# Netload Range Cost Curves for a Transmission-Aware Distribution System
Planning under DER Growth Uncertainty

Samuel Córdova, Alexandre Moreira, Miguel Heleno

###### Abstract

In the face of a substantial and uncertain growth of behind-the-meter
Distributed Energy Resources (DERs), utilities and regulators are currently in
the search for new network planning strategies for facilitating an efficient
Transmission & Distribution (T&D) coordination. In this context, here we
propose a novel distribution system planning methodology to facilitate
coordinated planning exercises with transmission system planners through the
management of long-term DER growth uncertainty and its impact on the
substation netload. The proposed approach is based on the design of a
transmission-aware distribution planning model embedding DER growth
uncertainty, which is used to determine a “menu” of secure distribution
network upgrade options with different associated costs and peak netload
guarantees observed from the transmission-side, referred here as Netload Range
Cost Curves (NRCCs). NRCCs can provide a practical approach for coordinating
T&D planning exercises, as these curves can be integrated into existing
transmission planning workflows, and specify a direct incentive for
distribution planners to evaluate peak netload reduction alternatives in their
planning process. We perform computational experiments based on a realistic
distribution network that demonstrate the benefits and applicability of our
proposed planning approach.

###### Index Terms:

Distribution Planning, Transmission and Distribution Coordination, Distributed
Energy Resources, Peak Netload, Long-term Uncertainty.

##  I Introduction

Distribution network upgrades constitute one of largest sources of capital
expenditures in power systems, accounting for more than 30% of the total
investments costs for investor-owned utilities in the U.S. [1]. In this
context, distribution utilities are required to periodically justify their
proposed investment plans to regulators, since these plans directly impact the
resulting end-consumer electricity rates [2]. Historically, investment plans
have been justified based on expected demand growth projections, for which
network reinforcement needs are identified using exhaustive powerflow
simulations; however, given the increasingly large integration of behind-the-
meter Distributed Energy Resources (DERs), such as Rooftop Photovoltaic (RPV)
and Electric Vehicles (EVs), utilities have recently started to integrate DER
growth forecasts into their planning processes [3, 2]. Despite the progress
made so far, necessary improvements are still needed to make the current
methods able to deal with DER growth uncertainty and to take into account the
impact of distribution system investments on the transmission expansion
planning process.

While several reports and papers have addressed the issue of incorporating DER
growth forecasts into the distribution planning process, less attention has
been given to DER growth uncertainty and its impact on power system planning
[4, 5]. In fact, as discussed in [6, 7, 8], the uncertainty in DER growth can
play a major role when determining possible investment options for the
distribution network, since different potential DER projects with diverse
sizes and connecting points can result in different distribution reinforcement
needs with varying locations. Furthermore, the impacts of DER growth
uncertainty can also propagate to the transmission side, as each DER growth
scenario can result in a different corresponding netload profile to be
observed at the substation connecting Transmission and Distribution (T&D).
From the transmission perspective, this new source of netload uncertainty may
potentially lead to under- or over-investments in transmission assets [9].

Ongoing discussions on T&D coordination indicate the possible advantages of
aligning distribution and transmission systems planning processes to enable a
cost-effective DER integration [2, 10, 11]. However, there are limitations to
a practical implementation of these:

  * •

Under existing regulatory frameworks, transmission and distribution grid
planning processes occur in silos, with low and non-standarized data exchange
between planning entities [10], and with desynchronized planning cycles (3-5
years for distribution [3] and 10-20 years for transmission [12]).

  * •

Distribution regulation does not provide sufficient incentives for reducing
the distribution system’s peak netload at a planning stage [13]. Worst-case
peak load conditions across distribution systems propagate upstream and tend
to produce costly transmission planning requirements. This phenomenon is
aggravated with the increasing levels of netload uncertainty at the planning
stage, due to DER adoption.

  * •

The relatively high investment cost for Non-Wire-Alternatives (NWAs), such as
Battery Energy Storage Systems (BESSs), which depend on multiple value streams
to be economically viable. For this purpose, an accurate valuation of the T&D
savings resulting from integrating NWAs into distribution planning (and
potential impact on transmission) is fundamental [11].

Therefore, there is a need for practical approaches that can induce, with
minimal regulatory changes, a desired level of transmission awareness when
optimizing investments for distribution grids, particularly under DER growth
uncertainty.

In this context, we propose a methodology to plan distribution system
investments under long-term DER growth uncertainty while providing planners at
the transmission level with Netload Range Cost Curves (NRCCs). Such curves
correspond to different distribution-level investments, with their
implications in terms of costs and netload uncertainty at the substation, that
can be presented as options to the transmission system planning.

###  I-A Literature Review

Several research efforts have been dedicated in the last years to integrate
DER uncertainty into distribution system planning. These efforts have resulted
in advanced planning models embedding relevant short-term uncertainties that
stem from DERs, including those associated with renewable generation [14], EV
user behavior [15] and demand response availability [16] among others (see
[17, 5] for a thorough literature review). However, less attention has been
given to the incorporation of the long-term DER growth uncertainty into
distribution system planning, with only a handful of works directly addressing
this issue in the context of RPV and EV growth [6, 18, 19]. Furthermore,
existing distribution system planning models that embed DER uncertainty [14,
15, 16, 6, 18, 19] have neglected the impact of distribution network
investments on transmission-level reinforcement needs, which leaves a research
gap in this regard.

A number of centralized planning models considering T&D coordination have been
proposed in the last years. These models have integrated relevant aspects such
as the uncertainty related to load growth and variable renewable generation
[20], the impact of distribution investment decisions on the transmission-
level energy prices [21], the installation of electric vehicle charging
stations [22], and the use of open-source data and tools [23]. In general,
these works are based on large-scale mathematical programs that can model the
operation of assets at both T&D levels, therefore allowing an overall
assessment of the impact of distribution-side investments on the transmission
level. However, as a major drawback, this centralized approach directly
conflicts with the current T&D independent planning workflows and data
exchange limitation imposed by the regulatory framework. While decentralized
T&D planning models have recently been proposed to address the above practical
issue through the use of iterative hierarchical optimization methods [24, 25],
they do not provide a clear design of incentives that would encourage T&D
systems to collaborate, while maintaining their independence in the planning
process. Furthermore, existing T&D planning models [20, 21, 22, 23, 24, 25]
have an additional pending research gap in terms of including long-term DER
growth uncertainty in their decision-making process.

In power systems operations, different T&D coordination schemes have also been
proposed in the literature [26, 27]. In this context, the idea of using
flexibility-vs-cost curves to describe the aggregate distribution-side
flexibility seen from the transmission-side has gained considerable attention
in the last years [28, 29, 30], as this approach allows T&D operators to
interact with each other in a direct yet secure fashion. However, to the best
of our knowledge, this framework has only been applied to operations so far,
thus leaving an untapped potential for T&D planning applications.

Hence, there is a need for practical network planning strategies that create a
link between distribution and transmission systems planners to deal with long-
term DER growth uncertainty and its impact on grid infrastructure needs at
both T&D levels.

###  I-B Contributions

To bridge the gaps discussed above, we propose a methodology for distribution
system planning that considers DER growth uncertainty and provides
transmission system planners with different options to control substation peak
netload conditions via distribution investments. These options are presented
in a form of Netload Range Cost Curves (NRCCs) that can be easily integrated
into transmission planning processes. The contributions of this paper are
threefold:

  1. 1.

The introduction of NRCCs as a form of expressing distribution system
investment options to the transmission planners, while ensuring independence
in the T&D planning process.

  2. 2.

A novel distribution system planning methodology that provides: (i) a simple
and compact representation of the distribution-side’s planning flexibility,
which can foster a collaboration between T&D system expansion planners; (ii) a
“menu” of options of distribution systems plans with associated long-term
netloads that can be used as an input for transmission studies.

  3. 3.

Detailed computational experiments based on a realistic distribution network.
Our results demonstrate the applicability and benefits of the proposed
transmission-aware distribution planning approach in the context of uncertain
netload growth and adoption of RPV.

The rest of the paper is organized as follows: Section II describes
conventional approaches to address the distribution system planning problem.
Section III presents our proposed transmission-aware distribution planning
strategy embedding DER growth uncertainty. Section IV reports computational
experiments and results, demonstrating the relevance and advantages of our
proposed distribution planning approach. Finally, we draw our main conclusions
in Section V.

##  II Conventional Cost Minimization Approaches for Distribution Planning

The decision making process to address distribution system planning can be
modeled via conventional optimization frameworks. One alternative is the use
of deterministic optimization, which would render network reinforcement plans
that are determined based on a cost-minimization strategy and ‘expected’ load
and DER growth projections. Another alternative is to formulate the problem in
a scenario-based manner, through which a set of possible load and DER growth
scenarios are incorporated so that a secure system operation can be guaranteed
for all considered scenarios.

###  II-A Deterministic Planning Model

Under an approach based on deterministic optimization, network upgrade plans
are obtained considering a single user-defined load and DER growth scenario.
Candidate investment options can include the reinforcement of existing network
corridors as well as the installation of new voltage regulators and/or BESSs
[4, 5]. The resulting deterministic planning model includes investment-related
decision variables $\bm{x}=\left[\bm{x}^{B},\bm{x}^{L},\bm{x}^{R}\right]$,
operation-related decision variables
$\bm{y}=\left[\bm{\rho}^{P},\bm{\rho}^{Q},\bm{p}^{B,D}\bm{p}^{B,C},\bm{q}^{B},\bm{f}^{P},\bm{f}^{Q},\bm{e}^{B},\bm{v}^{\dagger},\bm{\tilde{v}}^{R}\right]$,
and sets describing the hours of the day $h\in\mathcal{H}$, representative
days $d\in\mathcal{D}$, distribution lines $l\in\mathcal{L}$, candidate BESSs
$b\in\mathcal{B}$, and network buses $n\in\mathcal{N}$, as described next:

$\displaystyle\min_{\bm{x},\bm{y}}\;\sum_{l\in\mathcal{L}}C^{L}_{l}\,x_{l}^{L}+\sum_{b\in\mathcal{B}}C^{B}_{b}\,x^{B}_{b}+\sum_{n\in\mathcal{N}^{CR}}C^{V}_{n}\,x_{n}^{V}$
(1a) subject to:
$\displaystyle\rho^{P}_{h,d}+\sum_{b\in\mathcal{B}(PoI)}\left(p^{B,D}_{b,h,d}-p^{B,C}_{b,h,d}\right)=\sum_{l\in\mathcal{L}^{FR}(PoI)}f_{l,h,d}^{P}$
$\displaystyle\hskip 156.49014pt\forall h\in\mathcal{H},d\in\mathcal{D}$ (1b)
$\displaystyle\rho^{Q}_{h,d}+\sum_{b\in\mathcal{B}(PoI)}q^{B}_{b,h,d}=\sum_{l\in\mathcal{L}^{FR}(PoI)}f_{l,h,d}^{Q}$
$\displaystyle\hskip 156.49014pt\forall h\in\mathcal{H},d\in\mathcal{D}$ (1c)
$\displaystyle-
LD_{n,h,d}^{P}+\sum_{b\in\mathcal{B}(n)}\left(p^{B,D}_{b,h,d}-p^{B,C}_{b,h,d}\right)=\sum_{l\in\mathcal{L}^{FR}(n)}f_{l,h,d}^{P}$
$\displaystyle\hskip
42.67912pt-\sum_{l\in\mathcal{L}^{TO}(n)}f_{l,h,d}^{P}\quad\forall
h\in\mathcal{H},d\in\mathcal{D},n\in\mathcal{N}$ (1d) $\displaystyle-
LD_{n,h,d}^{Q}+\sum_{b\in\mathcal{B}(n)}q^{B}_{b,h,d}=\sum_{l\in\mathcal{L}^{FR}(n)}f_{l,h,d}^{Q}$
$\displaystyle\hskip
42.67912pt-\sum_{l\in\mathcal{L}^{TO}(n)}f_{l,h,d}^{Q}\quad\forall
h\in\mathcal{H},d\in\mathcal{D},n\in\mathcal{N}$ (1e)
$\displaystyle\Upsilon^{P}_{j}\,f_{l,h,d}^{P}+\Upsilon^{Q}_{j}\,f_{l,h,d}^{Q}\leq
x^{L}_{l}\,C_{l}^{L}$ $\displaystyle\hskip 99.58464pt\forall
h\in\mathcal{H},d\in\mathcal{D},l\in\mathcal{L},j\in\mathcal{J}$ (1f)
$\displaystyle
e^{B}_{b,h,d}=e^{B}_{b,h-1,d}-\frac{1}{\eta^{D}_{b}}p^{B,D}_{b,h,d}+\eta^{C}_{b}\,p^{B,C}_{b,h,d}$
$\displaystyle\hskip 128.0374pt\forall
h\in\mathcal{H},d\in\mathcal{D},b\in\mathcal{B}$ (1g) $\displaystyle
e^{B}_{b,h,d}\leq DR^{B}_{b}\;x^{B}_{b}\hskip 56.9055pt\forall
h\in\mathcal{H},d\in\mathcal{D},b\in\mathcal{B}$ (1h)
$\displaystyle\Upsilon^{P}_{j}\left(p^{B,D}_{b,h,d}-p^{B,C}_{b,h,d}\right)+\Upsilon^{Q}_{j}\,q^{B}_{b,h,d}\leq
x^{B}_{b}$ $\displaystyle\hskip 99.58464pt\forall
h\in\mathcal{H},d\in\mathcal{D},b\in\mathcal{B},j\in\mathcal{J}$ (1i)
$\displaystyle-M^{L}_{l}\left(1-x_{l}^{L}\right)\leq$ $\displaystyle
v^{\dagger}_{TO(l),h,d}-v^{\dagger}_{FR(l),h,d}+2\left(R_{l}\;f^{P}_{l,h,d}+X_{l}\;f^{Q}_{l,h,d}\right)$
$\displaystyle\hskip
14.22636pt\leq\;M^{L}_{l}\left(1-x_{l}^{L}\right)\quad\forall
h\in\mathcal{H},d\in\mathcal{D},l\in\mathcal{L}^{0}\subseteq\mathcal{L}$ (1j)
$\displaystyle-M^{L}_{l}\left(1-x_{l}^{L}\right)\leq$
$\displaystyle\tilde{v}^{R}_{TO(l),h,d}-v^{\dagger}_{FR(l),h,d}+2\left(R_{l}\,f^{P}_{l,h,d}+X_{l}\,f^{Q}_{l,h,d}\right)$
$\displaystyle\hskip
14.22636pt\leq\;M^{L}_{l}\left(1-x_{l}^{L}\right)\quad\forall
h\in\mathcal{H},d\in\mathcal{D},l\in\mathcal{L}^{R}\subseteq\mathcal{L}$ (1k)
$\displaystyle\underline{V}_{\,n}\leq
v^{\dagger}_{n,h,d}\leq\overline{V}_{n}\hskip 42.67912pt\forall
h\in\mathcal{H},d\in\mathcal{D},n\in\mathcal{N}$ (1l)
$\displaystyle\underline{\Phi}_{\,n}\,\tilde{v}^{R}_{n,h,d}\leq
v^{\dagger}_{n,h,d}\leq\overline{\Phi}_{n}\;\tilde{v}^{R}_{n,h,d}$
$\displaystyle\hskip 91.04872pt\forall
h\in\mathcal{H},d\in\mathcal{D},n\in\mathcal{N}^{R}\subseteq\mathcal{N}$ (1m)
$\displaystyle-M^{R}_{n}\;x^{R}_{n}\leq
v^{\dagger}_{n,h,d}-\tilde{v}^{R}_{n,h,d}\leq M^{R}_{n}\;x^{R}_{n}$
$\displaystyle\hskip 91.04872pt\forall
h\in\mathcal{H},d\in\mathcal{D},n\in\mathcal{N}^{CR}\subseteq\mathcal{N}$ (1n)
$\displaystyle\sum_{l\in\mathcal{L}^{C}(n,n^{\prime})}x_{l}^{L}=1\quad\forall(n,n^{\prime})\text{
already connected}$ (1o)
$\displaystyle\left(\,\bm{p}^{B,D},\bm{p}^{B,C},\bm{e}^{B},\bm{v}^{\dagger},\bm{\tilde{v}}^{R}\,\right)\in\mathbb{R}^{+}$
(1p)
$\displaystyle\bm{x}^{B}\in\mathbb{R}^{+},\quad\left(\bm{x}^{L},\bm{x}^{R}\right)\in\left\\{0,1\right\\},$
(1q)

where $\bm{x}^{L}$ and $\bm{x}^{V}$ are binary investment decisions for new
line reinforcements and voltage regulators, with associated costs $C^{L}$ and
$C^{V}$; $\bm{x}^{B}$ is the additional BESS capacity (in MW) to be installed,
with associated costs $C^{B}$; $\bm{\rho}^{P}$ and $\bm{\rho}^{Q}$ are the
active and reactive power drawn from the transmission system at the connecting
substation, respectively; $\bm{f}_{l}^{P}$ and $\bm{f}_{l}^{Q}$ are the active
and reactive lineflows; $\bm{p}^{B,D}$ and $\bm{p}^{B,C}$ are the battery
active power discharge and charge; $\bm{q}^{B}$ is the battery reactive power
injection; $\bm{LD}^{P}$ and $\bm{LD}^{Q}$ are the active and reactive bus
netloads; $\bm{C}^{L}$ is the apparent power line capacity; $\bm{e}^{B}$ is
the energy stored (in MWh) in the BESS; $\bm{\eta}^{D}$ and $\bm{\eta}^{C}$
are the discharging and charging efficiencies; $\bm{x}^{B}$ is the BESS power
capacity (in MW) to be installed; $\bm{DR}^{B}$ is the power-to-energy
duration ratio (in hours); $\bm{v}^{\dagger}$ is the bus squared voltage;
$\bm{R}$ and $\bm{X}$ are the line resistance and reactance;
$\bm{\tilde{v}}^{R}$ is the regulator’s inner squared voltage;
$\underline{\bm{V}}$ and $\overline{\bm{V}}$ are the minimum and maximum bus
voltage limits; $\underline{\bm{\Phi}}$ and $\overline{\bm{\Phi}}$ are the
minimum and maximum voltage regulation ratio; and $\bm{M}^{L}$ and
$\bm{M}^{R}$ are sufficiently large big-M parameters.

Thus, (1) is a mixed-integer linear programming model, whose objective
function (1a) aims to minimize the total investments costs resulting from line
reinforcements and new voltage regulators and BESSs. Active and reactive nodal
power balances are enforced through (1b)-(1c) for the substation that connects
the distribution system to the main transmission grid, and via (1d)-(1e) for
the rest of the buses. Apparent power capacity limits for lines are enforced
through (1f), in which an inner piece-wise linear representation is used based
on approximating hyperplanes $j\in\mathcal{J}$ with coefficients
$\Upsilon^{P}_{j}$ and $\Upsilon^{Q}_{j}$ following [31]. The operation of
BESSs is represented through constraints that describe the evolution of the
stored energy over time (1g), the maximum stored energy limit (1h) and the
apparent power capacity limit (1i). Voltage drops resulting from line
powerflows are described through (1j)-(1k), where $\mathcal{L}^{0}$ is the
subset of lines whose receiving end buses do not have (neither existing nor
candidate) voltage regulation, and $\mathcal{L}^{R}$ is the subset of lines
whose receiving end buses have (either existing or candidate) voltage
regulation. Bus voltage limits and voltage regulation capabilities and are
enforced through (1l) and (1m)-(1n), respectively, where $\mathcal{N}^{R}$ is
the subset of buses with either existing or candidate voltage regulators and
$\mathcal{N}^{CR}$ is a more restrict subset, which comprises the buses with
non-existing but candidate voltage regulators. Constraint (1o) ensures that
one of the line reinforcement options (including a ’no-upgrade’ option) is
chosen for each network corridor, where $\mathcal{L}^{C}(n,n^{\prime})$
indicates the set of candidate line reinforcements for each pair of connected
buses $(n,n^{\prime})$. Non-negative and binary decisions variables are
described by (1p)-(1q).

###  II-B Scenario-Based Planning Model

The main drawback of the deterministic planning model is the utilization of a
single deterministic growth scenario, which leaves all other possible
realizations of load and DER growth neglected in the decision-making process
for new investments. A conventional alternative to address this issue is to
formulate the investment problem within a scenario-based framework considering
a set of load and DER growth scenarios as follows:

$\displaystyle\min_{\bm{x},\hat{\bm{y}}}\;\sum_{l\in\mathcal{L}}C^{L}_{l}\,x_{l}^{L}+\sum_{b\in\mathcal{B}}C^{B}_{b}\,x^{B}_{b}+\sum_{n\in\mathcal{N}^{CR}}C^{V}_{n}\,x_{n}^{V}$
subject to: $\displaystyle\text{Operational constraints
\eqref{eq:pbal_PoI}--\eqref{eq:vreg_lim_bigM},
\eqref{eq:oper_vars}}\quad\forall k\in\mathcal{K}$ Other constraints (1o),
(1q)

where operational decision variables and constraints are defined for each
load/DER growth scenario $k\in\mathcal{K}=\left\\{1,\ldots,K\right\\}$, with
$\hat{\bm{y}}=\left[\bm{y}_{1},\ldots,\bm{y}_{k},\ldots,\bm{y}_{K}\right]$,
and bus netload $\bm{LD}^{P}$ in (1d) now depending on scenario realization
$\xi_{k}$, i.e., $\bm{LD}^{P}(\xi_{k})$. Thus, the above planning model
determines a set of reliable distribution network upgrade options that are
secure for the set of considered load/DER growth scenarios. Note, however,
that this planning approach fails to capture the impact of distribution
network investments on the transmission-side, which motivates the design of a
transmission-aware distribution planning strategy, as described next.

##  III Proposed Transmission-Aware Distribution Planning Strategy

As previously discussed in Sections I and II, one of the main disadvantages of
the conventional cost minimization distribution planning approaches is that
they do not simultaneously consider DER growth uncertainty and its impact on
the netload “seen” from the transmission-level, which limits their practical
application in coordinated T&D planning schemes. As an alternative to provide
a step towards overcoming this limitation, here, we propose a novel
distribution system planning strategy integrating these two aspects so as to
facilitate the dialog between distribution and transmission system planners.
First, in Section III-A, a novel transmission-aware distribution planning
model that embeds DER growth uncertainty is designed. This model can be used
to determine a range of secure and efficient network upgrade options with
different associated budget and peak netload intervals. Then, based on the
above planning model, in Section III-B, we introduce the concept of NRCCs and
the methodology to compute them to describe the distribution-side planning
flexibility. In addition, in Section III-B, we include a discussion of how
these curves can be implemented in current transmission planning workflows,
and other practical advantages associated to the proposed planning approach.

###  III-A Transmission-Aware Distribution Planning Model

The proposed distribution system planning model, given an available budget,
aims to optimize the investment plan so as to avoid peak netloads, under the
realization of different DER load growth scenarios, that largely deviate from
the peak netload that would have been a result of an optimization exercise
that determines investments while considering a single expected long-term DER
load growth scenario. This model is written as follows:

$\displaystyle\min_{\bm{x},\hat{\bm{y}},\bm{\lambda}}\;W\left[\lambda^{D}-\Lambda^{D}\right]^{+}+\left(1-W\right)\left[\lambda^{R}-\Lambda^{R}\right]^{+}$
(2a) subject to:
$\displaystyle-\lambda^{R}\leq\rho_{h,d,k}^{P}\leq\lambda^{D}\quad\forall
h\in\mathcal{H},d\in\mathcal{D},k\in\mathcal{K}$ (2b)
$\displaystyle\left(\bm{\lambda}^{D},\bm{\lambda}^{R}\right)\in\mathbb{R}^{+}$
(2c)
$\displaystyle\sum_{l\in\mathcal{L}}C^{L}_{l}\,x_{l}^{L}+\sum_{b\in\mathcal{B}}C^{B}_{b}\,x^{B}_{b}+\sum_{n\in\mathcal{N}^{CR}}C^{V}_{n}\,x_{n}^{V}\leq\Gamma$
(2d) $\displaystyle\text{Operational constraints
\eqref{eq:pbal_PoI}--\eqref{eq:vreg_lim_bigM},
\eqref{eq:oper_vars}}\quad\forall k\in\mathcal{K}$ $\displaystyle\text{Other
constraints \eqref{eq:one_line_cand}, \eqref{eq:invest_vars}},$

where $\bm{\lambda}=\left[\lambda^{D},\lambda^{R}\right]$, in the optimal
solution, represent maximum direct (D) and reverse (R) peak netloads among all
considered scenarios, which can indicate transmission-side’s reinforcement
needs resulting from considering multiple possible DER growth scenarios.
Moreover, $\Lambda^{D}$ and $\Lambda^{R}$ are the originally ‘expected’ direct
and reverse peak netloads resulting from a deterministic DER growth
projection; $\left[\,y\,\right]^{+}=\max\\{y,0\\}$ is the positive-part
function; $W\in[0,1]$ is a weighting factor for prioritizing the penalization
of direct or reverse peak netloads compared to the expected scenario; and,
finally, $\Gamma$ is a tunable parameter that indicates the available budget
to invest in the distribution grid.

Thus, (2) is a mixed-integer linear programming model. The objective function
to be minimized in (2a) penalizes the largest deviation in terms of peak
netload among all considered DER load growth scenarios compared to the
expected scenario, therefore aiming at minimizing the ‘unexpected’
transmission-side’s reinforcement needs resulting from DER growth uncertainty.
Constraints (2b)-(2c) link the active power drawn from the transmission system
and the largest values of direct and reverse power flows. Constraint (2d)
limits the total investment cost based on the available budget and the last
two sets of constraints ensure a secure system operation in all considered DER
growth scenarios.

The distribution planning model (2) allows for the simultaneous consideration
of multiple possible network upgrade options based on the budget parameter
$\Gamma$. In particular, by adjusting the value of $\Gamma$, different
possible network upgrade plans with corresponding associated peak netload
intervals and budget requirements can be obtained. This information can then
be used to build the proposed NRCCs for facilitating the dialog between
distribution and transmission system planners, as described next.

###  III-B Netload Range Cost Curves

Based on (2), NRCCs for characterizing the distribution-side planning
flexibility can be built using the procedure illustrated in Fig. 1. For this
purpose, first, our methodology requires, as an input, a set of projected DER
long-term growth scenarios $k\in\mathcal{K}$ with corresponding bus netloads
$\bm{LD}^{P}(\xi_{k})$. Although we describe a way to obtain this input in
Section IV, proposing an ideal methodology to generate scenarios is out of the
scope of this paper and our proposed approach assumes the required set of
scenarios is readily available. Given the bus netload scenarios, the
distribution planning model (2) is solved iteratively for different budget
values $\Gamma$, resulting in a range of possible direct and reverse peak
netloads $\lambda^{D}$ and $\lambda^{R}$. Finally, using the above values, the
NRCCs are obtained. These curves can then be shared with the transmission
planning entity for its independent network studies.

Figure 1: Workflow for building NRCCs to facilitate T&D planning coordination.

One of the main advantages of the proposed NRCCs lies in their compact
representation of the distribution-side’s planning flexibility in a way that
can be treated by the transmission counterpart. An example of the integration
of NRCCs in transmission planning is illustrated in Fig. 2, in which it can be
observed that instead of having a single peak netload value for Bus 2, now a
range of possible peak netload values for this bus is available, each of which
with an associated cost. Therefore, distribution grid investments are
presented as “cost vs netload range guarantees” at the substation.

From a transmission planner perspective, these curves describe the marginal
cost of decreasing the netload range in the substation. Therefore, they can be
treated equivalently to any other nodal capacity investment (such as generator
or storage asset) in the expansion planning process. From a procedural
perspective, the NRCCs are a non-invasive form of coordinating T&D planning,
adaptable to the two T&D ownership models that exist in current regulatory
frameworks:

  1. 1.

Integrated utilities: when T&D is managed by the same entity, NRCCs can be
used to communicate least-cost grid investment options from distribution to
transmission planning teams.

  2. 2.

Separated utilities: when T&D is managed by two distinct entities, NRCCs can
be seen as a marginal capacity product that can be offered by the distribution
company. Since marginal costs and netload range guarantees are clearly
identified in NRCCs, these products are verifiable and therefore can be
regulated.

Thus, NRCCs are a straightforward and transparent mechanism for quantifying
the contribution of distribution-side investments, in particular BESS and
other forms of NWAs, on transmission level.

Figure 2: Example of application of NRCCs in transmission planning

##  IV Computational Experiments

In this section, we present a realistic case study to evaluate the performance
of the proposed transmission-aware distribution planning approach. First, we
introduce the general settings considered for the computational experiments.
Then, we describe a potential method to generate realistic long-term DER
growth scenarios, which results in corresponding scenarios of distribution
grid nodal netloads to be considered in a planning process. It is worth
emphasizing that a scenario-generation technique is not a contribution of our
methodology, which assumes long-term DER growth scenarios as an available
input. Finally, we discuss the distribution planning results, starting with
the network upgrade plans that result from existing distribution planning
strategies, and followed by a comparison to the results obtained from
implementing the proposed transmission-aware planning approach based on NRCCs.

###  IV-A General Settings

Our computational experiments are based on the San Francisco distribution
network provided in the SmartDS dataset [32], which includes one full year
load timeseries with a 6.5 MW peak value and has a 12.47 kV base voltage, 126
existing lines, and 354 buses. This distribution system has been modified to
have an initial 1.68 MW RPV capacity and an existing voltage regulator near
the main substation. A 10-year planning horizon is considered in the
simulations, for which three load growth scenarios with different growth rates
per year are considered: Low (2%), Medium (3%) and High (4%); yielding a total
of 81 candidate line reinforcements, and 5 candidate buses to receive the
installation of new BESS and/or voltage regulators (see Fig. 3). Parameters
and costs associated with candidate line reinforcements, BESSs, and voltage
regulators are extracted from [33, 34].

Figure 3: San Francisco distribution network layout, including installed RPV
capacity and candidate reinforcements. The size of the circles indicate the
RPV capacity installed at each bus, ranging from 36 to 369 kW.

We consider a total of 100 RPV adoption scenarios to characterize realistic
DER growth patterns. These scenarios are obtained from an agent-based
extension of the NREL’s dGen model [35] (see Section IV-B for more details).
To reduce computational burden, 3 of the 100 DER growth scenarios are selected
based on the resulting aggregate RPV capacities, yielding the Maximum,
Average, and Minimum RPV adoption scenarios indicated in Figs. 4 and 5. These
scenarios are then combined with the 3 aforementioned load growth scenarios,
therefore a total of 3x3=9 netload scenarios $\bm{\xi}_{k}$ are considered for
this planning exercise. Such netload scenarios are further simplified by
picking the two representative days that are associated with the highest and
lowest peak netload values (see Fig. 6).

Figure 4: Aggregated RPV capacity evolution resulting from 100 agent-based
simulations of RPV adoption. Selected representative scenarios are highlighted
in black. Figure 5: Minimum, Average, and Maximum RPV adoption scenarios
resulting from an agent-based simulation. The size of the circles indicate the
installed RPV capacity at each bus, ranging from 36 to 1939 kW. The total
resulting installed RPV capacity is indicated in parenthesis for each
scenario. Figure 6: Netload scenarios for the two chosen representative days
to be used in distribution planning. Representative days were selected based
on the maximum and minimum peak netloads observed across the year.

For benchmarking purposes, we compare three different distribution planning
strategies in our experiments. The first one corresponds to the deterministic
planning model presented in Section II-A, which minimizes overall costs for
distribution network upgrades while taking into account a single deterministic
load/DER growth projection scenario [4, 5]. This deterministic scenario is
based on the previously described Medium load growth and Average RPV adoption
scenarios. The second planning strategy corresponds to the scenario-based
planning model described in Section II-B, in which all 9 load/DER growth
scenarios are considered in the planning process while maintaining a cost-
minimization approach focused on the distribution-side. The third one is the
proposed transmission-aware distribution planning strategy, described in
Section III, in which a transmission-aware planning model embeds DER growth
uncertainty and is used to build NRCCs that describe possible distribution
network upgrade options with different associated budgets and peak netload
intervals. Under this strategy, a symmetric weighting factor of $W=0.5$ is
considered for both direct and reverse peak netloads.

The conventional and proposed approaches are implemented in Python, with Pyomo
and Gurobi building and solving the optimization problems, and our
computational experiments have been performed on a PC with an Intel Core i7
2.20-GHz processor and 32 GB of RAM under a 64-bit Windows 11 operating
system. RPV projects are simulated using the PySAM package [36]. In addition,
we run AC powerflows for result validation using the OpenDSSDirect package
[37].

###  IV-B DER Growth Scenario Generation

Here we describe the methodology used to generate realistic DER growth
scenarios for distribution planning. This methodology is based on simulating
future RPV adoption patterns, while considering the uncertainty related to the
location and size of the new RPV projects in the distribution grid.

Modern RPV adoption models are typically based on Innovation Diffusion theory,
which characterizes how a new technology is adopted over time. Under this
approach, RPV adoption is characterized as follows [35, 38]:

$\displaystyle\frac{dF(t)}{dt}\frac{1}{1-F(t)}=p+q\,F(t),$ (3)

where $F(t)$ is the fraction of RPV adopters at time $t$, and $p$ and $q$ are
the coefficient of innovation and imitation, respectively. Model (3) has
successfully been implemented to describe aggregate RPV adoption trends in the
U.S. [35]; however, in its current version, is incapable of modeling the
geographical distribution of the new RPV projects in a distribution network,
as it only models aggregate dynamics. To address this issue, we modified (3)
for its application in an agent-based simulation framework, as described next.

As a first step, each bus $n\in\mathcal{N}=\\{1,\ldots,N\\}$ in the
distribution network is treated as an independent agent, for which an adoption
status $x_{n,t}$ at time $t$ is defined ($1$ if adopted, $0$ if not). Thus,
based on (3) and [38], the probability of adoption at time $t+\Delta t$ can be
computed as follows:

$\displaystyle Pr\left(x_{n,t+\Delta
t}=1\,\lvert\,x_{n,t}=0\right)\,=\left(p+\frac{q}{N}\sum_{n\in\mathcal{N}}x_{n,t}\right)\Delta
t$ (4)

which can be used to run Monte-Carlo simulations describing potential adoption
scenarios for each agent $n\in\mathcal{N}$. As an example, the results of
running 100 independent agent-based simulations is shown in Fig. 7, in which
the fraction of adopting agents over time for each simulated scenario is
presented, and compared to the one estimated by the aggregate continuous model
(3).

Figure 7: Evolution of fraction of adopting agents over time for aggregate
continous model (blue) and multiple agent-based simulations (grey).

Another relevant output of the modified agent-based simulation framework is
the size (in kW) of the RPV project assigned to each agent. To capture the
economic rationality of the agents, this aspect is modeled by solving an
optimization problem in which the RPV capacity that maximizes the Net Present
Value (NPV) of each potential project is calculated independently. Thus, when
a new RPV project in installed into the distribution grid (i.e., $x_{n,t}=1$),
this pre-determined RPV capacity value is used. Furthermore, similar to [35],
a maximum market share (MMS) dependent on the project’s NPV and payback period
is also considered here. Within the modified agent-based framework, this means
first running a Bernoulli trial (with a success probability equal to the MMS)
to determine the subset of agents $n\in\mathcal{N}^{\prime}\subset\mathcal{N}$
that will be considered for an eventual adoption in the future resulting from
(4).

The Minimum, Average, and Maximum RPV scenarios resulting from running 100
agent-based simulations and picking the 3 most representative cases (in terms
of aggregate RPV capacity) are illustrated in Figs. 4 and 5. Observe that not
only the total installed RPV capacity varies significantly across the 3
scenarios (from 3.97 to 10.60 MW), but also the spatial distribution of the
new RPV projects. Therefore there is a two-dimensional RPV growth uncertainty
related to the size and location of new projects. As described in Section
IV-A, the above 3 RPV adoption scenarios are the ones considered for
distribution planning.

###  IV-C Planning Results:Deterministic Planning Model

The results obtained from implementing the deterministic planning model are
presented in Fig. 8a. Under this approach, a total investment cost of 0.98
MUSD is obtained as a result of line reinforcements. Moreover, minimum and
maximum peak netload values for the single considered long-term scenario are
equal to -0.42 MW and +8.62 MW, respectively. These observed results stem from
the deterministic planning model’s objective to minimize investment costs
only. Hence, the cheapest network upgrade option is preferred (line
reinforcements in this case) without considering the potential impact of the
distribution system’s peak netload on the transmission-side.

The above planning results were tested under all 9 load and DER growth
scenarios by running detailed AC powerflow simulations, which revealed
significant line capacity violations (see Fig. 9). These violations result
from the fact that only a single expected scenario of load growth and RPV
adoption is considered under the framework of the deterministic planning
model, which underestimates the network upgrades required for most extreme
scenarios with higher load and RPV growth.

Figure 8: Network upgrade plan resulting from (a) the deterministic planning
model and (b) the scenario-based planning model. Figure 9: Line rating in per
unit (p.u.) observed during the maximum peak hour under a High load and DER
growth scenario. Line capacity threshold is indicated to identify limit
violations.

###  IV-D Planning Results: Scenario-Based Planning Model

The results obtained from implementing the scenario-based planning model are
presented in Fig. 8b, for which all selected 9 load/DER growth scenarios are
considered in the planning process. Observe that unlike indicated by the
solution attained via the deterministic planning model (see Fig. 8a),
additional distribution line reinforcements are required in this case to
ensure a secure system operation under the realization of all selected
scenarios. As a result, the total investment cost is $1.24>0.98$ MUSD.
Moreover, note that in this case, a wider peak netload range equal to [-3.66,
+9.50] MW is obtained, which is significantly larger than the one originally
estimated by the deterministic planning model, i.e., [-0.42, +8.62] MW. This
larger range of peak netloads is a natural consequence of considering
different long-term scenarios.

###  IV-E Planning Results: Proposed Transmission-Aware Distribution Planning
Strategy

The performance of the proposed transmission-aware distribution planning
strategy described in Section III is tested by solving planning model (2)
considering all 9 load/DER growth scenarios for different available budget
values $\Gamma$. A direct and reverse ‘expected’ transmission reinforcement
need of $\Lambda^{D}=8.62$ MW and $\Lambda^{R}=0.42$ MW is used in (2), based
on the peak netloads estimated by the deterministic planning model (see
Section IV-C).

As a first analysis, the planning model (2) is solved considering
$\Gamma=1.24$ MUSD, which corresponds to the total investment cost obtained
via the scenario-based planning model, as discussed in Section IV-D. The
planning results for this case are illustrated in Fig. 10a, which as
anticipated, are consistent with the ones obtained through the scenario-based
planning model (see Fig. 8b), including the resulting direct and reverse peak
netloads $\lambda^{D}=9.50$ and $\lambda^{R}=3.66$ MW.

Figure 10: Network upgrade plans resulting from proposed transmission-aware
distribution planning strategy for different available budget $\Gamma$ values

As a second analysis, the available budget was increased to $\Gamma=2$ MUSD,
which resulted in the network upgrade plan presented in Fig. 10b. Observe how
unlike the case with $\Gamma=1.24$ MUSD (see Fig. 10a), a 440kW BESS is
installed in this new case, which translates into reduced direct and reverse
peak netloads of $\lambda^{D}=9.06<9.50$ MW and $\lambda^{R}=3.22<3.66$ MW. In
addition this solution indicated the deferral of some of the originally
required distribution line upgrades due to the BESS smoothing dispatch. We
observe similar results by further raising the available budget to $\Gamma=3$
MUSD. As a consequence of this larger budget, the BESS capacity is increased
to $1007>440$ kW, and the peak netloads are further reduced to
$\lambda^{D}=8.62<9.06$ MW and $\lambda^{R}=2.65<3.22$ MW (see Fig. 10c),
therefore very likely decreasing the levels of needed investments in the
transmission system.

A summary of the resulting direct and reverse peak netloads $\lambda^{D}$ and
$\lambda^{R}$ for different available budgets $\Gamma$ is presented in Fig.
11, which corresponds to the NRCCs originally described in Section III-B. It
is worth noting how peak netloads $\lambda^{D}$ and $\lambda^{R}$ decrease as
a larger budget $\Gamma$ is made available for the distribution network,
ultimately converging to the ‘expected’ requirements $\Lambda^{D}$ and
$\Lambda^{R}$ originally estimated by the deterministic planning model. Such
reduction results from the installation of new BESSs in the distribution
network, which reduces the system’s minimum and maximum peak netloads across
all the selected 9 load/DER growth scenarios. Note that Fig. 11 also includes
dispersion bars to depict the peak netloads obtained by implementing and
assessing the performance of the attained planning decisions (which were
determined while considering 9 scenarios as previously mentioned) under the
realization of 300 scenarios of DER penetration and load growth. These 300
scenarios are a combination of the original 100 RPV adoption scenarios (see
Fig. 4) with the 3 load growth scenarios under consideration. As can be seen,
the dispersion bars indicate that the peak netload for all scenarios are
within the bounds of the NRCC, which were built using only the Maximum,
Average, and Minimum RPV adoption scenarios.

Figure 11: Netload Range Cost Curves (NRCC) resulting from distribution
planning for different costs (budget $\Gamma$). Dispersion bars resulting from
implementing the planning solutions across all 300 DER penetration and load
growth scenarios are included.

From the observation of Fig 11, it is possible to derive the three main
characteristics of the proposed NRCCs:

  1. 1.

The notion of distribution-side planning flexibility, i.e. the exploration of
feasible investment solutions that go beyond what is strictly necessary to
ensure distribution grid security to decrease the uncertainty “seen” from the
transmission side;

  2. 2.

The expression of this planning flexibility in the form of netload range
guarantees at the distribution feeder head or substation transformer;

  3. 3.

The ability to link these netload range guarantees to concrete tractable
distribution investments and corresponding incremental costs, which
facilitates the regulatory processes around T&D.

After disaggregating the planning solutions by investment type, it is possible
to observe the benefit of strategically locating new BESS investments to avoid
reconductoring some of the distribution’s network corridors (see Fig. 10).
This benefit can directly be quantified, as illustrated in Fig. 12, in which
it can be observed that efficiently-located BESS investments result in a
reduction of line reinforcement needs at a distribution level. In particular,
for $\Gamma=3$ MUSD, line savings equal to 0.25 MUSD are obtained as a result
of strategically locating a 1007kW BESS in the middle of the feeder. This
illustrates how, in our proposed framework, the value of strategically
locating BESS (and other possible NWAs) is captured from a T&D point of view.

Figure 12: Investment cost breakdown for different available budget $\Gamma$
values

Thus, from the case study above, it results clear that NRCCs provide a series
of practical benefits that facilitate their implementation in the context of a
coordinated T&D planning. First, they represent the distribution-side’s
planning flexibility in a simple and compact manner, which facilitates its
integration within existing transmission planning workflows. Second, from a
distribution grid perspective, NRCCs are able to distinguish assets strictly
required to ensure system security from those that can be installed to
additionally benefit the transmission side.

##  V Conclusion and Future Work

Driven by a large and uncertain growth of behind-the-meter DERs, utilities and
regulators are currently in the search for new practical network planning
strategies capable of guaranteeing an efficient yet reliable T&D system
expansion. Motivated by this challenge, we presented in this paper a novel
distribution system planning strategy for facilitating T&D coordination
through the management of DER growth uncertainty and its impact on
transmission-side reinforcement needs. The proposed strategy is based on a
novel transmission-aware distribution planning model, which introduces the
concept of NRCCs to describe the distribution-side planning flexibility that
can result from allocating additional resources to the distribution system so
as to reduce peak netload uncertainty, and potentially defer transmission-side
investments. Computational experiments on a realistic distribution network
demonstrated the benefits and applicability of the proposed planning approach,
which would require minimal changes to the current regulatory settings. These
benefits include:

  * •

A simple and compact description of distribution-side planning flexibility
through NRCCs, which allow a seamless integration within current transmission
planning workflows for harvesting the benefits of a coordinated T&D planning.

  * •

The provision of a direct incentive for distribution planners to evaluate peak
netload reduction alternatives, in which higher budgets are allocated to the
distribution-side in its role of supporting transmission planning.

  * •

A straightforward and transparent mechanism to capture the benefits of
distribution investments (including NWAs) that can be introduced in the
current regulatory distribution system planning processes, regardless of the
utility ownership mechanisms.

Potential future works could focus on a transmission planning with
consideration of NRCCs curves and quantifying the monetary savings resulting
from this additional source of planning flexibility.

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