(LABEL:)

# Distribution Locational Marginal Emission for Carbon Alleviation in
Distribution Networks: Formulation, Calculation, and Implication

Linwei Sang, Yinliang Xu, , Hongbin Sun, ,
Qiuwei Wu, , and Wenchuan Wu Linwei Sang, Yinliang Xu, and Qiuwei Wu are with
Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate
School, Tsinghua University, Shenzhen, China. (E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>qiuwu@sz.tsinghua.edu.cn.)Wenchuan Wu and Hongbin Sun are with the Department
of Electrical Engineering, Tsinghua University, Beijing, China.

###### Abstract

Regulating the proper carbon-aware intervention policy is one of the keys to
emission alleviation in the distribution network, whose basis lies in
effectively attributing the emission responsibility using emission factors.
This paper establishes the distribution locational marginal emission (DLME) to
calculate the marginal change of emission from the marginal change of both
active and reactive load demand for incentivizing carbon alleviation. It first
formulates the day-head distribution network scheduling model based on the
second-order cone program (SOCP). The emission propagation and responsibility
are analyzed from demand to supply to system emission. Considering the complex
and implicit mapping of the SOCP-based scheduling model, the implicit theorem
is leveraged to exploit the optimal condition of SOCP. The corresponding SOCP-
based implicit derivation approach is proposed to calculate the DLMEs
effectively in a model-based way. Comprehensive numerical studies are
conducted to verify the superiority of the proposed method by comparing its
calculation efficacy to the conventional marginal estimation approach,
assessing its effectiveness in carbon alleviation with comparison to the
average emission factors, and evaluating its carbon alleviation ability of
reactive DLME.

###### Index Terms:

Emission factor, second-order cone program, distributed energy resources,
implicit mapping, distribution locational marginal emission.

##  I Introduction

Decarbonizing the electrical energy system is urgent and essential to tackle
the increasingly visible climate changes [1]. Accelerating the carbon-neutral
transition requires massive renewables penetrated in the distribution networks
and the corresponding facilitation from technology, market, and policy areas
[2]. Regulating the proper carbon-aware intervention policy is one of the keys
to emission alleviation [3, 4], and its basis lies in attributing the emission
responsibility from the power supply to the load demand in the electrical
power networks temporally and spatially using emission factors.

Current emission factors can be categorized into average emission factors
(AEFs) and marginal emission factors (MEFs) [3, 5]. AEFs quantify the average
emission intensity per MWh of load consumption and can be calculated by the
popular carbon emission flow (CEF) models [6, 7], which measure emissions over
long periods; in contrast, MEFs quantify the ratio of the change of emission
intensity to the marginal change of load demand [8, 9], which measures
emission impacts to small local load change. Among the MEFs, locational
marginal emissions (LMEs) are proposed in [3, 5, 10] to quantify the emission
sensitivity at the nodal level, indicating the spatial heterogeneity in
marginal emission rates that emerge from network constraints for transmission
networks. LMEs can be calculated by the regression-based and model-based
approaches. Regression-based approaches utilize the emission and load data to
estimate the values of LME across the regions [3, 11], which require plenty of
existing data. In contrast, model-based approaches utilize the transmission
dispatching model to identify the marginal generators, whose emissions are the
marginal emissions [10, 8].

Intuitively, LMEs are the emission equivalent of locational marginal prices
(LMPs) in the power systems [5]. LMPs have been studied and applied
extensively to incentivize the economic operation of the transmission system
[12], and, similarly, LMEs can incentivize the low-carbon operation of the
transmission network. With the increase of renewables and demand flexibility
in distribution networks, distribution locational marginal price (DLMP) is
proposed in [13, 14, 15, 16] to extend the LMPs for distribution network
management. Liu _et al._ [13] propose the integrated DLMP for optimal
electrical vehicle management based on DC optimal power flow (OPF). Bai _et
al._ [14] leverage the linearized AC OPF to derive the DLMPs for the
active/reactive power and facilitate congestion management and voltage support
in day-ahead distribution market clearing. Unlike the transmission network,
the distribution network with the high R/X ratio cannot neglect the power loss
in system operation, which requires more precise power flow models, e.g., the
second-order cone program (SOCP) formulation [17]. These characteristics lead
to additional emissions from reactive load demand in the distribution
networks. The main reason is that, though generating reactive power does not
generate emissions directly, delivering reactive power will incur active power
loss, leading to additional emissions beyond the emissions from active load
demand. The above distribution network characteristics raise the challenges of
quantifying the emission factors for load demand temporally and spatially to
alleviate carbon emission effectively.

In analogy to extending LMPs to DLMPs, this paper extends the LME to establish
the distribution location marginal emissions (DLMEs) for active and reactive
power to incentivize carbon alleviation in the distribution network. It
formulates the SOCP-based distribution network scheduling model, analyzes the
emission propagation and responsibility from demand to generation to system
emission, and leverages the SOCP-based implicit derivation approach to
calculate the DLMEs effectively in a model-based way. Comprehensive numerical
studies are conducted to validate the proposed method, including comparing its
calculation efficacy to the conventional LME estimation approach, assessing
its effectiveness in carbon alleviation with comparison to the AEFs from CEF,
and evaluating its carbon alleviation ability of reactive DLME. So the main
contributions of this paper can be summarized as follows:

1) _Formulation:_ Inspired by DLMP, we, _firstly_ , formulate the distribution
locational marginal emission (DLME) for both active and reactive power to
incentivize the carbon alleviation in the SOCP-based distribution scheduling
model.

2) _Calculation:_ Based on optimal conditions of SOCP, we leverage the
implicit derivation of the SOCP solution mapping to calculate the DLME of
distribution networks in a model-based manner via emission propagation
analysis and emission responsibility attribution.

3) _Implication:_ Case studies are conducted to validate the advantages of
DLME by _i)_ its calculation efficacy over the conventional LME estimation
approach, _ii)_ its carbon alleviation effectiveness over comparison to the
AEFs from CEF, _iii)_ and its carbon alleviation ability of reactive DLME.

Significant differences can be observed when comparing this work with previous
work [17]. i) they focus on different problems: the former regulates the
carbon-aware intervention policy for the distribution system, and the latter
regulates the voltage devices; ii) they have different goals: the former aims
to incentivize the carbon alleviation effectively, and the latter achieves the
system safe and economic operation; iii) they are solved by different
techniques: the former is solved in the one step backward way, and the latter
is solved in a stochastic gradient descent way.

The rest of this paper is organized as follows. Section II prepares the SOCP-
based distribution network scheduling model with emerging distributed energy
resources formulation and develops the distribution locational marginal
emission formulation. Section III proposes the differential DLME calculation
approach via the emission propagation analysis and implicit gradient
derivation. Section IV verifies the effectiveness of the proposed formulation
and calculation approach. Section V concludes this paper.

##  II Distribution Locational Marginal Emission Formulation

This section firstly prepares the emerging distributed energy resources
formulation in II-A and the corresponding SOCP-based distribution network
scheduling model in II-B. Then, based on the above, it proposes the DLME
formulation by the emission propagation and responsibility analysis in II-C.

###  II-A Distributed Energy Resources Formulation

We consider comprehensive distributed energy resource (DER) formulation in the
SOCP-based distribution network scheduling model, including the distributed
generators, distributed energy resources, and the emerging EV aggregator model
sequentially.

####  II-A1 Distributed Generators

We formulate the distributed generators (DGs) with two common forms the
emerging inverter-based generators of eq: ib dg and the conventional
synchronous machine-based generators of eq: sm dg.

#####  II-A1a Inverter-based DG

Distributed wind turbines (WTs) and photovoltaics (PVs) are connected to the
distribution network through the power electronics inverters to provide the
active power supply and reactive power support. Due to technical requirements,
the power factors of the inverters are limited to a given range in eq: ib dg
coef.

$\displaystyle P^{inv}_{i,t,\min}\leq p^{inv}_{i,t}\leq P^{inv}_{i,t,\max},$
$\displaystyle\forall i\in\mathcal{IB},\forall t\in\mathcal{T}$ (1a)
$\displaystyle\frac{p^{inv}_{i,t}}{\sqrt{(p^{inv}_{i,t})^{2}+(q^{inv}_{i,t})^{2}}}\geq\kappa_{i,\min},$
$\displaystyle\forall i\in\mathcal{IB},\forall t\in\mathcal{T}$ (1b)

where $p^{inv}_{i,t}$ and $q^{inv}_{i,t}$ are the active and reactive power of
inverter-based DG $i$ in time $t$; $\kappa_{i,\min}$ is the minimum power
factor of inverter-based DG $i$; $P^{inv}_{i,t,\min}$ and $P^{inv}_{i,t,\min}$
are the minimum and maximum active power of inverter-based DG $i$ in time $t$.

The complex eq: ib dg coef can be further transformed into the linear
constraints in eq: ib dg trans.

$\displaystyle\begin{aligned}
&-\frac{p^{inv}_{i,t}\sqrt{1-(\kappa_{i,\min})^{2}}}{\kappa_{i,\min}}\leq
q^{inv}_{i,t}\leq\frac{p^{inv}_{i,t}\sqrt{1-(\kappa_{i,\min})^{2}}}{\kappa_{i,\min}}.\end{aligned}$
(2)

The maximum outputs of the above inverter-based DGs mainly depend on the solar
radiation for PVs and the wind speed for WTs in eq: ib dg limit.

#####  II-A1b Synchronous machine-based DG

Distributed synchronous generators rely on conventional fossil-based energy
resources for the active power supply and the excitation for the reactive
power supply for the distribution network in eq: sm dg. Unlike the inverter-
based DG, synchronous generators’ active power generation is further limited
by their physical constraints, as formulated as the ramping constraints in eq:
sm dg ramp.

$\displaystyle P^{sm}_{i,t,\min}\leq p^{sm}_{i,t}\leq P^{sm}_{i,t,\max},$
$\displaystyle\forall i\in\mathcal{SM},\forall t\in\mathcal{T}$ (3a)
$\displaystyle Q^{sm}_{i,t,\min}\leq q^{sm}_{i,t}\leq Q^{sm}_{i,t,\max},$
$\displaystyle\forall i\in\mathcal{SM},\forall t\in\mathcal{T}$ (3b)
$\displaystyle r^{RD}_{i}\leq P^{sm}_{i,t}-P^{sm}_{i,t-1}\leq r^{RU}_{i}$
$\displaystyle\forall i\in\mathcal{SM},\forall t\in\mathcal{T}$ (3c)

where $p^{sm}_{i,t}$ and $q^{sm}_{i,t}$ are the active and reactive power of
synchronous machine-based DG $i$ in time $t$; $P^{sm}_{i,t,\min}$ and
$P^{sm}_{i,t,\max}$ are the minimum and maximum active power;
$Q^{sm}_{i,t,\min}$ and $Q^{sm}_{i,t,\max}$ are the minimum and maximum
reactive power; $r^{RD}_{i}$ and $r^{RU}_{i}$ are the maximum ramping down and
ramping up values of synchronous machine $i$; $\mathcal{SM}$ is the set of
synchronous machine-based DGs.

####  II-A2 Distributed Energy Storage Model

Distributed energy storage (ES) models mainly provide the active power for
temporal energy transferring flexibility with the storage capacity, power, and
energy change limits in eq: es.

$\displaystyle 0\leq p^{cha}_{i,t}\leq P^{cha}_{i,\max},0\leq
p^{dis}_{i,t}\leq P^{dis}_{i,\max},$ $\displaystyle\forall
i\in\mathcal{ES},\forall t\in\mathcal{T}$ (4a) $\displaystyle
e_{i,t}=e_{i,t-1}+\eta^{cha}_{i}p^{cha}_{i,t}-(1/\eta^{dis}_{i})p^{dis}_{i,t},$
$\displaystyle\forall i\in\mathcal{ES},\forall t\in\mathcal{T}$ (4b)
$\displaystyle E_{i,\min}\leq e_{i,t}\leq E_{i,\max},$ $\displaystyle\forall
i\in\mathcal{ES},\forall t\in\mathcal{T}$ (4c)

where $p^{cha}_{i,t}$ and $p^{dis}_{i,t}$ are the charging and discharging
power of ES $i$ in time $t$; $e_{i,t}$ is the stored energy;
$P^{cha}_{i,\max}$ and $P^{dis}_{i,\max}$ are the maximum charging and
discharging power; $E_{i,\min}$ and $E_{i,\max}$ are the minimum and maximum
stored energy; $\eta^{cha}_{i}$ and $\eta^{dis}_{i}$ are the charging and
discharging coefficients.

To minimize operation cost, the simultaneous charging and discharging of ES in
eq: es power will not coincide at the same time due to the power losses in the
charging and discharging, so binary variables indicating charging/discharging
status can be eliminated in eq: es state.

####  II-A3 EV Aggregators with Flexibility

Based on [18], the aggregated charging power of electrical vehicle (EV)
aggregators can provide temporal load demand flexibility under the
satisfaction of each EV’s energy demand with the power bounds constraint of
eq: ev power and the energy bound constraint of eq: ev energy.

$\displaystyle p^{EV}_{i,t,lb}\leq p^{EV}_{i,t}\leq p^{EV}_{i,t,ub},$
$\displaystyle\forall i\in\mathcal{EV},\forall t\in\mathcal{T}$ (5a)
$\displaystyle e^{EV}_{i,t,lb}\leq\sum^{t}_{\tau=1}p^{EV}_{\tau,t}\leq
e^{EV}_{i,t,ub},$ $\displaystyle\forall i\in\mathcal{EV},\forall
t\in\mathcal{T}$ (5b)

where $p^{EV}_{i,t}$ is the active power of EV fleet $i$ in time $t$;
$p^{EV}_{i,t,lb}$ and $p^{EV}_{i,t,ub}$ are the corresponding low bound and
upper power bound; $e^{EV}_{i,t,lb}$ and $e^{EV}_{i,t,ub}$ are the
corresponding low bound and upper energy bound.

###  II-B SOCP-based Scheduling Formulation

On top of the DERs formulation, we formulate a SOCP-based day-ahead
distribution network management model in eq: dso da. The main difference
between the real-time and day-ahead models lies in considering the temporal
constraints of synchronous machine-based distributed generators, distributed
energy storage, and EV aggregators.

####  II-B1 Day-ahead Scheduling Model

SOCP-based day-ahead scheduling model is formulated in eq: dso da, which
considers the temporal relationship of energy devices with fossil-based
generator ramping and energy storage coupling constraints. The corresponding
DLME from the day-ahead scheduling model reflects the temporal and spatial
emission factors.

$\displaystyle\min_{q^{reg}}$
$\displaystyle\sum_{t\in\mathcal{T}}\Bigl{\\{}\sigma^{P}_{S,t}p^{S}_{t}+\sigma^{Q}_{S,t}q^{S}_{t}+\sum_{i\in\mathcal{DER}}(\sigma^{P}_{S,i}p^{G}_{i,t}+\sigma^{Q}_{S,t}q^{G}_{i,t})\Bigr{\\}}$
(6a) $\displaystyle\mathcal{DER}=\\{\mathcal{IB},\mathcal{SM},\mathcal{ES}\\}$
s.t.
$\displaystyle\sum_{jk\in\mathcal{E}}p_{jk,t}-\sum_{ij\in\mathcal{E}}(p_{ij,t}-r_{ij}l_{ij,t})={p}^{G}_{j,t}-{p}^{D}_{j,t}$
(6b)
$\displaystyle\sum_{jk\in\mathcal{E}}q_{jk,t}-\sum_{ij\in\mathcal{E}}(q_{ij,t}-x_{ij}l_{ij,t})={q}^{G}_{j,t}-{q}^{D}_{j,t}$
(6c) $\displaystyle
v_{j,t}=v_{i,t}+(r^{2}_{ij}+x^{2}_{ij})l_{ij,t}-2(r_{ij}p_{ij,t}+x_{ij}q_{ij,t})\forall
ij\in\mathcal{E}$ (6d) $\displaystyle||2p_{ij,t}\quad 2q_{ij,t}\quad
l_{ij,t}-v_{i,t}||\leq l_{ij,t}+v_{i,t}$ (6e)
$\displaystyle(V^{\min}_{i,t})^{2}\leq v_{i,t}\leq(V^{\max}_{i,t})^{2},\forall
i\in\mathcal{N}_{B}$ (6f) $\displaystyle(I^{\min}_{ij,t})^{2}\leq
l_{ij,t}\leq(I^{\max}_{ij,t})^{2},\forall ij\in\mathcal{E}$ (6g)
$\displaystyle\text{DER
constraints:}\originaleqref{eq:ibdg},\originaleqref{eq:smdg},\originaleqref{eq:es},\originaleqref{eq:ev}$
(6h)

where ${p}^{S}_{j,t}$ and ${q}^{S}_{j,t}$ are the active and reactive power
supply at substation; ${p}^{G}_{j,t}$ and ${q}^{G}_{j,t}$ are the active and
reactive power supply of DERs; ${p}^{D}_{j,t}$ and ${q}^{D}_{j,t}$ are the
active and reactive bus load demand; $v_{i,t}$ and $l_{ij,t}$ are the square
of voltage magnitude and current; $\sigma^{P}_{S,t}$ and $\sigma^{Q}_{S,t}$
are the active and reactive prices of the substation; $\sigma^{P}_{G,t}$ and
$\sigma^{Q}_{G,t}$ are the active and reactive prices of DERs; $r_{ij}$ and
$x_{ij}$ are the resistance and reactance of branch $ij$; $p_{ij,t}$ and
$q_{ij,t}$ are the active and reactive power of branches; $\mathcal{T}$ and
$\mathcal{E}$ are the set of operation periods and network branches. Equations
eq: da pf p and eq: socp c formulate the bus active and reactive balance, eq:
socp e and eq: socp e formulate the branch model and relax the nonlinear power
flow equations by the second-order cone constraints, and eq: socp f and eq:
socp g limit the voltage and branch power transmission. The detailed
formulation is shown in [14].

####  II-B2 Scheduling Solution Mapping

The above scheduling model eq: dso da maps from the load demand
$D=(p_{d},q_{d})\in\mathbb{R}^{2N_{B}\times T}$ to the optimal power supply
$G=(p_{g},q_{g})\in\mathbb{R}^{2N_{G}\times T}$ implicitly under the optimal
solution of SOCP. Let $(p_{g},q_{g})=f_{g}(D):\mathbb{R}^{2N_{B}\times
T}\rightarrow\mathbb{R}^{2N_{G}\times T}$ denote the above mapping, called the
_scheduling solution mapping_.

######  Remark 1.

For discrete variables of SOCP-based scheduling, such as the feeder status, we
first run the mixed-integer second order-cone programming (MISOCP)-based model
to determine the value of integer variables and then fix the integer value to
formulate the SOCP-based solution mapping for DLME calculation in the proposed
differential approach.

###  II-C Distribution Locational Marginal Emission Formulation

The day-ahead SOCP-based scheduling model eq: dso da achieves the optimal
active/reactive power allocation under the security requirement in the
distribution networks. At the same time, the emissions are generated,
including the bulk and the DER generation emission. Let $e_{g}$ denote the
emission of generator $g$, which is a function of $p_{g}$, denoted by
$f_{e}(p_{g})$. Then we analyze the emission from the demand side to the
generation side in section II-C1, allocate the emission responsibility from
the generation side to the demand side in section II-C2, and marginally
formulate the DLME in section II-C1.

####  II-C1 Emission Propagation Analysis

For each time $t$, in the emission propagation of eq: forward, the
active/reactive power $(p_{d},q_{d})$ in the load demand requires the
distributed generator energy supply $(p_{g},q_{g})$ under the operational
requirements via the optimal distribution network management eq: dso da,
denoted by _scheduling solution mapping_ , in eq: fg, $(p_{g},q_{g})$ leads to
the generator emission $e_{g}$ in eq: fe, and the total emission $e_{sum}$ can
be obtained by accumulating each generator’s emission in eq: esum.

$\displaystyle{(p_{d},q_{d})\Rightarrow(p_{g},q_{g})}\Rightarrow
e_{g}\Rightarrow e_{sum}$ (7a) $\displaystyle\text{Scheduling solution
mapping: }(p_{g},q_{g})=f_{g}(p_{d},q_{d})$ (7b) $\displaystyle\text{Emission
mapping: }e_{g}=f_{e}(p_{g}),g\in\mathcal{G}$ (7c)
$\displaystyle\text{Emission summing: }e_{sum}=\sum_{g\in\mathcal{G}}e_{g}$
(7d)

####  II-C2 Emission Responsibility Analysis

Load demand should take responsibility for the distribution network emission
by the corresponding emission factors. We consider the emission factors in the
distribution network from the marginal perspective. The slight change in
active/reactive load demand will lead to change the distribution system
emission change, denoted by the marginal emission. Current LME calculation
focuses on the transmission network ignoring the power loss, which leverages
the DC optimal power flow model for the model-based LME calculation in [19].
We borrow the idea of extending LMP to DLMP for the LME in the distribution
network. Unlike the DC power flow-based LME, for the distribution network of
eq: dso da, the active and reactive are coupled by the SOCP-based power flow
model, and reactive power will induce active power flow loss to generate
system emission. Active power generation of fossil-based will generate the
system emissions.

####  II-C3 Distribution Locational Marginal Emission

So, based on the above, we quantify the load emission factors for carbon
alleviation in the distribution networks by DLME as follows:

######  Definition 1 (Distribution locational marginal emission).

Distribution locational marginal emission (DLME) is formulated as the ratio of
the marginal change of the total emission in the distribution network emission
with respect to marginal active and reactive load demand change, as presented
in eq: DLME def and eq: DLME reactive.

$\displaystyle\begin{aligned} \text{DLME}=\frac{\partial e_{sum}}{\partial
p_{d}}=\lim_{\Delta_{p}\rightarrow
0}\frac{e_{sum}(p_{d},q_{d})-e_{sum}(p_{d}-\Delta_{p},q_{d})}{\Delta_{p}}\end{aligned}$
(8) $\displaystyle\begin{aligned} \text{DLME}_{q}=\frac{\partial
e_{sum}}{\partial q_{d}}=\lim_{\Delta_{q}\rightarrow
0}\frac{e_{sum}(p_{d},q_{d})-e_{sum}(p_{d},q_{d}-\Delta_{q})}{\Delta_{q}}\end{aligned}$
(9)

We note that DLME without subscript denotes active DLME for active load
demand, and $\text{DLME}_{q}$ denotes the reactive DLME for reactive load
demand. The relationship between the $e_{sum}$ and $p_{d}$/$q_{d}$ is analyzed
with the mapping of $f_{g}(\cdot)$ and $f_{e}(\cdot)$ in eq: forward. DLME and
$\text{DLME}_{q}$ cannot be calculated directly and explicitly due to the
complex composite relationship.

##  III Methodology

This section presents the calculation of DLME through the implicit derivation
approach by introducing the overall calculation framework in section III-A,
analyzing the backward emission responsibility in section III-B, and providing
the solution mapping with the derivative of SOCP in section III-C. We note
that the following calculation analysis focuses on the DLME, and
$\text{DLME}_{q}$ can be calculated in the same way.

###  III-A Overall Framework of the DLME Calculation

Based on the DLME formulation of section II-C, we formulate the overall
framework of the DLME calculation in the emission propagation and
responsibility analysis in Fig. 1.

Figure 1: Overall framework of the DLME framework.

Marginal demand leads to marginal power supply, and marginal supply further
leads to the system’s marginal emission (from right to left); DLME is the
composite of the derivatives of the emission to the supply and the derivatives
of the supply to the demand (from left to right) in Fig. 1. The calculated
DLME can incentivize effective demand response for carbon alleviation in the
bottom of Fig. 1.

###  III-B Backward Emission Responsibility Derivation

The key of DLME lies in computing the derivative of $e_{sum}$ concerning
$p_{d}$ in eq: backward. Based on the emission propagation process of eq:
forward and the chain rules, we decompose the composite derivative calculation
into three parts differentiating through total emission eq: grad e_sum,
through each node emission eq: grad e_g, and power generation eq: grad p_g.

$\displaystyle\frac{\partial e_{sum}}{\partial p_{d}}=\frac{\partial
e_{sum}}{\partial e_{g}}\frac{\partial e_{g}}{\partial p_{g}}\frac{\partial
p_{g}}{\partial p_{d}}$ (10a) $\displaystyle\text{Differentiate through total
emission:}\frac{\partial e_{sum}}{\partial e_{g}}$ (10b)
$\displaystyle\text{Differentiate through each emission:}\frac{\partial
e_{g}}{\partial(p_{g},q_{g})}$ (10c) $\displaystyle\text{Differentiate through
power generation:}\frac{\partial p_{g}}{\partial(p_{d},q_{d})}$ (10d)

The differentiation calculation of eq: grad e_sum and eq: grad e_g relies on
eq: esum and eq: fe, which features the explicit expressions; in contrast, the
calculation of eq: grad p_g relies on the optimal SOCP-based scheduling of eq:
fg, which the derivatives are embedded in the optimal condition in an implicit
form, leading to the main challenge of the differential DLME calculation
approach.

###  III-C The Solution Mapping with Its Derivative of SOCP via Implicit
Derivation

The main challenge of eq: backward lies in the differentiating through the
distribution network scheduling model of eq: grad p_g. We propose to leverage
the _implicit function theorem_ from [20] to calculate the derivative of eq:
grad p_g through the SOCP differentiation. Calculating eq: grad p_g cannot be
implemented explicitly due to the complex mapping relationship of eq: dso da.
But it is possible to differentiate through the SOCP-based scheduling model by
implicitly differentiating its optimality conditions. The main scheme of the
SOCP solution mapping and its derivative comprises the forward propagation for
optimal decision and the backward derivatives for LME, as presented in Fig. 2.

Figure 2: The solution map and its derivative of the general SOCP for DLME
calculation.

The parameterized SOCP-based _scheduling solution mapping_ is decomposed into
three stages: an affine mapping from input parameters to the skew matrix, a
solver, and an affine mapping from the solver solution to the original problem
solution. Then the gradients from the solution to the input parameters can be
back-propagated based on the chain rule. The rest elaborates on the above
three components under a general convex conic program formulation.

Consider the general form of the SOCP-based scheduling model of eq: dso da as
a conic program with the primal and dual forms:

$\begin{aligned} (\text{Primal})\min\text{ }&c^{T}x\\\ \text{s.t.}\text{
}&Ax+s=b\\\ &s\in\mathcal{K}\\\ \end{aligned}\quad\begin{aligned}
(\text{Dual})\min\text{ }&b^{T}y\\\ \text{s.t.}\text{ }&A^{T}y+c=0\\\
&y\in\mathcal{K}^{*}\end{aligned}$ (11)

where $x\in\mathbb{R}^{n}$ and $y\in\mathbb{R}^{m}$ are the primal and dual
variables of eq: dso da, where $p_{g}$ is included in $x$ and $(p_{d},q_{d})$
is included in $b$; $s$ is the primal slack variable;
$\mathcal{K}\in\mathbb{R}^{m}$ is the closed, convex cone with its dual cone
$\mathcal{K}^{*}\in\mathbb{R}^{m}$, including the second-order cone
constraints of the distribution power flow and distribution system operational
constraints.

According to the Karush-Kuhn-Tucker (KKT) conditions, the optimal
$(x^{*},y^{*},s^{*})$ of eq: general socp should satisfy:

$\displaystyle Ax+s=b,A^{T}y+c=0$ (12a) $\displaystyle
s\in\mathcal{K},y\in\mathcal{K}^{*},s^{T}y=0.$ (12b)

The solution mapping from the input parameters $(A,b,c)$ to the distribution
network optimal solution $(x^{*},y^{*},s^{*})$ is formulated as
$\mathcal{F}:\mathbb{R}^{m\times
n}\times\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{n+2m}$, and then the
$\mathcal{F}$ is decomposed into $\phi\circ s\circ Q$, where $\circ$ is the
function composition operator111For example, $h=f\circ g$ refers to
$h(x)=f(g(x))$. Specifically, $Q$ maps the input parameters $(A,b,c)$ to the
corresponding skew-symmetric matrix $Q$; $s$ takes $Q$ as input and solves the
homogeneous self-dual embedding for implicit differentiation with the
intermediate output variable $z$; $\phi$ maps the $z$ to the optimal primal-
dual solution $(x^{*},y^{*},s^{*})$.

Based on the composition, the corresponding total derivative of the solution
mapping $\mathcal{F}$ is derived as:

$\mathsf{D}\mathcal{F}(A,b,c)=\mathsf{D}\phi(z)\mathsf{D}s(Q)\mathsf{D}Q(A,b,c)$
(13)

where $\mathsf{D}$ is the derivative operator, and $\mathsf{D}f(x)$ is the
derivative of function $f$ with respect to $x$.

#####  III-C0a Skew-symmetric Mapping

The skew-symmetric mapping defines the Q matrix as:

$Q(A,b,c)=\begin{bmatrix}0&A^{T}&c\\\ -A&0&b\\\ -c^{T}&-b^{T}&0\\\
\end{bmatrix}\in\mathcal{Q}.$ (14)

The corresponding derivative of $Q(A,b,c)$ with respect to the $(A,b,c)$ is
formulated as:

$\mathsf{D}Q(A,b,c)=\begin{bmatrix}0&\mathsf{d}A^{T}&\mathsf{d}c\\\
-\mathsf{d}A&0&\mathsf{d}b\\\ -\mathsf{d}c^{T}&-\mathsf{d}b^{T}&0\\\
\end{bmatrix}.$ (15)

#####  III-C0b Homogeneous Self-dual Embedding

The homogeneous self-dual embedding finds a zero point of a certain residual
mapping by solving eq: general socp. It utilizes the variable $z$ as an
intermediate variable, partitioned as $(u,v,\omega)$, to formulate the
normalized residual map function from [21] as:

$\mathcal{N}(z,Q)=((Q-I)\Pi+I)(z/|\omega|)$ (16)

where $\Pi$ is the project operation onto $\mathbb{R}\times\mathcal{K}\times
R_{+}$.

If and only if $\mathcal{N}(z,Q)=0$ and $\omega>0$, the $z$ can construct the
solution of eq: general socp for given $Q$. Then we derive the derivatives of
$\mathcal{N}(z,Q)$ with respect to $z$ and $Q$:

$\displaystyle\mathsf{D}_{Q}\mathcal{N}(z,Q)=$ $\displaystyle\Pi(z/|\omega|)$
(17a) $\displaystyle\mathsf{D}_{z}\mathcal{N}(z,Q)=$
$\displaystyle((Q-I)\mathsf{D}\Pi(z)+I)/|\omega|$
$\displaystyle-\text{sign}(\omega)((Q-I)\Pi+I)(z/\omega^{2})e^{T}$ (17b)
$\displaystyle=$ $\displaystyle((Q-I)\mathsf{D}\Pi(z)+I)/\omega$ (17c) when
$z$ is the solution of eq: general socp and $\omega>0$

where $e\in\mathbb{R}^{n}$ is $(0,0,...,1)$.

######  Definition 2 (Implicit function theorem, [20]).

For a given $Q$, $z$ is the solution of the primal-dual pair problem of eq:
general socp, and $\Pi$ is differentiable at $z$. Then $\mathcal{N}$ is
differentiable at $z$ with $\mathcal{N}(z,Q)=0$ and $\omega>0$.

Based on the above implicit function theorem, there exists a neighborhood
$V\subseteq\mathcal{Q}$ such that the $z=s(Q)$ of $\mathcal{N}(z,Q)$ is
unique. So $\mathsf{D}_{Q}\mathcal{N}(z(Q),Q)$ is zero when $z$ is the
solution of eq: general socp. Then the derivative of $s(Q)$ with respect to Q
is derived based on the derivatives with respect to $z$ and $Q$ eq: DzN as:

$\displaystyle\mathsf{D}\mathcal{N}_{z}(s(Q,Q)\mathsf{D}s(Q)+\mathsf{D}_{Q}\mathcal{N}(z,Q)=0$
(18a) $\displaystyle\Rightarrow$
$\displaystyle\mathsf{D}s(Q)=-\mathsf{D}_{z}\mathcal{N}_{z}(s(Q,Q)^{-1}\mathsf{D}_{Q}\mathcal{N}(z,Q)).$
(18b)

#####  III-C0c Solution Construction

The solution construction function $\phi(z)$ construct the optimal solution
$(x^{*},y^{*},s^{*})\in\mathbb{R}^{n+2m}$ from the primal-dual pair eq:
general socp from $z=(u,v,\omega)\in\mathbb{R}^{N}$ as:

$(x^{*},y^{*},s^{*})=\phi(z)=(u,\Pi_{\mathcal{K}^{*}}(v),\Pi_{\mathcal{K}^{*}}(v)-v)/\omega$
(19)

where $\Pi_{\mathcal{K}^{*}}$ is projection operator onto the dual cone
$\mathcal{K}^{*}$.

So the corresponding $\phi(z)$ is differentiable with the derivatives as

$\mathsf{D}\phi(z)=\begin{bmatrix}I&0&-x^{*}\\\
0&\mathsf{D}\Pi_{\mathcal{K}^{*}}(v)&-y^{*}\\\
0&\mathsf{D}\Pi_{\mathcal{K}^{*}}(v)-I^{*}&-s^{*}\\\ \end{bmatrix}/\omega.$
(20)

Applying the derivative of $\mathsf{D}\mathcal{F}(A,b,c)$ to the perturbation
of $(\mathsf{d}A,\mathsf{d}b,\mathsf{d}c)$ is formulated by stacking the above
three derivative components of eq: Q derivative, eq: DsQ, and eq: phi z:

$\displaystyle\begin{aligned}
(\mathsf{d}x^{*},\mathsf{d}y^{*},\mathsf{d}s^{*})=\mathsf{D}\mathcal{F}(A,b,c)(\mathsf{d}A,\mathsf{d}b,\mathsf{d}c)\end{aligned}$
(21)

The above implicit derivative of the SOCP solution calculation method eq:
apply can obtain the backward propagation derivatives from the implicit
function theorem for the derivation of eq: grad p_g.

##  IV Case Study

We utilize the modified IEEE 33-bus distribution networks with multiple PVs to
verify the carbon alleviation effectiveness of the proposed DLME and the
calculation efficacy of the corresponding implicit derivation calculation
approach. We construct the day-ahead SOCP-based scheduling models of eq: dso
da with CVXPY for detailed implementation. All the above models are coded by
Python 3.10 and deployed in a MacBook Pro with RAM of 16 GB, CPU Intel Core i7
(2.6 GHz). The other detailed data are attached in Ref. [22].

For more precise delivery, we present the organization of the case study with
data and experiment setting in section IV-A, calculation efficacy analysis in
comparison with RODM in section IV-B, the carbon alleviation effectiveness of
the DLME in comparison with CEF in section IV-C, the emerging reactive demand
response in section IV-D, and application to an extensive system in section
IV-E.

###  IV-A Data Processing and Experiment Setting

####  IV-A1 Experiment Setting

We set _i)_ the different carbon emission factors for the synchronous machine-
based DGs eq: sm dg and the main grid of the 33-bus system _ii)_ and other
essential SOCP-based scheduling model parameters eq: dso da in Tab. I.

TABLE I: The key parameters of DLME calculation in day-ahead scheduling model. Hyperparameter | Value | Hyperparameter | Value
---|---|---|---
Coal generator $e$ | 0.875 tCO2/MWh | PV capacity | 50MW
Gas generator $e$ | 0.520 tCO2/MWh | PV number | 5
Charging depth | 0.5 | Discharging depth | 0.5
$\eta_{dis}$ | 0.92 | $\eta_{ch}$ | 0.90
$\Delta_{t}$ | 1 hour |  |

We note that the emission factors from DLME and CEF are normalized into p.u.
with the unit of tCO2/MWh.

####  IV-A2 Data Processing

We leverage the implicit derivation to compute the DLME for two distribution
network systems under various renewable and load scenarios. Fig. 3 presents
the typical four average load and PV values in the normalized form (p.u.) via
the k-mean++ algorithm.

Figure 3: Average load and PV in p.u. under various scenarios with the solid
lines for the average load and the dotted lines for the average PV.

As shown in Fig. 3, different colors denote different typical scenarios, the
solid lines denote the average load, and the dotted lines denote the average
PV. We determine the number of clusters according to the metrics of the sum of
the squared Euclidean (SSE) distance.

####  IV-A3 Comparison Setting

To verify the calculation efficacy and carbon alleviation effectiveness, we
compare the proposed DLME calculation against _i)_ the reduced-order dispatch
model (RODM) described in [10] for the distribution network to verify the
calculation efficacy of the proposed approach _ii)_ and the carbon emission
flow (CEF) model proposed in [6] to verify the effectiveness of the proposed
DLME, which are detailed as follows.

The core of the RODM is a merit order-based dispatch process where generators
are dispatched in ascending order of cost. After dispatching generators via
the merit order, the marginal generator—the generator next in line to modify
its output to meet an increase in demand—is identified to find the marginal
emissions rate of the system. The corresponding experiment is conducted in
section IV-B.

CEF quantifies the power emission responsibility from the power supply to the
load demand based on the power flow model formulated in appendix -A. The node
carbon intensity (NCI) index in CEF describes the average carbon emission of
each bus per unit injected power flow, which can provide the average carbon
emission to incentivize low carbon system operation for the system operator.
We denote the NCI index of CEF in the distribution network as the distribution
locational average emission (DLAE), distinguished from the DLME. The
corresponding comparison experiment is conducted in section IV-C.

####  IV-A4 Budget-based Demand Response Model

We leverage the budget-based demand response (DR) model to verify the
alleviation effectiveness of the proposed DLME under the implicit derivation
approach by comparing the carbon reduction performance under various
approaches in eq: dr, inspired from [23]. It _i)_ receives the carbon signals
from the various models (DLME, RODM, and DLAE), _ii)_ maximizes the carbon
reduction benefits under the response budget via eq: dr budget for each
period, _iii)_ and implements the optimized response values for optimal
dispatching via eq: dr.

$\displaystyle\allowdisplaybreaks\begin{aligned} \max\text{
}e^{(\cdot)}_{i}p^{dr}_{i}\quad\text{s.t. }\sum
p^{dr}_{i}=P^{dr}_{\Delta}\end{aligned}$ (22)

where $p^{dr}_{i}$ is the response power of bus $i$; $P^{dr}_{\Delta}$ is
total demand response budget; $e^{(\cdot)}_{i}$ is the bus emission signal by
various calculation approaches, $(\cdot)$ can be the DLME from implicit
derivation and RODM or DLAE from the CEF.

$\displaystyle\allowdisplaybreaks\begin{aligned}
&{p}^{D,ca}_{i}=p^{D}_{i}-p^{dr}_{i}\end{aligned}$ (23)

where ${p}^{D,ca}_{i}$ is the power consumption after the demand response, and
${p}^{D,ca}_{i}$ can be taken into the eq: dso da for redispatching and carbon
alleviation calculation. Moreover, the demand response capacity of
$P^{dr}_{\Delta}$ is set as 1% of the total system maximum load demand.

###  IV-B Calculation Effciency Analysis

This part compares the proposed implicit derivation-based approach with the
RODM approach to verify the calculation efficacy of DLME in day-ahead
scheduling of eq: dso da from the perspectives of results and response
analysis. Concretely, result analysis focuses on the temporal DLME
distribution under various PV and load scenarios; response analysis focuses on
the carbon alleviation performance through the budget-based DR model under
different calculation approaches numerically.

####  IV-B1 Results Analysis

We leverage the implicit derivation of eq: backward to calculate the DLME for
SOCP-based scheduling model eq: dso da under typical four scenarios of Fig. 3
and present the corresponding hourly distribution of DLME via violin plot in
Fig. 4.

(a) Hourly DLME analysis in case 1.

(b) Hourly DLME analysis in case 2.

(c) Hourly DLME analysis in case 3.

(d) Hourly DLME analysis in case 4.

Figure 4: Temporal DLME distribution under different scenarios in violin plot.

As shown in Fig. 4, the calculated DLME features various spatial and temporal
change patterns under various scenarios, demonstrating different carbon
reduction potentials.

For calculation efficacy, we compare the DLME from the conventional RODM and
the implicit derivation approach in the following Fig. 5.

Figure 5: Temporal distribution comparison of DLME from the implicit
derivation and RODM in violin plot.

As shown in Fig. 5, DLMEs under RODM and implicit derivation show a similar
pattern with low values during the 9-10 and 21-24 periods. However, DLME under
implicit derivation features more temporal and spatial variance to capture the
system carbon dynamics, which are more sensitive than the RODM model.

Fig. 6 further compares the distributions of DLME under the implicit
derivation approach and RODM in the kernel density estimate plot, where the
emission factors are more concentrated to low values in an implicit derivation
way. The carbon alleviation effectiveness of calculated DLME is evaluated via
the budget-based DR program.

Figure 6: Distribution comparison of DLME from the implicit derivation and
RODM in density plot.

####  IV-B2 Response Analysis

The above results analysis illustrates the temporal and spatial difference of
DLME from the RODM and implicit derivation _intuitively_ , where more
fluctuation implies high sensitivity. We further verify the carbon reduction
effectiveness of DLME from implicit derivation _numerically_ based on the
budget-based DR model of eq: dr budget and eq: dr. So Tab. II compares the
carbon emission reduction performance via the response model under different
DLME calculation approaches.

TABLE II: Comparison of carbon emission reduction performance under various
scenarios in the day-ahead scheduling.

Cases | Initial/tCO2 | DLME /tCO2 | RODM /tCO2 | Enhance
---|---|---|---|---
Case1 | 34.006 | 33.660 | 33.815 | 81.525%
Case2 | 17.106 | 16.688 | 16.871 | 78.047%
Case3 | 30.987 | 30.563 | 30.781 | 106.158%
Case4 | 26.099 | 25.852 | 25.906 | 24.761%

We note that “Enhance” refers to the improving percentage of carbon
alleviation under DLME over other models.

As shown in Tab. II, DLME from implicit derivation achieves lower carbon
emission for different cases, verifying its carbon alleviation effectiveness.
For carbon alleviation performance, we improve the carbon reduction from RODM
with a maximum of 106.16% and a minimum of 24.76% enhancement.

###  IV-C Comparison of Marginal and Average Emissions

This part analyzes the difference between the marginal and average carbon
signals by comparing the DLME and DLAE. DLME from implicit derivation eq:
backward and DLAE from CEF eq: cef node quantify the carbon emission of the
demand side from both the marginal and average perspectives. Similarly, we
utilize the result analysis for intuitive analysis and response analysis for
carbon alleviation numerical analysis for SOCP-based scheduling of eq: dso da.

####  IV-C1 Results Analysis

We compare the temporal distribution of DLME and DLAE via the violin plot in
the following Fig. 7.

Figure 7: Temporal distribution comparison of DLME and DLAE in violin plot.

As shown in Fig. 7, DLAE fluctuates less than DLME across the scheduling
period, indicating it may not capture the carbon reduction dynamics timely.
The different patterns imply the different temporal characteristics of DLAE
and DLME. The violin plot of DLAE at each period indicates similar values,
and, in contrast, the violin plot of DLME at each period indicates the
concentrated values from the 1-12 and 17-24 periods and diffused values from
the 13-16 periods. The different patterns imply the different spatial
characteristics of DLAE and DLME.

Concretely, we compare the temporal change of DLAE and DLME of typical buses
(bus 2, 11, 14, and 15) in the following Fig. 8. The change of DLAE in these
buses is relatively placid, but the DLAE values among these buses are
distinguished. In contrast, the change of DLME in these buses is quite
varying, but the DLME values among these buses are similar from 1-12 and 17-24
periods. The main reason behind the phenomenon is that DLAE captures the
average emission responsibility, and DLME captures the marginal emission
responsibility.

Figure 8: Typical bus temporal comparison of DLAE and DLME.

Fig. 9 further compares the distribution of DLAE and DLME in the density plot
and indicates the different incentivizing between the average and marginal
emission quantification perspectives, whose carbon alleviation effectiveness
is verified in Tab. III.

Figure 9: Distribution comparison of DLME and DLAE in density plot.

####  IV-C2 Response Analysis

We further verify the carbon reduction effectiveness of DLME over DLAE based
on the budget-based DR model of eq: dr budget and eq: dr _numerically_. Tab.
III compares the carbon emission reduction performance under DLAE and DLME via
the response model.

TABLE III: Comparison of carbon emission reduction performance under various scenarios in the day-ahead scheduling. Cases | Initial/tCO2 | DLME /tCO2 | DLAE /tCO2 | Enhance
---|---|---|---|---
Case1 | 32.202 | 31.878 | 31.943 | 25.380%
Case2 | 17.460 | 17.106 | 17.253 | 70.949%
Case3 | 31.326 | 30.987 | 31.052 | 23.946%
Case4 | 26.573 | 26.099 | 26.436 | 244.459%

As shown in Tab. III, DLME from implicit derivation achieves lower carbon
emission than DLME for different cases, verifying its carbon alleviation
effectiveness. For carbon alleviation performance, we improve the carbon
reduction from DLAE with a maximum of 244.46% and a minimum of 23.95%
enhancement.

For case 1, we further compare the temporal carbon reduction performance
comparison under DLME and DLAE in the following Fig. 10. Under the same DR
budget, different from DLAE, the DLME leverages DR in 5-7 and 14 periods for
effective carbon reduction.

Figure 10: Temporal carbon reduction performance comparison under DLME and
DLAE.

####  IV-C3 DLME under various RES Penetration

With the increase of PV, the penetration of renewables in the distribution is
increasing correspondingly. The penetration rate will affect the distribution
of the corresponding calculated DLME. So we set the low (20%), medium (50%),
and high (70%) PV capacity penetration scenarios for penetration analysis,
where the only difference is the PV capacity. The DLME distributions under
these scenarios are compared by the density plot in the following Fig. 11.

Figure 11: DLME distribution comparison under low, medium, and high renewable
penetration in density plot.

As shown in Fig. 11, DLME features the same density distribution pattern but
different density values. Higher PV penetration will lead to the concentration
of lower DLME values.

###  IV-D Reactive DLME for Reactive Demand Response

Beyond the active power emission, the reactive power demand also generates
emission through the active power loss in distributing reactive power of eq:
dso da. Future reactive power markets are considered in [9] to unlock the
regulation potential of inverter-based DERs to support the system’s safe
operation. Inspired by [9], we discuss the carbon reduction potential from the
reactive demand response by formulating the reactive DLME in eq: DLME
reactive, which is an analogy to active DLME of eq: DLME def and can also be
solved via the implicit derivation approach of eq: backward.

####  IV-D1 Results Analysis

We analyze the temporal distribution of reactive DLME via the violin plot in
Fig. 12.

Figure 12: Temporal distribution of reactive DLME in violin plot.

As shown in Fig. 12, different from active DLME, the reactive DLME can be
negative during the 5-7 periods with different change patterns.

####  IV-D2 Reactive Response Analysis

To verify the reduction effectiveness, we formulate the budget-based reactive
DR program in eq: dr reactive, similar to eq: dr.

$\displaystyle\allowdisplaybreaks\begin{aligned} \max\text{
}e^{(\cdot)}_{i}q^{dr}_{i}\quad\text{s.t. }\sum
q^{dr}_{i}=Q^{dr}_{\Delta}\end{aligned}$ (24)

As shown in Tab. IV, compared with active DR, the reactive DR performance for
carbon alleviation is less effective with lower emission reduction with
maximum 0.00262 tCO2 and minimum 0.00121 tCO2 reductions.

TABLE IV: Comparison of carbon emission reduction performance under various scenarios for reactive DLME. Cases | Initial/tCO2 | DLMEq /tCO2 | Reduction /tCO2
---|---|---|---
Case1 | 33.65974 | 33.65711 | 0.00262
Case2 | 16.68775 | 16.68650 | 0.00125
Case3 | 30.56316 | 30.56172 | 0.00144
Case4 | 25.85825 | 25.85704 | 0.00121

###  IV-E Scalability to Large Systems

We further verify the scalability of the DLME, and its distribution differs
from the DLAE in the modified IEEE 69-bus system, indicating different carbon
alleviation incentivizing, as shown in Fig. 13.

Figure 13: DLME distribution comparison of DLME and DLAE.

Tab. V further verifies the carbon alleviation enhancement of DLME.

TABLE V: Comparison of carbon emission reduction performance under various scenarios for the 69-bus system. Cases | Initial/tCO2 | DLME /tCO2 | DLAE /tCO2 | Enhance
---|---|---|---|---
Case1 | 2515.470 | 2341.532 | 2366.841 | 17.028%
Case2 | 1181.552 | 1035.591 | 1080.098 | 43.870%
Case3 | 2124.439 | 1978.422 | 1994.626 | 12.483%
Case4 | 1867.345 | 1780.961 | 1798.437 | 25.361%

As shown in Tab. V, DLME achieves a lower carbon emission with a maximum of
43.870% reduction and a minimum of 12.483% enhancement, verifying the
scalability of DLME for carbon alleviation in the distribution network.

##  V Conclusion

In analogy to extending LMPs to DLMPs, this paper establishes the DLME for
both active and reactive power to incentivize carbon alleviation in the
distribution network. It formulates the SOCP-based day-ahead distribution
network scheduling model, analyzes the emission propagation and responsibility
from power supply to load demand, and leverages the SOCP-based implicit
derivation approach to and calculates the DLMEs effectively in a model-based
way. Comprehensive numerical studies are conducted to validate the proposed
method, comparing its calculation efficacy to the conventional LME estimation
approach, assessing its effectiveness in carbon alleviation compared to the
AEFs from CEF, and evaluating its carbon alleviation ability of reactive DLME.

###  -A Carbon Emission Flow Model

CEF calculation is based on the generation, network, node, and storage side
carbon analysis. Based on the Ref. [6], the node carbon emission intensity
(NCI) with ES is further derived as the weighted average of injected branch
flow emission (BCI) and generation (GCI), as illustrated in eq: cef node and
eq: cef branch.

$\displaystyle{e}_{i,t}=\frac{\sum_{i\in\mathcal{N}^{G}_{i}}{e}^{G}_{i}{p}^{G}_{i,t}+\sum_{j\in\mathcal{N}^{l+}_{i}}{\rho}_{ij,t}|{p}_{ij,t}|+\sum_{i\in\mathcal{N}^{ES}_{i}}p^{dis}_{i,t}e^{dis}_{i,t}}{\sum_{i\in\mathcal{N}^{G}_{i}}{p}^{G}_{i,t}+\sum_{j\in\mathcal{N}^{l+}_{i}}|{p}_{ij,t}|+\sum_{i\in\mathcal{N}^{ES}_{i}}p^{dis}_{i,t}}$
(25)

where $\mathcal{N}^{G}_{i}$, $\mathcal{N}^{l+}_{i}$, and
$\mathcal{N}^{ES}_{i}$ are the sets of generation, flow-in branches, and
energy storage with bus $i$; $e_{i,t}$ is the nodal carbon emission intensity
for bus $i$ in time $t$.

$\displaystyle{\rho}_{ij,t}=\left\\{\begin{matrix}{e}_{i,t}&\quad{P}_{ij,t}\geq
0\\\ {e}_{j,t}&\quad{P}_{ij,t}<0\end{matrix}\right.$ (26)

where $p_{ij,t}$ and $\rho_{ij,t}$ are the active power flow and corresponding
BCI from bus $i$ to bus $j$ in time $t$. BCI calculation follows the
proportional sharing principle from Ref. [6].

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