# Stochastic Operation of Energy Constrained Microgrids Considering Battery
Degradation

Per Aaslid12, Magnus Korpås2, Michael M Belsnes1 and Olav B Fosso2
1Department of Energy Systems, SINTEF Energy Research, Trondheim, Norway
2Department of Electrical Power Engineering, Norwegian University of Science
and Technology, Trondheim, Norway

###### Abstract

Power systems with high penetration of variable renewable generation are
vulnerable to periods with low generation. An alternative to retain high
dispatchable generation capacity is electric energy storage that enables
utilization of surplus power, where the electric energy storage contributes to
the security of supply. Such systems can be considered as energy-constrained,
and the operation of the electric energy storage must balance operating cost
minimization and the risk of scarcity. In this matter, operation dependent
storage characteristics such as energy storage degradation are a complicating
factor. This paper proposes a linear approximation of battery state-of-charge
degradation and implements it in a stochastic dual dynamic programming based
energy-management model in combination with cycling degradation. The
implications of degradation modelling are studied for a small Norwegian
microgrid with variable renewable power generation and limited dispatchable
generation capacity as well as battery and hydrogen storage to balance supply
and demand. Our results show that the proposed strategy can prolong the
expected battery lifetime by more than four years compared to the naive
stochastic strategy but may cause increased degradation for other system
resources. It is also evident that a stochastic strategy is crucial to retain
low risk of scarcity in energy-constrained systems.

###### Index Terms:

Energy Management, Electric Energy Storage, Multi-Stage Stochastic
Programming, Battery Degradation

This work has been funded by the Norwegian Research Council under grant number
272398.

##  I Introduction

This paper addresses important operational challenges of future electricity
systems with high variable renewable energy source (VRES) penetration and
electric energy storage (EES), where new types of technological constraints
will influence the optimal scheduling decisions. New methodologies that
capture risk of scarcity, consequence of uncertainty, flexible demand, energy
storage, and degradation are needed. The presented work investigates how
degradation impacts operational costs, strategy, and the expected lifetime of
batteries. These are crucial mechanisms and should be included in both the
planning and operation of the future power system, where the installed solar
photovoltaic (PV) and wind capacity must be quadrupled by 2030 to follow the
net zero pathway [1].

Systems with high penetration of VRESs rely on sufficient dispatchable
generation capacity to meet the peak demand and also in the hours with low
VRES generation. An alternative to dispatchable thermal generation capacity is
to utilize EES flexibility. A challenge with EES is that the current decision
also affects the future energy content and the capability of providing
capacity in the future. The decisions here and now must be taken while
accounting for future power generation and load under uncertainty, and needs
to balance the risk of generation curtailment versus the risk of scarcity. The
operation of EES in these situations can be seen as a precaution or arbitrage
against extreme prices [2, 3].

Different EES technologies have complementary properties with respect to power
and energy scalability. Lithium-ion batteries have gained high attention both
in the research community as well as for power system applications due to
their ability to deliver and absorb very high power almost instantaneously
with low losses. They also have a relatively high energy to weight ratio
compared to similar battery technologies. However, they are expensive to scale
up with respect to energy compared to hydrogen, which can be stored in large
tanks and scaled up at a relatively low cost. However, the cost of fuel cells
and electrolyzers are still very high, and the round-trip efficiency is poor
compared to batteries [4].

Degradation characteristics also differ for batteries and hydrogen systems.
The aging of hydrogen fuel cells are largely affected by start, stop and rapid
ramping. However, the degradation has often been studied for vehicles that
exhibit several cycles each hour [5], while a grid connected fuel cell will
operate with less frequent cycling. Moreover, degradation of fuel cells can
also be related to dry membranes caused by limited operation, and modest
operation can extend the expected lifetime compared to low operation [6]. The
degradation cost of the hydrogen system has therefore been neglected.

The degradation of lithium-ion batteries is closely related to operating
conditions like charge/discharge power, depth-of-discharge (DOD), state-of-
charge (SOC), temperature, and ampere throughput [7]. Energy management of
VRES typically involves daily cycles, and the battery will rarely operate
close to its maximum power capabilities. Moreover, the battery temperature
will be controlled to ensure optimal operating conditions and minimal
degradation. This paper will therefore consider degradation caused by DOD and
SOC.

Experimental results show that the degradation rate of lithium-ion batteries
increases with increasing DOD. Moreover, the degradation rate is also higher
for high SOC [8, 9], but very low SOC will also cause high degradation [10,
11, 12, 13]. The aging is therefore influenced by the operational pattern, and
the optimal power dispatch largely depends on the battery’s aging model [14].
Previous studies of microgrid (MG) economic dispatch often neglect the cost
associated with degradation [15, 16]. Single factor models, considering
degradation either as a function of power, SOC, DOD, or ampere hour
throughput, are also widely adopted [14]. For example, references [17, 18, 19,
20] assume the degradation is proportional with energy throughput. More
sophisticated models capture non-linear effects, either as a single factor
model [21] or combined models [22, 23].

However, non-linear models are often computationally intensive, especially for
large-scale systems and stochastic problems. Convex and linear problem
formulations reduce the computational burden, and enable utilization of
decomposition techniques such as dual decomposition. Reference [24] proposes a
piece-wise linear relaxation of the the non-linear DOD degradation, and shows
that a cost reduction can be achieved by considering the cyclic degradation in
the market clearing of a battery. However, linear approximation of the non-
linear SOC degradation has gained less attention.

This paper proposes a piece-wise linear approximation of the battery SOC
degradation effect, and demonstrates it in combination with linear DOD
degradation [24] on a real MG from the EU project REMOTE [25, 26]. The system
is operating islanded, and is energy constrained since the backup generator is
too small to cover the peak demand and the EES must be operated to prevent
load shedding in extreme situations. The optimal operation is considered using
rolling horizon [27] and stochastic dual dynamic programming (SDDP) [28]. The
system is simulated for a whole year with rolling horizon using real
observations from the MG and scenarios generated from historical weather
forecasts. Infinite horizon is embedded using cyclic Markov chains [29], and
the implication of including battery degradation will be studied with respect
to the costs, VRES and EES utilization as well as expected battery lifetime.

The remainder of the paper is organized as follows: section II describes the
rolling horizon simulation method as well as the SDDP algorithm, and derives
the linear power system model including the SOC degradation model; section III
presents how the proposed method is implemented and the numerical values of
the cases; section IV shows and discusses the results; and section V draws the
conclusions and suggests future work.

##  II Method

This section presents the rolling horizon stochastic energy-management model
where the goal is to find the optimal control decisions $u_{s}$ for each stage
$s$ for the in-going state $x_{s}$ and the uncertainty $\omega_{s}$.

###  II-A Rolling horizon simulation

123456Prediction horizonRoll horizonPrediction horizonRoll horizonPrediction
horizonRoll horizon$u_{1}$$u_{2}$$x_{2}$$x_{3}$stages Figure 1: Rolling
horizon optimization.

Generation forecasts are based on weather forecasts issued with fixed
intervals, and the stochastic model is trained each time a new forecast is
available for the prediction horizon as illustrated in Fig. 1. The actual
generation and demand is observed and evaluated for the roll horizon interval
using the trained stochastic model yielding the optimal control $u_{s}$ that
is implemented. Finally, the optimization horizon is moved forward and the
procedure is repeated using the end state $x_{s+1}$ from previous optimization
as the initial value.

###  II-B Multi-stage stochastic programming

This paper considers multi-stage stochastic programming (MSSP) for solving the
proposed energy management problem. MSSP captures that energy management is a
sequential decision making process and recognizes that decisions can be
updated stage-wise as uncertainty is revealed. The MSSP formulation in 1a, 1b
and 1c is divided into several stages $s$, each representing a discrete moment
in time. The goal is to minimize the current and future operating costs. State
variables $x_{s}$ represent variables connected across stages, such as EES
SOC. Control variables $u_{s}$ are decisions, both implicit and explicit, and
must be within the technical limitations of the system given by the set of
admissible controls 1c. The state transition function 1b describes the
relation between the state variables across stages. The random variable
$\omega_{s}$ represents the uncertainty in demand and VRES generation.

$\min_{u_{t}}\Bigg{\\{}C_{1}(x_{1},u_{1},\omega_{1})+\underset{\omega_{2}|\omega_{1}}{\mathbb{E}}\bigg{[}\underset{u_{2}}{\min}\Big{(}C_{2}(x_{2},u_{2},\omega_{2})+\dots{\\\
}+\underset{\omega_{S}|\omega_{S-1},\dots,\omega_{2}}{\mathbb{E}}\Big{[}\underset{u_{S}}{\min}\big{(}C_{S}(x_{S},u_{S},\omega_{S})\big{)}\Big{]}\Big{)}\bigg{]}\Bigg{\\}}$
(1a) $\displaystyle\text{s.t.}\quad x_{s+1}$
$\displaystyle=T_{s}(x_{s},u_{s},\omega_{s})$ (1b) $\displaystyle u_{s}$
$\displaystyle\in U_{s}(x_{s},\omega_{s})$ (1c)

By assuming the random variable is stage-wise independent and has a discrete
set of realizations for each stage, the problem can be formulated on extended
form. Instead of solving the intractable extended problem, SDDP [28]
decomposes the problem into sub-problems for each stage $s\in S$ and
realization of the random variable $\omega_{s}\in\Omega_{s}$ as shown in 2a,
2b, 2c, 2d and 2e. The algorithm is divided into two phases: forward pass and
backward recursion. The forward pass samples a random variable for each stage
and solves the sequence of sub-problems using the outgoing state of stage $s$
as the in-going state to stage $s+1$. When the final stage is reached, the
backward recursion starts by solving the final stage for all the random
variables. From convexity, the dual variables $\lambda_{s}$ of the state
variable $x_{s}$ 2b can be used to generate a linear cutting plane 2e that
acts as a lower bound for the previous stage problem, and the procedure is
repeated all the way back to the first stage. The whole procedure is repeated,
adding new cuts for each iteration $k$, until the convergence criteria is met
[28, 29].

$\displaystyle\underset{u_{s},x_{s},x_{s+1},\theta_{s}}{\min}$ $\displaystyle
C_{s}(x_{s},u_{s},\omega_{s})+\theta_{s}$ (2a) $\displaystyle\text{s.t.
}x_{s}$ $\displaystyle=\bar{x_{s}},\quad[\lambda_{s}]$ (2b) $\displaystyle
x_{s+1}$ $\displaystyle=T_{s}(x_{s},u_{s},\omega_{s})$ (2c) $\displaystyle
u_{s}$ $\displaystyle\in U_{s}(x_{s},\omega_{s})$ (2d)
$\displaystyle\theta_{s}$
$\displaystyle\geq\alpha_{s}^{k}+\beta_{s}^{k}x_{s+1},\quad
k\in\\{1,2,\dots,K\\}$ (2e)

###  II-C Power system model

The stage-wise power system model including the battery degradation model will
be expressed using the following sets and indices:

  * •

$t\in\mathcal{T}$: Time index $t$ and the set of time steps $\mathcal{T}$.

  * •

$g\in\mathcal{G}$: Generator $g$ and the set of dispatchable generators
$\mathcal{G}$.

  * •

$r\in\mathcal{R}$: VRES $r$ and the set of VRESs $\mathcal{R}$.

  * •

$d\in\mathcal{D}$: Consumer $d$ and the set of consumers $\mathcal{D}$.

  * •

$e\in\mathcal{E}$: EES $e$ and the set of EESs $\mathcal{E}$.

  * •

$k_{\delta}\in\mathcal{K}_{\delta}$: Battery cycling segment $k_{\delta}$ in
the set of segments $\mathcal{K}_{\delta}$.

  * •

$k_{\sigma}^{up}\in\mathcal{K}_{\sigma}^{up}/k_{\sigma}^{dn}\in\mathcal{K}_{\sigma}^{up}$:
Battery SOC segment $k_{\sigma}^{up}/k_{\sigma}^{dn}$ direction up/down in the
set of segments $\mathcal{K}_{\sigma}^{up}/\mathcal{K}_{\sigma}^{up}$
respectively.

The following parameters have been used:

  * •

$\Delta T_{t}$: Timestep $t$ step length.

  * •

$PG_{g}^{max}$: Generator $g$ maximum power dispatch.

  * •

$C_{g}$: Generator $g$ marginal operating cost.

  * •

$PR_{r,t}^{max}$: VRES $r$ maximum power at time $t$.

  * •

$PD_{d,t}$: Demand of consumer $d$ at time $t$.

  * •

$C_{d}$: Consumer $d$ marginal load shedding cost.

  * •

$SOC_{e}^{min}/SOC_{e}^{max}$: EES $e$ minimum and maximum SOC.

  * •

$PS_{e}^{c}/PS_{e}^{d}$: EES $e$ maximum charge/discharge power.

  * •

$C_{e,k_{\delta}}$: EES $e$ DOD marginal degradation cost segment $k$.

  * •

$SOC_{e}^{ref}$: EES $e$ SOC degradation reference value.

  * •

$C_{e,k_{\sigma}}^{up}/C_{e,k_{\sigma}}^{dn}$: EES $e$ SOC marginal
degradation cost up/down segment $k_{\sigma}^{up}/k_{\sigma}^{dn}$.

  * •

$\eta_{e}^{c}/\eta_{e}^{d}$: EES $e$ charge/discharge efficiency.

  * •

$R$: Battery replacement cost.

The functions and variables are:

  * •

$p_{g,t}$: Generator $g$ power dispatch at time $t$.

  * •

$p_{g,t}$: VRES $r$ power dispatch at time $t$.

  * •

$p_{d,t}$: Consumer $d$ demand at time $t$.

  * •

$pls_{d,t}$: Consumer $d$ load shedding at time $t$.

  * •

$ps_{e,t(,k_{\delta})}^{c}/ps_{e,t(,k_{\delta})}^{d}$: EES $e$ power
charge/discharge DOD (segment $k_{\delta}$) at time $t$.

  * •

$soc_{e,t,k_{\delta}}$: EES $e$ SOC DOD segment $k_{\delta}$ at time $t$.

  * •

$soc_{e,t,k_{\sigma}}^{up}/soc_{e,t,k_{\sigma}}^{dn}$: EES $e$ SOC degradation
segment $k_{\sigma}^{up}/k_{\sigma}^{dn}$ above/below reference value at time
$t$.

  * •

$\delta_{t}$: Unitless EES cycle depth at time $t$.

  * •

$\sigma_{t}$: Unitless EES SOC at time $t$.

  * •

$\sigma^{ref}$: Unitless EES reference SOC.

  * •

$f_{\delta}(\delta_{t})$: Incremental battery fade as a function of cycle
depth $\delta_{t}$ at time $t$.

  * •

$f_{\sigma}(\sigma_{t})$: Incremental battery fade as a function of SOC
$\sigma_{t}$ at time $t$.

The resulting model is summarized in 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13.
The objective is to minimize the dispatchable generation costs, load shedding,
and EES degradation associated with both DOD and SOC 3. The total power
injections and withdrawals must balance at all time steps 4. The generation,
both dispatchable and VRES, must respect the maximum generation 5 and 6.
Demand that can not be met causes load shedding 7. The battery
charge/discharge must respect the maximum limits, both per segment 8 and 9 and
the sum of the segments 10 and 11. The EES energy balance is expressed per
segment 12, where the segments are divided into equal sizes 13.

$\min\sum_{t\in\mathcal{T}}\Bigg{[}\sum_{g\in\mathcal{G}}C_{g}p_{g,t}+\sum_{d\in\mathcal{D}}C_{d}pls_{d,t}+\sum_{e\in\mathcal{E}}\sum_{j\in\mathcal{J}}C_{e,k_{\delta}}ps_{e,t,k_{\delta}}^{d}\\\
+\sum_{e\in\mathcal{E}}\bigg{(}\sum_{k_{\sigma}\in\mathcal{K}_{\sigma}^{up}}C_{e,k_{\sigma}}^{up}soc_{e,t,k_{\sigma}}^{up}+\sum_{k_{\sigma}\in\mathcal{K}_{\sigma}^{dn}}C_{e,k_{\sigma}}^{dn}soc_{e,t,k_{\sigma}}^{dn}\bigg{)}\Bigg{]}$
(3)

subject to

$\sum_{g\in\mathcal{G}}p_{g,t}+\sum_{r\in\mathcal{R}}p_{r,t}+\sum_{e\in\mathcal{E}}ps_{e,t}^{d}=\sum_{d\in\mathcal{D}}p_{d,t}+\sum_{e\in\mathcal{E}}ps_{e,t}^{c}$
(4) $\displaystyle 0$ $\displaystyle\leq p_{g,t}\leq PG_{g}^{max}$ (5)
$\displaystyle 0$ $\displaystyle\leq p_{r,t}\leq PR_{r,t}^{max}$ (6)
$\displaystyle p_{d,t}$ $\displaystyle=PD_{d,t}-pls_{d,t}$ (7) $\displaystyle
0$ $\displaystyle\leq ps_{e,t,k_{\delta}}^{c}\leq PS_{e}^{c}$ (8)
$\displaystyle 0$ $\displaystyle\leq ps_{e,t,k_{\delta}}^{d}\leq PS_{e}^{d}$
(9) $\displaystyle ps_{e,t}^{c}$
$\displaystyle=\sum_{j\in\mathcal{J}}ps_{e,t,k_{\delta}}^{c}\leq PS_{e}^{c}$
(10) $\displaystyle ps_{e,t}^{d}$
$\displaystyle=\sum_{j\in\mathcal{J}}ps_{e,t,k_{\delta}}^{d}\leq PS_{e}^{d}$
(11) $soc_{e,t,k_{\delta}}=soc_{e,t-1,k_{\delta}}\\\ +\Delta
T_{t}\left(\eta_{e}^{c}ps_{e,t,k_{\delta}}^{c}-\frac{1}{\eta_{e}^{d}}ps_{e,t,k_{\delta}}^{d}\right)$
(12) $\displaystyle 0$ $\displaystyle\leq
soc_{e,t,k_{\delta}}\leq\frac{1}{|\mathcal{K}_{\delta}|}(SOC_{e}^{max}-SOC_{e}^{min})$
(13)

Note that restrictions to prevent simultaneous charging and discharging have
not been included, which implies that dumping of energy from the battery is
accepted. This is not a problem when the VRES generation can be curtailed at
no cost. However, by introducing SOC degradation cost, situations where
dumping of energy is beneficial might arise.

The state $x_{s}$ comprises the initial SOC variable $soc_{e,t,k_{\delta}}$ at
each stage, and the final SOC variable at the outgoing state $x_{s+1}$. The
random variable $\omega$ comprises the renewable generation $PR_{r,t}^{max}$
and the demand $PD_{d,t}$ for all the steps in the stage. The remaining
variables are decisions $u_{s}$, either explicit or implicit.

###  II-D EES degradation model

The proposed model assumes the degradation due to DOD and SOC are decoupled.
For an arbitrary convex DOD capacity fade function $f_{\delta}(\delta_{t})$,
the EES SOC is divided into equally sized segments $K_{\delta}$ yielding cost
coefficients $C_{\delta k}$ 14 [24].

$\displaystyle C_{\delta
k}=\frac{R}{\eta^{d}SOC^{max}}|\mathcal{K}_{\delta}|\left[f_{\delta}\left(\frac{k}{|\mathcal{K}_{\delta}|}\right)-f_{\delta}\left(\frac{k-1}{|\mathcal{K}_{\delta}|}\right)\right],k\in\mathcal{K}_{\delta}$
(14)

Each energy level $soc_{t}$ and charge/discharge $ps_{t}^{c}/ps_{t}^{d}$ is
divided into $K_{\delta}$ segments. Since the marginal cost curve is convex,
the cheapest available segment will always be discharged, and the suggested
method will therefore count cycles in a similar manner as the Rainflow
counting algorithm [30].

For an arbitrary convex SOC capacity fade function $f_{\sigma}(\sigma)$,
$\sigma^{ref}$ represents the SOC level where the SOC degradation is lowest as
shown in 15.

$\displaystyle\sigma^{ref}=\underset{\sigma}{\mathrm{argmin}}\frac{\partial
f_{\sigma}(\sigma)}{\partial\sigma}$ (15)

The incremental capacity fade as a function of SOC can be found by taking the
derivative of $f_{\sigma}(\sigma)$ with respect to $soc_{t}$.

$\displaystyle\frac{\partial f_{\sigma}(\sigma_{t})}{\partial
soc_{t}}=\frac{df_{\sigma}(\sigma_{t})}{d\sigma_{t}}\frac{\partial\sigma_{t}}{\partial
soc_{t}}=\frac{1}{SOC^{max}}\frac{df_{\sigma}(\sigma_{t})}{d\sigma_{t}}$ (16)

The $soc_{e,t}$ variable is divided into $K_{\sigma}^{up}$ and
$K_{\sigma}^{dn}$ equally sized segments $soc_{e,t,k}^{up}$ and
$soc_{e,t,k}^{dn}$ for up and down direction, respectively, as shown in 17,
18, 19 and 20 such that $soc_{e,t,k}^{up}$ and $soc_{e,t,k}^{dn}$ represent
the distance from the reference value in both directions.

$\displaystyle\sum_{k\in\mathcal{K}_{\sigma}^{up}}soc_{e,t,k}^{up}$
$\displaystyle\geq soc_{e,t}-SOC_{e}^{ref}$ (17)
$\displaystyle\sum_{k\in\mathcal{K}_{\sigma}^{dn}}soc_{e,t,k}^{dn}$
$\displaystyle\geq SOC_{e}^{ref}-soc_{e,t}$ (18) $\displaystyle 0\leq
soc_{e,t,k}^{up}$
$\displaystyle\leq\frac{1}{|\mathcal{K}_{\sigma}^{up}|}\left(SOC^{max}-SOC_{e}^{ref}\right)$
(19) $\displaystyle 0\leq soc_{e,t,k}^{dn}$
$\displaystyle\leq\frac{1}{|\mathcal{K}_{\sigma}^{dn}|}SOC_{e}^{ref}$ (20)

Let $C_{\sigma,k}^{up}$ and $C_{\sigma,k}^{dn}$ denote the incremental aging
cost with respect to each segment in either direction $soc_{e,t,k}^{up}$ and
$soc_{e,t,k}^{dn}$. The resulting cost coefficients are expressed in 21 and
22.

$C_{\sigma
k}^{up}=\frac{R}{SOC^{max}}|\mathcal{K}_{\sigma}^{up}|\bigg{[}f_{\sigma}\left(\sigma^{ref}+\frac{k}{|\mathcal{K}_{\sigma}^{up}|}(1-\sigma^{ref})\right)\\\
-f_{\sigma}\left(\sigma^{ref}+\frac{k-1}{|\mathcal{K}_{\sigma}^{up}|}(1-\sigma^{ref})\right)\bigg{]},k\in\mathcal{K}_{\sigma}^{up}$
(21) $C_{\sigma
k}^{dn}=\frac{R}{SOC^{max}}|\mathcal{K}_{\sigma}^{dn}|\bigg{[}f_{\sigma}\left(\sigma^{ref}-\frac{k}{|\mathcal{K}_{\sigma}^{dn}|}\sigma^{ref}\right)\\\
-f_{\sigma}\left(\sigma^{ref}-\frac{k-1}{|\mathcal{K}_{\sigma}^{dn}|}\sigma^{ref}\right)\bigg{]},k\in\mathcal{K}_{\sigma}^{dn}$
(22)

The cheapest segments will always be used first, and the correct segment will
be used given a convex cost function.

##  III Implementation and numerical values

The proposed method is implemented in Julia (1.4.2) with SDDP.jl (0.3.14) [31]
and Gurobi (9.1). The models were trained with 50 SDDP iterations.

###  III-A Microgrid

The model has been tested on the Rye microgrid in central Norway that is
partly funded by the Horizon 2020 project REMOTE [25]. The MG comprises a few
farms and houses, and is supplied by a wind turbine, solar PV, and a diesel
generator that serves as backup in case of insufficient VRES generation. The
system is equipped with a battery and hydrogen storage to balance supply and
demand [26]. Load shedding costs occur if the supply is unable to meet the
demand. Numerical values for the MG are shown in Table I, and the EES in Table
II. Note that the wind and diesel generator sizes in this case differs from
the actual system. Fig. 2 shows the four day average VRES generation and
demand for the whole period.

TABLE I: Microgrid numerical values Description | Unit | Value
---|---|---
Wind turbine capacity | [kW] | 135
Solar PV capacity | [kW] | 86
Diesel generator capcity | [kW] | 25 / 75
Diesel generation cost | [€/MWh] | 100
Load shedding cost | [€/MWh] | 5000
TABLE II: Numerical values for microgrid ees. Description | Unit | Lithium-ion | Hydrogen
---|---|---|---
Charge power | [kW] | 500 | 55
Discharge power | [kW] | 500 | 100
Size | [kWh] | 500 / 1000 | 3300 / -
Charge efficiency | [%] | 96 | 64
Discharge efficiency | [%] | 96 | 50
Replacement cost | [€/kWh] | 100 | NA
$\displaystyle{0}$$\displaystyle{100}$$\displaystyle{200}$$\displaystyle{300}$$\displaystyle{365}$Time
[days]$\displaystyle{0}$$\displaystyle{20}$$\displaystyle{40}$Power
[kW]WindSolarDemand Figure 2: Four day average generation and demand for the
optimization period.

###  III-B Battery degradation

This paper uses a quadratic DOD capacity fade function 23 [13, 22]. The upper
range of the SOC stress function 24 is exponential [32], while the lower part
is defined to capture the potential collapse associated with operating at very
low SOC [23, 12]. Note that any convex fade function can be used.

$\displaystyle f_{\delta}(\delta)$ $\displaystyle=k_{\delta}\ \delta^{2}$ (23)
$\displaystyle f_{\sigma}(\sigma)$ $\displaystyle=\begin{cases}k_{\sigma
1}e^{k_{\sigma 2}(\sigma-\sigma^{ref})}&0.2\leq\sigma\leq 1\\\
f_{\sigma}(1.0)+\frac{\sigma}{0.1}(f_{\sigma}(0.2)-f_{\sigma}(1.0))&0\leq\sigma<0.1\\\
f_{\sigma}(0.2)&0.1\leq\sigma<0.2\end{cases}$ (24)

Assuming the battery fade is 85% higher at 90% compared to 10% SOC [33],
yields $k_{\sigma 2}=0.769$. The battery is assumed to reach end of life after
10 years with 3,000 cycles at 80% DOD with 50% average SOC and after 20 years
with no cycling at 50% SOC. These assumptions yield
$k_{\delta}=\text{$3.09210-4$}$ and $k_{\sigma 1}=\text{$5.70810-6$}$,
respectively. The resulting SOC loss function and the corresponding linearized
segments are shown in Fig. 3. The SOC loss curve is assumed to be flat between
10 and 20%, and the SOC degradation is equal at 0 and 100% SOC to also capture
capacity fade at low SOC.

$\displaystyle{0}$$\displaystyle{20}$$\displaystyle{40}$$\displaystyle{60}$$\displaystyle{80}$$\displaystyle{100}$SOC
[%]$\displaystyle{0.0006}$$\displaystyle{0.0008}$Capacity loss [%]Loss
curveSegmented curve upSegmented curve down Figure 3: Continuous and
linearized SOC capacity loss functions.

###  III-C Scenarios and stages

The time-series for wind and solar PV power and demand are based on actual
values from the Rye MG through year 2020 and is available online [34]. The
wind power has been scaled down to 60% of the original size. Wind, solar PV
and demand have each been forecasted with a low, medium, and high scenario
with probability 20, 60 and 20% respectively, based on the 0.2, 0.5, and 0.8
quantiles, yielding a combination of 27 scenarios. These have been ordered
based on accumulated net production and reduced by selecting the median
scenario of the percentiles 0-10, 10-30, 30-70, 70-90, and 90-100 with the
corresponding probabilities 0.1, 0.2, 0.4, 0.2, and 0.1. More sophisticated
scenario generation methods could have been used but are outside the scope of
this paper. The individual percentiles are published online [35]. The four
first stages are 6 hours each, the next is 24 hours, while the final is 72
hours and a repeated cyclic with discount factor 0.7 as described in reference
[29]. The final stage is beyond the meterological forecast, and historical
daily mean values are applied as scenarios using the same quantiles as
previous stages.

###  III-D Cases

The model has been simulated with three different variations of the system
with respect to generator and EES sizes as shown in Table III. The systems in
cases 1 and 3 are energy constrained since the diesel generator is too small
to meet the peak demand, hence load shedding depends on how the EESs are
scheduled. However, the diesel generator in case 2 is large enough to always
meet the peak demand, hence there is never a risk of scarcity unless the
generator fails.

TABLE III: Overview of cases the proposed model has been tested on. Case | Diesel generator [kW] | Battery [kWh] | Hydrogen [kWh]
---|---|---|---
1. | 25 | 500 | 3300
2. | 75 | 500 | 3300
3. | 25 | 1000 | -

Each case will be analyzed with perfect forecast, deterministic forecast, and
stochastic optimization, both with and without degradation in the optimization
model as shown in Table IV.

TABLE IV: Overview of methods the proposed cases have been analyzed with. Method | Forecast | DOD | SOC
---|---|---|---
a. | Perfect | X | X
b. | Deterministic |  |
c. | Stochastic |  |
d. | Stochastic | X |
e. | Stochastic |  | X
f. | Stochastic | X | X

##  IV Results and discussion

The main objective is to always meet the demand in the most cost effective way
by using as much VRES generation as possible, and only using diesel if
necessary to avoid load shedding. EES must also be utilized to maximize the
VRES utilization and to minimize the diesel consumption and the load shedding.
However, the battery degradation is a complicating element. Although cycling
the battery is less expensive than generating power from the diesel generator,
even for deep cycles, it is difficult due to the uncertainty in determining if
the power charged now is needed later or if it can be consumed directly from
VRES generation. It is therefore necessary to balance the cost of cycling the
battery toward the expected diesel generation reduction. Additionally, there
is an increasing cost associated with staying at high SOC. It can therefore be
cost effective to keep the SOC low in periods with a stable high VRES
generation to extend the battery’s lifetime.

TABLE V: Overview of operating costs, degradation costs, battery expected lifetime, VRES generation and EES charge/discharge for cases in Table III and methods in Table IV. |  | Operating cost $[€]$ | Degradation cost $[€]$ | $[year]$ | Energy $[MWh]$
---|---|---|---|---|---
Case | Method | Total | Load shedding | Diesel | DOD | SOC up | SOC down | Lifetime | VRES | H2 ch/dch | Batt ch/dch
1 | a | 3719.7 | 0.00 | 3076 | 514 | 117 | 12 | 21.68 | 185.03 | 61.52 / 20.19 | 39.83 / 36.90
1 | b | 8541.1 | 4054.93 | 2840 | 1041 | 435 | 171 | 17.81 | 171.41 | 40.74 / 13.54 | 49.34 / 45.67
1 | c | 6011.1 | 968.94 | 2860 | 1328 | 731 | 123 | 16.26 | 166.73 | 32.87 / 11.02 | 52.67 / 48.73
1 | d | 5579.5 | 1086.00 | 2934 | 627 | 882 | 50 | 18.09 | 181.56 | 57.16 / 18.79 | 40.84 / 37.83
1 | e | 5804.1 | 1108.53 | 2861 | 1488 | 310 | 37 | 17.23 | 181.25 | 53.03 / 17.47 | 63.36 / 58.58
1 | f | 5256.8 | 1414.40 | 2964 | 582 | 268 | 29 | 20.62 | 183.48 | 60.64 / 19.90 | 39.71 / 36.79
2 | a | 3719.7 | 0.00 | 3076 | 514 | 117 | 12 | 21.68 | 185.03 | 61.52 / 20.19 | 39.83 / 36.90
2 | b | 4508.7 | 0.00 | 2917 | 998 | 445 | 149 | 17.98 | 171.41 | 40.86 / 13.58 | 47.87 / 44.31
2 | c | 5093.3 | 0.00 | 2864 | 1285 | 723 | 220 | 16.13 | 166.46 | 32.50 / 10.90 | 50.46 / 46.69
2 | d | 4575.3 | 0.00 | 2892 | 650 | 848 | 185 | 17.69 | 181.75 | 56.55 / 18.60 | 40.43 / 37.45
2 | e | 4704.9 | 0.00 | 2883 | 1494 | 313 | 16 | 17.27 | 181.26 | 53.10 / 17.49 | 62.37 / 57.63
2 | f | 3814.1 | 0.00 | 3000 | 556 | 253 | 6 | 20.90 | 183.46 | 60.79 / 19.95 | 38.50 / 35.63
3 | a | 4631.9 | 0.00 | 3459 | 771 | 348 | 53 | 19.45 | 141.21 | \- / - | 56.93 / 52.66
3 | b | 9718.3 | 3822.67 | 3071 | 1228 | 1258 | 338 | 14.72 | 143.96 | \- / - | 63.49 / 58.71
3 | c | 7202.8 | 775.38 | 3064 | 1436 | 1741 | 186 | 13.64 | 143.42 | \- / - | 59.47 / 55.00
3 | d | 6968.8 | 1061.56 | 3091 | 885 | 1745 | 186 | 14.74 | 143.05 | \- / - | 59.03 / 54.59
3 | e | 8467.5 | 1107.20 | 3215 | 3133 | 945 | 67 | 12.32 | 161.68 | \- / - | 181.47 / 167.43
3 | f | 6188.1 | 1236.83 | 3321 | 776 | 789 | 66 | 17.86 | 140.56 | \- / - | 56.43 / 52.20

The results in Table V show that accounting for DOD and SOC degradation
increases the load shedding case 1 and 3, and the diesel cost for all cases.
However, the reduction in degradation surpasses the increase in diesel and
load shedding costs and indicates more than four years of increase in expected
lifetime for all cases when comparing methods c and f. A very common way to
reduce battery degradation is to apply fixed operating limits, such as
enforcing a permanent operating range between 10 and 90%. However, in
situations where the only alternative is load shedding, it is optimal to
utilize the full battery range since the cost reduction associated with
reducing load shedding outperforms the cost accrued by degradation.

The energy balance in cases 1 and 2 show that the hydrogen system replaces
some of the battery cycling, since it has no degradation costs. However, case
3, which has no hydrogen in the system, also shows a significant cost
reduction associated with battery degradation. However, SOC degradation
without DOD degradation (method e) causes significant cycling and energy
dumping through simultaneous charging and discharging.

Fig. 4a shows that the hydrogen SOC on average is lower when including the DOD
degradation in methods d and f. The wind turbine peak capacity is 135 kW while
the electrolyzer charge capacity is only 55 kW. In periods with very high wind
power, the battery can act as a buffer for the hydrogen tank that is unable to
absorb all the wind power. However, DOD degradation makes this less
profitable, resulting in lower hydrogen filling.

Fig. 4b shows that the battery SOC for methods e and f, where SOC degradation
is included, is stable low in mid-year when the demand is low. These periods
require relatively low stored energy to secure the supply, especially when the
solar PV is delivering substantial energy due to many hours of sunlight
through the summer. However, SOC degradation will, in general, lower the
battery SOC, and DOD degradation will lower the hydrogen SOC, which in turn
increases the risk of scarcity. The results in Table V shows a modest increase
in both load shedding and diesel consumption when accounting for battery
degradation.

$\displaystyle{0}$$\displaystyle{20}$Energy [MWh](a)
Hydrogen1a1b1c1d1e1f$\displaystyle{0}$$\displaystyle{100}$$\displaystyle{200}$$\displaystyle{300}$$\displaystyle{365}$Time
[days]$\displaystyle{0}$$\displaystyle{2}$$\displaystyle{4}$Energy [MWh](b)
Battery Figure 4: Four day average SOC for case 1.

##  V Conclusions

Battery degradation is strongly connected to the operational pattern. In this
study, the expected battery lifetime was prolonged by more than four years by
properly accounting for degradation effects caused by DOD and SOC. Moreover,
the total operating costs were reduced by up to 25% compared to the naive
stochastic model without representation of degradation. However, battery
degradation minimization also influences the operational pattern for the
remaining resources in the system. The operational costs for generators and
other EES technologies can, in the worst case scenario, increase even more
than the savings for battery degradation. Degradation costs and inefficiencies
associated with the operational pattern should therefore be considered for the
whole system.

Dedicated battery degradation minimization can be contradictory to maximizing
the security of supply, and the risk of scarcity must be balanced against the
potential reduction in degradation. Realistic uncertainty models are therefore
highly important.

The optimal operation of the future power system will to a greater extent be
influenced by technology prices rather than fuel price, and energy adequacy
rather than power adequacy. Future research should therefore give more
attention to both the degradation of all the flexible resources in the system
as well as precise uncertainty modeling to capture the future risk of scarcity
accurately.

## References

  * [1] International Energy Agency, “Net Zero by 2050 A Roadmap for the Global Energy Sector,” Paris, 2021. [Online]. Available: https://www.iea.org/reports/net-zero-by-2050
  * [2] J. Geske and R. Green, “Optimal Storage, Investment and Management under Uncertainty: It is Costly to Avoid Outages!” _The Energy Journal_ , vol. 41, no. 2, 4 2020.
  * [3] P. Aaslid, M. Korpås, M. M. Belsnes, and O. B. Fosso, “Pricing electricity in constrained networks dominated by stochastic renewable generation and electric energy storage,” _Electric Power Systems Research_ , vol. 197, p. 107169, 8 2021.
  * [4] M. A. Pellow, C. J. Emmott, C. J. Barnhart, and S. M. Benson, “Hydrogen or batteries for grid storage? A net energy analysis,” _Energy and Environmental Science_ , vol. 8, no. 7, pp. 1938–1952, 7 2015.
  * [5] P. Ahmadi, S. H. Torabi, H. Afsaneh, Y. Sadegheih, H. Ganjehsarabi, and M. Ashjaee, “The effects of driving patterns and PEM fuel cell degradation on the lifecycle assessment of hydrogen fuel cell vehicles,” _International Journal of Hydrogen Energy_ , vol. 45, no. 5, pp. 3595–3608, 1 2020.
  * [6] B. Bidoggia and S. K. Kær, “Estimation of membrane hydration status for standby proton exchange membrane fuel cell systems by complex impedance measurement: Constant temperature stack characterization,” _International Journal of Hydrogen Energy_ , vol. 38, no. 10, pp. 4054–4066, 4 2013.
  * [7] J. Wang, H. Zhong, W. Tang, R. Rajagopal, Q. Xia, C. Kang, and Y. Wang, “Optimal bidding strategy for microgrids in joint energy and ancillary service markets considering flexible ramping products,” _Applied Energy_ , vol. 205, pp. 294–303, 11 2017.
  * [8] P. Keil, S. F. Schuster, J. Wilhelm, J. Travi, A. Hauser, R. C. Karl, and A. Jossen, “Calendar Aging of Lithium-Ion Batteries,” _Journal of The Electrochemical Society_ , vol. 163, no. 9, p. A1872, 7 2016.
  * [9] K. Liu, T. R. Ashwin, X. Hu, M. Lucu, and W. D. Widanage, “An evaluation study of different modelling techniques for calendar ageing prediction of lithium-ion batteries,” _Renewable and Sustainable Energy Reviews_ , vol. 131, p. 110017, 10 2020.
  * [10] J. Vetter, P. Novák, M. R. Wagner, C. Veit, K. C. Möller, J. O. Besenhard, M. Winter, M. Wohlfahrt-Mehrens, C. Vogler, and A. Hammouche, “Ageing mechanisms in lithium-ion batteries,” _Journal of Power Sources_ , vol. 147, no. 1-2, pp. 269–281, 9 2005.
  * [11] Y. Gao, J. Jiang, C. Zhang, W. Zhang, and Y. Jiang, “Aging mechanisms under different state-of-charge ranges and the multi-indicators system of state-of-health for lithium-ion battery with Li(NiMnCo)O2 cathode,” _Journal of Power Sources_ , vol. 400, pp. 641–651, 10 2018.
  * [12] J. Zhu, M. Knapp, D. R. Sørensen, M. Heere, M. S. Darma, M. Müller, L. Mereacre, H. Dai, A. Senyshyn, X. Wei, and H. Ehrenberg, “Investigation of capacity fade for 18650-type lithium-ion batteries cycled in different state of charge (SoC) ranges,” _Journal of Power Sources_ , vol. 489, p. 229422, 3 2021.
  * [13] I. Laresgoiti, S. Käbitz, M. Ecker, and D. U. Sauer, “Modeling mechanical degradation in lithium ion batteries during cycling: Solid electrolyte interphase fracture,” _Journal of Power Sources_ , vol. 300, pp. 112–122, 12 2015.
  * [14] S. Wang, D. Guo, X. Han, L. Lu, K. Sun, W. Li, D. U. Sauer, and M. Ouyang, “Impact of battery degradation models on energy management of a grid-connected DC microgrid,” _Energy_ , vol. 207, p. 118228, 9 2020.
  * [15] H. Shuai, J. Fang, X. Ai, Y. Tang, J. Wen, and H. He, “Stochastic optimization of economic dispatch for microgrid based on approximate dynamic programming,” _IEEE Transactions on Smart Grid_ , vol. 10, no. 3, pp. 2440–2452, 5 2019.
  * [16] W. Gil-González, O. D. Montoya, E. Holguín, A. Garces, and L. F. Grisales-Noreña, “Economic dispatch of energy storage systems in dc microgrids employing a semidefinite programming model,” _Journal of Energy Storage_ , vol. 21, pp. 1–8, 2 2019.
  * [17] M. Elkazaz, M. Sumner, and D. Thomas, “Energy management system for hybrid PV-wind-battery microgrid using convex programming, model predictive and rolling horizon predictive control with experimental validation,” _International Journal of Electrical Power and Energy Systems_ , vol. 115, p. 105483, 2 2020.
  * [18] T. A. Nguyen and M. L. Crow, “Stochastic Optimization of Renewable-Based Microgrid Operation Incorporating Battery Operating Cost,” _IEEE Transactions on Power Systems_ , vol. 31, no. 3, pp. 2289–2296, 5 2016.
  * [19] D. Fioriti, D. Poli, P. Duenas-Martinez, and I. Perez-Arriaga, “Multi-year stochastic planning of off-grid microgrids subject to significant load growth uncertainty: overcoming single-year methodologies,” _Electric Power Systems Research_ , vol. 194, p. 107053, 5 2021.
  * [20] W. Su, J. Wang, and J. Roh, “Stochastic energy scheduling in microgrids with intermittent renewable energy resources,” _IEEE Transactions on Smart Grid_ , vol. 5, no. 4, pp. 1876–1883, 7 2014.
  * [21] C. Ju, P. Wang, L. Goel, and Y. Xu, “A two-layer energy management system for microgrids with hybrid energy storage considering degradation costs,” _IEEE Transactions on Smart Grid_ , vol. 9, no. 6, pp. 6047–6057, 11 2018\.
  * [22] M. Koller, T. Borsche, A. Ulbig, and G. Andersson, “Defining a degradation cost function for optimal control of a battery energy storage system,” _2013 IEEE Grenoble Conference PowerTech, POWERTECH 2013_ , 2013.
  * [23] B. Xu, A. Oudalov, A. Ulbig, G. Andersson, and D. S. Kirschen, “Modeling of lithium-ion battery degradation for cell life assessment,” _IEEE Transactions on Smart Grid_ , vol. 9, no. 2, pp. 1131–1140, 3 2018.
  * [24] B. Xu, J. Zhao, T. Zheng, E. Litvinov, and D. S. Kirschen, “Factoring the Cycle Aging Cost of Batteries Participating in Electricity Markets,” _IEEE Transactions on Power Systems_ , vol. 33, no. 2, pp. 2248–2259, 3 2018\.
  * [25] “Remote EU project,” 2021. [Online]. Available: https://www.remote-euproject.eu/remote-project/
  * [26] P. Marocco, D. Ferrero, M. Gandiglio, and M. Santarelli, “Remote area Energy supply with Multiple Options for integrated hydrogen-based TEchnologies - Deliverable number 2.2,” 2018. [Online]. Available: https://www.remote-euproject.eu/remote18/rem18-cont/uploads/2019/03/REMOTE-D2.2.pdf
  * [27] J. M. Maciejowski, _Predictive Control with Constraints_. Pearson education, 2002.
  * [28] M. V. F. Pereira and L. M. V. G. Pinto, “Multi-stage stochastic optimization applied to energy planning,” _Mathematical Programming_ , vol. 52, no. 1-3, pp. 359–375, 5 1991.
  * [29] O. Dowson, “The policy graph decomposition of multistage stochastic optimization problems,” _Networks_ , vol. 76, no. 1, pp. 3–23, 2020.
  * [30] C. Amzallag, J. P. Gerey, J. L. Robert, and J. Bahuaud, “Standardization of the rainflow counting method for fatigue analysis,” _International Journal of Fatigue_ , vol. 16, no. 4, pp. 287–293, 6 1994.
  * [31] O. Dowson and L. Kapelevich, “SDDP.jl: A julia package for stochastic dual dynamic programming,” _INFORMS Journal on Computing_ , vol. 33, no. 1, pp. 27–33, 12 2021.
  * [32] A. Millner, “Modeling lithium ion battery degradation in electric vehicles,” _2010 IEEE Conference on Innovative Technologies for an Efficient and Reliable Electricity Supply, CITRES 2010_ , pp. 349–356, 2010.
  * [33] D. I. Stroe, M. Swierczynski, A. I. Stroe, R. Teodorescu, R. Laerke, and P. C. Kjaer, “Degradation behaviour of Lithium-ion batteries based on field measured frequency regulation mission profile,” _2015 IEEE Energy Conversion Congress and Exposition, ECCE 2015_ , pp. 14–21, 10 2015.
  * [34] TrønderEnergi, “AI Hackathon Challenge - Optimal Control of Microgrid,” 2021. [Online]. Available: http://doi.org/10.5281/zenodo.5500209
  * [35] P. Aaslid, “Rye microgrid historical weather forecasts and stochastic scenarios [Data set],” 2021. [Online]. Available: https://doi.org/10.5281/zenodo.5526241


11institutetext:  Purple Mountain Observatory, Chinese Academy of Sciences,
No. 10 Yuanhua Road, Nanjing 210033, PR China
11email<EMAIL_ADDRESS>22institutetext:  School of Astronomy and Space
Science, University of Science and Technology of China, No. 96 Jinzhai Road,
Hefei, Anhui 230026, PR China 33institutetext:  Xinjiang Astronomical
Observatory, Chinese Academy of Sciences, 150 Science 1-Street, Urumqi,
Xinjiang 830011, PR China 44institutetext:  Xingming Observatory, Urumqi,
Xinjiang 830000, PR China 55institutetext:  School of Landscape Architecture,
Beijing Forestry University, 35 Qinghua E Rd, Haidian, Beijing 100083, PR
China 66institutetext:  Nanjing Institute of Astronomical Optics & Technology,
Chinese Academy of Sciences, No. 188 Bancang Street, Nanjing 210042, PR China
77institutetext:  CAS Key Laboratory of Astronomical Optics & Technology,
Nanjing Institute of Astronomical Optics & Technology, Nanjing 210042, PR
China 88institutetext:  University of Chinese Academy of Sciences, No.1
Yanqihu East Rd, Huairou District, Beijing 101408, PR China 99institutetext:
FuJian Astronomical Society, 3a-2303, phase 2, Shimao cuican Tiancheng, Jimei
District, Xiamen, Fujian 361000, PR China 1010institutetext:  Xiamen Dolphin
Sounthern Astronomy Technology Co., Ltd, Room 511-59, No. 28, Song Yang Yili,
Fengxiang Street, Xiang’an District, Xiamen 361000, PR China
1111institutetext:  Beijing Amateur Astronomer Sodality, Beijing 100000, PR
China 1212institutetext:  Dongguan Foreign Language School, No. 2 Hengtang
Road, Dongguan, Guangdong 523413, PR China 1313institutetext:  Dongguan
Science Museum, No. 38 Xinfen Road, Dongguan, Guangdong 523000, PR China
1414institutetext:  Amateur Astornomrs Association Of Maoming, Room 101-1,Unit
4 of Building 50, North Renmin Road, Maoming, Guangdong 525000, PR China
1515institutetext:  JiNan Amateur Astronomy Association, Jinan, Shandong
250000, PR China 1616institutetext:  Qixing Observatory, Jinan, Shandong
250000, PR China 1717institutetext:  College of Physics and Elctronic
Engineering, Sichuan Normal University, No. 1819, Section 2, Chenglong Avenue,
Longquan District, Chengdu 610101, PR China 1818institutetext:  Xiamen Minli
Primary School, No. 1 Shengping Road, Xiamen, Fujian 361001, PR China
1919institutetext:  Duzhou Primary School, Shipai Town, Dongguan, Guangdong
523000, PR China 2020institutetext:  School of Physics and Astronomy, Sun Yat-
Sen University, No. 135, Xingang Xi Road, Guangzhou, 510275, PR China
2121institutetext:  Foshan Xingkai Technology Co., Ltd, Foshan, 528200, PR
China 2222institutetext:  Dalian Bootes Astronomical Society, Dalian, Liaoning
116000, PR China 2323institutetext:  BCTO, No. 5 Chaogui South Road, Foshan,
Guangdong 528300, PR China 2424institutetext:  XingYuan College, Ocean
University of China, No. 238 Songling Road, Qindao, Shandong 266100, PR China
2525institutetext:  Quzhou Astronomical Association, Quzhou, Zhejiang 324000,
PR China 2626institutetext:  Hunan Astronomical Association, Changsha, Hunan
410000, PR China 2727institutetext:  Nanjing Amateur Astronomers Association,
Nanjing 210000, PR China 2828institutetext:  Shenzhen Astronomical
Observatory, Tianwen Road, Shenzhen, Guangdong 518040, PR China
2929institutetext:  Department of Space Sciences and Astronomy, Hebei Normal
University, No. 20 Road East. 2nd Ring South, Shijiazhuang, Hebei 050024, PR
China

# New constraints on Triton’s atmosphere from the 6 October 2022 stellar
occultation

Ye Yuan (袁烨) New constraints on Triton’s atmosphere from the 6 October 2022
stellar occultationNew constraints on Triton’s atmosphere from the 6 October
2022 stellar occultation 0000-0002-4686-8548 Chen Zhang (张晨) New constraints
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stellar occultation 0000-0002-5845-4369 Yixing Gao (高伊星) New constraints on
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from the 6 October 2022 stellar occultationNew constraints on Triton’s
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stellar occultation Donghua Chen (陈栋华) New constraints on Triton’s atmosphere
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Triton’s atmosphere from the 6 October 2022 stellar occultationNew constraints
on Triton’s atmosphere from the 6 October 2022 stellar occultation
0009-0003-1624-7024 Jiajun Lin (林家骏) New constraints on Triton’s atmosphere
from the 6 October 2022 stellar occultationNew constraints on Triton’s
atmosphere from the 6 October 2022 stellar occultation 0009-0009-8378-9425 Wei
Tan (谭巍) New constraints on Triton’s atmosphere from the 6 October 2022
stellar occultationNew constraints on Triton’s atmosphere from the 6 October
2022 stellar occultation 0009-0001-5905-6143 Jia Zhou (周嘉) New constraints on
Triton’s atmosphere from the 6 October 2022 stellar occultationNew constraints
on Triton’s atmosphere from the 6 October 2022 stellar occultation
0009-0007-5423-6480 Jun Xu (许军) New constraints on Triton’s atmosphere from
the 6 October 2022 stellar occultationNew constraints on Triton’s atmosphere
from the 6 October 2022 stellar occultation 0000-0002-8086-1665 Jun He (何骏)
New constraints on Triton’s atmosphere from the 6 October 2022 stellar
occultationNew constraints on Triton’s atmosphere from the 6 October 2022
stellar occultation 0009-0009-3492-7174 Jiahui Ye (叶嘉晖) New constraints on
Triton’s atmosphere from the 6 October 2022 stellar occultationNew constraints
on Triton’s atmosphere from the 6 October 2022 stellar occultation
0009-0003-3602-731X Delai Li (李德铼) New constraints on Triton’s atmosphere from
the 6 October 2022 stellar occultationNew constraints on Triton’s atmosphere
from the 6 October 2022 stellar occultation 0009-0002-7208-8200 Shuai Zhang
(张帅) New constraints on Triton’s atmosphere from the 6 October 2022 stellar
occultationNew constraints on Triton’s atmosphere from the 6 October 2022
stellar occultation 0000-0003-2413-9587 Qingyue Qu (曲卿乐) New constraints on
Triton’s atmosphere from the 6 October 2022 stellar occultationNew constraints
on Triton’s atmosphere from the 6 October 2022 stellar occultation

The atmosphere of Triton was probed directly by observing a ground-based
stellar occultation on 6 October 2022. This rare event yielded 23 positive
light curves collected from 13 separate observation stations contributing to
our campaign. The significance of this event lies in its potential to directly
validate the modest pressure fluctuation on Triton, a phenomenon not
definitively verified by previous observations, including only five stellar
occultations, and the Voyager 2 radio occultation in 1989. Using an approach
consistent with a comparable study, we precisely determined a surface pressure
of $14.07_{-0.13}^{+0.21}\leavevmode\nobreak\ \mathrm{\upmu bar}$ in 2022.
This new pressure rules out any significant monotonic variation in pressure
between 2017 and 2022 through direct observations, as it is in alignment with
the 2017 value. Additionally, both the pressures in 2017 and 2022 align with
the 1989 value. This provides further support for the conclusion drawn from
the previous volatile transport model simulation, which is consistent with the
observed alignment between the pressures in 1989 and 2017; that is to say, the
pressure fluctuation is modest. Moreover, this conclusion suggests the
existence of a northern polar cap extended down to at least
$45\degr$N–$60\degr$N and the presence of nitrogen between $30\degr$S and
$0\degr$.

###### Key Words.:

planets and satellites: individual: Triton – planets and satellites:
atmospheres – planets and satellites: physical evolution – occultations –
techniques: photometric

##  1 Introduction

Triton, Neptune’s largest moon, has a tenuous atmosphere primarily composed of
nitrogen, with trace amounts of methane and carbon monoxide (Lellouch et al.,
2010, 2017; Merlin et al., 2018). The abundances of the two minor components
are a few ten-thousandths of that of nitrogen (Lellouch et al., 2010, 2017;
Merlin et al., 2018). Triton’s climate is heavily influenced by volatile
condensation–sublimation cycles and is subject to the effects of tidal heating
(Bertrand et al., 2022).

Despite its intriguing characteristics, Triton remains relatively unexplored.
It received a brief visit from Voyager 2 in 1989, during which its atmosphere
was directly probed using the radio occultation method (Tyler et al., 1989;
Gurrola, 1995). Subsequently, stellar occultation, another technique for
directly probing planetary atmospheres, has played a crucial role in studying
Triton’s atmospheric structure, composition, and evolution (Olkin et al.,
1997; Elliot et al., 1998, 2000a, 2000b, 2003; Marques-Oliveira et al., 2022,
referred to as MO22 hereafter). However, observable stellar occultations by
Triton are rare. Even with the inclusion of the new event on 6 October
2022—our focus in the present paper—merely six such events have been observed
and analyzed. The five pre-2022 events, as detailed in MO22, occurred on 14
August 1995, 18 July 1997, 4 November 1997, 21 May 2008, and 5 October 2017.
The infrequency of occultations highlights the significance of studying the
2022 event. This event provides a new opportunity to validate Triton’s global
climate models (GCMs) through volatile transport models (VTMs) and to reveal
any possible short-term variations not accounted for in the existing models.
Furthermore, deviations from these models could prompt revisions of Triton’s
fundamental physical parameters, such as ground thermal inertia, nitrogen ice
albedo, and the latitudinal distributions of its northern and southern polar
caps (Bertrand et al., 2022).

According to the previously measured pressures presented in MO22, Triton’s
atmospheric pressure exhibited a possible surge in the 1990s and a return to
its 1989 level by 2017. However, given the limited high-quality data and
incomplete reanalysis, this surge is debatable. MO22 suggests that no
definitive conclusions can be drawn regarding pressure changes between 1989
and 2008. In summary, either no surge occurred between 1989 and 2017, or if it
did, pressures had returned to 1989 levels by 2017. Moreover, the surge
contradicts the pressure trends predicted by the MO22 VTM (VTM22) simulations,
which indicate that only a modest fluctuation in pressure is consistent with
the observed return in 2017. While MO22 did not definitively observe a modest
fluctuation, the 2022 event holds the potential to confirm this phenomenon.

The multichord observations of the 2022 event obtained by our observation
campaign are presented in Section 2. The light-curve fitting results are
presented in Section 3, with a brief description of the fitting method. The
atmospheric pressure evolution on Triton is discussed in Section 4 based on
our new pressure measurement from the 2022 event. Conclusions and
recommendations are presented in Section 5.

##  2 Observed light curves

An observation campaign for the stellar occultation by Triton on 6 October
2022 was organized in China as described in Appendix A. Table 1 lists the
general circumstances of this campaign. Figure 1(a) presents the reconstructed
path of Triton’s shadow on Earth during this event. Precise time references,
based on GPS data, were obtained by the QHY174M-GPS
cameras111https://www.qhyccd.com mounted on XMC8 and QXO telescopes, and the
PMO-GPSBOX devices222The PMO-GPSBOX is a remote event synchronization device
developed by the Purple Mountain Observatory, and is designed to provide high-
precision GPS/Beidou hardware timescales. This allows cameras operating in
internal trigger mode to acquire exposure-time information with a time error
accuracy of approximately $100\leavevmode\nobreak\ \mathrm{ns}$. More detailed
information can be found in the patent description (CN: No. CN108881727B,
2021-03-16) available at http://epub.cnipa.gov.cn/cred/CN108881727B. on the
YAHPT and YACHES telescopes.

A total of 29 separate stations attempted to observe the rare event. The
geometric relationships between these stations and the shadow are displayed in
Figures 1(a) and 1(b). In total, 21 telescopes on 13 stations achieved
positive detections. Using the data-processing method outlined in Appendix B
of Yuan et al. (2023), a total of 23 effective light curves were extracted,
along with $1\sigma$ uncertainties on flux data points. Moreover, to speed up
the light-curve fitting procedure, we binned the data points for each station,
resulting in a time resolution of just over one second. The binned light-curve
observations are presented in Figure 1(c). Test calculations show that this
data binning has very little effect on light-curve fitting results.

Table 1: Circumstances and light-curve fitting results of the 6 October 2022
occultation.

Occulted star
---
Identification (Gaia DR3aa$a$Gaia Collaboration et al. (2023), with the
astrometric, photometric, and astrophysical parameters of the star obtained
from VizieR online gaiadr3 and paramsup tables of the Gaia DR3 Part 1. Main
source.
) | $2639239368824994944$
Geocentric astrometric position | $\alpha_{\mathrm{s}}=23^{\mathrm{h}}36^{\mathrm{m}}52\aas@@fstack{s}4514$
at observational epoch (ICRFbb$b$International Celestial Reference Frame.
) | $\delta_{\mathrm{s}}=-03\degr 50^{\prime}09\aas@@fstack{\prime\prime}796$
Parallax (mas) | $2.2717$
G-mag/BP-mag/RP-mag |  $11.55$/$11.85$/$11.08$
Radius, $R_{\mathrm{s}}$ (in solar radius) | $1.526$
Triton’s bodycc$c$All the input parameters of Triton’s body and atmosphere are
the same as in MO22, with nitrogen gas ($\mathrm{N}_{2}$) assumed to be the
only constituent in the light-curve model. This is mainly to ensure that
comparisons are free from systematic discrepancies attributable to differences
in ray-tracing codes.

Mass, $GM_{\mathrm{b}}$ ($\mathrm{km}^{3}\cdot\mathrm{s}^{-2}$) | $1428$
Radius, $R_{\mathrm{b}}$ (km) | $1353$
Triton’s atmospherecc$c$All the input parameters of Triton’s body and
atmosphere are the same as in MO22, with nitrogen gas ($\mathrm{N}_{2}$)
assumed to be the only constituent in the light-curve model. This is mainly to
ensure that comparisons are free from systematic discrepancies attributable to
differences in ray-tracing codes.

$\mathrm{N}_{2}$ molecular mass, $\mu$ ($\mathrm{kg}$) | $4.652\times 10^{-26}$
$\mathrm{N}_{2}$ molecular | $1.091\times 10^{-23}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
refractivity, $K$ ($\mathrm{cm}^{3}$) | $+6.282\times 10^{-26}/\lambda_{\mathrm{\upmu m}}^{2}$
Boltzmann constant, $k_{\mathrm{B}}$ ($\mathrm{J}\cdot\mathrm{K}^{-1}$) | $1.380626\times 10^{-23}$
Given reference radius, $r_{\mathrm{ref}}$ ($\mathrm{km}$) | $1400$
Template temperature profile ($\mathrm{K}$) |  $T(r)$dd$d$Using parameters in Table B.1 of MO22.

Other information on Triton
Ephemerides | DE440/NEP097
Apparent magnitude | $13.45$
Geocentric distance, $D$ (km) | $4.3341\times 10^{9}$
Projected $R_{\mathrm{s}}$ at $D$ (km) | $0.3362$
Light-curve fitting results (with $1\sigma$ error bars)
Pressureee$e$Using a ratio $p_{\mathrm{surf}}/p_{\mathrm{ref}}=12.0$ given in
MO22.
at $r_{\mathrm{ref}}$, $p_{\mathrm{ref}}$ ($\mathrm{\upmu bar}$) | $1.173_{-0.011}^{+0.018}$
Surface pressureee$e$Using a ratio $p_{\mathrm{surf}}/p_{\mathrm{ref}}=12.0$
given in MO22.
at $R_{\mathrm{b}}$, $p_{\mathrm{surf}}$ ($\mathrm{\upmu bar}$) | $14.07_{-0.13}^{+0.21}$
Geocentric C/A distanceff$f$C/A mean closest approach to shadow axis. Positive
(negative) distance values mean that the shadow center passes north (south) of
the geocenter.
, $\rho_{\mathrm{cag}}$ ($\mathrm{km}$) | $4068.98\pm 0.64$
Geocentric C/A timegg$g$Timings by QHY174M-GPS cameras and PMO-GPSBOX devices
are used as time references, considering their reliability and accuracy.
, $t_{\mathrm{cag}}$ (UTC) | $14\mathrm{:}39\mathrm{:}46.223\pm 34\leavevmode\nobreak\ \mathrm{ms}$
333

(a)

(b)

(c)

(d)

(e)

Figure 1: Results of the 6 October 2022 occultation. Panel (a): Reconstructed
occultation map. Panel (b): Reconstructed occultation chords relative to
Triton. Panel (c): Simultaneous fit of 23 light curves. Panel (d): $\chi^{2}$
map for the simultaneous fit of 23 light curves. Panel (e): Pressure
measurements compared with the VTM22 simulations of Triton’s surface pressure
cycle. The simulations are denoted by the blue and pink curves, which are
obtained from Figure 23 of MO22. The $1\sigma$ and $3\sigma$ uncertainties on
pressure measurements are denoted by the thicker and thinner error bars,
respectively, which are derived from the $\chi^{2}$ maps in MO22.

##  3 Light-curve fitting results

In order to obtain light-curve fitting results that can be compared with MO22,
we use the same light-curve model, that is, the DO15 model (Dias-Oliveira et
al., 2015; Meza et al., 2019), with the temperature profile, $T(r)$,
constructed by adopting the parameters of Table B.1 in MO22. Yuan et al.
(2023) implemented this model for their analysis of the atmosphere of Pluto
from occultations, with a ray-tracing code described in Appendix C of their
paper. In the present paper, this code is also used to analyze Triton’s
atmosphere. Moreover, as mentioned in Appendix B, we confirm that this code is
fully consistent with that used in MO22.

The light-curve model for a given station $i$ is formally expressed as

$\phi_{i}(t)=A_{i}\cdot\left(s_{i}\cdot\psi_{i}(t;\Delta
t_{i},\Delta\tau,\Delta\rho,p_{\mathrm{ref}})+(1-s_{i})\right),$ (1)

where the subscript $i$ indicates the quantities associated with the given
station; $t$ is the recorded timing of an observation; in cases where the star
is not occulted, $A$ is the total observed flux of the star and Triton’s
system, and $s$ the flux ratio of the star to this total flux; $\psi$ is the
normalized (between zero and unity) flux of the occulted star, which is
formulated as the total flux of the primary (sometimes called the near-limb)
and secondary (or far-limb, when available) images produced by Triton’s
spherical atmosphere (Sicardy, 2023); $\Delta t$ is the camera time recording
offset; $p_{\mathrm{ref}}$ is the atmospheric pressure at the reference radius
$r_{\mathrm{ref}}=1400\leavevmode\nobreak\ \mathrm{km}$, to which the ratio of
the surface pressure $p_{\mathrm{surf}}$ is $12.0$ as used in MO22; and
$\Delta\tau$ and $\Delta\rho$ are the two ephemeris offset parameters
representing the corrections to the epoch $t_{\mathrm{cag}}$ and distance
$\rho_{\mathrm{cag}}$ of the calculated geocentric closest approach to
Triton’s shadow center, respectively. In the calculation, the reference
ephemerides are the
NEP097444https://ssd.jpl.nasa.gov/ftp/eph/satellites/bsp/nep097.bsp satellite
ephemerides (Brozović & Jacobson, 2022) for the orbit of Triton with respect
to the Neptunian system barycenter, and the
DE440555https://ssd.jpl.nasa.gov/ftp/eph/planets/bsp/de440.bsp planetary
ephemerides (Park et al., 2021) for the orbits of the Earth and the Neptunian
system barycenter with respect to the Solar System barycenter. Also, the
reference star catalog is Gaia DR3 (Gaia Collaboration et al., 2023), from
which the data of the occulted star are obtained.

The fitting process, as detailed in Yuan et al. (2023), involves generating a
$\chi^{2}$ map for either $(\Delta\rho,p_{\mathrm{ref}})$ or
$(\rho_{\mathrm{cag}},p_{\mathrm{surf}})$ (refer to Figure 1(d)). From this
map, we can further derive the best-fitting solution and the confidence
limits. The $\chi^{2}$ value is

$\chi^{2}=\sum_{i}\chi_{i}^{2},$ (2)

with each component $\chi_{i}^{2}$ for station $i$ calculated as

$\chi_{i}^{2}=\sum_{j=1}^{N_{i}}\left(\frac{\phi_{i}(t_{ij})-f_{i}(t_{ij})}{k_{i}\cdot\sigma_{i}(t_{ij})}\right)^{2},$
(3)

where $f_{i}$, $\sigma_{i}$, and $N_{i}$ represent the observed light curve,
its $1\sigma$ uncertainty, and the number of data points for station $i$,
respectively; $t_{ij}$ denotes the mid-exposure time of the $j$-th data point
for this station; and $k_{i}$ is a weighting tuning factor designed to adjust
for potentially underestimated errors and to balance $\chi_{i}^{2}$ values
across different stations. Initially, all the $k_{i}$ factors are set to one
to determine the preliminary best-fitting solution. Subsequently, they are
adjusted to bring the root mean square (RMS) of weighted residuals,
$\chi_{i}^{2}/N_{i}$, that exceed one as close to one as possible. Finally,
our light-curve fitting results are obtained and evaluated as presented in
Figure 1(d) and listed in Table 1, with the simultaneously fitted light curves
displayed in Figure 1(c).

Our measurement of the atmospheric pressure at
$r_{\mathrm{ref}}=1400\leavevmode\nobreak\ \mathrm{km}$ is
$p_{\mathrm{ref}}=1.173_{-0.011}^{+0.018}\leavevmode\nobreak\ \mathrm{\upmu
bar}$, and when extrapolated to the surface, the pressure is
$p_{\mathrm{surf}}=14.07_{-0.13}^{+0.21}\leavevmode\nobreak\ \mathrm{\upmu
bar}$. Additionally, we obtain an astrometric position for Triton with a
formal precision of approximately $1\leavevmode\nobreak\ \mathrm{mas}$,
considering the positional uncertainties of the ephemeris offset and the
occulted star at the occultation epoch. This high-precision position holds
significant potential for refining the ephemerides for Neptune and Triton in
the future, as in the case of Pluto (Desmars et al., 2019).

##  4 Pressure evolution of Triton’s atmosphere

Figure 1(e) presents our new pressure measurement in 2022 represented by the
red point, along with the six previously measured pressures presented in MO22.
The black points represent the pressures measured by MO22, of which the
earliest corresponds to the Voyager 2 radio occultation in 1989. The yellow
and blue points represent the measurements from Olkin et al. (1997) and Elliot
et al. (1998, 2000b, 2003), respectively. It should be pointed out that, only
the red and black points can be directly compared, as they have been verified
to be free from systematic discrepancies attributable to differences in ray-
tracing codes (see Appendix A). In addition, Sicardy et al. (2024) also
independently observed and analyzed the same occultation event in 2022 and
reported results that are roughly consistent with the findings of our study.

As shown in Figure 1(e), the new surface pressure in 2022 aligns closely with
the 2017 value of $14.1\pm 0.4\leavevmode\nobreak\ \mathrm{\upmu bar}$. The
relative difference in surface pressures between 2017 and 2022 is only
$-0.2_{-3.0}^{+3.2}\%$, which rules out any significant monotonic variations
between 2017 and 2022 through direct observations. Moreover, both the
pressures in 2017 and 2022 align with the 1989 value. This alignment, in
conjunction with other comparable measurements, suggests that either no
significant surge in pressure occurred between 1989 and 2022, or if it did,
the variation since its return to the 1989 level by 2017 has been small,
remaining near the 1989 level.

Despite the inability of MO22 to definitively observe a modest pressure
fluctuation, their preference for such a variation is supported by their VTM22
simulation, which is consistent with the observed alignment between the
pressures in 1989 and 2017. This simulation is represented by the most modest
blue curve in Figure 1(e), which implies the existence of a northern polar cap
extended down to at least $45\degr$N–$60\degr$N and the presence of nitrogen
between $30\degr$S and $0\degr$. According to VTM22, from 1989 to 2022,
Triton’s atmospheric pressure experienced a modest increase and a modest
decrease successively, with a peak occurring in the 2000s and a return to its
1989 level by 2017. Although a potential surge in pressure in the 1990s is
suggested by the measurements between 1989 to 1997, it is debatable given the
limited high-quality data and incomplete reanalysis, as mentioned in Section
1. Instead, our new pressure confirms the modest fluctuation.

##  5 Conclusion

In the present paper, we provide a new measurement of Triton’s atmospheric
pressure based on multichord observations of the stellar occultation on 6
October 2022. Our observation campaign resulted in 23 positive light curves
observed from 13 separate stations. An approach consistent with MO22 was used
for a global fit to these positive light curves, providing a pressure of
$p_{\mathrm{ref}}=1.173_{-0.011}^{+0.018}\leavevmode\nobreak\ \mathrm{\upmu
bar}$ at the reference radius of $1400\leavevmode\nobreak\ \mathrm{km}$, as
well as an extrapolated surface pressure of
$p_{\mathrm{surf}}=14.07_{-0.13}^{+0.21}\leavevmode\nobreak\ \mathrm{\upmu
bar}$ at the body radius of $1353\leavevmode\nobreak\ \mathrm{km}$. This new
pressure in 2022 is in alignment with the 1989 and 2017 values from MO22.
Notably, it rules out any significant monotonic variation in pressure from
2017 to 2022 through direct observations and supports a modest pressure
fluctuation from 1989 to 2022 as indicated by the VTM22 simulations.
Additionally, an astrometric position for Triton is obtained with a formal
precision of approximately $1\leavevmode\nobreak\ \mathrm{mas}$, which can be
used to refine the ephemerides for Neptune and Triton.

However, our knowledge of Triton is still limited. Continuous stellar
occultation observations are important for detecting any significant pressure
fluctuations, especially those not considered in the VTM22 model; they play a
crucial role in advancing our understanding of Triton’s seasonal and short-
term atmospheric evolution. Nevertheless, in the following decades, there will
be very few observable stellar occultations by Triton. A recent study by
French & Souami (2023), who limited the near-infrared $K$-band magnitudes of
candidate occulted stars to $K\leq 15$, identified that only 22 ground-based
stellar occultations are expected to occur between 2023 and 2050. Further
restricting occulted stars to Gaia $G$-band magnitudes of $G\leq 15$ leaves
only $40\%$ of these occultations as viable opportunities for exploration in
this band. This highlights the importance of maximizing efforts to observe
these rare events.

In addition to arranging as many ground observations as possible, it is
essential to explore alternative observational opportunities via space-based,
airborne, and balloon-borne platforms, such as the Hubble Space Telescope
observation of the stellar occultation by Triton on 4 November 1997 (Elliot et
al., 1998, 2000b, 2003). These platforms also offer the potential to enhance
our observational capabilities, improving both observing geometry and data
quality.

###### Acknowledgements.

We would like to thank the anonymous referee for taking the time to review and
helping us to improve the manuscript. This work was supported by the Young
Scientists Fund of the National Natural Science Foundation of China (Grant
Nos. 12203105 and 12103091), the Strategic Priority Research Program of the
Chinese Academy of Sciences (Grant No. XDA0350300), the Major Program of the
National Natural Science Foundation of China (Grant No. 62394351), the General
Program of the National Natural Science Foundation of China (Grant No.
12073008), and the Joint Funds of the National Natural Science Foundation of
China (Grant No. U2031144). We acknowledge the science research grants from
the China Manned Space Project with NO.CMS-CSST-2021-A12 and NO.CMS-
CSST-2021-B10. We acknowledge the support of the staff of the CAS Key
Laboratory of Space Objects and Debris Observation, Purple Mountain
Observatory, Chinese Academy of Sciences. This work has made use of data from
the 80-cm Yaoan High Precision Telescope (YAHPT) and the 80-cm CHES-800
telescope at Yaoan Astronomical Observation Station. We acknowledge the
support of the staff of the Xinglong 2.16m telescope. This work was partially
supported by the Open Project Program of the CAS Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences.
We acknowledge the support of the staff of the Lijiang 2.4m telescope. Funding
for the telescope has been provided by Chinese Academy of Sciences and the
People’s Government of Yunnan Province. We acknowledge the support of the
staff of the Nanshan 1.2-m and 0.25-m telescopes, Xinjiang Astronomical
Observatory, Chinese Academy of Sciences. We particularly acknowledge Jie
Zheng, Feng Xiao, Yuguang Sun, and Yue Sun for their supports of the Xinglong
126-cm, 85-cm, 80-cm, and 60-cm telescopes. We especially ackonwledge the
support of Chinese amateur astronomers from FuJian Astronomical Society,
Xiamen Southern Astronomy Team, Xingming Observatory Team, Beijing Amateur
Astronomer Sodality, Dongguan Science Museum, Maoming Amateur Astronomer
Association, JiNan Amateur Astronomy Association, Dalian Bootes Astronomical
Society, Chengdu Normal University, Duzhou Primary School, Ocean University of
China, Quzhou Astronomical Association, Hunan Astronomical Association,
Nanjing Amateur Astronomers Association, Shanghai Astronomical Museum,
Shenzhen Astronomical Observatory Team, and Wuyi University.

## References

  * Bertrand et al. (2022) Bertrand, T., Lellouch, E., Holler, B. J., et al. 2022, Icarus, 373, 114764
  * Brozović & Jacobson (2022) Brozović, M. & Jacobson, R. A. 2022, The Astronomical Journal, 163, 241
  * Desmars et al. (2019) Desmars, J., Meza, E., Sicardy, B., et al. 2019, A&A, 625, A43
  * Dias-Oliveira et al. (2015) Dias-Oliveira, A., Sicardy, B., Lellouch, E., et al. 2015, ApJ, 811, 53
  * Elliot et al. (1998) Elliot, J. L., Hammel, H. B., Wasserman, L. H., et al. 1998, Nature, 393, 765
  * Elliot et al. (2000a) Elliot, J. L., Person, M. J., McDonald, S. W., et al. 2000a, Icarus, 148, 347
  * Elliot et al. (2003) Elliot, J. L., Person, M. J., & Qu, S. 2003, AJ, 126, 1041
  * Elliot et al. (2000b) Elliot, J. L., Strobel, D. F., Zhu, X., et al. 2000b, Icarus, 143, 425
  * French & Souami (2023) French, R. G. & Souami, D. 2023, PSJ, 4, 202
  * Gaia Collaboration et al. (2023) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1
  * Gurrola (1995) Gurrola, E. M. 1995, PhD thesis, Stanford University, California
  * Lellouch et al. (2010) Lellouch, E., de Bergh, C., Sicardy, B., Ferron, S., & Käufl, H. U. 2010, A&A, 512, L8
  * Lellouch et al. (2017) Lellouch, E., Gurwell, M., Butler, B., et al. 2017, Icarus, 286, 289
  * Marques-Oliveira et al. (2022) Marques-Oliveira, J., Sicardy, B., Gomes-Júnior, A. R., et al. 2022, A&A, 659, A136
  * Merlin et al. (2018) Merlin, F., Lellouch, E., Quirico, E., & Schmitt, B. 2018, Icarus, 314, 274
  * Meza et al. (2019) Meza, E., Sicardy, B., Assafin, M., et al. 2019, A&A, 625, A42
  * Olkin et al. (1997) Olkin, C. B., Elliot, J. L., Hammel, H. B., et al. 1997, Icarus, 129, 178
  * Park et al. (2021) Park, R. S., Folkner, W. M., Williams, J. G., & Boggs, D. H. 2021, AJ, 161, 105
  * Sicardy (2023) Sicardy, B. 2023, Comptes Rendus Physique, 23, 213
  * Sicardy et al. (2024) Sicardy, B., Tej, A., Gomes-Júnior, A. R., et al. 2024, A&A, 682, L24
  * Tyler et al. (1989) Tyler, G. L., Sweetnam, D. N., Anderson, J. D., et al. 1989, Science, 246, 1466
  * Yuan et al. (2023) Yuan, Y., Li, F., Fu, Y., et al. 2023, A&A, 680, A9

##  Appendix A The occultation observation campaign

We originally organized the observation campaign of the 6 October 2022 stellar
occultation by Triton based on the prediction generated by our code using
DE440 and NEP097 ephemerides and the Gaia DR3 catalog, with ephemeris errors
in both ICRS coordinates set to $0\aas@@fstack{\prime\prime}05$. No
significant difference was found when comparing our prediction with that
generated by the OCCULT software666http://www.lunar-
occultations.com/iota/occult4.htm.

Two weeks before this event, we used the prediction
777https://lesia.obspm.fr/lucky-star/occ.php?p=109326 provided by the ERC
Lucky Star project 888https://lesia.obspm.fr/lucky-star, which had been
refined by applying ephemeris offsets as in MO22. The offsets in right
ascension and declination are $0\aas@@fstack{\prime\prime}011$ and
$-0\aas@@fstack{\prime\prime}015$, respectively, shifting the shadow path
about $350\leavevmode\nobreak\ \mathrm{km}$ to the south in the sky.

Table 2 lists the circumstances of stations with positive detections. Table 3
lists the circumstances of stations that encountered weather or overexposure
issues.

Table 2: Stations with positive detections on 6 October 2022

Station | Longitude (E) | Telescope | Exposure (s) | Observers
---|---|---|---|---
| Latitude (N) | Camera | Circle (s) |
| Altitude (m) | Filter |  |
NSCF | $87\degr 10^{\prime}40\aas@@fstack{\prime\prime}0$ | 0.25 m | $2.0$ | Chunhai Bai, Tuhong Zhong
| $43\degr 28^{\prime}27\aas@@fstack{\prime\prime}0$ | QHY4040 | $4.1$ |
| $2070$ | V |  |
NSLZ | $87\degr 10^{\prime}36\aas@@fstack{\prime\prime}1$ | 1.2 m | $0.5$ | Yong Wang
| $43\degr 28^{\prime}32\aas@@fstack{\prime\prime}2$ | ASI1600MM Pro | $4.0$ |
| $2070$ | clear |  |
XMO | $87\degr 10^{\prime}23\aas@@fstack{\prime\prime}5$ | NEXT RC 0.6 m | $1.0$ | Xing Gao, Yixing Gao
| $43\degr 28^{\prime}18\aas@@fstack{\prime\prime}2$ | FLI 230-42 | $1.9$ |
| $2098$ | i’ |  |
ZFT | $114\degr 56^{\prime}11\aas@@fstack{\prime\prime}0$ | TOA130FT | $8.0$ | Weitang Liang, Tianrui Sun
| $41\degr 35^{\prime}41\aas@@fstack{\prime\prime}0$ | ASI2600MM Pro | $8.9$ |
| $1300$ | L |  |
ZNFB | $114\degr 56^{\prime}11\aas@@fstack{\prime\prime}0$ | TOA130NFB | $8.0$ | Weitang Liang, Tianrui Sun
| $41\degr 35^{\prime}41\aas@@fstack{\prime\prime}0$ | ASI2600MM Pro | $8.9$ |
| $1300$ | L |  |
BAAS | $116\degr 36^{\prime}40\aas@@fstack{\prime\prime}7$ | 0.254 m | $0.3$ | Ding Liu, Jun Cao, Shupi Zhang
| $40\degr 53^{\prime}21\aas@@fstack{\prime\prime}1$ | QHY268M | $0.3$ | Xiangdong Yin
| $483$ | L |  |
XL60 | $117\degr 34^{\prime}38\aas@@fstack{\prime\prime}0$ | 0.6 m | $2.0$ | Yue Sun
| $40\degr 23^{\prime}45\aas@@fstack{\prime\prime}0$ | ASI Camera | $3.3$ |
| $900$ | V |  |
XL80 | $117\degr 34^{\prime}38\aas@@fstack{\prime\prime}0$ | 0.8 m | $0.5$ | Yuguang Sun
| $40\degr 23^{\prime}45\aas@@fstack{\prime\prime}0$ | PI VersArray 1300B LN | $2.0$ |
| $900$ | R |  |
XL85 | $117\degr 34^{\prime}38\aas@@fstack{\prime\prime}0$ | 0.85 m | $0.5$ | Yue Sun
| $40\degr 23^{\prime}45\aas@@fstack{\prime\prime}0$ | Andor CCD | $2.2$ |
| $900$ | V |  |
XL126 | $117\degr 34^{\prime}38\aas@@fstack{\prime\prime}0$ | 1.26 m/TRIPOL | $10.0$ | Yuguang Sun
| $40\degr 23^{\prime}45\aas@@fstack{\prime\prime}0$ | 3 SBIG STT-8300M CCDs | $14.9$ |
| $900$ | g’, r’, i’ |  |
XL216 | $117\degr 34^{\prime}38\aas@@fstack{\prime\prime}0$ | 2.16 m | $0.5$ | Jie Zheng, Feng Xiao
| $40\degr 23^{\prime}45\aas@@fstack{\prime\prime}0$ | BFOSC | $2.1$ |
| $900$ | I |  |
DBAS | $122\degr 36^{\prime}59\aas@@fstack{\prime\prime}0$ | TOA130NFB | $5.0$ | Chengcheng Zhu, Fan Li
| $39\degr 55^{\prime}25\aas@@fstack{\prime\prime}0$ | QHY16200A | $7.3$ |
| $110$ | L |  |
QXAO aa$a$The light curve for the QXAO site in Figure 1(c) shows a significant variation in the exposure cycle during the observation. This is mainly because, at some point, the camera’s power supply failed, causing cooling issues as well as stuttering and dropped frames. | $117\degr 19^{\prime}05\aas@@fstack{\prime\prime}9$ | C11 Hyperstar | $1.0$ | Guihua Niu
| $36\degr 29^{\prime}15\aas@@fstack{\prime\prime}3$ | QHY174M GPS | $4.0$ |
| $688$ | clear |  |
YLAO | $100\degr 37^{\prime}32\aas@@fstack{\prime\prime}0$ | C14HD | $0.3$ | Yixing Zhang
| $25\degr 47^{\prime}01\aas@@fstack{\prime\prime}5$ | ASI432MM | $0.6$ |
| $1850$ | clear |  |
YAHPT | $101\degr 10^{\prime}52\aas@@fstack{\prime\prime}0$ | ASA RC 0.8 m | $0.13$ | Jian Chen, Ye Yuan
| $25\degr 31^{\prime}43\aas@@fstack{\prime\prime}0$ | Andor DU888 GPS | $0.13$ |
| $1943$ | I |  |
YACHES | $101\degr 10^{\prime}53\aas@@fstack{\prime\prime}0$ | ASA 0.8 m | $0.5$ | Chen Zhang
| $25\degr 31^{\prime}36\aas@@fstack{\prime\prime}0$ | QHY461M GPS | $2.2$ |
| $2009$ | L |  |
XMC8 | $118\degr 04^{\prime}17\aas@@fstack{\prime\prime}5$ | C8HD TV2X | $1.0$ | Wenpeng Xie, Xiansheng Zheng
| $24\degr 35^{\prime}52\aas@@fstack{\prime\prime}7$ | QHY174M GPS | $1.0$ | Donghua Chen
| $8$ | clear |  |
XMC11 | $118\degr 04^{\prime}17\aas@@fstack{\prime\prime}4$ | C11HD | $0.115$ | Yang Zhang, Yuchen Lin, Feng Xia
| $24\degr 35^{\prime}52\aas@@fstack{\prime\prime}9$ | ASI174MM | $0.115$ |
| $31$ | clear |  |
DGSM | $113\degr 58^{\prime}59\aas@@fstack{\prime\prime}6$ | C925XLT | $0.1$ | Jin Liu, Peiyu Ye
| $23\degr 05^{\prime}08\aas@@fstack{\prime\prime}6$ | ASI533MM Pro | $0.1$ |
| $22$ | clear |  |
DGFLS | $113\degr 48^{\prime}20\aas@@fstack{\prime\prime}0$ | C11HD | $0.5$ | Xiaojun Mo, Xinru Han
| $23\degr 00^{\prime}52\aas@@fstack{\prime\prime}0$ | ASI432MM | $0.8$ |
| $20$ | clear |  |
MAAA | $110\degr 50^{\prime}45\aas@@fstack{\prime\prime}8$ | GSO RC8 | $1.0$ | Tong Liu, Jinjie Xiao, Xiang Zeng
| $21\degr 35^{\prime}59\aas@@fstack{\prime\prime}1$ | ASI174MM | $1.0$ | Lihua Zhu, Yuqiang Chen, Zhendong Gao
| $9$ | clear |  | Jianming Xu, Hongyu Lin, Haolong Lin
999

Table 3: Stations that encountered weather or overexposure issues on 6 October 2022 Station | Longitude (E) | Latitude (N) | Altitude (m) | Observers
---|---|---|---|---
CCO | $126\degr 19^{\prime}49\aas@@fstack{\prime\prime}7$ | $43\degr 49^{\prime}27\aas@@fstack{\prime\prime}8$ | $320$ | Zhe Kang
LHS | $93\degr 53^{\prime}46\aas@@fstack{\prime\prime}0$ | $38\degr 36^{\prime}24\aas@@fstack{\prime\prime}5$ | $4200$ | Chen Zhang
HNU | $114\degr 30^{\prime}58\aas@@fstack{\prime\prime}0$ | $37\degr 59^{\prime}55\aas@@fstack{\prime\prime}0$ | $69$ | Shuai Zhang, Qingle Qu
TGD | $104\degr 56^{\prime}27\aas@@fstack{\prime\prime}6$ | $37\degr 48^{\prime}44\aas@@fstack{\prime\prime}6$ | $1451$ | Wei Tan, Jia Zhou
QHOS | $97\degr 33^{\prime}36\aas@@fstack{\prime\prime}0$ | $37\degr 22^{\prime}24\aas@@fstack{\prime\prime}0$ | $3200$ | Wei Zhang
OUC | $120\degr 29^{\prime}53\aas@@fstack{\prime\prime}4$ | $36\degr 09^{\prime}31\aas@@fstack{\prime\prime}5$ | $74$ | Chenyang Guo, Yue Lu
XYOS | $118\degr 27^{\prime}50\aas@@fstack{\prime\prime}0$ | $32\degr 44^{\prime}03\aas@@fstack{\prime\prime}7$ | $227$ | Wei Zhang, Bin Li
PMOXL | $118\degr 57^{\prime}30\aas@@fstack{\prime\prime}0$ | $32\degr 07^{\prime}33\aas@@fstack{\prime\prime}0$ | $20$ | Jian Chen, Yue Chen
NJXD | $118\degr 25^{\prime}17\aas@@fstack{\prime\prime}5$ | $32\degr 04^{\prime}40\aas@@fstack{\prime\prime}8$ | $74$ | Jun Xu, Jun He
SAM | $121\degr 55^{\prime}26\aas@@fstack{\prime\prime}8$ | $30\degr 54^{\prime}57\aas@@fstack{\prime\prime}7$ | $26$ | Song Yao
QZAA | $118\degr 35^{\prime}24\aas@@fstack{\prime\prime}0$ | $29\degr 06^{\prime}28\aas@@fstack{\prime\prime}8$ | $233$ | Jiajun Lin
DWM | $114\degr 06^{\prime}44\aas@@fstack{\prime\prime}5$ | $28\degr 25^{\prime}28\aas@@fstack{\prime\prime}0$ | $1388$ | Wei Tan, Jia Zhou
GMG | $100\degr 01^{\prime}51\aas@@fstack{\prime\prime}6$ | $26\degr 42^{\prime}33\aas@@fstack{\prime\prime}1$ | $3185$ | Jianguo Wang
BCTO | $113\degr 19^{\prime}29\aas@@fstack{\prime\prime}0$ | $22\degr 46^{\prime}18\aas@@fstack{\prime\prime}0$ | $3$ | Jianguo He, Zhiqiang Hu
WYU | $113\degr 05^{\prime}12\aas@@fstack{\prime\prime}1$ | $22\degr 35^{\prime}55\aas@@fstack{\prime\prime}9$ | $2$ | Boyu Li, Yehua Pang
SZAO | $114\degr 33^{\prime}21\aas@@fstack{\prime\prime}0$ | $22\degr 28^{\prime}56\aas@@fstack{\prime\prime}0$ | $220$ | Jiahui Ye, Delai Li

##  Appendix B Method consistencies

As mentioned in MO22, when comparing our results with theirs, it should be
made sure that the ray-tracing code used in the present paper is fully
consistent with that used in MO22 in order to rule out any systematic
discrepancy caused by differences in the codes. The upper panel of Figure 2
compares the ray-tracing results of MO22 and the present paper, relevant to
the 5 October 2017 stellar occultation by Triton. They match at a high-
accuracy level and therefore eliminate any systematic discrepancies that might
have arisen from differences in the ray-tracing codes.

(a)

(b)

Figure 2: Comparisons of the ray-tracing results of MO22 and the present
paper, relevant to the 5 October 2017 stellar occultation by Triton. Panel
(a): Comparison of normalized single-image stellar flux. Panel (b): Comparison
of the total normalized primary- and secondary-image fluxes. The ray-tracing
results of MO22 are obtained from Figure B.1 of MO22.