(7)
$\displaystyle\frac{c_{\pi}}{2}\frac{1}{\omega_{b}}\frac{d\boldsymbol{v_{2}}}{dt}$
$\displaystyle=(\boldsymbol{i}-\boldsymbol{i_{out}})-y_{\pi}\boldsymbol{v_{2}}$
(8)

Similar to $statpi$, $dynpi$ only captures the effect of the distributed line
parameters in steady state.

####  II-B3 Multi-segment single-branch $\pi$ model ($MSSB$)

We define a line with multiple ‘segments’ as one divided into a discrete set
of $N$ identical length $\pi$ components in series. The parameters for each
segment are given by:

$\displaystyle r_{seg}$ $\displaystyle=r_{km}\ell_{seg}$ (9) $\displaystyle
l_{seg}$ $\displaystyle=l_{km}\ell_{seg}$ (10) $\displaystyle c_{seg}$
$\displaystyle=c_{km}\ell_{seg}$ (11)

where $\ell_{seg}=\frac{\ell}{N}$. Further, $z_{seg}=r_{seg}+j\omega l_{seg}$
and $y_{seg}=j\omega c_{seg}$. This model is shown in Fig. 1(b). The $i^{th}$
segment of an $N$-segment $MSSB$ model is defined by the following equations:

$\displaystyle\frac{l_{seg}}{\omega_{b}}\frac{d\boldsymbol{i_{i}}}{dt}$
$\displaystyle=(\boldsymbol{v_{i}}-\boldsymbol{v_{i+1}})-z_{seg}\boldsymbol{i_{i}}$
(12)
$\displaystyle\frac{c_{seg}}{2}\frac{1}{\omega_{b}}\frac{d\boldsymbol{v_{i}}}{dt}$
$\displaystyle=(\boldsymbol{i_{i-1}}-\boldsymbol{i_{i}})-y_{seg}\boldsymbol{v_{i}}$
(13)
$\displaystyle\frac{c_{seg}}{2}\frac{1}{\omega_{b}}\frac{d\boldsymbol{v_{i+1}}}{dt}$
$\displaystyle=(\boldsymbol{i_{i}}-\boldsymbol{i_{i+1}})-y_{seg}\boldsymbol{v_{i+1}}$
(14)

Note that for $i=1$, $i_{i-1}=i_{in}$, and for $i=N$, $i_{i+1}=i_{out}$. As
$N$ is increased, the $MSSB$ model more closely approximates the equivalent
$\pi$ model in steady state frequency response. The advantage of explicitly
representing segments is that unlike the equivalent $\pi$ model (Eqn.
(1)–(3)), it captures the distributed nature of the line parameters in both
transient and steady state responses.

###  II-C Line parameters

To obtain line parameters, we start with the frequency dependent line
impedance parameters from Table 3 in [10]. From this data, we build a
frequency dependent model of $z_{km}$ via vector fitting [11, 7], a form of
parameter estimation that incorporates real transmission line data. We then
evaluate this model at nominal frequency to obtain $z_{km}$. In combination
with a constant value for $y_{km}$, we use this value in Eqns. (1)–(3) to
obtain the lumped parameter model parameters for $statpi$ and $dynpi$. This
approach is useful because it allows frequency dependent models to be built if
needed, such as those in [7] and [12].

###  II-D Aggregate system model

All the models described above can be written as a system of differential-
algebraic equations (DAEs). Linking the devices and network, we arrive at a
mathematical form as follows:

$\begin{bmatrix}\dot{x}\\\ 0\end{bmatrix}=\begin{bmatrix}f(x,y,u)\\\
g(x,y,u)\end{bmatrix}$ (15)

Here, $x\in\mathbb{R}^{n}$ are the system’s dynamic states,
$y\in\mathbb{R}^{m}$ are the system’s algebraic states, $u\in\mathbb{R}^{p}$
are the system inputs.
$f:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{n}$
and
$g:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{m}$
are the vector equations associated to the dynamics of the network and the
algebraic constraints.

We are interested in studying the small-signal stability of such a system. In
general, $f$ is a nonlinear vector field. Therefore, to study small-signal
stability we find an equilibrium point $(x^{\star},y^{\star},0)$ by setting
$\dot{x}=0$ and solving the nonlinear system of equations, then linearize
around that point, and arrive at a set of linear dynamics that characterize
the behavior of the system in the vicinity of that equilibrium point. The
resulting equations will then be of the following form:

$\Delta\dot{x}=J(x^{\star},y^{\star})\Delta x$ (16)

Here, $J(x^{\star},y^{\star})$ is the reduced system Jacobian matrix. By
studying the eigenvalues of this matrix, we can determine if
$(x^{\star},y^{\star},0)$ is a stable or unstable operating condition for the
network. See [13] or [14] for further details on the linearization process in
PSID.

##  III Small-Signal Stability Analysis

###  III-A Test case

We use the IEEE WSCC 9 Bus Test case, shown in Fig. 2 to perform our analysis.
We verify that the operating conditions we study in the same network, where
each generator is a SM.

Figure 2: Modified IEEE 9 bus test system. SM at bus 1 (reference bus), GFL at
bus 2, VSM GFM at bus 3.

####  III-A1 Parameters

We fix the parameters for the GFL, with outer loop PI gains shown in Table I.
We use the parameters from [3] for the inner loop, filter, and PLL.

Parameter | Symbol | Value [p.u.]
---|---|---
Active power control P gain | $k_{pp}$ | 0.0059
Active power control I gain | $k_{ip}$ | 7.36
Reactive power control P gain | $k_{pq}$ | 0.0059
Reactive power control I gain | $k_{iq}$ | 7.36
Table I: GFL parameters

Table II summarises the GFM gain parameters we chose to vary. Ranges are
constructed from values reported in [3], [2], and [4], although we note that a
much wider range of parameters is reported across the literature.

Parameter | Symbol | Range | Unit
---|---|---|---
VSM inertia time constant | $T_{a}$ | 0.5 – 2 | s
VSM damping coefficient | $k_{d}$ | 100 – 400 | p.u.
Reactive Power Droop gain | $k_{q}$ | 0.05 – 0.2 | p.u.
Inner voltage loop P gain | $k_{p,v}$ | 0.5 – 0.6 | p.u.
Inner voltage loop I gain | $k_{i,v}$ | 400 – 800 | p.u.
Inner current loop P gain | $k_{p,c}$ | 0.74 – 1.27 | p.u.
Inner current loop I gain | $k_{i,c}$ | 1.19 – 14.3 | p.u.
Table II: GFM parameter ranges

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: Boxplots for different operating conditions, showing the real part
of the least stable eigenvalue for different line models. Top row is $load\
scale=0.4$, bottom row is $load\ scale=1.0$. Columns correspond to Case 1
through 4, respectively. All columns except for (b)/(f) have the same y-axis
range (difference between maximum and minimum value of the eigenvalue).

Not all combinations of parameters are desirable from an engineering
perspective. For example, separation of time scales is an important attribute
of GFM and GFL control architecture design. Therefore, we filter the
parameters based on engineering considerations.

For the GFM controls, we require that that $k_{pc}$ is $10\times$ larger than
$k_{pv}$. The same requirement is applied to the GFL: $k_{pc}$ must be
$10\times$ larger than $k_{pp}$ and $k_{pq}$. This will help achieve time
scale separation. We also require that the corresponding integral gains for
all these controllers be $10\times$ larger than the proportional gain to help
with driving steady state errors to zero.

Case Name | $\mathbf{\eta_{ibr}}$ | $\mathbf{\eta_{gfl}}$ | $\mathbf{\eta_{gfm}}$
---|---|---|---
Case 1 | 70% | 14% | 56%
Case 2 | 70% | 42% | 28%
Case 3 | 90% | 18% | 72%
Case 4 | 90% | 54% | 36%
Table III: Operating conditions

Lastly, we construct operating scenarios where inverters supply a high (70%)
and very high (90%) share of total active power. We also vary which inverter
is injecting more active power. Table III summarises the operating conditions
analysed – IBR refers to inverter-based resources. The values quoted in the
table refer to the fraction of active power load supplied by the GFM and GFL.
Losses are picked up by the SM at the slack bus, so in reality, these
fractions are slightly smaller than what is quoted. Note that
$\eta_{ibr}=\eta_{gfm}+\eta_{gfl}$.

Finally, we choose two $load\ scale$ values, to simulate light and heavier
loading. We conduct experiments by first discretizing the gain choices
according to our filtering scheme, then randomly sampling 1000 parameter
choices from a possible 7000 combinations. Then, for each set of sampled
parameter choices, and for each case and $load\ scale$, we perform small-
signal stability analysis of the system using all three line models. We
examine the real part of the least stable eigenvalue to evaluate stability at
that operating point.

###  III-B Results and Discussion

(a) $k_{d}$

(b) $T_{a}$

(c) $k_{q}$

Figure 4: Boxplots for outer loop GFM gains, with $load\ scale=0.4$, case 2
(corresponding to Fig. 3(b)).

####  III-B1 Effect of generation share

Fig. 3 presents results for the eight operating conditions selected. Firstly,
we observe that high GFL share cases (column 2 and 4) are less stable for a
fixed $\eta_{ibr}$ (i.e., comparing column 1 to 2, and 3 to 4). This effect is
more pronounced for larger $\eta_{ibr}$. These results agree with our
expectations and prior results [2].

####  III-B2 Effect of loading

We observe that the range in the eigenvalues’ real part is much larger for
heavier loading. Furthermore, for some gain choices, increasing the loading
actually results in a more stable system. For example, comparing Fig. 3(c) and
3(g) we see that Fig. 3(g) (higher loading) can be either more or less stable
than Fig. 3(c) (lower loading). This suggests that – unsurprisingly – under
high loading gains must be chosen carefully.

####  III-B3 Effect of line models

Including line dynamics has either no effect or a stabilizing effect on the
equilibrium point, and this effect is most pronounced for high $\eta_{gfl}$.
This means $statpi$ provides a conservative estimate of system stability. This
is seen clearly in Fig. 3(b), where using $statpi$ may result in an incorrect
assessment of stability. In other cases (e.g., Fig. 3(f)) line dynamics do not
always provide sufficient stabilizing to move the eigenvalues to the LHP.
Echoing our conclusions on generation share, since $statpi$ and $dynpi$/$MSSB$
are most different at high $\eta_{gfl}$, and $MSSB$ is the highest fidelity
line model, this suggests that $statpi$ should be treated with caution at high
GFL shares.

####  III-B4 Effect of gains

We explore Fig. 3(b) to understand why $statpi$ and $dynpi$/$MSSB$ are
different under high GFL shares, and if there are any trends in the gains that
might give insight. Fig. 4 shows boxplots for the three outer loop GFM gains
that were varied in this analysis ($k_{d}$, $T_{a}$, and $k_{q}$). The inner
loop gains ($k_{pv}$, $k_{iv}$, $k_{pc}$, and $k_{ic}$) were also swept, but
they didn’t have a strong trend/influence on the overall stability of the
system so are not included here.

In Fig. 4(c) we see a strong upward trend, which suggests that, for the ranges
of parameters sampled, $k_{q}$ has the most influence on small-signal
stability. This result could be a consequence of the parameter ranges we chose
– other gains may exhibit trends over a larger range of values.

Lastly, Fig. 5 examines how the eigenvalues in Fig. 3(b) change for two
different parameter sets, one where $k_{q}=0.05$ and one where $k_{q}=0.2$.
For the set where $k_{q}=0.05$, the system is stable for $statpi$ and $dynpi$
(solid points). For the set where $k_{q}=0.2$, the system is unstable for
$statpi$ but stable for $dynpi$. The eigenvalue pair that cross into the RHP
are strongly participated in by the the GFL $\sigma_{q}$ state, as measured by
participation factors. This state corresponds to the GFL outer loop reactive
power control. The unstable eigenvalues are also complex, which has
implications for appropriate measures for estimating stability (e.g., a
damping ratio may be appropriate).

Figure 5: Eigenvalues for $load\ scale=0.4$, case 2 (Fig. 3(b)), for two
different sets of parameters. Solid points correspond to $k_{q}=0.05$ (both
line models stable), light points correspond to $k_{q}=0.2$ ($statpi$
unstable, $dynpi$ stable).

##  IV Conclusions and Future Work

We find that transmission line model choice can alter a modeler’s conclusion
about a system’s small-signal stability in inverter-dominated grids. This is
especially true for GFL-dominated systems. In the cases we studied we found
that lower fidelity line models (those without dynamics) tended to be more
conservative with respect to small-signal stability. This suggests that using
low fidelity line models will not lead to false positive characterizations of
system stability. In this work we also see that the operating condition
significantly affects the overall stability of the system, regardless of line
models. We also see that the parameter choice for GFM gains matters more under
heavier loading, in particular, the reactive power outer loop gain $k_{q}$. We
also observed that unstable eigenvalues are complex, motivating further
exploration of appropriate measures for estimating stability in inverter-
dominated grids.

Future work includes integrating GFL gains into the parameter ranges. We also
saw a large range of reported parameters for inverters, including parameters
that do not adhere to the engineering principles discussed in Section III-A1.
This large range, as well as the diversity of test systems studied in the
literature, makes comparing results across studies challenging. Further work
should focus on establishing sets of parameters and test systems that can
enable better benchmarking.

## Acknowledgment

This research was supported by the U.S. Department of Energy’s Solar Energy
Technologies Office through award 38637 (UNIFI Consortium).

## References

  * [1] N. Hatziargyriou, J. Milanovic, C. Rahmann, V. Ajjarapu, C. Canizares, I. Erlich, D. Hill, I. Hiskens, I. Kamwa, B. Pal _et al._ , “Definition and classification of power system stability–revisited & extended,” _IEEE Transactions on Power Systems_ , vol. 36, no. 4, pp. 3271–3281, 2020.
  * [2] U. Markovic, O. Stanojev, P. Aristidou, E. Vrettos, D. S. Callaway, and G. Hug, “Understanding small-signal stability of low-inertia systems,” _IEEE Transactions on Power Systems_ , 2021.
  * [3] S. D’Arco, J. A. Suul, and O. B. Fosso, “A virtual synchronous machine implementation for distributed control of power converters in smartgrids,” _Electric Power Systems Research_ , vol. 122, pp. 180–197, 2015.
  * [4] “Virtual synchronous machine grid-forming inverter model specification (regfm_b1),” NREL, Tech. Rep.
  * [5] S. D’Arco, J. Beerten, and J. Suul, “Cable model order reduction for hvdc systems interoperability analysis,” in _11th IET International Conference on AC and DC Power Transmission_. Birmingham, UK: Institution of Engineering and Technology, 2015, pp. 026 (10 .)–026 (10 .). [Online]. Available: https://digital-library.theiet.org/content/conferences/10.1049/cp.2015.0039
  * [6] F. Milano, _Power system modelling and scripting_. Springer Science & Business Media, 2010.
  * [7] J. Beerten, S. D’Arco, and J. A. Suul, “Frequency‐dependent cable modelling for small‐signal stability analysis of vsc‐hvdc systems,” _IET Generation, Transmission & Distribution_, vol. 10, no. 6, p. 1370–1381, Apr 2016.
  * [8] A. Yazdani and R. Iravani, _Voltage-sourced converters in power systems: modeling, control, and applications_. John Wiley & Sons, 2010.
  * [9] R. W. Kenyon, A. Sajadi, A. Hoke, and B.-M. Hodge, “Open-source pscad grid-following and grid-forming inverters and a benchmark for zero-inertia power system simulations,” in _2021 IEEE Kansas Power and Energy Conference (KPEC)_. IEEE, 2021, pp. 1–6.
  * [10] H. Dommel, “Overhead line parameters from handbook formulas and computer programs,” _IEEE Transactions on Power Apparatus and Systems_ , vol. PAS-104, no. 2, p. 366–372, Feb. 1985.
  * [11] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” _IEEE Transactions on power delivery_ , vol. 14, no. 3, pp. 1052–1061, 1999.
  * [12] G. E. Colon-Reyes, R. Kravis, S. Sharma, and D. Callaway, “Transmission line dynamics on inverter-dominated grids: analysis and simulations,” 2023. [Online]. Available: https://arxiv.org/abs/2310.08553
  * [13] R. Henriquez-Auba, J. D. Lara, C. Roberts, and D. S. Callaway, “Grid forming inverter small signal stability: Examining role of line and voltage dynamics,” in _IECON 2020 The 46th Annual Conference of the IEEE Industrial Electronics Society_. IEEE, 2020, pp. 4063–4068.
  * [14] J. D. Lara, R. Henriquez-Auba, M. Bossart, D. S. Callaway, and C. Barrows, “Powersimulationsdynamics.jl – an open source modeling package for modern power systems with inverter-based resources,” 2023. [Online]. Available: https://arxiv.org/abs/2308.02921


Further author information:
A.Drescher.: E-mail<EMAIL_ADDRESS>
M.Fabricius<EMAIL_ADDRESS>
# GRAVITY+ Wide: Towards hundreds of z $\sim$ 2 AGN

Antonia Drescher  Max-Planck-Institut für Extraterrestrische Physik (MPE),
Gießenbachstraße, Garching bei München, Germany  Felix Widmann  Max-Planck-
Institut für Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei
München, Germany  Stefan Gillessen  Max-Planck-Institut für Extraterrestrische
Physik (MPE), Gießenbachstraße, Garching bei München, Germany  Thomas Ott
Max-Planck-Institut für Extraterrestrische Physik (MPE), Gießenbachstraße,
Garching bei München, Germany  Taro Shimizu  Max-Planck-Institut für
Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei München,
Germany  Frank Eisenhauer  Max-Planck-Institut für Extraterrestrische Physik
(MPE), Gießenbachstraße, Garching bei München, Germany  Maximilian Fabricius
Max-Planck-Institut für Extraterrestrische Physik (MPE), Gießenbachstraße,
Garching bei München, Germany  Julien Woillez  European Southern Observatory,
Karl-Schwarzschild-Str. 2, 85748 Garching, Germany  Frederic Gonté  European
Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany
Matthew Horrobin  1\. Physikalisches Institut, Universität zu Köln, Zülpicher
Str. 77, 50937 Köln, Germany  Jingyi Shangguan  Max-Planck-Institut für
Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei München,
Germany  Senol Yazici  Max-Planck-Institut für Extraterrestrische Physik
(MPE), Gießenbachstraße, Garching bei München, Germany  Guy Perrin  LESIA,
Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités,
UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cité, 92195 Meudon
Cedex, France  Thibaut Paumard  LESIA, Observatoire de Paris, PSL Research
University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris
Diderot, Sorbonne Paris Cité, 92195 Meudon Cedex, France  Wolfgang Brandner
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
Laura Kreidberg  Max-Planck-Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany  Christian Straubmeier  1\. Physikalisches Institut,
Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany  Karine Perraut
Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  Jean-Baptiste Le
Bouquin  Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  Paulo
Garcia  CENTRA and Universidade de Lisboa - Faculdade de Ciencias, Campo
Grande, 1749-016 Lisboa, Portugal  Faculdade de Engenharia, Universidade do
Porto, rua Dr. Roberto Frias, 4200-465 Porto, Portugal  Sebastian Hönig
School of Physics & Astronomy, University of Southampton, Southampton, SO17
1BJ, UK  Denis Defrère  Institute of Astronomy, KU Leuven, Celestijnenlaan
200D, B-3001, Leuven, Belgium  Guillaume Bourdarot  Max-Planck-Institut für
Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei München,
Germany  Helmut Feuchtgruber  Max-Planck-Institut für Extraterrestrische
Physik (MPE), Gießenbachstraße, Garching bei München, Germany  Reinhard Genzel
Max-Planck-Institut für Extraterrestrische Physik (MPE), Gießenbachstraße,
Garching bei München, Germany  Michael Hartl  Max-Planck-Institut für
Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei München,
Germany  Dieter Lutz  Max-Planck-Institut für Extraterrestrische Physik (MPE),
Gießenbachstraße, Garching bei München, Germany  Nikhil More  Max-Planck-
Institut für Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei
München, Germany  Christian Rau  Max-Planck-Institut für Extraterrestrische
Physik (MPE), Gießenbachstraße, Garching bei München, Germany  Sinem Uysal
Max-Planck-Institut für Extraterrestrische Physik (MPE), Gießenbachstraße,
Garching bei München, Germany  Ekkehard Wieprecht  Max-Planck-Institut für
Extraterrestrische Physik (MPE), Gießenbachstraße, Garching bei München,
Germany  GRAVITY+ Collaboration

###### Abstract

As part of the GRAVITY+project, GRAVITY and the VLTI are currently undergoing
a series of significant upgrades to further improve the performance and sky
coverage. The instrumental changes will be transformational, and for instance
uniquely position GRAVITY to observe the broad line region of hundreds of
$z\sim 2$ Active Galactic Nuclei (AGN). The increased sky coverage is achieved
by enlarging the maximum angular separation between the celestial science
object (SC) and the off-axis fringe tracking (FT) object from currently 2
arcseconds (arcsec) up to unprecedented 30 arcseconds (limited by the
atmospheric conditions), now demonstrated at the 8m telescopes for the first
time.

###### keywords:

interferometry, wide-angle fringe tracking, VLTI, GRAVITY

##  1 INTRODUCTION

GRAVITY [2017A&A...602A..94G] at the VLTI has delivered groundbreaking results
in different areas in modern astronomy in its first five years of science
operation. In order to stay at the forefront of discovery and to go even
further, a new combination of upgrades to the VLTI and GRAVITY has been
commissioned. This project is called GRAVITY+. GRAVITY+aims for all-sky and
high contrast interferometry with milli-arcsecond resolution imaging and faint
science with a limiting magnitude of m${}_{K}=22$[Eisenhauer2019]. GRAVITY+is
implemented in several phases, where some of them are already concluded, such
as the implementation of two new grisms in 2019 to increase the instrument
throughput by a factor $>2$ [Yazici2021]. Further phases which are currently
under development are the implementation of new deformable mirrors and state-
of-the-art adaptive optics wavefront sensors, laser guide stars on all four 8m
UTs, an improved vibration control and the reduction of existing noise sources
in GRAVITY. For the latter, a new observing mode has been developed which
suppresses the noise from the metrology laser and is called GRAVITY-
faintAddFelixWSPIEpaperreference. Finally, GRAVITY+includes the implementation
of wide-field off-axis fringe tracking, called GRAVITY Wide, which is the
topic of this paper. GRAVITY Wideallows for increasing the separation between
the SC and FT from currently 2 arcsec up to about 30 arcsec, which
tremendously improves the sky coverage.

This paper presents the implementation of GRAVITY Wide, an upgrade of the near
in- frared beam combiner GRAVITY at the Very Large Telescope Interferometer.
GRAVITY Wide is one specific part of the GRAVITY+ upgrades which allow for
increasing the sensitivity and sky coverage in order to go into all-sky
interferometric imaging at milli-arcsecond resolution and towards faint
science on targets with m K ¿ 22 mag. The sky coverage is improved by the
implementation of wide-field off-axis fringe tracking which allows for
increasing the angular separation between science object and fringe tracking
star where the latter is a star used to stabilize the optical path difference
between the telescopes. This implementation is called GRAVITY Wide.

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Examples of the format to be followed are given at the end of this document.

The reference list at the end of this document is created using BibTeX, which
looks through the file report.bib for the entries cited in the LaTeX source
file. The format of the reference list is determined by the bibliography style
file spiebib.bst, as specified in the `\bibliographystyle{spiebib}` command.
Alternatively, the references may be directly formatted in the LaTeX source
file.

For books[Lamport94, Alred03, Goossens97], the listing includes the list of
authors, book title, publisher, city, page or chapter numbers, and year of
publication. A reference to a journal article[Metropolis53] includes the
author list, title of the article (in quotes), journal name (in italics,
properly abbreviated), volume number (in bold), inclusive page numbers, and
year. By convention[Lamport94], article titles are capitalized as described in
Sec. 2.1. A reference to a proceedings paper or a chapter in an edited
book[Gull89a] includes the author list, title of the article (in quotes),
volume or series title (in italics), volume number (in bold), if applicable,
inclusive page numbers, publisher, city, and year. References to an article in
the SPIE Proceedings may include the conference name (in italics), as shown in
Ref. Hanson93c. For websites[Lees-Miller-LaTeX-course-1] the listing includes
the list of authors, title of the article (in quotes), website name, article
date, website address either enclosed in chevron symbols (’$<$’ and ’$>$’),
underlined or linked, and the date the website was accessed.

If you use this formatting, your references will link your manuscript to other
research papers that are in the CrossRef system. Exact punctuation is required
for the automated linking to be successful.

Citations to the references are made using superscript numerals, as
demonstrated in the above paragraph. One may also directly refer to a
reference within the text, e.g., “as shown in Ref. Metropolis53 …”

###  2.7 Footnotes

Footnotes111Footnotes are indicated as superscript symbols to avoid confusion
with citations. may be used to provide auxiliary information that doesn’t need
to appear in the text, e.g., to explain measurement units. They should be used
sparingly, however.

Only nine footnote symbols are available in LaTeX. If you have more than nine
footnotes, you will need to restart the sequence using the command
`\footnote[1]{Your footnote text goes here.}`. If you don’t, LaTeX will
provide the error message Counter too large., followed by the offending
footnote command.

##  3 SECTION FORMATTING

Section headings are centered and formatted completely in uppercase 11-point
bold font. Sections should be numbered sequentially, starting with the first
section after the Abstract. The heading starts with the section number,
followed by a period. In LaTeX, a new section is created with the `\section{}`
command, which automatically numbers the sections.

Paragraphs that immediately follow a section heading are leading paragraphs
and should not be indented, according to standard publishing style[Lamport94].
The same goes for leading paragraphs of subsections and sub-subsections.
Subsequent paragraphs are standard paragraphs, with 14-pt. (5 mm) indentation.
An extra half-line space should be inserted between paragraphs. In LaTeX, this
spacing is specified by the parameter `\parskip`, which is set in spie.cls.
Indentation of the first line of a paragraph may be avoided by starting it
with `\noindent`.

###  3.1 Subsection Attributes

The subsection heading is left justified and set in 11-point, bold font.
Capitalization rules are the same as those for book titles. The first word of
a subsection heading is capitalized. The remaining words are also capitalized,
except for minor words with fewer than four letters, such as articles (a, an,
and the), short prepositions (of, at, by, for, in, etc.), and short
conjunctions (and, or, as, but, etc.). Subsection numbers consist of the
section number, followed by a period, and the subsection number within that
section.

####  3.1.1 Sub-subsection attributes

The sub-subsection heading is left justified and its font is 10 point, bold.
Capitalize as for sentences. The first word of a sub-subsection heading is
capitalized. The rest of the heading is not capitalized, except for acronyms
and proper names.

##  4 FIGURES AND TABLES

Figures are numbered in the order of their first citation. They should appear
in numerical order and on or after the same page as their first reference in
the text. Alternatively, all figures may be placed at the end of the
manuscript, that is, after the Reference section. It is preferable to have
figures appear at the top or bottom of the page. Figures, along with their
captions, should be separated from the main text by at least 0.2 in. or 5 mm.

Figure captions are centered below the figure or graph. Figure captions start
with the figure number in 9-point bold font, followed by a period; the text is
in 9-point normal font; for example, “Figure 3. Original image…”. See Fig. 1
for an example of a figure caption. When the caption is too long to fit on one
line, it should be justified to the right and left margins of the body of the
text.

Tables are handled identically to figures, except that their captions appear
above the table.

---
Figure 1:  Figure captions are used to describe the figure and help the reader
understand it’s significance. The caption should be centered underneath the
figure and set in 9-point font. It is preferable for figures and tables to be
placed at the top or bottom of the page. LaTeX tends to adhere to this
standard.

##  5 MULTIMEDIA FIGURES - VIDEO AND AUDIO FILES

Video and audio files can be included for publication. See Tab. 3 for the
specifications for the mulitimedia files. Use a screenshot or another .jpg
illustration for placement in the text. Use the file name to begin the
caption. The text of the caption must end with the text
“http://dx.doi.org/doi.number.goes.here” which tells the SPIE editor where to
insert the hyperlink in the digital version of the manuscript.

Here is a sample illustration and caption for a multimedia file:

---
Figure 2:  A label of “Video/Audio 1, 2, …” should appear at the beginning of the caption to indicate to which multimedia file it is linked . Include this text at the end of the caption: http://dx.doi.org/doi.number.goes.here Table 3: Information on video and audio files that must accompany a manuscript submission. Item | Video | Audio
---|---|---
File name | Video1, video2… | Audio1, audio2…
Number of files | 0-10 | 0-10
Size of each file | 5 MB | 5 MB
File types accepted | .mpeg, .mov (Quicktime), .wmv (Windows Media Player) | .wav, .mp3

##  Appendix A MISCELLANEOUS FORMATTING DETAILS

It is often useful to refer back (or forward) to other sections in the
article. Such references are made by section number. When a section reference
starts a sentence, Section is spelled out; otherwise use its abbreviation, for
example, “In Sec. 2 we showed…” or “Section 2.1 contained a description…”.
References to figures, tables, and theorems are handled the same way.

###  A.1 Formatting Equations

Equations may appear in line with the text, if they are simple, short, and not
of major importance; e.g., $\beta=b/r$. Important equations appear on their
own line. Such equations are centered. For example, “The expression for the
field of view is

$2a=\frac{(b+1)}{3c}\,,$ (1)

where $a$ is the …” Principal equations are numbered, with the equation number
placed within parentheses and right justified.

Equations are considered to be part of a sentence and should be punctuated
accordingly. In the above example, a comma follows the equation because the
next line is a subordinate clause. If the equation ends the sentence, a period
should follow the equation. The line following an equation should not be
indented unless it is meant to start a new paragraph. Indentation after an
equation is avoided in LaTeX by not leaving a blank line between the equation
and the subsequent text.

References to equations include the equation number in parentheses, for
example, “Equation (1) shows …” or “Combining Eqs. (2) and (3), we obtain…”
Using a tilde in the LaTeX source file between two characters avoids unwanted
line breaks.

###  A.2 Formatting Theorems

To include theorems in a formal way, the theorem identification should appear
in a 10-point, bold font, left justified and followed by a period. The text of
the theorem continues on the same line in normal, 10-point font. For example,

Theorem 1. For any unbiased estimator…

Formal statements of lemmas and algorithms receive a similar treatment.

###### Acknowledgements.

This unnumbered section is used to identify those who have aided the authors
in understanding or accomplishing the work presented and to acknowledge
sources of funding.