# Towards a Sustainable Power Grid: Stochastic Hierarchical Planning for High
Renewable Integration

Semih Atakan<EMAIL_ADDRESS>Department of Industrial and Systems Engineering,
University of Southern California, Los Angeles, CA  Harsha Gangammanavar
<EMAIL_ADDRESS>Department of Operations Research and Engineering Management,
Southern Methodist University, Dallas TX  Suvrajeet Sen<EMAIL_ADDRESS>Department of Industrial and Systems Engineering, University of Southern
California, Los Angeles, CA

( (First version: 09/2019)111The first version was entitled “Operations
planning experiments for power systems with high renewable resources.”)

###### Abstract

Driven by ambitious renewable portfolio standards, large scale inclusion of
variable energy resources (such as wind and solar) are expected to introduce
unprecedented levels of uncertainty into power system operations. The current
practice of operations planning with deterministic optimization models may be
ill-suited for a future with abundant uncertainty. To overcome the potential
reliability and economic challenges, we present a stochastic hierarchical
planning (SHP) framework. This framework captures operations at day-ahead,
short-term and hour-ahead timescales, along with the interactions between the
stochastic processes and decisions. In contrast to earlier studies where
stochastic optimization of individual problems (e.g., unit commitment,
economic dispatch) have been studied, this paper studies an integrated
framework of _planning under uncertainty_ , where stochastic optimization
models are stitched together in a hierarchical setting which parallels the
deterministic hierarchical planning approach that is widely adopted in the
power industry. Our experiments, based on the NREL-118 dataset, reveal that
under high renewable integration, significant operational improvements can be
expected by transitioning to the SHP paradigm. In particular, the
computational results show that significant improvements can be achieved in
several metrics, including system reliability, environmental sustainability,
and system economics, solely by making a strategic choice to adopt the new SHP
paradigm.

##  1 Introduction

Lawmakers throughout the U.S. have mandated that renewable resources meet a
significant percentage of the electricity supply. Each state has set its own
goals, with California being the most aggressive, requiring 50% renewables by
2026, 60% by 2030, and 100% by 2045 (see Proctor, 2018). European countries
have also set ambitious targets both at the level of individual nations, as
well as the European Union as a whole (International Renewable Energy Agency,
2018). State and local authorities (e.g., Independent System Operators (ISOs)
in the U.S. and Transmission System Operators (TSOs) in Europe) have
commissioned studies to assess operational considerations such as system
reliability, market design, incorporation of storage technologies, and other
avenues. A recent simulation study (Olson et al., 2015), commissioned by
California ISO (CAISO), suggests that for renewable-integration levels beyond
33%, one can expect a fair amount of over-generation and renewable curtailment
during the daytime, and perhaps, load-shedding around sundown. Higher levels
of renewable integration exacerbate these issues.

Power systems operations planning processes are mainly based on a hierarchical
strategy. This strategy leverages layers of optimization models solved in
sequence using a rolling horizon approach, to improve the reliability and
economics of power systems operations. Typically, at the top layers of this
hierarchy, long-horizon and low-resolution models are executed to determine
commitment plans for generators with slower-ramping capabilities (e.g.,
nuclear). These plans are then passed to downstream layers of the hierarchy,
where the decision space is gradually refined with higher-resolution models.
The higher-resolution models use more accurate forecasts for demand and
renewable generation to determine the commitment plans of most generators.
Finally, a few minutes before dispatch, the economic dispatch model is solved
to refine the generation plan for the committed generators. Electricity supply
in this setup can be thought of as an aggregation of several products supplied
to the consumer via a “middle-man”, the ISO/TSO. These entities use a
portfolio of power contracts that obligates a producer to provide electricity
at a cleared price during the corresponding time window. The market time
windows are aligned with the operations in a similar manner. For instance, a
low-resolution market in the day-ahead is followed by a higher resolution
real-time market closer to the time of actual dispatch. The operational
elements (generator commitments, generation amounts, etc.) and the market
elements (electricity prices) are outputs of the rolling-horizon optimization
models.

Today’s power system planning process use _deterministic_ optimization models
in all layers of the hierarchy. Such a process can be termed as deterministic
hierarchical planning (DHP). The adoption of DHP for planning power systems
operations predates the introduction of vast amounts of renewable energy into
the grid. The significant levels of volatility of renewable resources pose one
of the major challenges to maintaining reliable supply under DHP. Indeed, a
recent study points to a significant increase in costs to maintain a fully
renewable power system while meeting today’s level of system adequacy (Zappa
et al., 2019).

A popular illustration of the issues with renewable integration is captured by
the so-called “duck-chart” of CAISO (see Fig. 1). This figure depicts the
daily net-load (total electricity load minus generation from “must-run” units)
across successive years with increasing levels of solar added to the
generation mix. A surplus of solar energy during the daytime leads to a dip in
the net load, followed by a significant upward ramp around the sundown. In a
grid with limited storage capabilities, excess supply during daytime poses
significant challenges as utilities will be required to procure sufficient
ramp-up capabilities to meet the electric load during evening hours. The
absence of substantial ramping capabilities can push the loss-of-load
probability to unacceptable levels, and may even cause load-shedding in
certain areas, jeopardizing system reliability and performance. On the other
hand, over-generation during daytime could lead to negative prices in the
market, resulting in, for instance, large shipments of energy to neighboring
states (e.g., from California to Arizona) while paying these states to accept
the surplus (see Penn, 2017). Additionally, it is important to recognize that
Fig. 1 is based on point forecasts and overlooks the uncertainty in demand and
renewable generation. During actual operations, the trajectory of the net-load
is unknown to the operators and can significantly deviate from the point
forecast. Therefore, plans based on deterministic models that use point
forecasts as input will result in immense reliability challenges.

Figure 1: CAISO’s duck chart, predicting four emerging ramping patterns with
increased renewable integration California ISO (2016).

To overcome the challenges of renewable integration and “tame the duck”, so to
speak, a recent U.S. Department of Energy (DOE) report (Bergman et al., 2016)
has distilled a myriad of operational guidelines (for maintaining reliability)
into four specific rules: (1) Power generation and transmission capacity must
be sufficient to meet the peak demand for electricity; (2) Power systems must
have adequate flexibility to address variability and uncertainty in demand and
generation resources; (3) Power systems must be able to maintain a steady
frequency; (4) Power systems must be able to maintain a steady voltage at
various points on the grid. The focus of these rules is on expected changes
over the next several years due to the inclusion of a new generation of
resources, especially variable energy resources (VER). The first two rules
address operations planning, whereas the last two rules pertain to operations
control. It is fair to suggest that technological solutions for operations
control, which permit high renewable integration, are well underway.

As for operations planning, we posit that it is necessary to reassess the
structure of the DHP paradigm to accommodate the volatility associated with
renewable energy that affects every layer of this hierarchy. Such an
assessment involves studying alternative setups that allow combinations of
contemporary (deterministic) planning models and novel (stochastic) planning
models. Given this focus, the main contributions of this paper are as follows.

  1. 1.

Overarching Goal: Comparing Stochastic and Deterministic Hierarchical
Planning. We study the implications of transitioning from DHP towards
_stochastic hierarchical planning_ (SHP) in a system with a centralized
planner. We investigate whether it is possible to fully realize the positive
impacts of VER integration (i.e., reduced greenhouse gas (GHG) emissions)
without deterioration in system reliability and significant increases in
production and distribution costs. The novelty of our work is in an integrated
assessment of combinations of models that are arranged in a hierarchy, and
aware of system volatility due to VERs.

  2. 2.

A Decision Evolution Architecture. The decision processes used in this paper
are tightly connected with the NREL-118 dataset (Pena et al., 2017), which not
only provides the grid data, but also the renewable energy environment for our
experiments. In particular, the evolution of decisions and uncertainty allowed
by this dataset lets us investigate the goal set in item 1. Our architecture
allows us to draw conclusions regarding costs, GHG emissions, load shedding,
etc., based on the specific decision-modeling choices.

  3. 3.

Experimental Study. We perform a comprehensive computational study of the SHP
under different models (deterministic v. stochastic) at individual layers of
the hierarchy, varying renewable integration levels, and system reliability
requirements. Our computational study demonstrates the potential of SHP when
compared to the current DHP strategy and acts as a precursor to an ISO-scale
study.

It is important to emphasize that this study focuses on the NREL-118 dataset,
and while similar studies can be undertaken using other datasets, the
conclusions of this paper pertain only to those power systems which have
similar VER levels and weather patterns. Our thesis is that the choice of the
decision hierarchy of a system should be made in a manner that reflects the
renewable integration levels because systems with high VERs may be better
served using SHP, while those with lower VERs may continue to adopt the DHP
setting. To give the reader a brief preview of our results, it turns out that
the current DHP framework may not be as effective for decision-making under
uncertainty. We present strong evidence that for NREL-118, SHP provides
superior metrics for operational costs, lost-load, as well as GHG emissions.

Studies of using stochastic optimization for power system operations can be
traced back at least forty years to Murphy et al. (1982). While there are many
studies of stochastic optimization within any one layer of the hierarchical
system, no other study pits the standard hierarchy of deterministic models
against a hierarchy of stochastic ones in a system operated by a centralized
planner. It is such a comparison that provides a preview of the potential
advantages and disadvantages of alternative hierarchies (DHP v. SHP) on
electricity production planning. Thus, this study examines whether a system-
wide overhaul that introduces stochastic optimization and coordination among
one or more layers of the hierarchy can mitigate difficulties associated with
high integration of renewable energy into the grid.

The remainder of this paper is organized as follows. We begin with an overview
of today’s power system operations in §2 and discuss relevant literature in
§3. In §4 we present a detailed description of the SHP framework, including
optimization models and solution algorithms used to simulate system
performance. In §5 we present our experimental results. Finally, we conclude
with a brief discussion of how one might proceed to the next phase of tests,
which would require ISO-size networks and technologies in §6. The overall
structure of each model used in the hierarchy is discussed in Appendix A, and
details on solar and wind inputs are given in Appendix B. In the interest of
enhancing reproducibility of our study, the code and data associated with our
experiments are available at USC 3D Lab (2021).

##  2 An Overview of Power Systems Operations Planning

Electric power systems are very large-scale networks interconnecting many
sources of electric power to points of consumption. The entire network is
arranged at several voltage levels, converted from one to the other by step-up
or step-down transformers. Minimizing total cost while ensuring reliable power
delivery to customers is the overarching goal of the system operations. Given
the continuous and very large-scale nature of operations, the implementation
of this objective is complex when viewed as a single decision-making problem.
Therefore, system operators use a reformulation involving a hierarchy of
optimization models defined over overlapping horizons with different time
resolutions for decisions and constraints.

Our hierarchical planning paradigms assume coordination via a central planning
authority (e.g., ISO or TSO). Most ISOs/TSOs currently implement some form of
a DHP framework, which divide the daily planning activities into three
principal layers: a) _day-ahead unit commitment_ based on a daily forecast of
load and generation limits leading to a production and transmission plan, b)
_short-term unit commitment_ over a shorter planning window (typically three
to four hours), producing some commitment decisions and updated transmission
plans, and c) _hour-ahead economic dispatch_ where the production and
transmission plans are finalized and necessary reserve capacities are
committed. There are some variations of this multi-layer hierarchy, such as
updating the dispatching plan in 15-minute intervals to accommodate high
levels of VER. However, all layers in this hierarchical setup adopt some form
of deterministic optimization models, and as such, all forecasts used in these
models are _point forecasts_. In what follows, we give further details on the
three principal layers mentioned above, and provide notes on how our setup
(described in greater depth later) incorporates these layers.

#### Day-ahead (DA) Operations

This phase begins by estimating demand and renewable-supplies, as well as
collecting generation and demand bids via electricity markets. This
information is used in simultaneous co-optimization for the next operating
day, using security-constrained unit commitment (UC) and security-constrained
economic dispatch (ED) models. In most settings (including ours), these models
are formulated over a $24$-hour horizon, with decisions and constraints
specified at an hourly resolution. The UC model commits and schedules
resources for regulation. The amount of resources (mainly spinning operating
reserves) scheduled in DA are based on estimates generated using historical
data, as well as ISO-specific practices. DA planning also involves committing
resources for reliability assessment and emergency operations. For simplicity,
we do not consider these in our setup. The UC optimization model involves
continuous as well as binary decision variables, resulting in a mixed-integer
program (MIP).

The UC decisions are used to instantiate the DA security-constrained ED model.
This model determines generation, regulating and spinning reserve amounts for
all committed resources, as well as ex-ante DA prices. While the ED model in
the DA phase is often solved separately for each hour of the day, we formulate
the ED model as a single optimization model defined over an entire day at an
hourly resolution. In our models, we do not allow for generator self-
scheduling and do not consider system operations under contingency/emergency.

#### Short-term (ST) Operations

Some of the advanced ISOs use additional instruments that commit fast-start
resources to ensure schedules meet all reliability requirements. The
associated models are solved independently against DA transactions and
generation bids. At certain ISOs (e.g., NYISO), these operations are
considered to be part of the real-time (RT) markets and are referred to as the
RT-UC. Following the terminology at CAISO, we will refer to these operations
as Short-Term UC (ST-UC). These models are formulated at finer resolution
(e.g., $15$ mins), allow adaptive (de)commitment decisions, and solved over a
horizon of few hours (e.g., $4.5$ hrs at CAISO and $2.5$ hrs in NYISO). In our
setup, we define these models at a resolution of $15$ minutes and a horizon of
$4$ hours, solved every $3$ hours.

#### Real-time (RT) Operations

There is always some RT deviation of actual generation and load from what was
scheduled during DA planning. One of the key functions of the ISO is to
perform real-time balancing of loads and generation. RT balance is maintained
through a combination of spinning and ancillary services along with the units
providing regulation reserves, which are managed by automatic generation
control (AGC). The non-AGC units are dispatched every few minutes (usually
$5$-$15$ mins), while the regulation units are used only to respond
instantaneously to system imbalances. The planning of dispatch amounts for RT
operations are done using ED models with 5 to 15 minute intervals. Our setup
will consider ED models at a 15-minute resolution with 75-minute horizon, but
set aside tasks associated with downstream models (managing actual dispatch
and transmission control) as being outside the scope of this paper.

##  3 Literature Review

Even with the decomposition of operations planning into DA, ST and RT phases,
generation scheduling and dispatch problems are truly stochastic optimization
problems, therefore they are computationally very challenging. Under a DHP
setting, system operators approximate the random variables with their point
forecasts, converting these stochastic problems into large-scale deterministic
(mixed-integer) problems, which can be handed off to commercial solvers. Note
that electricity production and distribution are governed by nonlinear
equations which can lead to non-convex problems. For the sake of scalability,
we focus on linear approximations of all such non-linear model components
(similar to the literature cited in this section).

Both deterministic UC and ED problems are fundamental to power system
operations planning, and both have been studied extensively in the literature.
Comprehensive treatments of state-of-the-art UC formulations can be found in
Knueven et al. (2019), where the latter also includes a detailed computational
study. The surveys Frank et al. (2012); Chowdhury and Rahman (1990) provide an
overview of deterministic ED models.

Typical UC formulations studied in the literature (such as Ostrowski et al.,
2012; Atakan et al., 2018; Knueven et al., 2019) do not include a transmission
network, and model dispatch at an aggregate level. In the interest of greater
accuracy with transmission constraints, we adopt the dispatch model studied in
Gangammanavar et al. (2015). This formulation involves a linear objective
function that captures production costs and over-generation/load-shedding
penalties, as well as constraints corresponding to generation capacities,
ramping, flow balance, linearized power flow (DC approximation), operating
reserve utilization, bounds on bus angles, and line capacities.

The traditional approach for addressing uncertainty has been, in essence,
over-producing, which creates a reserve of electricity that is used as a
buffer against unexpected volatility. Some ISOs (e.g., New England ISO) have
already recognized the shortcomings of deterministic planning within the
context of renewable integration, and recommend certain deterministic policies
(Zhao et al., 2015). Over the last decades, developments in stochastic
modeling and optimization have allowed uncertainty to be explicitly handled
within the decision process. In particular, stochastic programming (SP) has
played a prominent role to enable decision-making under uncertainty in real-
scale problems across many application domains including power systems
(Wallace and Ziemba, 2005). Two-stage stochastic programs (2-SPs), including
models with discrete first-stage variables, have gained acceptance among both
the power systems research community as well as practitioners.

Stochastic UC models have been studied to better accommodate wind uncertainty
(Ruiz et al., 2009; Papavasiliou et al., 2011); or other sources of
stochasticity such as generator/transmission failures and demand uncertainty
(Zheng et al., 2013; Cheung et al., 2015). For a review paper on stochastic
UC, we refer the reader to Zheng et al. (2015). To account for uncertainty
closer to the time of dispatch, stochastic ED models have been investigated
using stochastic programming (e.g., Gangammanavar et al., 2015; Gu and Xie,
2016) or robust optimization approaches (e.g., Lorca and Sun, 2014).

In addition to deterministic v. stochastic comparisons of individual models,
longer term impact of stochastic models are assessed in Tuohy et al. (2009);
Sturt and Strbac (2012). In contrast to earlier citations, these studies
capture the temporal dynamics of system operations by solving sequences of
stochastic planning models over a rolling time horizon (i.e., model predictive
control as in Lopez et al., 2003). Our setup is similar in this respect, as it
also involves (multiple types of) models solved over a rolling time horizon.
Under (supply) uncertainty, studies were conducted to bridge the gap between
DA planning and RT execution (e.g., Reddy and Bijwe, 2016; Choi, 2018). Both
of these studies leverage deterministic linear optimization models and their
focus is on coordinating DA and RT operations as opposed to assessing the
impact of VER integration.

In contrast, Ilic (2007) calls for considering a hierarchical framework
involving _stochastic_ dynamic problems evolving at different timescales.
While their paper provides high-level (big-picture) assessment, it does not
provide a quantitative assessment. On the other hand, Schulze and McKinnon
(2016) investigate what advantages might accrue from using stochastic
optimization in stochastic DA and intra-day (ST) UC problems over a multi-year
time horizon. However, it is important to highlight some of the differences
between their setup, and ours. First, their study does not consider ED as part
of the planning process, and moreover, their models and data may be described
as a somewhat lower resolution representation which includes a much smaller
set of scenarios in their stochastic programs. The dataset used in our study
(Pena et al., 2017) was specifically introduced by NREL for the purpose of
investigating systems with high VERs. Due to the level of detail provided in
this dataset, as well as our focus on methods which allow a very large
collection of scenarios, our conclusions happen to be different from that
reported by Schulze and McKinnon (2016), whose experiments observed only
modest improvements resulting from stochastic models with a very coarse
description of scenarios and time resolutions.

It is worthwhile to note that prices in power systems are settled through
hierarchically arranged markets which involve generator companies, consumers
(i.e., loads) as well as transmission owners as their participants. Each
participant (e.g., a generator company) bids their price and capacity, and the
ISO/TSO is given the responsibility to clear the market. We emphasize that our
study focuses on operational models which are instantiated by such centralized
planners (e.g., ISO/TSO) using bids submitted via the market. Moreover, our
study inherently assumes competitive markets, risk-neutral market players, and
cost-optimal bid prices for all market participants. We refer the reader to
Conejo et al. (2010) for a detailed treatment of electricity markets under
uncertainty; and Ralph and Smeers (2015) for treatment of risk-aware market
participants.

In the context of markets, there have been recent attempts to capture multiple
layers at different timescales. Dorostkar-Ghamsari et al. (2019) consider the
IEEE-33 instance and adopt a two-layer hierarchy with DA and RT layers.
Aasgard et al. (2019) provide a conceptual study where they argue the benefits
of a multi-market (hierarchical) bidding process (mainly for hydro-
generators). Ottesen et al. (2018) consider an optimal bidding problem over a
three-layer energy market, where stochastic programs under price uncertainty
are adopted at each layer. Finally, Morales et al. (2014) consider a two-layer
energy market, analyze shortcomings of conventional and stochastic market
clearing models and provide a bi-level programming solution.

Despite the significant attention paid to address the challenges of
integrating large-scale renewable resources into power systems, the scope of
the models considered is limited to a single layer in the hierarchy. The only
exceptions are the studies noted above that mostly focus on electricity
markets. Even when 2-SP models are used, the evaluation (using Monte Carlo
simulations or against historical data in a rolling horizon setting) is
limited to the first-stage decisions. In contrast, our setup allows us to
evaluate decisions at all layers considered in the hierarchy. In this regard,
this setup closely mimics the actual operations of a power system. We
reiterate that this is the whole point behind using NREL-118 for our
experimental design.

##  4 Stochastic Hierarchical Planning

In this section, we introduce the details of the SHP framework illustrated in
Fig. 2. We provide an abstract description of SP models used in each
individual layer and defer the detailed description to Appendix A. In the
models that constitute the DHP framework, the expectation-valued objective
functions are replaced by their deterministic counterparts.

[width=1]./figures/operatingFramework

Figure 2: Hierarchical structure of Operating Framework.

###  4.1 The Multilayer Framework

Our setup will capture operations of the power systems across multiple days.
We use three subscripts, $[i]$, $[j]$, and $[k]$, corresponding to the DA-UC,
ST-UC, and HA-ED models, respectively. Following the timescales presented in
Table 1, the subscript $[i]$ is used to index the twenty four (hourly)
decision epochs of the DA-UC model instance corresponding to the $i$th day.
Since a ST-UC instance is solved every three hours, eight ST-UC model
instances are solved on a given day, each with a four-hour horizon. The
subscript $[j]$ is used to index the sixteen ($15$-minute) decision epochs
corresponding to the $j$th ST-UC instance. Finally, since a HA-ED instance is
solved every $15$ minutes, ninety six HA-ED instances are solved on a given
day. For the $k$th instance, the five decision epochs are captured by $[k]$.
To make our notation clear, assume that the operations of the day begin at
12:00 am. The DA operations of the first day will be captured by DA-UC
instance indexed by $i=1$; the ST operations between 9:00 am - 12:00 pm will
be captured by the ST-UC instance indexed by $j=4$; and the RT operations for
10:00 am will be captured by the HA-ED indexed by $k=11$.

Let ${\mathbf{x}}$ and ${\mathbf{y}}$ be the vector variables that model the
generators’ on/off statuses and production levels, respectively. We will use
superscripts $d$, $s$, and $r$ on objective functions, constraint sets, and
parameters associated with DA, ST, and HA, respectively. A subscript
$[\bullet]$ (as in, $z_{[\bullet]}$) is used to refer to the decision vector
corresponding the specific instances. The randomness associated with renewable
supplies and electricity demand is captured by random vectors which evolve
over time. We denote these random vectors by $\tilde{\xi}_{[i]}^{d}$,
$\tilde{\xi}_{[j]}^{s}$, and $\tilde{\xi}_{[k]}^{r}$, for the $i^{th}$,
$j^{th}$, and $k^{th}$ instances of the DA-UC, ST-UC, and HA-ED models,
respectively.

Any model instance is built using optimal solutions obtained by solving
previous model instances at all three levels of the hierarchy. We summarize
these optimal solutions as a state vector $\mathcal{H}_{[\bullet]}$. For
instance, the $j$th instance of ST-UC is built using (i) the optimal solution
of DA-UC instance corresponding to time periods $[j]$, (ii) the solutions of
the previous ST-UC instances, viz. $[j-1],[j-2],$ etc., and (iii) the
solutions of the HA-ED for time periods immediately preceding $[j]$. These
will be collectively referred to as $\mathcal{H}_{[j]}$.

| Horizon | Resolution | Solution Frequency
---|---|---|---
DA | 24 hours | 60 minutes | 24 hours
ST | 4 hours | 15 minutes | 3 hours
RT | 75 minutes | 15 minutes | 15 minutes
Table 1: Model Timescale used for experiments

Given the above notation, for a given day $i$, we define the DA-UC model as a
two stage stochastic program. The first-stage involves commitment decisions
and the second-stage has a security-constrained ED defined over the entire
model horizon. This program can be abstracted as follows:

$\displaystyle\text{DA-
UC}\,\left(\mathcal{H}_{[i]},\,\tilde{\xi}_{[i]}^{d}\right)=\min~{}f^{d}_{[i]}\left(\mathbf{x}_{[i]},\mathbf{y}_{[i]}\right)$
$\displaystyle\text{subject
to:}~{}\left({\mathbf{x}}_{[i]},\,{\mathbf{y}}_{[i]}\right)\in\mathcal{X}^{d}_{[i]}\left(\mathcal{H}_{[i]},\,\tilde{\xi}_{[i]}^{d}\right).$
(1)

Above, the function $\text{DA-UC}\,(\cdot)$ uses the history of the generators
(i.e., $\mathcal{H}_{[i]}$) and a representation of renewable supplies and
demand stochastic process $\tilde{\xi}_{[i]}^{d}$ to determine commitment
schedules for the DA generators. The feasible set of this model is denoted as
$\mathcal{X}^{d}_{[i]}\left(\cdot\right)$, and the expectation-valued function
$f^{d}_{[i]}(\cdot)$ captures the combined commitment and dispatch costs. We
denote the optimal solution of this model as
$(\mathbf{x}^{\star,d}_{[i]},\,\mathbf{y}^{\star,d}_{[i]})$. The optimal
solution acts as an input to the lower level optimization problems, viz. ST-UC
and HA-ED. Since the resolution of the lower level optimization problems is
higher ($15$ minutes) when compared to the DA-UC ($60$ minutes), appropriate
adjustments are made to keep time consistency.

Using the optimal decisions obtained from previously solved model instances,
i.e., $\mathcal{H}_{[j]}$, the $j$th instance of the two-stage stochastic
program for ST-UC is given as follows:

$\displaystyle\text{ST-
UC}\,\left(\mathcal{H}_{[j]},\,\tilde{\xi}_{[j]}^{s}\right)=\min~{}f^{s}_{[j]}\left({\mathbf{x}}_{[j]},\,{\mathbf{y}}_{[j]}\right)$
subject to:
$\displaystyle\left({\mathbf{x}}_{[j]},{\mathbf{y}}_{[j]}\right)\in\mathcal{X}^{s}_{[j]}\left(\mathcal{H}_{[j]},\,\tilde{\xi}_{[j]}^{s}\right),$
$\displaystyle{\mathbf{x}}_{[j]}^{d}={\mathbf{x}}_{[j]}^{\star,d},$ (2a)
$\displaystyle|{\mathbf{y}}_{[j]}^{d}-{\mathbf{y}}_{[j]}^{\star,d}|\leq\epsilon_{j}.$
(2b)

The $\text{ST-UC}\,(\cdot)$ and $\text{DA-UC}\,(\cdot)$ are similar in nature,
except for (2a) and (2b). The former constraint ensures that the DA commitment
decisions are respected in the ST-UC model for generators that participate
only in the DA market, and the latter allows for their generation levels to be
updated only within a bound defined by the parameter $\epsilon_{j}$. Such
bounds are placed to avoid myopic solutions of ST models as they have shorter
horizons than the DA-UC model. The impact of alternative constraints, such as
imposing only an end-state requirement that agrees with the generation amounts
obtained in higher level planning, could also be investigated. In contrast,
the decisions in our ST-UC model are restricted to be within $\epsilon_{j}$ of
corresponding higher layer decisions across all time periods in the model
horizon. Our choice for $\epsilon_{j}$ is to set them to generators’ ramping
limits. This ensures that generators can always ramp up/down to the production
levels that DA-UC recommends, while still being able to deviate from them to a
certain extent.

Using all the commitment decisions and generation levels (captured by
$\mathcal{H}_{[k]}$) prescribed by higher-level UC models, the $k$th instance
of the two-stage HA-ED program is instantiated as shown below:

$\displaystyle\text{ED}\,\left(\mathcal{H}_{[k]},\tilde{\xi}_{[k]}^{r}\right)=\min~{}f^{r}_{[k]}\left({\mathbf{x}}_{[k]},\,{\mathbf{y}}_{[k]}\right)$
subject to:
$\displaystyle({\mathbf{x}}_{[k]},{\mathbf{y}}_{[k]})\in\mathcal{X}_{[k]}^{r}\left(\mathcal{H}_{[k]},\,\tilde{\xi}_{[k]}^{r}\right),$
$\displaystyle{\mathbf{x}}_{[k]}^{d}={\mathbf{x}}^{\star,d}_{[k]},~{}{\mathbf{x}}_{[k]}^{s}={\mathbf{x}}^{\star,s}_{[k]}$
(3a)
$\displaystyle|{\mathbf{y}}_{[k]}-{\mathbf{y}}_{[k]}^{\star,s}|\leq\epsilon_{k}.$
(3b)

The first-stage of HA-ED captures generation for the first decision epoch,
whereas, the second-stage involves the remaining decision epochs in the
planning horizon. Note that, only the first-stage decision (the here-and-now)
decision is implemented. Since the dispatch decisions are fixed in (3a), the
resulting model only has continuous decision variables. As in the case of ST-
UC, the constraint (3b) ensures that the HA generation does not deviate beyond
$\epsilon_{k}$ from the generation decision from the upper layer optimization
problem (ST-UC, in this case). This is done to overcome the myopic nature of
the HA-ED model resulting from its shorter horizon when compared to the UC
models at higher levels of the hierarchy.

The deterministic variants of the models defined above use only point
forecasts $\bar{\xi}_{[\cdot]}$ and the objective function is defined as the
cost associated with both unit commitment decisions and dispatch under such
forecast. The main distinction between the models used in the DHP and SHP
hierarchies is that the objective functions of the latter are defined with a
deterministic first-stage cost and an expected recourse (second-stage) cost.
The recourse cost is the optimal objective of a second-stage optimization
model that is instantiated by the first-stage decisions $x$ and a realization
of the random variable $\tilde{\xi}$.

###  4.2 Modeling and Solving Individual Optimization Problems

The deterministic UC and ED models are solved using MIP and LP algorithms
available in off-the-shelf solvers (e.g., CPLEX, Gurobi). In the 2-SP models
for UC, we use a finite set of scenarios to represent the uncertainty (100
scenarios, to be specific). Even with modest numbers of scenarios, the
resulting deterministic equivalent models could be very large and cannot be
handled by off-the-shelf solvers. Given the recourse problems are LPs and can
be decoupled by scenarios, we use the well-known L-shaped method (also known
as Benders decomposition) to solve the stochastic UC problems (Benders, 1962;
Van Slyke and Wets, 1969). The 2-SP formulation of HA-ED problem has LPs in
both stages. We solve these models using a sequential sampling method called
regularized stochastic decomposition (SD) algorithm (Higle and Sen, 1994).
Table 2 provides a summary of model characteristics and the adopted
algorithmic choices.

Operational | Deterministic | Stochastic
---|---|---
Layer | Problem type | Solver | Problem type | Algorithms
DA-UC | MIP | CPLEX default | Stochastic MIP | Benders/L-Shaped
ST-UC | MIP | CPLEX default | Stochastic MIP | Benders/L-Shaped
HA-ED | LP | CPLEX default | Stochastic LP | Regularized SD

Table 2: Problem class and solution algorithms used for individual instances

The solution quality of an SP is directly influenced by the number of
scenarios considered. We label a solution to an SP model to be statistically-
optimal (for a given confidence level $\alpha$) if the optimal value reported
by the algorithm falls within an acceptable confidence interval computed using
an independent validation process. We refer the reader to Sen and Liu (2016)
for details regarding statistical optimality and the validation process. In
our setup, the SD algorithm can achieve statistical optimality by increasing
its sample size on the fly, whereas a fixed sample size used in stochastic
mixed-integer UC models makes statistical optimality is too demanding a
requirement to impose for the current state-of-the-art.

In our framework, our knowledge of the underlying stochastic process improves
as we get closer to the time of dispatch. This leads to more accurate point
forecasts (from DA to ST, and eventually HA forecasts) of wind and solar
generation amounts. Deterministic models are set up using the point-forecast
time series available just before their execution. On the other hand, the
scenarios used to instantiate and solve stochastic UC and ED problems are
generated using separate time series simulators for solar and wind. The
simulators are based on a vector autoregressive (VAR) model that captures the
spatio-temporal correlation of the stochastic processes governing generator
outputs. The VAR models are trained using the point-forecast time series
available immediately before the execution of the respective stochastic
models. We refer the reader to Appendix B for an extensive discussion on the
prediction and scenario generation aspects of our study.

###  4.3 Evaluating the Performance of Hierarchical Planning Frameworks

We evaluate the performance of both DHP and SHP frameworks by solving the
individual model instances in a rolling horizon fashion over multiple days
against the historical observations of demand and renewable generation (see
Fig. 2). The historical commitment decisions, summarized in
$\mathcal{H}_{[i]}$ with $i=1$, serve as input to our simulation setup. The
DA-UC and ST-UC instances use a representation of the stochastic process that
is built based on the forecast data. On the other hand, the first-stage of HA-
ED uses the actual observations, while the scenarios in the second-stage are
based on forecast available at the time when the model is instantiated. Once
the HA-ED problem is solved, the generation amounts are recorded and all
metrics (such as load shedding, cost) are computed based on these values. The
simulation clock is then moved forward and the next problem (possibly from the
upstream layers of the hierarchy) is solved. Notice that only the HA-ED
problem’s first-stage generation solutions (and the associated commitment
decisions) constitute our final outputs, while the solutions of the UC
problems can only _influence_ these outcomes.

##  5 Experimental Study

The NREL-118 instance includes 327 generators (75 solar and 17 wind), 118
buses, and 186 transmission lines, along with DA forecasts and real-time
outputs of renewable generators and demand. This dataset has a power system
that is rich in solar, wind, and hydro resources, and may be considered as
forward-looking. We assess three factors that have a significant impact on
power system operations. These are solar and wind integration, reserve
requirements, and the planning strategy adopted for handling the UC and ED
problems. By varying these factors, we analyze their impact on certain
reliability metrics, as well as economic and environmental ones, such as unmet
demand, operating costs, and GHG emissions.

In Table 5, we summarize the values considered for these factors in our
experiments. We use a three-letter code to identify specific types of models
(i.e., Deterministic or Stochastic) used within the three-layer planning
hierarchy (DA, ST, RT). Following this notation, the DDD setting is the
benchmark deterministic planning framework (i.e., DHP), whereas both DDS and
SDS can be considered as examples of the SHP framework.

Note that in actual operations, both the ST-UC and HA-ED problems are solved
during the dispatch-day and optimal solutions from both must be computed in
very short time-intervals. This is especially important for ST-UC, as these
problems commit fast-ramp resources to respond to reliability concerns which
may emerge in real time. For the HA-ED solution obtained from stochastic
linear programs, the state-of-the-art algorithms are capable of delivering
statistically-optimal solutions rapidly using very detailed representations of
the stochastic processes. For instance, in our experiments the SD algorithm
that solves the HA-ED instances used a minimum of $256$ simulated multivariate
time series of wind and solar stochastic processes ($75$ solar and $17$ wind
locations resulting in a $92$ dimensional time series). However, the solver
technology for stochastic mixed-integer programs is nowhere near the same
level of maturity to deliver high quality solutions within tight computational
time bounds. On these grounds, academic studies on these models are typically
conducted with only a handful of scenarios, and their solutions are seldom
validated to ensure that the coarse representation of uncertainty indeed
provides a prediction of the optimal objective function which lies within an
upper bound confidence interval of the true SP problem. In contrast, our study
adopts the prudent step of following the standard industry practice today,
namely, solving ST-UC problems as deterministic mixed-integer programs. In
subsequent sections, our experiments will reveal promising results even with a
deterministic ST-UC, providing lower bounds on achievable gains through the
SHP framework with fully-stochastic layers222For a given input state, the
stochastic ST-UC will outperform the deterministic ST-UC due to the well known
non-negativity of the value of stochastic solution (VSS) (Birge and Louveaux,
1997).. This also reinforces the need for further research on two aspects of
renewable integration: a) reasonably accurate probabilistic forecasts of wind
energy over short intervals of time (less than six hours; Carroll et al.,
2018) and b) specialized (fast) algorithms for large scale ST-UC problems
which can deliver near-optimal solutions for real-scale models with binary
(start-up/shut-down) variables. For this reason, our study does not include
the following combinations: ($\bullet$, S, $\bullet$).

Category |  Label |  Description
---|---|---
Solar & Wind Integration |  Low SW: |  Original solar & wind outputs in the NREL-118 dataset.
Med. SW: |  Twice the original values.
High SW: |  Thrice the original values.
Reserve  Requirements |  Very Low: |  5% for UCs, 1.25% for ED.
Low: |  10% for UCs, 2.5% for ED.
Med. |  15% for UCs, 5% for ED.
High: |  20% for UCs, 10% for ED.
Planning Setting |  DDD: |  Deterministic DA-UC, ST-UC, ED.
DDS: |  Deterministic DA-UC, ST-UC; stochastic ED.
SDS: |  Deterministic ST-UC; stochastic DA-UC, ED.

Table 3: Investigated factors and their levels.

In all experiments, we consider a total planning horizon of 7 days, and a
maximum resolution of 15 minutes. Recall that evaluations of the hierarchical
frameworks are carried out in a rolling horizon manner. We assume that the DA-
UC and ST-UC problems commit distinct sets of generators, and ST-UC problems
cannot decommit DA resources (see Appendix A for details). The reported
metrics are based on generator heat rates, fuel consumption rates, and
emission rates given in the NREL-118 dataset, as well as first-period
generation decisions of HA-ED problems that are based on actual observations.
Note that both the deterministic and the stochastic models are evaluated based
on the same actual observations time series. In what follows, we present the
results from these evaluations.

###  5.1 Reliability Impact

Significant amounts of unmet demand revealed in the planning process may
result in blackouts with damaging economic consequences for system
constituents. Due to its importance to ISOs as well as customers, we start our
discussion by presenting the average and maximum unmet demand values in Table
4.

| Avg. Unmet Demand | Max. Unmet Demand
---|---|---
Planning | Reserve | Solar & Wind Integ. | Solar & Wind Integ.
Setting | Req. | Low | Med. | High | Low | Med. | High
DDD | V Low | 1.9 | 4.4 | 17.0 | 320.0 | 595.1 | 956.1
| Low | 0.5 | 3.0 | 1.5 | 246.9 | 336.5 | 274.3
| Med. | 0.0 | 0.4 | 1.2 | 0.0 | 252.6 | 552.5
| High | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
DDS | V Low | 1.2 | 0.7 | 3.9 | 172.0 | 162.8 | 494.3
| Low | 0.0 | 0.0 | 0.5 | 0.0 | 0.8 | 217.8
| Med. | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| High | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
SDS | V Low | 0.0 | 1.9 | 0.6 | 19.2 | 182.0 | 356.6
| Low | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| Med. | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| High | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
Table 4: Average and maximum unmet demand amounts (MW)

In general, we notice higher unmet demand when more solar and wind resources
are introduced to the system with fully deterministic planning (DDD). More
conservative reserve requirements substantially reduce these values, but
possibly comes with an additional monetary cost. On the other hand, adopting
stochastic planning models into the modeling framework can zero out unmet
demand, even at less conservative reserve requirements. For instance, to
completely eliminate unmet demand from the planning process, DDD, DDS, and SDS
necessitate high, medium, and low reserve requirements, respectively, under
Medium SW and High SW settings. This observation supports the use of
stochastic planning models to accommodate the uncertainty associated with VERs
and reduce reliance on (manually-imposed) reserve restrictions. More
importantly, it suggests the possibility of a more economic way of operating
the system.

During our planning process, certain fast-ramping generators can be committed
by the ST-UC problems to recover from unexpected supply shortages during the
day. We evaluate the reliance on ST-UC problems by looking at the percentage
of time that these generators were active (see Table 5). We observe a
consistent trend where the DDD setting heavily relies on ST-UC problems to
maintain reliability. In contrast, DDS and SDS settings substantially reduce
these requirements even under higher renewable-integration settings.

Planning Setting | Low SW | Med. SW | High SW
---|---|---|---
DDD | 11.0 | 11.7 | 11.8
DDS | 9.7 | 8.3 | 7.8
SDS | 8.7 | 6.5 | 5.3
Table 5: Average percentage of time ST-UC generators were active.

Fig. 3 shows the average amounts of over-generation (by conventional
generators) estimated under all settings. Recall that in the absence of
significant storage capacity in the system (such as today), vast swings in
renewable supply may result in over-generation by conventional generators
(Olson et al., 2015). This, in turn, leads to negative electricity prices or
costly exchanges between neighboring ISOs to ensure supply and demand matches
without damaging physical infrastructure or posing financial concerns for
market participants (Penn, 2017).

We observe the smallest over-generation amounts under the DDD setting. This is
not surprising as it would never be optimal to over-produce in a deterministic
optimization model provided that ramping capabilities are sufficient to cover
ramping needs within the model’s horizon. In contrast, stochastic optimization
(i.e., DDS and SDS) compensates for the variability in future time periods by
over-generating in significantly larger amounts, thereby preventing situations
where upwards-ramping capabilities may not be sufficient under certain
settings. This may actually mitigate the Duck-Chart phenomenon discussed in
the introduction.

Figure 3: Average over-generation amounts by conventional generators.

In terms of solar and wind curtailment, Fig. 4 shows a significant trend where
higher renewable integration leads to substantial amounts of curtailment,
providing support for the need to introduce greater energy storage. In
addition, we still observe that both DDS and SDS lead to slightly more
curtailment than that in DDD.

Figure 4: Average curtailed solar and wind energy.

Overall, the outcomes we observed so far suggests that a transition to SHP
leads to reduced loss of load, through conventional over-generation and
renewable curtailment that is dynamically adjusted across the buses based on
the uncertainty lying ahead. This indicates that reliability can be improved
by better planning models, as opposed to increasing the reserve requirements.
In subsequent sections, we will see promising results on the economic and
environmental impacts of operating the network with lower reserve
requirements, which will reveal the contributions of stochastic modeling for
grids with high renewable integration.

Fig. 5 illustrates intra-day generation profiles under DDD and SDS settings.
Notice that the duck-chart is clearly visible. Another phenomenon to notice is
that unmet demand, over-generation, and renewable curtailment may all occur
simultaneously (at different buses), with the first occurring typically at
day-time, when solar generators are active, and transmission capacity
constitutes the bottleneck. This underscores the importance of accounting for
the transmission networks in power system experiments. As seen from the figure
under discussion, SDS leads to more over-generation but reduces unmet demand
from 16.9 MW to 0.7 MW. Furthermore, we observe higher variability in hydro-
based generation under SDS (coefficient of variation of hydro-based generation
is 0.13 in SDS vs. 0.05 in DDD). Hydro generators have better ramping
capabilities, making them better suited to accommodate uncertainty. SDS
naturally leverages this fact seamlessly.

Figure 5: Generation mix, unmet-demand, and over-generation in DDD and SDS
settings (very low reserves, High SW, and a sample day).

###  5.2 Economic Impact

Fig. 6 demonstrates the average daily operating costs recorded in our
experiments. The reported amounts exclude the potential costs associated with
the consequences of operating a grid with low reliability (e.g., penalties
associated with load-shedding). In line with expectations, increased renewable
integration leads to lower costs as these resources have negligible generation
costs. On the other hand, increased reserve requirements have the opposite
effect, since more (and possibly expensive) resources need to be committed to
maintain these requirements. Table 6 illustrates this, where there is a
consistent trend of growing number of committed generators as the reserve
requirements are increased.

Figure 6: Average daily generation cost of the power system under different reserve requirements and operations planning strategies. Reserve Req: | V Low | Low | Med. | High
---|---|---|---|---
DDD | 162.9 | 170.7 | 180.3 | 186.4
DDS | 157.6 | 162.8 | 168.0 | 174.2
SDS | 154.5 | 159.8 | 166.3 | 174.6
Table 6: Average number of committed generators in each framework across
different reserve requirement levels.

Table 6 also reveals another trend. The number of committed generators under
DDS and SDS settings are consistently lower than that of DDD. A particular
reason for this is the number of ST generators that are committed near
dispatch time, which has already been illustrated in Table 5. Overall,
however, we posit that the stochastic models can mitigate concerns over
reliability without committing an abundance of resources.

We now turn our attention to cases where reliability is achieved with the
smallest reserves. Specifically, we look at reserve requirement levels at
which the network demand is seamlessly fulfilled. Table 7 presents the daily
operating costs corresponding to the minimum reserve requirements that must be
set at each level of the hierarchy in order to ensure zero unmet demand. The
cost figures indicate that, with stochastic optimization, reserve requirements
can be relaxed as the models are able to dynamically adjust production levels
by accounting for uncertainty in the future. As a result, the operating cost
of the network can be reduced by up to $10.4\%$ (e.g., compare $\$11.23$M with
DDD to $\$10.06$M with SDS, under high renewable integration). This is a
promising finding as it mitigates financial concerns regarding the use of
higher shares of VER in the grid (e.g., Zappa et al., 2019).

| DDD | DDS | SDS
---|---|---|---
Low SW | 12.42 (Med.) | 11.56 (Low) | 11.60 (Low)
Med. SW | 12.11 (High) | 11.00 (Med.) | 10.66 (Low)
High SW | 11.23 (High) | 10.34 (Med.) | 10.06 (Low)
Table 7: Avg. daily operating cost of the system corresponding to the minimum
reserve requirements leading to zero unmet demand (in million $; Reserve
requirements in parenthesis).

For reserve requirements higher than those cited in Table 7, conflicting
outcomes can be observed. For instance, the operating costs of DDS is better
than that of SDS under medium and high reserve requirements, and high
renewable integration. Clearly, this would not be a suggested manner of
operations, as it costs more compared to what Table 7 reports. However, it
shows that just the stochastic HA-ED can accommodate fluctuations in VERs well
with high-enough reserve requirements. Specifically, the deterministic DA-UC
provides a (short-sighted) cost-optimal generator mix for a given forecast,
and stochastic HA-ED ensures that not many additional ST generators are
necessary during dispatch time (see Table 5). In comparison, the stochastic
DA-UC provides a generator mix that is more conservative, therefore ready for
scenarios that can jeopardize reliability, ensuring the sufficiency of lower
reserve requirements.

###  5.3 Environmental Impact

To assess the environmental impact of increasing renewable energy in the power
system, we estimate the daily GHG emissions using the recorded generation
amounts and mixes. Analogous to Table 7, Fig. 7 demonstrates daily CO2
emission estimates under the minimum reserve requirements that lead to zero
unmet demand. These estimates are based on generators’ heat and emission
rates, which are given in the NREL-118 dataset, as well as their generation
levels, which is determined by the optimization. Similar observations were
made for the NO${}_{\text{x}}$ and SO2 emissions.

Figure 7: Average daily CO2 emissions of the power system corresponding to the
minimum reserve requirements leading to zero unmet demand (reserve
requirements are noted on top of the bars).

Concerning CO2 emissions, we have two observations. First, in the experimented
power system, higher renewable integration leads to lower levels of CO2
emissions. While this sounds intuitive, opponents of this intuition typically
suggest that the duck-chart phenomenon could lead to more emissions. This
increase is attributed to over-generation and reliance on significant amount
of gas-fired fast generators to overcome insufficient ramping capabilities and
volatility of VERs. Our experiments suggest that this is not the case for
power systems with similar characteristics. Second, while stochastic modeling
(i.e., DDS, SDS) also leads to over-generation and renewable curtailment (see
Fig. 3-4), their impact can largely be reversed by the lower reserve
requirements necessary to achieve the same level of reliability. In this
regard, a concurrent optimization-simulation approach to obtain statistically
appropriate measure of reserve requirement is presented in Wang et al. (2020).

##  6 Conclusions and Discussion

We presented the SHP framework for power systems with large-scale VER
integration. While the call for a framework comprising of stochastic dynamic
problems evolving at different timescales has been made before (e.g., Ilic,
2007), this is the first study to conduct comprehensive computational
experiments with such a framework under a centralized planner. Our framework
captures the operations and their interactions across day-ahead, short-term,
and hour-ahead timescales.

The optimization models in the SHP framework are solved using stochastic
programming (SP) algorithms, some of which have been studied rigorously for
over thirty years (Higle and Sen, 1991, 1994) with more recent versions in Sen
and Liu (2016); Sen and Zhou (2014). In this paper, we coalesce our research
from SP, including stochastic mixed-integer programming (Atakan and Sen,
2018), with the work in power systems research for economic dispatch (ED)
(Gangammanavar et al., 2015; Gangammanavar and Sen, 2018) and unit commitment
(UC) (Atakan et al., 2018).

Our experiments indicate that the SHP framework overcomes many of the
shortcomings of the DHP approach currently in practice. We observed that the
SHP framework typically outperforms DHP in terms of reliability: Even at lower
reserve requirements levels, SHP is more effective in eliminating unmet
demand. Moreover, reliance on ST-UC problems (to avoid unmet demand) reduces
by using SHP. On the other hand, the SHP framework leads to more conventional
over-generation and renewable curtailment. The introduction of utility-scale
storage devices can mitigate both of these drawbacks. The considered NREL-118
dataset is a reasonable approximation of networks with high renewable
integration. Remarkably, our study provides hope that the grid of the future
may be able to operate reliably at lower levels of reserve requirements, while
simultaneously reducing both operating costs, as well as GHG emissions. While
our paper has made a strong case for a transition to SHP, we have only begun
the discussion, and there is much to be gained by refining our methods to
incorporate sophisticated SMIP algorithms (e.g., Gade et al., 2014; Yuan and
Sen, 2009). These algorithms are better suited to include start-up/shut-down
variables in the second-stage of stochastic UC models, such as those necessary
for advanced ST-UC modeling. We suspect that specialized versions of these
algorithms may bear fruit in future studies of the SHP framework.

Given the current climate-change concerns, (i) efficient generator designs and
power electronics, (ii) market design, and (iii) optimization software used in
planning and operations can help address the challenges associated with
meeting ambitious renewable portfolio standards. To “tame the duck”, advances
along all three fronts will be critical. Efforts in this paper lay the
groundwork for addressing (iii) through the SHP framework and the use of
stochastic optimization tools. These results point to the next steps that
should involve experiments with actual ISO data. We hope to undertake this
research as part of our future endeavors.

## Acknowledgments

This research was supported by the National Science Foundation (Grant No: CMMI
1822327). We thank the referees for their thorough reading, their
recommendations, and even seeking clarifications which helped bring greater
focus on aspects which are likely to promote greater renewable-energy
integration.

##  7 Detailed Mathematical Models

Sets
---
$\mathcal{B}$: buses. $\mathcal{L}$: transmission lines. $\mathcal{G}$:
generators. $\mathcal{G}_{r}$: solar and wind generators. $\mathcal{G}_{c}$:
conventional generators
($\mathcal{G}_{c}=\mathcal{G}\setminus\mathcal{G}_{r}$). $\mathcal{G}_{j}$:
generators located in bus $j\in\mathcal{B}$. $\mathcal{G}^{DA}$: Slow-ramp DA
generators. $\mathcal{G}^{ST}$: Fast-ramp ST generators. $[t{]}$: time
periods.
Parameters
$G^{\max}_{g}$: generation capacity of $g\in\mathcal{G}_{c}$. $G^{\min}_{g}$:
minimum generation requirement for $g\in\mathcal{G}_{c}$. $\Delta
G^{\max}_{g}$: ramp up limit for $g\in\mathcal{G}_{c}$. $\Delta G^{\min}_{g}$:
ramp down limit for $g\in\mathcal{G}_{c}$. $UT_{g}$: minimum uptime
requirement of $g\in\mathcal{G}_{c}$. $DT_{g}$: minimum downtime requirement
of $g\in\mathcal{G}_{c}$. $B_{ij}$: susceptance of arc $(i,j)\in\mathcal{L}$.
$D_{jt}$: load in bus $j\in\mathcal{B}$, in period $t\in[t]$. $R_{jt}$:
reserve requirement in bus $j\in\mathcal{B}$ and period $t\in[t]$.
$F_{ij}^{\max}$: maximum permitted flow through arc $(i,j)\in\mathcal{L}$.
$c^{s}_{g}$: start up cost of $g\in\mathcal{G}$. $c^{f}_{g}$: no-load cost of
$g\in\mathcal{G}$ (i.e., the intercept of the cost curve). $c^{p}_{g}$:
variable generation cost of $g\in\mathcal{G}$ (i.e., the slope of the cost
curve). $\theta^{\max}_{j}$: upper bound on the voltage-angle at bus
$j\in\mathcal{B}$. $\theta^{\min}_{j}$: lower bound on the voltage-angle at
bus $j\in\mathcal{B}$. $\phi^{o}_{g}$: penalty for over-generation by
$g\in\mathcal{G}_{c}$. $\phi^{c}_{g}$: penalty for renewable curtailment in
$g\in\mathcal{G}_{r}$. $\phi^{u}_{j}$: penalty for unmet demand in bus
$j\in\mathcal{B}$.
Decision Variables
$s_{gt}$: 1 if $g\in\mathcal{G}$ is turned on in $t\in[t]$, 0 otherwise.
$x_{gt}$: 1 if $g\in\mathcal{G}$ is operational in $t\in[t]$, 0 otherwise.
$z_{gt}$: 1 if $g\in\mathcal{G}$ is turned off in $t\in[t]$, 0 otherwise.
$G^{+}_{gt}$: generation amount of $g\in\mathcal{G}$, in $t\in[t]$, which is
consumed by the grid. $G^{-}_{gt}$: over-generation amount by
$g\in\mathcal{G}_{c}$, in $t\in[t]$. $G^{-}_{gt}$: renewable curtailment in
$g\in\mathcal{G}_{r}$, in $t\in[t]$. $F_{ij,t}$: electricity flow through
$(i,j)\in\mathcal{L}$, in $t\in[t]$. $\theta_{jt}$: voltage angle at
$j\in\mathcal{B}$, in $t\in[t]$. $D^{\text{shed}}_{jt}$: amount of unmet load
at $j\in\mathcal{B}$, in $t\in[t]$.
Table 8: Nomenclature for mathematical formulations.

In this section we outline the mathematical models of DA-UC, ST-UC, and HA-ED
problems. We first provide the definition of the decision variables and
describe the constraints, then, refer to these while constructing individual
models. All models are presented for an arbitrary planning interval. Our
notation is summarized in Table 8. Recall that $[t]$ denotes the decision
epochs of the problem being considered.

We denote by $\mathcal{G}$ the set of all generators in the system. This set
includes the both the conventional generators $\mathcal{G}_{c}$ and renewable
generators $\mathcal{G}_{r}$. Furthermore, we assume that only a subset of
conventional generators (typically gas-fired) are capable of providing fast-
start services. We denote this set of generators by
$\mathcal{G}^{ST}_{c}\subset\mathcal{G}_{c}$. Therefore, in our setting we
commit a subset $\mathcal{G}_{c}^{DA}\subset\mathcal{G}_{c}$ in the DA-UC and
the remaining generators are committed in the ST-UC. We assume that ST-UC
cannot decommit DA resources. We also assume that renewable generators are
always committed, however, their production levels can be curtailed.

Generator commitment decisions are often modeled using three sets of binary
variables $(x_{gt}$, $s_{gt}$, $z_{gt})$ that indicate whether $g$ is
operational, turned on, and turned off in period $t$, respectively (Garver,
1962). These variables are linked with the following
constraints333$\mathcal{G}_{c}^{*}$ refers to $\mathcal{G}_{c}^{DA}$ or
$\mathcal{G}_{c}^{ST}$ based on whether the constraint appears in DA-UC and
ST-UC models, respectively.:

$x_{gt}-x_{gt-1}=s_{gt}-z_{gt},\qquad\forall
g\in\mathcal{G}_{c}^{*},\,t\in[t].$ (4)

To model the minimum uptime and downtime requirements of generators, we use
the turn on and off inequalities of Rajan and Takriti (2005):

$\displaystyle\sum_{j=t-UT_{g}+1}^{t-1}s_{gt}\leq x_{gt},\qquad\forall
g\in\mathcal{G}_{c}^{*},\,t\in[t],$ (5)
$\displaystyle\sum_{j=t-DT_{g}}^{t}s_{gt}\leq 1-x_{gt},\qquad\forall
g\in\mathcal{G}_{c}^{*},\,t\in[t].$ (6)

We use two sets of variables to model generation levels. The $G^{+}_{gt}$
variable denotes the amount of electricity produced by generator $g$ in period
$t$ _and consumed by the grid_. The second variable $G^{-}_{gt}$ can assume
two different meanings, depending on the type of the generator. For solar and
wind generators (i.e., $\forall g\in\mathcal{G}_{r}$), these variables capture
the amount of renewable supply that is curtailed, whereas for all other
generators (i.e., $\forall g\in\mathcal{G}_{c}$), they represent the amount of
electricity that is over-generated in period $t$. Here, we assume the
existence of a mechanism that can consume over-generation at the buses where
conventional generators are located in. In more realistic settings, the over-
generated electricity should be accounted for at certain locations where a
consumer (e.g., a neighboring grid or energy-storage facilities) exists. Such
information is not available in the NREL-118 dataset.

All conventional generators (including hydro) must obey certain physical
requirements for attaining feasible production schedules. The generator
capacities and minimum generation requirements are given by

$G_{g}^{\min}x_{gt}\leq G_{gt}\leq G_{g}^{\max}x_{gt},\qquad\forall
g\in\mathcal{G}_{c},\,t\in[t],$ (7)

whereas ramping requirements are modeled as follows:

$-\Delta G_{g}^{\min}\leq G_{gt}-G_{gt-1}\leq\Delta G_{g}^{\max},\qquad\forall
g\in\mathcal{G}_{c},\,t\in[t].$ (8)

Above (and in the ensuing discussion) $G_{gt}$ is used to simplify exposition
and defined as $G_{gt}=G^{+}_{gt}+G^{-}_{gt}$. Ramping constraints (8) can be
strengthened with binary variables to enhance the computational performance of
MIP solvers. Our study incorporated some of the developments made in Damcı-
Kurt et al. (2015) and Atakan et al. (2018). For the purpose of conciseness,
we do not present them in here.

Electricity transmission is modeled using three sets of variables which
represent the electricity flow $(F_{ij,t})$, bus voltage angles
$(\theta_{jt})$, and the amount of unmet demand $(D^{\text{shed}}_{jt})$. We
begin with the flow-balance equations:

$\displaystyle\sum_{i\in\mathcal{B}:(i,j)\in\mathcal{L}}F_{ij,t}-\sum_{i\in\mathcal{B}:(j,i)\in\mathcal{L}}F_{ji,t}+\sum_{g\in\mathcal{G}_{j}}G^{+}_{gt}+D^{\text{shed}}_{jt}=D_{jt}+R_{jt},~{}j\in\mathcal{B},\,t\in[t].$
(9)

Note that the above constraint also involves reserve considerations
$(R_{jt})$, which are modeled as the sum of contingency and regulation
requirements.

We consider linear (direct-current) approximations of power-flows in our
models. These are given, in terms of the bus voltage-angles, as follows:

$\displaystyle F_{ij,t}=B_{ij}(\theta_{it}-\theta_{jt}),$
$\displaystyle\forall(i,j)\in\mathcal{L},t\in[t],$ (10)
$\displaystyle\theta_{j}^{\min}\leq\theta_{jt}\leq\theta_{j}^{\max},$
$\displaystyle\forall j\in\mathcal{B},\,t\in[t].$ (11)

Finally, transmission capacities are given by

$\displaystyle F^{\min}_{ij}\leq F_{ij,t}\leq
F_{ij}^{\max},\qquad\forall(i,j)\in\mathcal{L},\,t\in[t].$ (12)

We use $\tilde{\xi}=\\{\tilde{\xi}_{t}\\}_{t\in[t]}$ to represent stochastic
process corresponding to the solar and wind output. The realization of
$\tilde{\xi}_{t}$ under scenario $s\in\Xi$ is given by $\xi_{t}^{s}$. The
components $\xi^{s}_{gt}$ denote the realization of solar/wind availability
under scenario $s$, for generator $g\in\mathcal{G}_{r}$, in period $t\in[t]$.
For solar and wind generators, the forecast and actual supply time-series are
assumed to be capturing the physical requirements that these generators are
subject to. Accordingly, we only need to impose a solar/wind availability
constraints, which are given as follows:

$\displaystyle G_{gt}^{+}+G_{gt}^{-}=\xi_{gt}^{s}x_{gt},\qquad\qquad\forall
g\in\mathcal{G}_{r},\,t\in[t].$ (13)

This constraint implies that the amount of available renewable generation is
either consumed or curtailed. Note that, since renewable generators are
assumed to be committed, we have $x_{gt}=1$.

In all models, we set $\phi_{g}^{c}=\phi_{g}^{o}=\$25$ per MW $\forall
g\in\mathcal{G}$, and $\phi_{j}^{o}=\$5000$ per MW $\forall j\in\mathcal{B}$.
This setup is inspired by the cost figures discussed in Penn (2017) and Olson
et al. (2015), respectively. Note that conventional over-generation also
involves production cost, therefore our models prioritize load-shedding,
conventional over-generation, and renewable curtailment, in the given order.

###  7.1 Day-ahead and Short-term Unit Commitment

A DA-UC model is solved once every day with a $24$-hour horizon and one hour
time resolution with decision epochs indexed by $[i]$. On the other hand, the
ST-UC model is solved eight times a day with a horizon of four hours and
decision epochs indexed by $[j]$. In either case, the stochastic UC model is
given by:

$\displaystyle\min~{}$
$\displaystyle\sum_{t\in[t^{\prime}]}\sum_{g\in\mathcal{G}_{c}^{*}}\big{(}c_{g}^{s}s_{gt}+c_{g}^{f}x_{gt}\big{)}+\mathbb{E}\big{[}ED(\mathbf{x},\tilde{\xi})\big{]}$
subject to:
$\displaystyle\eqref{con:state_equations}-\eqref{con:minimum_down_requirements},$
$\displaystyle(x_{gt},s_{gt},z_{gt})\in\\{0,1\\}^{3},\quad\forall
g\in\mathcal{G}_{c}^{*},\,t\in[t^{\prime}].$

Here, consolidated decision vector $\mathbf{x}=(x_{gt},s_{gt},z_{gt})_{\forall
g\in\mathcal{G},t\in[t^{\prime}]}$ and

$\displaystyle ED(\mathbf{x},\xi^{s})=\min~{}$
$\displaystyle\sum_{t\in[t^{\prime}]}\Bigg{(}\sum_{g\in\mathcal{G}_{c}^{*}}c_{g}^{v}G^{+}_{gt}+\sum_{j\in\mathcal{B}}\bigg{(}\sum_{g\in\mathcal{G}_{j}\cap\mathcal{G}_{c}^{*}}\phi^{o}_{g}G^{-}_{gt}$
$\displaystyle\qquad\qquad+\sum_{g\in\mathcal{G}_{j}\cap\mathcal{G}_{r}}\phi^{c}_{g}G^{-}_{gt}+\phi^{u}_{j}D^{\text{shed}}_{jt}\bigg{)}\Bigg{)}$
subject to:
$\displaystyle\eqref{con:generation_capacity}-\eqref{con:generation_availability},$
$\displaystyle(G^{+}_{gt},G^{-}_{gt})\in\mathbb{R}_{+}^{2},\qquad\forall
g\in\mathcal{G},\,t\in[t^{\prime}],$ $\displaystyle
F_{ij,t}\in\mathbb{R},\qquad\forall(i,j)\in\mathcal{L},\,t\in[t^{\prime}],$
$\displaystyle\theta_{jt}\in\mathbb{R},\,D^{\text{shed}}_{jt}\in\mathbb{R}_{+},\qquad\forall
j\in\mathcal{B},\,t\in[t^{\prime}].$

The consolidated decision vector $\mathbf{y}$ used in the abstract model in
§4.1 is defined as a concatenation of real-valued decision variables in the
above model. That is, $\mathbf{y}=((G_{gt}^{\pm},)_{\forall
g\in\mathcal{G}},(F_{ijt})_{\forall(i,j)\in\mathcal{L}},(D_{it}^{\text{shed}},R_{it},\theta_{it})_{\forall
i\in\mathcal{B}}))_{\forall t\in[t^{\prime}]}$. Note that the ST-UC model will
additionally include constraints (2a) and (2b) that ensure that the DA-UC
commitments are honored in the ST-UC model and the difference in the
generation amounts is bounded. The latter restrictions are imposed only on the
generation decision variables, i.e., $(G_{gt}^{\pm})_{\forall
g\in\mathcal{G},t\in[t^{\prime}]}$. The second-stage problem captures the
dispatch corresponding to solar/wind availability under scenario $\xi^{s}$
that affects right-hand side of constraint (13). The input $\mathbf{x}$ only
affects constraints (7) and (13).

###  7.2 Hour-ahead Economic Dispatch

Next, we describe the stochastic ED model. The first-stage consists of all the
decisions associated with the first epoch, whereas the decisions for the
remaining epochs in the horizon are made in the second-stage. The resulting
formulation is given below.

$\displaystyle\min~{}$
$\displaystyle\sum_{g\in\mathcal{G}}c_{g}^{v}G^{+}_{g1}+\sum_{j\in\mathcal{B}}\bigg{(}\sum_{g\in\mathcal{G}_{j}\cap\mathcal{G}_{c}}\phi^{o}_{g}G^{-}_{g1}+\sum_{g\in\mathcal{G}_{j}\cap\mathcal{G}_{r}}\phi^{c}_{g}G^{-}_{g1}+\phi^{u}_{j}D^{\text{shed}}_{j1}\bigg{)}$
$\displaystyle\qquad\qquad+\mathbb{E}\big{[}ED^{\prime}((G_{g1}^{\pm})_{\forall
g},\xi)\big{]}$ subject to:
$\displaystyle\eqref{con:generation_capacity}-\eqref{con:line_capacity},\eqref{eqn:ed-
fixedcommitments},\eqref{eqn:ed_generationDeviation}\text{ ($t=1$)},$
$\displaystyle(G^{+}_{g1},G^{-}_{g1})\in\mathbb{R}_{+}^{2},\qquad\forall
g\in\mathcal{G},$ $\displaystyle
F_{ij,1}\in\mathbb{R},\qquad\forall(i,j)\in\mathcal{L},$
$\displaystyle\theta_{j1}\in\mathbb{R},\,D^{\text{shed}}_{j1}\in\mathbb{R}_{+},\qquad\forall
j\in\mathcal{B},$

where,

$\displaystyle ED^{\prime}((G_{g1})_{\forall g},\xi^{s})=\min~{}$
$\displaystyle\sum_{t\in[k]\setminus\\{1\\}}\Bigg{(}\sum_{g\in\mathcal{G}}c_{g}^{v}G^{+}_{gt}+\sum_{j\in\mathcal{B}}\bigg{(}\sum_{g\in\mathcal{G}_{j}\cap\mathcal{G}_{c}}\phi^{o}_{g}G^{-}_{gt}$
$\displaystyle\qquad\qquad+\sum_{g\in\mathcal{G}_{j}\cap\mathcal{G}_{r}}\phi^{c}_{g}G^{-}_{gt}+\phi^{u}_{j}D^{\text{shed}}_{jt}\bigg{)}\Bigg{)}\qquad\qquad\qquad$
subject to:
$\displaystyle\eqref{con:generation_capacity}-\eqref{con:line_capacity},\eqref{eqn:ed-
fixedcommitments},\eqref{eqn:ed_generationDeviation}\text{ (excluding
$t=1$)},$ $\displaystyle G_{gt}=\xi^{s}_{gt}x_{gt},\qquad\forall
g\in\mathcal{G}_{r},\,t\in[k]\setminus\\{1\\},$
$\displaystyle(G^{+}_{gt},G^{-}_{gt})\in\mathbb{R}_{+}^{2},\quad\forall
g\in\mathcal{G},\,t\in[k]\setminus\\{1\\},$ $\displaystyle
F_{ij,t}\in\mathbb{R},\qquad\forall(i,j)\in\mathcal{L},\,t\in[k]\setminus\\{1\\},$
$\displaystyle\theta_{jt}\in\mathbb{R},\,D^{\text{shed}}_{jt}\in\mathbb{R}_{+},\qquad\forall
j\in\mathcal{B},\,t\in[k]\setminus\\{1\\}.$

Within the hierarchical frameworks, note that the HA-ED model also includes
constraints (3a) and (3b) to ensure that upstream commitment decisions are
honored and the differences in the generation amounts are bounded.

##  8 Predicting Solar and Wind Energy

The accuracy of solar and wind forecasts depends on the quality and abundance
of information collected from operation sites, and the ability of the
prediction models to harness them. Modern ISOs/TSOs already receive reliable
streams of data from solar and wind farms, and new mechanisms are being
deployed to monitor behind-the-meter generation (e.g., by rooftop solar
panels; see, for instance, New York ISO, 2017). The prediction models may
directly process physical input (e.g., weather, satellite images) and convert
it into power forecasts, use real-time and historical supply information to
make statistical predictions, or use a combination of both (Monteiro et al.,
2009; Tuohy et al., 2015). The produced forecasts may have a look-ahead
ranging from a few minutes to several days, depending on the nature and
purpose of the forecasting model.

The NREL-118 instance involves DA supply forecasts actual observations for
each wind and solar generator. However, our framework still needs a scheme to
model the distributions of these forecast supply time-series, so that
scenarios can be sampled and used for stochastic programs. Additionally, we
need a mechanism to dynamically update the DA forecasts (and the associated
scenarios). Ultimately, as time progresses, the information gained should lead
to more accurate and precise distributions of the renewable supply. This
suggests, for instance, that the forecast used within the ED model for a given
time period should be more reliable than that used within the DA-UC for the
same time-period.

We address the first necessity with an $m$-th order vector auto-regression
(VAR) model, which can be stated as

$\displaystyle\text{VAR}(m)=\bar{y}_{t+i}=\sum_{j=1}^{m}\Phi_{j}\bar{y}_{t-j+1}+\epsilon_{t+i},\qquad\forall
t,\,i=1,2,\ldots$

where, $\bar{y}_{t}$ is the day-ahead supply forecast for a vector of
generators, $\epsilon_{t}$ is a white-noise process with a time-invariant
positive-definite covariance matrix; and $(\Phi_{j})_{j=1}^{m}$ are
autoregression coefficient matrices. In contrast to a univariate
autoregressive process, a VAR model can capture both temporal and spatial
correlations of the stochastic processes considered. This is especially
important for modeling renewable output, as weather systems can move over
large areas, simultaneously impacting multiple generators. We refer the reader
to Lutkepohl (2007) for further details on multivariate autoregressive
processes, Hering et al. (2015) for more advanced VAR models, and
Gangammanavar and Sen (2018) for a similar application of VAR models to power-
system planning problems. A general discussion incorporating predictive
statistical models in SP appears in Deng and Sen (2021).

In our study, separate VAR models are used for wind and solar generators, both
based on two weeks of normalized past data. For creating scenarios, we sample
from the distribution of the white-noise process $\epsilon_{t}$, and add these
sampled-deviations to the forecasts corresponding to the day ahead point
forecast $(\bar{y}_{t})$. This procedure is repeated from scratch for every
tested day. For alternative scenario generation procedures, the reader is
referred to Rios et al. (2015); Woodruff et al. (2018) for solar, and Morales
et al. (2010) for wind outputs, among others. We used the vars and MTS
packages in R (Pfaff and Stigler, 2018; Tsay and Wood, 2021) for fitting the
VAR models (with a maximum lag of 3) and sampling from the noise
distributions, respectively, where further details can be found in our code
repository USC 3D Lab (2021).

Next, we describe our scheme to capture the evolution of uncertainty across
the hierarchy and time. Note that the accuracy and precision of DA supply
forecasts (and scenarios) typically deteriorate as the prediction-horizon
grows. Therefore, it is crucial to evolve these to prevent unlikely inputs
from becoming inputs to optimization models, thereby leading to low-quality
plans. Moreover, the selected update scheme should be fast and lightweight, as
it will need to be invoked, on the fly, in between optimization model runs. In
this study, we consider the following update rule:

$\displaystyle\tilde{y}_{t+i}=\alpha_{i}\big{(}y_{t}+(\bar{y}_{t+i}-\bar{y}_{t+i-1})\big{)}+(1-\alpha_{i})\bar{y}_{t+i},\qquad\forall
t,\,i=1,2,\ldots$

where, $\tilde{y}_{t}$ and $y_{t}$ are the updated supply forecast and actual
supply for a vector of generators, respectively, and $\\{\alpha_{i}\\}$ is a
sequence of coefficients with $\alpha_{i}\in[0,1],\,\forall i$, and
$\alpha_{i}\searrow 0$. The above equation adds the forecasted increases
$(\bar{y}_{t+i}-\bar{y}_{t+i-1})$ to the most-recently observed generator
outputs $(y_{t})$, and sets the updated forecast as a combination of these and
the original DA supply forecasts $(\bar{y}_{t+i})$. The latter ensures that
the updated forecasts can get close to day-ahead forecast levels, despite
prediction inaccuracies around exact timings (e.g., peak wind output may be
achieved at a different hour than the DA forecast).

In our study, the DA-UC uses DA supply forecasts (or scenarios, if it is a
stochastic model), while ST-UC and ED use updated DA forecasts according to
the above scheme, based on the time period that they are solved. We note that
the first period of the ED model is based on actual observations, as it is
recorded for metric calculations in the computational section. Similar to
solar and wind output, the demand at each bus can be considered as a
stochastic process, and the above models can be used as a basis for decision-
making. However, ISOs/TSOs can often obtain very accurate load forecasts
during the day, and the errors in DA load forecasts are less severe than that
of the renewable resources. Due to this observation, we disregard the
uncertainty associated with the loads in the ST-UC and ED problems. In the DA-
UC problem, however, we consider the DA forecasts of the loads. Finally, when
necessary, we use spline-interpolation to obtain time-series with finer
resolution.

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