# A Computational Efficient Pumped Storage Hydro Optimization in the Look-
ahead Unit Commitment and Real-time Market Dispatch Under Uncertainty

Bing Huang, Arezou Ghesmati, Yonghong Chen, and Ross Baldick Bing Huang,
Arezou Ghesmati and Y. Chen are with Midcontinent Independent System Operator,
Inc. (MISO), Carmel, IN, 46032 USA.R. Baldick is an emeritus professor at the
Department of Electrical and Computer Engineering, University of Texas,
Austin, TX, 78712 USA.This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

###### Abstract

Pumped storage hydro units (PSHU) are great sources of flexibility in power
systems. This is especially valuable in modern systems with increasing shares
of intermittent renewable resources. However, the flexibility from PSHUs,
particularly in the real-time market, has not been thoroughly studied. The
storage optimization in a real-time market hasn’t been well addressed. To
enhance the use of PSH resources and leverage their flexibility, it is
important to incorporate the uncertainties, properly address the risks and
avoid increasing too much computational burdens in the real-time market
operation. To provide a practical solution to the daily operation of a PSHU in
a single day look-ahead commitment (LAC) and real-time market, this paper
proposes two pumped storage hydro (PSH) models that only use probabilistic
price forecast to incorporate uncertainties and manage risks in the LAC and
real-time market operation. The price forecast scenarios are formulated only
on PSHUs that minimizes the computational challenges to the Security
Constrained Unit Commitment (SCUC) problem. Numerical studies in Mid-continent
Independent System Operator (MISO) demonstrate that the proposed models
improves market efficiency. Compared to traditional stochastic and robust unit
commitment, the proposed methods only moderately increase the solving time
from current practice of deterministic LAC. Probabilistic forecast for Real
Time Locational Marginal Price (RT-LMP) on PSH locations is created and
embedded into the proposed stochastic optimization model, an statistical
robust approach is used to generate scenarios for reflecting the temporal
inter-dependence of the LMP forecast uncertainties.

###### Index Terms:

Pumped Storage Hydro Market Integration, Security Constrained Unit Commitment,
Uncertainty and Risk Management.

## Nomenclature

Indices and sets:

$t\in\mathcal{T}$

time $t$ in the set of time intervals;

$g\in\mathcal{G}_{psh}$

unit $g$ in the set of all PSHUs in the system;

$g\in\mathcal{G}_{psh,r}$

unit $g$ in the set of PSHUs that share the same reservoir $r$;

$g\in\mathcal{G}$

unit $g$ in the set of the generating units besides PSHUs in a system;

$s\in\mathcal{S}$

scenario $s$ in the set of probabilistic scenarios;

$r\in\mathcal{R}$

reservoir $r$ in the set of reservoirs.

Data [units]:

$D_{t}$

system net load at period $t$ [MW];

$\eta_{g}^{gen}$

generating efficiency of the PSHU $g$ [NA];

$\eta_{g}^{pump}$

pumping efficiency of the PSHU $g$ [NA];

$E_{r,t_{1}}$

initial energy level of the reservoir $r$ [MWh];

$E_{r,T+1}$

final energy level of the reservoir $r$ [MWh];

$\overline{E_{r}}$

maximum energy level of the reservoir $r$ [MWh];

$\underline{E_{r}}$

minimum energy level of the reservoir $r$ [MWh];

$P_{s}$

weighted probability of scenario $s$ [NA];

$LMP_{g,s,t}^{t_{0}}$

Location Marginal Price forecast made at $t_{0}$ for unit $g$ in scenario $s$
at time $t$ [$/MW];

$Q^{gen,DA}_{g,t}$

generation of PSH unit $g$ at time $t$ in day-ahead solution [MW];

$Q^{pump,DA}_{g,t}$

pumping of PSH unit $g$ at time $t$ in day-ahead solution [MW];

$dT$

interval length [one hour].

Variables [units]:

$e_{r,t}$

continuous variable, energy stored in the reservoir $r$ at time $t$ [MWh];

$e_{r,s,t}$

continuous variable, energy stored in the reservoir $r$ in scenario $s$ at
time $t$ [MWh];

$u_{g,t}$

binary variable, commitment variable of unit $g$ during time interval $t$
[NA];

$p_{g,t}$

continuous variable, amount of generation at unit $g$ during time interval $t$
[MW];

$q_{g,t}^{gen}$

continuous variable, amount of generation at a PSHU $g$ during time interval
$t$ [MW];

$q_{g,t}^{pump}$

continuous variable, amount of pumping load at a PSHU $g$ during time interval
$t$ [MW];

$q_{g,s,t}^{gen}$

continuous variable, amount of generation at a PSHU $g$ in scenario $s$ during
time interval $t$ [MW];

$q_{g,s,t}^{pump}$

continuous variable, amount of pumping load at a PSHU $g$ in scenario $s$
during time interval $t$ [MW];

Auxiliary Variables [units]:

$C(q,u)$

dispatch and commitment cost function of a generating unit [$];

$W_{r}$

robust auxiliary variable [$].

##  I Introduction

###  I-A Background and Motivation

Pumped storage hydro (PSH) can mitigate the increasing variations and
uncertainties in a modern power system. A PSH unit (PSHU) not only has a large
power capacities, but also can switch between modes and ramp very fast. In
addition, a PSHU normally has a large storage capacity and can continuously
charge and discharge in a long duration in comparison to the other available
electric storage technologies. The combination of all of these characteristics
makes a PSHU capable of providing a wide range of services to the electric
grid such as weekly and daily smoothing of loads, spinning reserve and
voltage/frequency control [1]. What is more, the PSHU has been integrated and
operated in the power system for decades with valuable operational
experiences. There are 22.9 GW of PSH generating capacity in the US [2]. Due
to the increasing integration of intermittent renewable resources and the
growing active demand responses, more uncertainties and variations are
introduced to the modern power system. PSHUs can be used to absorb variations
and help system operators to mitigate some of the system uncertainties.
Therefore, it can be critical to efficiently use the PSHUs and leverage their
flexibility in the operation of the power systems.

Due to the increasing uncertainties partially driven by the integration of
renewable energy resources, there are increasing opportunities for storage
units to actively participate in the real-time market operation. However, in
the current Mid-continent Independent System Operator (MISO) system, PSH
participants try to stick to their day-ahead (DA) plan in the real-time (RT)
market to avoid the risk of volatile prices in the RT market. DA and RT
arbitrage revenue have been studied for Pennsylvania-New Jersey-Maryland
Interconnection (PJM) and California Independent System Operator (CAISO) in
[3] and [4] respectively. Both studies conclude that it could be more
beneficial to have storage unit actively participating in both DA and RT
markets. A study in multiple US markets shows that PSH is one of the storage
technologies that have the greatest potential for DA and RT market arbitrage
[5].

A rolling three hours look-ahead commitment (LAC) is applied in MISO system to
assist operators to make commitment decisions in the RT market. Based on the
previous work on the PSH DA model, this paper proposes two PSH models in the
LAC to explore the potential of having the PSHUs optimized in MISO real-time
(RT) market clearing software. It is assumed that the PSHUs participate in the
DA market, establishing a DA financial position, and also participate in the
RT market and can adjust the positions in the first interval of each of the
LAC optimizations. The challenge for including PSH in a LAC formulation is two
folds. First, the three hour LAC window is too short for the the charging and
discharging cycle of a PSHU. Second, the real-time market operation requires
the security constrained unit commitment (SCUC) to be solved in a few minutes.
The key issues are the incorporation of the real-time uncertainties beyond the
LAC window, the appropriate representation of the value of the stored energy
and the efficient stochastic modeling that does not impose major computational
burden.

###  I-B Literature Review

The research in the hydro/pumped storage hydro in the multistage market has
been active globally. The previous research works in the related area can be
summarized in two categories. The first group of work solve the profit
maximization in the RT market or in the joint DA and RT markets and look for
the optimal bidding strategy from the storage market participant stand point.
The second group of work solve the unit commitment and economic dispatch for
the system and leverage the storage units through the optimization.

Different approaches have been used in the first category where the profit
maximization is solved for the storage unit. The DA and RT markets are
considered and jointly solved in several works. A bi-level optimization is
presented in [6] where the RT market is formulated as the lower-level problem.
The conditional value at risk (CVaR) is applied to manage the uncertainty. A
stochastic optimization model is proposed in [7] that shows the significant
improvement of stochastic approach benchmarked with the deterministic
benchmark. In [8], a linear programming approach is proposed to dispatch a
solar-storage unit in DA and RT market. The storage unit is used in the RT
market to respond to the solar and load forecast error from DA. The co-
optimization of energy and regulation reserve as a price taker is studied for
variable speed pumped storage hydropower plants with the Iberian power system
[9]. Bidding strategy in sequential electricity markets has been studied with
the Nordic system [10].

A RT rolling optimization is used in the studies of the operation of the
storage unit in the RT market. In [11], a RT rolling look-ahead optimization
model is proposed for Compressed Air Energy Storage (CAES). The unit
commitment decision made for the CAES in the DA market is enforced in the RT
market. The RT optimization of a storage in the Ontario ISO is studied in
[12]. The price forecast is updated only in a three hour rolling look-ahead
window and the price forecast for the rest of the day is based on the DA price
forecast. In [13], DA and RT rolling profit maximization is investigated. This
work investigated optimal reactive power dispatch in a distribution system. In
[14], based on the realized actual Location Marginal Price (LMP) in RT market,
the pumped storage hydro unit is applied with a policy function to react to
the wind forecast error.

A dynamic programming approach is used to study the wind-storage hybrid model
in the RT market dispatch in [15]. A deterministic LMP forecast is used in the
RT simulation.

Several different approaches have been used in the second group of studies to
leverage the storage units in the RT unit commitment and economic dispatch
problem. In [16], multi-stage deterministic simulation are used to study the
impact of different combination of reserve products provided in the system on
the system costs and the profit of PSHUs.

Stochastic programming model is used to incorporate uncertainties in the
operation of storage units in a RT or close to RT market. The reserve capacity
procurement has been studied for a hydropower scheduling problem with
Norwegian watercourse [17]. A large-scale stochastic programming model that is
used for weekly hydrothermal dispatch and spot pricing of the Brazilian power
system is presented in [18]. In [19], the operation of storage devices is
studied in a rolling stochastic unit commitment model using both single point
and probabilistic forecast on wind and load. In [20], a multistage stochastic
programming formulation is proposed to optimize storage dispatch in RT
operations.

Policy function has been used to address this challenge. In [21], the
operation range of a battery storage in a Look-Ahead Commitment (LAC) is
determined by the stochastic DA unit commitment (UC) solution that is closest
to the realized renewable scenario. Generic multi-stage recourse policies are
investigated for the rolling unit commitment and dispatch problem in [22]. An
affine dispatch policy for renewable and storage units is studied in the
multi-stage unit commitment problem with the Polish system [23].

###  I-C Contributions

There are two research questions this paper tries to answer. The first
question is how uncertainties can be efficiently incorporated in optimizing
PSHU in a RT LAC rolling window without introducing major computational
burdens. The existing literature in the unit commitment and economic dispatch
(UCED) problem point to the stochastic programming direction [19], [20], [22],
[21]. However, it is computationally very expensive to incorporate all the
uncertainties from load and renewable resources and take the input scenarios
in a LAC especially for a large system like MISO. The second question is how
to address the risk of the uncertainties when dispatching a PSH in the RT
market. While many of the previous work on storage optimization consider
uncertainties [7], [14] and [8], only a few address the risk [6] and none of
them considers the risk management from the system stand point.

Although price forecast is used in the study of storage profit maximization
[12], [14], [15], a probabilistic scenario based price forecast hasn’t been
used in a SCUC problem from the system point of view.

The contribution of this paper is summarized below:

  * •

A novel price forecast based stochastic model is proposed for the PSHU LAC
optimization. In the proposed formulation, only PSHU is included in the
periods after the LAC which is much simpler and computationally efficient than
including the full system model for the entire day in each LAC.

  * •

An easier to implement scenario-based price forecast is used in the proposed
PSHU LAC models.

  * •

A novel formulation of robust model that reflects the actual risk-averseness
of PSH owner is proposed for the PSH LAC optimization. It is demonstrated that
market efficiency can be improved even with the risk-averse model.

  * •

The proposed models are prototyped in a MISO system and benchmarked with three
other models including the current practice. The case studies provide
realistic references.

##  II LAC Formulations for PSH

In this section, we propose two models that use probabilistic price forecasts
in the LAC formulation to capture the real-time market uncertainty beyond the
end of the LAC window. It is assumed the unit commitment decision of a PSHU
can be changed in the LAC with no impacts from the DA solution. The DA
commitments for other resources are respected in LAC. The DA dispatch solution
for a PSHU is used only in the robust model as a reference point as is
discussed in section II-B. Notice that the scope of the study is the single
day LAC and RT market. Therefore, both models are focused on the daily
operation of the PSHU in the LAC.

Figure 1: Mode transition diagram of a PSHU in two consecutive periods [24].

The LMP forecast is used to provide guidance to the PSHU in a LAC. LAC in MISO
contains system status and short term load forecast for the next three hours.
However, that is not enough for a PSHU which typically can operate a daily
cycle. The key is to find a good way to reflect the system information from
the future (after the LAC window) to the present (inside the current LAC
window) so that the LAC could optimize the SOC of the PSHU with the rest of
the system while being cognizant of the conditions in the future intervals.
Therefore, we propose to use LMP forecasts in the intervals post to a LAC
window to optimize a PSHU. The LMP forecast methodology is discussed in detail
in section III. In this section, we assume a probabilistic LMP forecast is
available.

The LAC formulations for a PSH is developed based on a configuration-based
modeling of PSHU that represents all feasible operation modes and the state-
of-charge (SOC) of a PSHU as described in [24]. A pumped storage hydro plant
can contain multiple units and each of them will be modeled individually;
however, there are only three operation modes in a PSHU, namely generating,
pumping, and offline, which are mutually exclusive as shown in Fig 1.
Transitions are allowed between each pair of these modes indicated as the
double-headed arrows.

###  II-A Stochastic PSH Model

A two-stage stochastic PSH Model in LAC is proposed in this subsection. The
first stage decisions are the unit commitment and dispatch decisions in a LAC
problem. The second stage decisions are the PSHU commitment and dispatch
decisions in the future intervals that starts from the first interval after a
LAC until the last interval in the day. The commitment and dispatch of the
PSHU in the intervals after the LAC is optimized using the LMP forecast. The
revenue of the PSHU after the LAC window are subtracted in the objective as
shown in the second term in (1). As illustrated in Fig 2, this formulation
combines 1) market wide production cost plus violation cost minimization
within the LAC window that take a deterministic net-load as input and 2) PSHU
profit maximization after the LAC window that takes price forecast scenarios
as input. Notice that, for each LAC window, the problem is formulated and
solved as a single Mixed-Integer linear programming (MILP) model.

Figure 2: Risk-neutral Stochastic PSH LAC Illustration

####  II-A1 Objective Function

The formulation of the stochastic PSH model in LAC is presented in (1).

$\begin{split}&\underset{p,q,u}{\min}[\sum_{g\in\mathcal{G}}\sum_{t=t_{1}}^{t_{end}}C(p_{g,t},q_{g,t},u_{g,t})\\\
&-\sum_{s\in\mathcal{S}}\sum_{t=t_{end}+1}^{T}\sum_{g\in\mathcal{G}_{psh}}P_{s}LMP_{g,s,t}^{t_{0}}(q^{gen}_{g,s,t}-q^{pump}_{g,s,t})].\end{split}$
(1)

The first term in (1) is the objective function for a LAC problem. The
production cost $C(p_{g,t},q_{g,t},u_{g,t})$ is minimized in a LAC window in
intervals that start at $t_{1}$ and end at $t_{end}$. It is assumed, except
for start-up costs, the operation and maintenance cost is negligible for a
PSHU. The second term without the negative sign reflects the expected revenue
from dispatching the PSHU under the forecasted pricing scenarios in the future
intervals post to LAC. Different PSHU generation and pumping values are
allowed in each scenario. It is acknowledged that, strictly speaking,
causality is violated by the implicit assumption that the generation and
pumping values can be chosen for all intervals in a given scenario. With the
weighted probability $P_{s}$, the probabilistic LMP forecast
$LMP_{g,s,t}^{t_{0}}$ is provided for each interval after the LAC and the
forecast is updated at $t_{0}$ that is one interval before the start of each
LAC window $t_{1}$.

####  II-A2 Power Balance Constraints

$\begin{split}\sum_{g\in\mathcal{G}}p_{g,t}+\sum_{g\in\mathcal{G}_{psh}}q_{g,t}^{gen}=D_{t}+\sum_{g\in\mathcal{G}_{psh}}q_{g,t}^{pump},\forall
t\in[t_{1},t_{end}],\end{split}$ (2)

The PSHU is fully optimized within the LAC window given a deterministic
forecast of the demand within the LAC window. In the power balance constraint
within the LAC window $\forall t\in[t_{1},t_{end}]$, the deterministic
generation of the PSHU, $q_{g,t}^{gen}$, is included on the left hand side of
power balance constraint (2) and the deterministic pumping load of the PSHU,
$q_{g,t}^{pump}$, is considered as demand on the right hand side of the power
balance constraint (2).

####  II-A3 The Private Constraints for a PSHU in the LAC

The private constraints of a PSHU model, such as the state transition
constraints, SOC constraints and mutually exclusive constraints, are the same
as the DA model described in (3)-(11),(13),(14) in [24]. The PSHU private
constraints are modeled in the intervals from the start of the LAC window
$t_{1}$ until the end of the operating day $T$. The private constraints for a
PSHU model are deterministic within the LAC intervals and they are defined for
each scenario in the intervals after the LAC. In the following, the SOC
constraints are explained in detail as an example. Notice that the time
interval is taken as hourly in this paper, therefore hourly time interval $dT$
is timed with the generation and pumping capacity in equations (3)-(5). The
rest of the private constraints are formulated similarly. The detailed
description of each of the constraints can be found in (3)-(11),(13),(14) in
[24].

$\begin{split}e_{r,t+1}=e_{r,t}+\sum_{g\in\mathcal{G}_{psh,r}}\eta_{g}^{pump}q_{g,t}^{pump}dT\\\
-\sum_{g\in\mathcal{G}_{psh,r}}\frac{q_{g,t}^{gen}}{\eta_{g}^{gen}}dT,\forall
r\in\mathcal{R},\ \ \forall t\in[t_{1},t_{end}-1],\end{split}$ (3)
$\begin{split}e_{r,s,t_{end}+1}=e_{r,t_{end}}+\sum_{g\in\mathcal{G}_{psh,r}}\eta_{g}^{pump}q_{g,t_{end}}^{pump}dT\\\
-\sum_{g\in\mathcal{G}_{psh,r}}\frac{q_{g,t_{end}}^{gen}}{\eta_{g}^{gen}}dT,\forall
r\in\mathcal{R},\ \ \forall s\in\mathcal{S},\end{split}$ (4)
$\begin{split}e_{r,s,t+1}=e_{r,s,t}+\sum_{g\in\mathcal{G}_{psh,r}}\eta_{g}^{pump}q_{g,s,t}^{pump}dT\\\
-\sum_{g\in\mathcal{G}_{psh,r}}\frac{q_{g,s,t}^{gen}}{\eta_{g}^{gen}}dT,\forall
r\in\mathcal{R},\ \ \forall s\in\mathcal{S},\ \ \forall
t\in[t_{end}+1,T].\end{split}$ (5)

In the intervals within the LAC window, the deterministic SOC constraints are
formulated in (3). The energy stored in the PSH system is linked between every
consecutive time interval. Notice that there can be more than one PSHU sharing
a reservoir in the model. Parameters $\eta_{g}^{gen}$ and $\eta_{g}^{pump}$
are the efficiencies of generating and pumping modes that indicates the energy
loss on both modes. In the intervals after the LAC window, starting at
$t_{end}+1$ until the end of the operating day $T$, the SOC constraints are
formulated for each scenario $s$ in (5). For the inter-temporal SOC
constraint, we need to specifically address the constraint when it crosses
between the interval within a LAC window and the interval after the LAC
window. In (4), the SOC changes from the last interval of the LAC, $t_{end}$,
and the first interval after the LAC, $t_{end}+1$, are defined for each
scenario $s$. Notice that the SOC variable and generation and pumping
variables at the last interval of LAC namely
$e_{r,t_{end}},q_{g,t_{end}}^{gen},q_{g,t_{end}}^{pump}$ are deterministic and
the SOC in the first interval after LAC is defined for each scenario,
$e_{r,s,t_{end}}$. Therefore, by (4), every scenario based SOC variable in the
intervals after LAC is linked to the last deterministic SOC variable within
the LAC.

$e_{r,t_{1}}=E_{r,t_{1}},\ \ \forall r\in\mathcal{R},$ (6)
$e_{r,s,T+1}=E_{r,T+1},\ \ \forall r\in\mathcal{R}\ \ \forall
s\in\mathcal{S},$ (7) $\underline{E_{r}}\leq e_{r,t}\leq\overline{E_{r}},\ \
\forall r\in\mathcal{R},\ \ \forall t\in[t_{1},t_{end}],$ (8)
$\underline{E_{r}}\leq e_{r,s,t}\leq\overline{E_{r}},\ \ \forall
r\in\mathcal{R},\ \ \forall s\in\mathcal{S},\ \ \forall t\in[t_{end}+1,T].$
(9)

The initial SOC in the LAC is given by the previous SOC solution indicated as
$E_{r,t_{1}}$ in (6). The SOC starting point in the first LAC is given by the
PSHU and it is the same as the starting SOC in the DA problem. The SOC
variable at the end of the day, $e_{r,s,T+1}$, is fixed to the given target
$E_{r,T+1}$, that is the SOC at the end of the day in the DA solution, in each
scenario in (7). The end of the day SOC target is calculated from the
historical production data, equation (7), so as to make a fair comparison
between the proposed models and the current practice. The upper and lower
limit are enforced on deterministic SOC variables within LAC and scenario
based SOC variables post to LAC for each scenario in (8) and (9) respectively.

###  II-B Robust PSH Model

Current PSH owners are usually concerned about the exposure to uncertain RT
price. Therefore, they typically stay with the DA positions. To address this
risk-averse concern in the existing practice, a robust risk-management
formulation is developed and it can be applied to the stochastic PSHU LAC
formulation described in section II-A.

####  II-B1 Objective Function and Risk Management Constraint

The formulation of the robust PSH model in LAC is presented in (10) and (11).

$\begin{split}\underset{p,q,u,v}{\min}(\sum_{g\in\mathcal{G}}\sum_{t=t_{1}}^{t_{end}}C(p_{g,t},q_{g,t},u_{g,t})+\sum_{r\in\mathcal{R}}W_{r}),\end{split}$
(10)

$\begin{split}W_{r}\geq&-\sum_{t=t_{end}+1}^{T}\sum_{g\in\mathcal{G}_{psh,r}}LMP_{g,s,t}^{t_{0}}[(q_{g,s,t}^{gen}-q_{g,s,t}^{pump})-\\\
&(Q_{g,t}^{gen,DA}-Q_{g,t}^{pump,DA})],\ \ \forall r\in\mathcal{R},\forall
s\in\mathcal{S}.\end{split}$ (11)

In the risk-management formulation, the objective is updated in (10). Notice
that the first term is the system production costs and that is the same to the
first term in the stochastic objective in (1). The difference is in the second
term of the objective function. In (1), the cost/negative profit of the PSHU
in the intervals after the LAC is weighted by the probability of each
scenario. Therefore, the model presented in (1) is a risk-neutral formulation.
However, in (10), the cost of each PSH plant $r$ is represented by an
auxiliary variable $W_{r}$ which represents the worst case scenario that is
defined in (11). The right-hand side of (11) is the negative profit of the
PSHU in the RT market after the LAC intervals in each scenario. The RT profit
is calculated as RT LMP forecast at each scenario $LMP_{g,s,t}^{t_{0}}$ times
with gen/pump difference between its solution in RT market in a scenario
$(q_{g,s,t}^{gen}-q_{g,s,t}^{pump})$ and the solution in the DA market
$(Q_{g,t}^{gen,DA}-Q_{g,t}^{pump,DA})$. Constraint (11) limits each cost
variable $W_{r}$ to be the cumulative cost of the PSHU from the first interval
after LAC, $t_{end}+1$, to the end of the day, $T$, in the worst-case scenario
(that is the largest cost to the system or the lowest RT profits to the PSHU)
based on the probabilistic LMP forecast. Therefore, since the worst-case PSHU
cost is minimized in the objective, it is a robust or risk averse formulation.
The rest of the stochastic PSHU model remains unchanged from II-A. With the
proposed risk-management formulation in (10) and (11), the solution for a PSHU
will only deviate from the DA solution if it is profitable in every post-LAC
price scenario. Therefore, (10) and (11) address the industry concern of
financial loss in the RT market. As in Section II-A, it is acknowledged that,
strictly speaking, causality is violated by the implicit assumption that the
generation and pumping values can be chosen for all intervals in a given
scenario. It is noted that equation (7) is enforced so as to make a fair
comparison between the proposed models and the current practice. In a
production setting, this constraint can be relaxed and thereby would prevent
the robust model being overly strict.

##  III Price Forecast Methodology

RT LMP is challenging to forecast due to its volatility that is caused by
factors such as generation availability and uncertainty, fuel prices, load
forecast uncertainty, weather condition, and market participant unpredictable
behavior, and that makes RT price forecasting a challenging problem [25]. This
has motivated numerous efforts to provide more information about the
uncertainty associated with the single point price forecast rather than just a
simple statistical summary. A general seasonal periodic regression model with
ARIMA and Fractional ARIMA (FARIMA) is considered in [26]. For model
identification and estimating optimum parameters for Auto-Regressive (AR) and
Moving Average (MA) terms, partial autocorrelation function (PACF) and
Autocorrelation function (ACF) is used in all ARIMA family time series models.
In the context of electricity market, for the purpose of energy system
planning and operations, the need for probabilistic electricity price forecast
(EPF) becomes more prominent. Reference [27] provides a thorough review of
probabilistic electricity price forecasting (EPF). The main approaches
discussed to deal with probabilistic EPF includes creating Prediction
Intervals [28], [29], distribution based probabilistic forecasts [30] [31],
boot-strapped PIs, which is commonly used in neural network EPF studies [32],
and Quantile Regression Averaging (QRA) which combines multiple point
forecasts of individual time series models with the concept of quantile
regression [33]

In this paper we first use a single point forecast namely ARIMAX model for RT-
LMP. Based on the single point forecast, a probabilistic forecast with
statistical scenarios is proposed. The method used for point forecasting is
Auto Regressive Integrated Moving Average with Exogenous variables ARIMAX,
where the idea of using covariates $X$ into the forecasting model is for
increasing the accuracy of forecasting the variable of interest. Since a
single quantile forecast does not provide enough information for most
optimization and decision-making processes, there is a potential economic
benefit from good estimates of the uncertainty associated with RT price
forecasts. The non-parametric probabilistic forecasts is used to cover the
uncertainty range in the prediction of RT-LMP. However, the general form of
probabilistic LMP forecasts do not reflect the interdependence structure of
errors coming from previous time horizons. The non parametric forecasts using
a state-of-the-art statistical scenario generation methodology that considers
the interdependence structure of errors used for wind production [34] is
adopted.

###  III-A ARIMAX- Point Forecast Methodology for RT-LMP

ARIMAX is a type of linear regression model that uses past observation of the
target variable (RT-LMP) along with some covariates called exogenous variables
(Day-ahead-LMP) to forecast future values. In this model, Auto-Regressive (AR)
means the currently observed value of target variable is some linear
combinations of its past values. Moving Average (MA) uses the past errors to
forecast the target variable. Integrated, means that the future changes in the
target variable is linear function of its past change, which are evaluated by
applying a differencing step to data. Given time series data (RT-LMP) and
exogeneous data (Day-ahead-LMP), represented by the variable $x_{(}m,t)$,
where $p$ is the number of auto-regressive lags, $d$ is the degree of
differencing, and $q$ is the number of moving average lags, the point
forecasting model is written as:

$y_{t}=\sum_{i=1}^{p}\phi{{}_{i}}y_{t-i}+\sum_{j=1}^{q}\theta_{j}\epsilon_{t-j}+\sum_{m=1}^{M}\beta_{m}x_{m,t}+\epsilon_{t},\hskip
4.0pt\epsilon\sim N(0,\sigma^{2}),$ (12)

Mean Absolute Percentage Error (MAPE) is used to evaluate the performance of
the ARIMAX model vs ARIMA and Seasonal ARIMA (SARIMAX). Considering one whole
year data as our test set, in 95% of test days ARIMAX outperformed ARIMA and
in 87% of test days ARIMAX surpassed the performance of Seasonal-ARIMAX
(SARIMAX) by reporting lower error number using MAPE as the performance
measurement. For the scenario based probabilistic forecast, the quality is
assessed by how effective the forecast scenarios could guide the PSHU in the
RT market dispatch. Therefore, a profit maximization simulation is used to
assess the quality in comparison to the after the fact LMP in [36]. The
average ratio of profit gained from the price forecast scenarios to the profit
using the after the fact LMP is $13.69\%$ with standard deviation of $25.4$.
In addition, the comparison of the results between the stochastic model and
the perfect LAC model in Table I and Table II in the manuscript can also be
served as a quality assessment for the price forecast scenarios.

###  III-B Probabilistic Forecast in the form of Statistical Scenarios for RT-
LMP

For most optimization operations and decision-making processes, a single
quantile forecast as shown by the thick red line in Fig. 4 is not sufficient
for making an optimal decision for a given time horizon. Assuming $p_{(t+k)}$
is the RT-LMP forecast at time $t+k$. If we do not have a certain assumption
on the shape of distribution function then we can use the results of a
quantile regression model of price data to provide a forecast for Probability
Density Function (PDF) shown in $f_{t+k}$ for any look-ahead time $t+k$.
Therefore, to reconstruct the conditional distributions for the time series
price values at any given time and forecast the Probability Density Function
(PDF) we can directly use the quantiles out of fitting the Quantile-Regression
model as follows:

$\hat{f}_{t+k}=\\{\hat{q}^{\alpha_{i}}_{(t+k)}\hskip
2.0pt0\leq\alpha_{1}\leq\cdots\leq\alpha_{m}\leq 1,\hskip 3.0pt,0\leq m\leq
1.\\}$ (13)

where $\hat{q}_{(t+k)}$ are the quantile functions. then the random variable
$W_{k}$ whose realization $W_{k}^{t}$ at time $t$ is defined by

$W_{k}^{t}=\hat{F}_{t+k}(p_{t+k})\hskip 3.0pt,\forall t$ (14)

is uniformly distributed on the unit interval $U[0,1]$. In order to transform
the variable $W_{k}^{t}$ with uniform distribution to a random variable
$X_{k}^{t}$ with standard normal distribution we use the probit function to
apply the transformation: $X_{k}^{t}=\Phi^{-1}(W_{k}^{t}),\forall t$, where
$\Phi^{-1}$ is the inverse of the Gaussian cumulative distribution function or
probit function.
For any look-ahead time, the vector $X$ as a transformed random vector of
forecasted price in 24-hours look ahead horizon

$X=(X_{1},X_{2},\cdots,X_{24})^{T}\sim N(\mu_{0},\Sigma)$ (15)

follows a multivariate Gaussian distribution where $\Sigma$ is Covariance
matrix and is defined recursively. To estimate the sample covariance Matrix of
the forecast error we follow [34] and use a recursive algorithm, where
$\lambda\in[0,1)$ is the forgetting factor, and the covariance matrix is
initialized by setting it equal to the Identity matrix.

$\sum_{t}=\lambda\sum_{t-1}+(1-\lambda)X^{t}{X^{t}}^{T}$ (16)

Figure 3: Covariance matrix of the multivariate normal random variable RT-LMP,
for one sample day.

In Fig. 3 each pixel shows the covariance of forecast errors between two
different forecast times and the diagonal of the matrix demonstrates the
variance of each RT-LMP random variables. This visualization helps in
estimating and capturing the dependence of forecast errors over the time.
Following the steps of generating statistical scenarios proposed in [34], we
will generate a series of trajectory lines. which collectively represent a
range of potential RT-LMP predictions over the forecast horizon, with
associated probabilities. Fig. 4 shows the associated scenarios reflecting
both the prediction uncertainty and the interdependence structure of
predictions errors.

Figure 4: RT-LMP point forecast and its associated statistical scenarios; The
scenarios show the expected range of forecasting uncertainty.

##  IV LAC Simulation

To implement and test the LMP forecasts and the PSH LAC models described in
the previous sections a high-performance unit commitment software, HIPPO, is
used and further developed to perform the LAC simulations. HIPPO is co-
developed by Pacific Northwest National Laboratory (PNNL), MISO and MIP solver
vendor Gurobi to solve large-scale security constrained unit commitment (SCUC)
and economic dispatch (SCED) problem for a day ahead (DA) market HIPPO [35].
The LAC rolling window simulations is developed based on the existing HIPPO.
The framework of the LAC simulation is included in this subsection.

Figure 5: LAC Framework

The framework of the LAC rolling window simulation in HIPPO is illustrated in
Fig 5. The system variables include the unit commitment and dispatch variables
for generators and PSHUs $\mathcal{Q}$, $\mathcal{U}$, which are defined for
each interval for the entire horizon in study:
$v_{t}=\\{\mathcal{Q},\mathcal{U}\\}$,$\forall t\in T$. The LAC windows are
highlighted by the dashed blue lines in Fig 5. As an example, there are three
intervals included in each LAC window in the figure but the number of
intervals in a LAC window is a parameter and can be changed to any integer
value between $1$ and the total number of intervals $T$.

Although LAC has sub-hour intervals in practice, we perform the LAC simulation
with hourly intervals as a simplification. The hourly intervals allow
straightforward comparison of results with DA solutions and it is easier to
validate. An illustration of the LAC simulation is provided in Fig 5. The
first LAC problem starts at $t_{1}=1$ and it is indicated by the first row of
the boxes representing variables in each of the intervals in Fig 5. After the
first LAC problem is solved, the solutions to the variables of the first
interval inside the LAC window, that is $v_{1}$ written in white font and
highlighted in the box filled with blue background, is saved and set as the
fixed value $V_{1}$ to the variables in interval $1$ in the next and following
LAC problems shown in the dot dashed black circle. The second LAC window
starts at $t_{1}=2$ with the fixed solution from the previous window $V_{1}$.
The LAC window slides forward one interval to $t\in[2,4]$. The LAC simulation
rolls forward one interval at a time in the same way until the last interval
inside the LAC window reaches the last interval of the entire horizon $T$.

##  V Numerical Studies

In this section, first the system and PSHU data used in the LAC simulation is
introduced. Then, four MISO production days are studied with five models
including the stochastic PSH LAC model described in section II-A, the robust
model described in section II-B and three benchmark models are presented.

While it is difficult to exactly reproduce the real-time (RT) system
condition, we assembled the LAC simulation data from the production RT data
and made necessary adjustment. The after-the-fact RT system wide demand,
generator, security transmission constraints for each time interval is updated
in each of the rolling LAC windows. It is noted that approximations are
necessary to resolve some inconsistency and to attain feasible solutions. Two
PSH plants are included in this study. The parameters of the units are matched
with production data. The capacities of the PSHU ranges from a few hundred MW
to GW. Because the confidentiality of the production data, the detailed
parameters of the PSH units cannot be presented in this paper.

###  V-A Current PSH Practice Model

The first benchmark is current PSH practice in the LAC. In current operation,
MISO does not optimize state of charge for PSH in LAC. The PSHU usually stay
with their DA position to avoid risks in the RT market. Therefore, in this
study, the proposed models are first compared to the current practice model.

###  V-B Perfect PSH LAC Model

The second benchmark is the perfect LAC model where the PSHU is fully
optimized in the day. That means all the unit constraints are fully
represented in the system wide optimization including the unit output limits,
ramp limits and SOC limits etc. The end of the day SOC is set to meet the end
of the day target in the DA solution. The after the fact RT system conditions
are known in the perfect model. Therefore, the perfect LAC model should
provide the best performance benchmark for the other models.

###  V-C Deterministic PSH LAC Model

The third benchmark is a deterministic PSH model. The deterministic PSH model
is exactly the same as the stochastic model described in section II-A except
there is only one scenario of price forecast instead of many. The single point
LMP forecast used in the deterministic PSH model is given by the methodology
described in section III-A. This benchmark is developed to study the impacts
of including probabilistic price forecast in the PSH LAC simulation.

###  V-D Simulation Results

TABLE I: System Objective Compared to The Current Practice Model | Day 1 | Day 2 | Day 3 | Day 4
---|---|---|---|---
Perfect LAC | $-0.57\%$ | $-0.13\%$ | $-0.08\%$ | $-0.38\%$
Deterministic | $-0.47\%$ | $+0.16\%$ | $-0.03\%$ | $-0.007\%$
Stochastic | $-0.55\%$ | $+0.1\%$ | $-0.08\%$ | $-0.3\%$
Robust | $-0.33\%$ | $+0.09\%$ | $-0.08\%$ | $-0.29\%$
TABLE II: PSHU RT Profit as Percentage of DA Profit |  Model
---
Day 1 | Day 2 | Day 3 | Day 4
| PSHU1 | PSHU2 | PSHU1 | PSHU2 | PSHU1 | PSHU2 | PSHU1 | PSHU2
Perfect LAC | $+16\%$ | $+9\%$ | $+2.0\%$ | $+0.9\%$ | $+14.76\%$ | $+2.9\%$ | $+19.3\%$ | $+9.5\%$
Deterministic | $+11\%$ | $+7\%$ | $-0.92\%$ | $-2.5\%$ | $+4.0\%$ | $+1.17\%$ | $+1.1\%$ | $+7.3\%$
Stochastic | $+13\%$ | $+9\%$ | $-2.1\%$ | $-1.1\%$ | $+13.6\%$ | $+2.0\%$ | $+2.3\%$ | $+5.6\%$
Robust | $+8\%$ | $+4\%$ | $+0.19\%$ | $-1.6\%$ | $+10.88\%$ | $+4.15\%$ | $+6.8\%$ | $+3.0\%$

Shown in Table I, the system objective value of the studied models are
compared. To show the comparison, the objective value of each of the rest of
the models is compared to the current practice in Table I. Each cell gives the
differences in percentage to the objective value of the current practice. A
negative sign indicates a reduction of objective value compared to the current
practice. Because the PSHU stays with the DA solution in LAC in the current
practice instead of adapting to the RT system condition, it results in sub-
optimality and high objective values. The Perfect LAC model uses the after the
fact RT system information, therefore, as expected, it gives the lowest
objective among all models for each of the studied days. Compared to the
Deterministic model, the Stochastic model gives a lower system objective for
every studied day. While the Robust model gives a higher objective than the
Stochastic model in the Day $1$, the objective values of the two models are
close in the rest of the studied days. It is observed that, except for Day
$2$, the system objective is improved for all models. Due to the
confidentiality of the system data, the actual system objective is not
displayed. The range of the system objective value reduction is from multi-
thousands dollars to multi-tens of thousands dollars.

The PSHU profit realized in each model is presented in Table II. After the
LAC, if the PSHU deviates from its DA position, the PSHU would gain (or lose)
profits from the RT market. The LAC profit is calculated as shown in (17).

$\begin{split}Profits_{g}^{LAC}=&LMP_{g,t}^{RT}[(Q_{g,t}^{gen,LAC}-Q_{g,t}^{pump,LAC})\\\
&-(Q_{g,t}^{gen,DA}-Q_{g,t}^{pump,DA})],\ \ \forall
g\in\mathcal{G}_{psh},\end{split}$ (17)

where $LMP_{g,t}^{RT}$ is the post simulation RT LMP, $Q_{g,t}^{gen,LAC}$ and
$Q_{g,t}^{pump,LAC}$ are the LAC dispatch solution for generation and pump
modes. Because the PSHU stay with the DA solution in the Current Practice
Model, it would not incur any non-zero profits in LAC. Therefore, the profit
results of the Current Practice Model are not presented.

In Table II, the LAC profit of each of the models in a day is shown as a
percentage to the DA profit of the unit in the same day. It can be observed
that the Perfect LAC model always results in positive and the largest profits
across all models in each of the studied days. The profit results from the
Perfect LAC serve as indications of how much room left for the PSHU to adjust
their positions and provide improvement in RT market. In Day $2$, the profit
from the Perfect LAC is significantly less than the other three days. That
gives a tight upper bound of how good each of the models can do in that day.
In Day $3$ and Day $4$, compared to the Deterministic model, the Stochastic
and Robust models generate significantly more profits for both PSHUs. In Day
$2$, while the Stochastic models causes profit loss to both PSHUs, the Robust
model could still generates a small amount of positive profit for PSHU1 and
causes a relatively small profit loss to PSHU2.

Based on the observation from both Table I and Table II, we can conclude that
in the days (Day $1$,$3$,$4$) where there is a larger potential improvement in
RT market, except for the Perfect LAC, the Stochastic model perform
consistently better than all the tested models. The Robust model provide less
but close to the profits from the Stochastic model. In Day $2$ when there is
less potential improvement in the RT market, the Robust model out performs the
other models.

As an example, the post simulation DA LMP and LAC LMP in Day $2$ with less RT
potential improvement and in Day $4$ with more RT potential improvement are
illustrated in Fig. 6. In Day $2$, it can be observed that although the LAC
LMP is lower than the DA LMP, the peak hours and the valley hours are the
same. In comparison, in Day $2$, the evening peak shifted from hour $18$ in DA
to hour $20$ in LAC. More importantly, the lowest LMP occurs at hour $15$ in
LAC instead of hour $2$ in DA. Thus, in the days where the peak and/or valley
hours are different in RT than they are in DA, it opens more opportunity to
the PSHUs to adapt to that changes in LAC.

Figure 6: Simulated DA and LAC LMP in Day 2 and Day 4

The price forecast results made at hour $0$ are illustrated in Fig. 7. The
single point forecast is shown as the bold red line, the after the fact RT LMP
is shown as the bold blue line. The colored thin lines are the probabilistic
price forecast scenarios.

Figure 7: RT Price Forecast Results in Day 4 Figure 8: DA and LAC Dispatch
Results for PSHU2 in Day 4

The dispatch results of PSHU2 in Day 4 from Deterministic, Stochastic and
Robust models are illustrated in Fig. 8. Shown in Fig. 7, the single point
forecast doesn’t predict the lower LMP in the afternoon, therefore the
Deterministic model is not able to lead the PSHU to reduce the pump load in
the early LAC windows. In contrast, in Fig. 7, it can be observed that there
are a good coverage of probabilistic price scenarios in the actual valley time
window around hour $15$. The effect of the probabilistic price forecast is
demonstrated by the Stochastic and the Robust model reducing the pump load
before hour $5$ from its DA position significantly. As a result, the
Stochastic model successfully adjusts the PSHU’s position in the series of LAC
windows and shifts a good amount of pump load from the DA schedule in the
early of the day to the afternoon around hour $15$ where the actual valley of
the day appears. Due to the effect from the risk constraints, the output of
the PSHU in Robust model is pushed closer to its DA position and it doesn’t
pump as much as the Stochastic model does in the actual valley hours. But the
Robust model at least catches the lowest LMP of the day at hour $15$ to pump
to its full capacity.

###  V-E Computational Time and MIP Size

TABLE III: Computational Time of the First LAC window Number of Scenarios | 10 | 20 | 50 | 75
---|---|---|---|---
Stochastic [sec] | $4.4$ | $5.92$ | $11.88$ | $15.38$
Robust [sec] | $6.11$ | $7.52$ | $10.72$ | $16.86$

In this section, we present the computational results with different number of
price forecast scenarios on Day 3. With the proposed PSH models, the first LAC
window includes the largest number of intervals that applies the price
forecast. Thus, the problem size and the computational time of the first LAC
window will be impacted the most by the price forecast scenarios. Therefore,
we only compare the computational time and problem size for the first LAC
window.

The first LAC in current practice that does not use any price forecast is
solved in $3.13$ sec. Shown in Table III, the robust model solves slower than
the stochastic model and both model takes longer time to solve with more
scenarios. Shown in IV, the problem size increase as the result of increased
number of scenarios and that is consistent with the computational time
results. However, even with the $75$ scenarios, both models can be solved
within $20$ seconds and can meet the computational needs for real time LAC.

TABLE IV: Increase Rate on the Problem Size in the First LAC window Number of Scenarios | 10 | 20 | 50 | 75
---|---|---|---|---
Rows | $4.2\%$ | $9.1\%$ | $23.9\%$ | $36.2\%$
Columns | $3.2\%$ | $6.8\%$ | $17.8\%$ | $26.8\%$
Non-zeros | $5.7\%$ | $12.5\%$ | $32.8\%$ | $49.7\%$

##  VI Conclusion

Due to the risk concerns, PSH owners currently avoid active participation in
the RT market. To explore the potential opportunity to enhance market
efficiency by including PSH in the RT market optimization, we propose a novel
and practical Stochastic and a Robust model that uses probabilistic price
forecast to adjust PSHU’s positions in the RT rolling LAC. The proposed model
only requires probabilistic price scenarios for PSHU in the intervals post to
a LAC. This modeling design provides two major benefits. First, by avoiding
taking stochastic scenarios input from load and renewable energy resources, it
is computationally very efficient and does not impose challenges in the real-
time operation. Second, only limited system parameters are required to change
from the current production LAC model and that greatly simplify the potential
testing and implementation.

The findings from the MISO case studies can be summarized in three aspects.
First, using probabilistic price forecast in the RT rolling LAC windows can
help to account for uncertainties in future intervals outside of LAC window
when adjusting PSHU’s charging/discharging positions. Those adjustments can
make significant improvement in the days when the RT system condition deviates
from the forecast in DA. Second, the proposed robust model shows the effect of
risk reduction particularly in the days when the RT system condition aligns
better with the forecast in DA. Third, it is computationally efficient to use
price forecast scenarios for the PSHUs in the LAC model. Both the stochastic
and robust models performs well and can solve within $20$ seconds.

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