# Managing Power Balance and Reserve Feasibility in the AC Unit Commitment
Problem

Robert Parker and Carleton Coffrin  Los Alamos National Laboratory
Los Alamos, New Mexico, USA
<EMAIL_ADDRESS>
###### Abstract

Incorporating the AC power flow equations into unit commitment models has the
potential to avoid costly corrective actions required by less accurate power
flow approximations. However, research on unit commitment with AC power flow
constraints has been limited to a few relatively small test networks. This
work investigates large-scale AC unit commitment problems for the day-ahead
market and develops decomposition algorithms capable of obtaining high-quality
solutions at industry-relevant scales. The results illustrate that a simple
algorithm that only seeks to satisfy unit commitment, reserve, and AC power
balance constraints can obtain surprisingly high-quality solutions to this AC
unit commitment problem. However, a naive strategy that prioritizes reserve
feasibility leads to AC infeasibility, motivating the need to design
heuristics that can effectively balance reserve and AC feasibility. Finally,
this work explores a parallel decomposition strategy that allows the proposed
algorithm to obtain feasible solutions on large cases within the two hour time
limit required by typical day-ahead market operations.

###### Index Terms:

AC power flow, Optimization, Reserve products, Unit commitment

## Nomenclature

### Sets

$T$

Set of time periods $t$

$J$

Set of dispatchable devices $j$

$J^{\rm\\{cs,pr\\}}$

Set of {consuming, producing} devices

$J^{\rm\\{cs,pr\\}}_{i}$

Set of {consuming, producing} devices at bus $i$

$J^{\rm ac}$

Set of AC transmission lines

$I$

Set of buses $i$

$J^{\rm\\{fr,to\\}}_{i}$

Set of AC transmission lines {from, to} bus $i$

$N^{p}$

Set of active reserve zones $n$

$J_{n}^{\rm pr,cs}$

Set of producing and consuming devices in active reserve zone $n$

### Parameters

$d_{t}$

Duration of time period $t$

$p^{\rm min}_{jt}$

Minimum online power of device $j$ at $t$

$p^{\rm max}_{jt}$

Maximum online power of device $j$ at $t$

$p^{\rm ru}_{jt}$

Maximum online ramp-up rate of device $j$ at $t$

$p^{\rm ru,su}_{jt}$

Maximum start-up ramp rate of device $j$ at $t$

$p^{\rm rd}_{jt}$

Maximum online ramp-down rate of device $j$ at $t$

$p^{\rm rd,sd}_{jt}$

Maximum shut-down ramp rate of device $j$ at $t$

$g_{j}^{\rm sr}$

Series conductance of line $j$

$g_{j}^{\rm\\{fr,to\\}}$

“{From, To}”-side conductance of line $j$

$b_{j}^{\rm sr}$

Series susceptance of line $j$

$b_{j}^{\rm\\{fr,to\\}}$

“{From, To}”-side susceptance of line $j$

$b_{j}^{\rm ch}$

Charging susceptance of line $j$

$p^{\rm rgu,req}_{nt}$

Regulation up reserve requirement in zone $n$ at $t$

$p^{\rm scr,req}_{nt}$

Synchronous reserve requirement in zone $n$ at $t$

### Binary variables

$u^{\rm on}_{jt}$

On status of device $j$ at $t$

$u^{\rm\\{su,sd\\}}_{jt}$

{Start-up, Shut-down} status of device $j$ at $t$

### Continuous variables

$p_{jt}$

Dispatched real power of device $j$ at $t$

$p_{it}$

Real power mismatch at bus $i$ at $t$

$q_{it}$

Reactive power mismatch at bus $i$ at $t$

$p_{jt}^{\rm\\{fr,to\\}}$

Real power through the “{from, to}” side of line $j$

$q_{jt}^{\rm\\{fr,to\\}}$

Reactive power through the “{from, to}” side of line $j$

$z_{jt}$

Operating cost of device $j$ at $t$

$v_{it}$

Voltage magnitude of bus $i$ at $t$

$\theta_{it}$

Voltage angle of bus $i$ at $t$

$p^{\rm rgu}_{jt}$

Regulation up reserve provided by device $j$ at $t$

$p^{\rm scr}_{jt}$

Synchronous reserve provided by device $j$ at $t$

$p^{\rm rru,on}_{jt}$

Online ramp-up reserve provided by device $j$ at $t$

$p^{\rm rgu,+}_{nt}$

Regulation up reserve shortfall in zone $n$ at $t$

$p^{\rm scr,+}_{nt}$

Synchronous reserve shortfall in zone $n$ at $t$

##  I Introduction

The unit commitment problem is a power system scheduling task that is solved
in day-ahead and real-time power markets to balance supply with demand at low
cost and ensure feasibility with respect to physical, reliability, and
operational constraints [1]. Due to the size of networks that grid operators
consider (thousands to tens of thousands of buses), discrete commitment
decisions, nonlinear alternating current (AC) physics that must be respected,
and strict time limits within which commitment decisions must be made, solving
the unit commitment problem is a challenging optimization task. The current
standard practice is for market operators to use a direct current (DC)
approximation of the power flow physics, which allows the unit commitment
problem to be formulated as a mixed-integer linear program (MIP) and solved
with established and performant commercial solvers such as CPLEX, Gurobi, and
Xpress [2].

While MIP technology has proven effective in saving hundreds of millions of
dollars annually in U.S. markets compared to previously used less accurate
Lagrangian relaxation approaches [2, 3], MIP-based unit commitment with a DC
approximation requires post-market clearing corrective action to ensure AC
feasibility. In a study on the 24-bus IEEE RTS test network, Castillo et al.
estimate a 1% cost increase due to these corrections compared to the solution
of the unit commitment problem with a full AC power flow model [4]. However,
unit commitment with AC power flow constraints (AC Unit Commitment, or AC-UC)
on industry-scale networks with practical time limits have not been widely
demonstrated.

To investigate whether AC-UC problems can be solved for large-scale networks
within realistic time limits, the U.S. Department of Energy’s Advanced
Research Projects Agency-Energy (ARPA-E) has developed the Grid Optimization
Competition (GOC), in which teams compete to develop solvers for the AC-UC
problem as specified in the GOC Challenge 3 problem formulation [5]. This
problem formulation strives to be representative of modern power market
optimization models and includes unit commitment and ramping constraints, AC
power flow, variable loads, real and reactive reserve products,
startup/shutdown trajectories, line switching, and contingency constraints.
The full optimization problem as written in [5] is a large-scale nonconvex,
multi-period, multi-scenario mixed-integer nonlinear program (MINLP). To
realize the benefits of using this AC-UC problem in real-world electricity
markets, high-quality solutions to large-scale instances of this problem must
be computed within realistic time limits.

Using the Grid Optimization Competition challenge 3 problem as foundation,
this work aims to develop the simplest algorithm capable of obtaining high-
quality solutions to the AC-UC problem for the day-ahead market within the two
hour time limit prescribed by the competition rules. This “benchmark
algorithm” is intended to help inform researchers and market software
providers about what features of the AC-UC problem are necessary for a viable
solution approach. To this end, the results presented in this paper indicate
that: (1) off-the shelf optimization solvers are incapable of solving the full
GOC challenge 3 problem within specified time limits; (2) it is possible with
current optimization methods (e.g., Gurobi and Ipopt) to develop high-quality
heuristics for the AC-UC problem, which can solve industry-scale instances
within the reasonable time limits; (3) one of the key challenges in developing
such heuristics is to manage the competing requirements of the AC power
balance and reserve allocation constraints; (4) problem decomposition and
parallelization across multiple cores is essential to achieving the runtime
requirements in large datasets with thousands of buses.

The next section begins by introducing the core features of the AC-UC problem
that is considered in this work and motivates the aspects of the problem that
make it challenging in practice.

##  II Problem formulation and scale

The unit commitment formulation considered maximizes market surplus by
scheduling dispatchable devices (generators and dispatchable loads) over a 48
hour time horizon (i.e. $T=\\{1,\dots,48\\}$). A simplified version of this
unit commitment problem is given by Equation (1).

$\begin{array}[]{cll}\underset{p_{jt},u_{jt}}{\max}&\displaystyle\sum_{t\in
T}d_{t}\left(\sum_{j\in J^{\rm cs}}z_{jt}-\sum_{j\in J^{\rm
pr}}z_{jt}\right)\\\ \text{s.t.}&\displaystyle\sum_{j\in J^{\rm
cs}}p_{jt}=\displaystyle\sum_{j\in J^{\rm pr}}p_{jt}&\forall t\in T\\\
&\text{Constraints }(\ref{eqn:semicontinuous})-(\ref{eqn:ramp-down})\\\
&z_{jt}=f_{z_{j}}(p_{jt})&\forall t\in T,j\in J\\\
&u_{jt}^{\text{on}},u_{jt}^{\text{su}},u_{jt}^{\text{sd}}\in\\{0,1\\},p_{jt}\in\mathbbm{R}&\forall
t\in T,j\in J\\\ \end{array}$ (1)

This formulation contains a copper-plate real power balance, semi-continuous
real power constraints (2), ramping constraints (3) and (4), and a convex
piecewise-linear function $f_{z_{j}}$ for the operating cost of each device.
Here, binary variables $u^{\rm on}_{jt}$, $u^{\rm su}_{jt}$, and $u^{\rm
sd}_{jt}$ encode a device’s status as either “on”, “starting up”, or “shutting
down”. For example, $u^{\rm on}_{jt}=1$ means that device $j$ is scheduled to
be online at time period $t$. In this case, the device’s real power $p_{jt}$
must be within its bounds $p^{\text{min}}_{jt}$ and $p^{\text{max}}_{jt}$. If
$u^{\rm on}_{jt}=0$, the device’s real power must be zero. Binary variables
for start-up and shut-down are used to encode different maximum ramp rates for
devices in these states, as implemented in (3) and (4). For more information
on unit commitment problems and formulations, see [1, 6, 7, 8].

$u^{\text{on}}_{jt}p_{jt}^{\text{min}}\leq p_{jt}\leq
u^{\text{on}}_{jt}p_{jt}^{\text{max}}~{}\forall t\in T,j\in J\\\ $ (2)
$p_{jt}-p_{j,t-1}\leq\begin{array}[]{l}d_{t}\left(p^{\text{ru}}_{j}\left(u^{\text{on}}_{jt}-u^{\text{su}}_{jt}\right)\right.\\\
~{}~{}\left.+p^{\text{ru,su}}_{j}\left(u^{\text{su}}_{jt}+1-u^{\text{on}}_{jt}\right)\right)\\\
\end{array}~{}\forall t\in T,j\in J$ (3) $p_{jt}-p_{j,t-1}\geq-
d_{t}\left(p^{\text{rd}}_{j}u^{\text{on}}_{jt}+p^{\text{rd,sd}}_{j}\left(1-u^{\text{on}}_{jt}\right)\right)~{}\forall
t\in T,j\in J$ (4)

Binary variables $u^{\rm on}_{jt}$, $u^{\rm su}_{jt}$, and $u^{\rm sd}_{jt}$
are the primary source of discrete variables in the GOC AC-UC problem, and
ramping constraints (3) and (4) are the primary source of intertemporal
linking. The full formulation of this problem can be found in the ARPA-E Grid
Optimization Competition Challenge 3 Problem Formulation [5]. Including all
specifications on input data, this formulation contains over 300 equations and
is omitted here for brevity. The notation in this work is consistent with the
problem formulation, which may be consulted for further reference. The
remainder of this section summarizes key features of the problem formulation
that differentiate it from previous work in the area.

###  II-A AC network constraints

The unit commitment problem in this work considers AC transmission lines and
the associated power flow (derived from Ohm’s law) and power balance (derived
from Kirchhoff’s first law) using the polar power-voltage formulation. The
power flow equations are defined with $i_{j}^{\rm fr}$ and $i_{j}^{\rm to}$
denoting the buses on either side of line $j$. Real and reactive power on each
side of an AC transmission line are given by (5)-(8). For more information on
this power flow formulation, see [9]. The power flow equations are nonlinear,
nonconvex constraints and add significant complexity when incorporated in the
unit commitment problem. The GOC problem formulation also considers DC lines,
transformer branches, and control thereof [5]. The algorithms presented in
this work but keep transformer operating settings fixed but are free to adjust
the power flow along a DC line.

$p^{\rm fr}_{jt}=u^{\rm on}_{jt}\left(\left(g_{j}^{\rm sr}+g^{\rm
fr}_{j}\right){v_{it}^{2}}\right.\\\ +\left(-g_{j}^{\rm
sr}\cos\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right.\left.\left.-b_{j}^{\rm
sr}\sin\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right){v_{it}v_{i^{\prime}t}}\right),\\\
\forall t\in T,j\in J^{\rm ac},i=i_{j}^{\rm fr},i^{\prime}=i_{j}^{\rm to}$ (5)
$q^{\rm fr}_{jt}=u^{\rm on}_{jt}\left(\left(-b_{j}^{\rm sr}-b_{j}^{\rm
fr}-b^{\rm ch}_{j}/2\right){v_{it}^{2}}\right.\\\ +\left(b_{j}^{\rm
sr}\cos\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right.\left.\left.-g_{j}^{\rm
sr}\sin\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right){v_{it}v_{i^{\prime}t}}\right),\\\
\forall t\in T,j\in J^{\rm ac},i=i_{j}^{\rm fr},i^{\prime}=i_{j}^{\rm to}$ (6)
$p^{\rm to}_{jt}=u^{\rm on}_{jt}\left(\left(g_{j}^{\rm sr}+g^{\rm
to}_{j}\right){v_{i^{\prime}t}^{2}}\right.\\\ +\left(-g_{j}^{\rm
sr}\cos\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right.\left.\left.-b_{j}^{\rm
sr}\sin\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right){v_{it}v_{i^{\prime}t}}\right),\\\
\forall t\in T,j\in J^{\rm ac},i=i_{j}^{\rm fr},i^{\prime}=i_{j}^{\rm to}$ (7)
$q^{\rm to}_{jt}=u^{\rm on}_{jt}\left(\left(-b_{j}^{\rm sr}-b_{j}^{\rm
to}-b^{\rm ch}_{j}/2\right){v_{i^{\prime}t}^{2}}\right.\\\ +\left(b_{j}^{\rm
sr}\cos\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right.\left.\left.-g_{j}\sin\left(\theta_{it}-\theta_{i^{\prime}t}\right)\right){v_{it}v_{i^{\prime}t}}\right),\\\
\forall t\in T,j\in J^{\rm ac},i=i_{j}^{\rm fr},i^{\prime}=i_{j}^{\rm to}$ (8)

Real and reactive power balance equations are given by (9) and (10). Bus power
variables $p_{it}$ and $q_{it}$ are mismatch slack variables, or power balance
violations, that are penalized in the objective function. In this way, the
power balance equations are soft constraints that always have a feasible
solution.

$\sum_{j\in J_{i}^{\rm cs}}p_{jt}+\sum_{j\in J_{i}^{\rm fr}}p_{jt}^{\rm
fr}+\sum_{j\in J_{i}^{\rm to}}p_{jt}^{\rm to}=\sum_{j\in J_{i}^{\rm
pr}}p_{jt}+p_{it}\\\ \forall t\in T,i\in I$ (9) $\sum_{j\in J_{i}^{\rm
cs}}q_{jt}+\sum_{j\in J_{i}^{\rm fr}}q_{jt}^{\rm fr}+\sum_{j\in J_{i}^{\rm
to}}q_{jt}^{\rm to}=\sum_{j\in J_{i}^{\rm pr}}q_{jt}+q_{it}\\\ \forall t\in
T,i\in I$ (10)

In the GOC problem formulation [5] the AC power flow constraints also include
shunt devices which are controlled via a collection of discrete “steps”.
However, shunt steps are considered fixed by all algorithms in this work, so
these terms are omitted for brevity.

###  II-B Reserve products

To ensure a power network has sufficient capacity to balance small deviations
in supply and demand and to provide backup in case of an unexpected
disturbance or shut-down, dispatchable devices provide reserve products in
addition to actual (i.e., planned) generation or consumption [10, 11]. The
unit commitment formulation considered in this work contains reserve products
for both real and reactive power capacity of all dispatchable devices, which
includes both generators and loads. Reserve products are categorized as “up”
or “down” according to whether they are provided by a device’s capacity to add
or remove net power from the system. Reserves must be satisfied within active
and reactive reserve zones, each of which partition the dispatchable devices
based on which network buses they are connected to. The reserve requirements
for each zone are implemented as soft constraints with a slack variable that
is penalized in the objective if the requirement is not satisfied. The reserve
balance equations for regulation up reserve and synchronous reserve are given
as examples in (11) and (12). Reserves for each device are constrained by the
“headroom” to the device’s upper or lower bounds depending on the type of
reserve and the type of device. For example, regulation up, synchronous, and
online regulation ramp-up reserves are constrained by the headroom to a
producer’s upper bound, as in (13), or the headroom to a consumer’s lower
bound, as in (14). In all, the GOC problem formulation considers eight active
reserve products and two reactive reserve products. In this work the vectors
$p^{\rm RES}$ and $q^{\rm RES}$ are used to denote an assignment of all active
and reactive reserve products for all dispatchable devices across all time
periods. Higher-quality reserve products, such as regulation up, can be used
simultaneously for lower-quality products such as synchronized reserve, as
implemented in (11) and (12).

$\sum_{j\in J_{n}^{\rm pr,cs}}p^{\rm rgu}_{jt}+p_{nt}^{\rm rgu,+}\geq p^{\rm
rgu,req}_{nt},\hskip 5.69046pt\forall t\in T,n\in N^{p}$ (11) $\sum_{j\in
J_{n}^{\rm pr,cs}}\left(p_{jt}^{\rm rgu}+p_{jt}^{\rm scr}\right)+p^{\rm
scr,+}_{jt}\geq p_{nt}^{\rm rgu,req}+p_{nt}^{\rm scr,req},\\\ \forall t\in
T,n\in N^{p}$ (12) $p+p^{\rm rgu}_{jt}+p^{\rm scr}_{jt}+p^{\rm
rru,on}_{jt}\leq p^{\rm max}_{jt}u^{\rm on}_{jt},\hskip 5.69046pt\forall t\in
T,j\in J^{\rm pr}$ (13) $p-p^{\rm rgu}_{jt}-p^{\rm scr}_{jt}-p^{\rm
rru,on}_{jt}\geq p^{\rm min}_{jt}u^{\rm on}_{jt},\hskip 5.69046pt\forall t\in
T,j\in J^{\rm cs}$ (14)

In addition to penalties incurred if reserve requirements are not satisfied,
devices may incur additional cost by providing reserves. As reserve
constraints are linear and reserve variables are continuous, the problem of
balancing reserve shortfall penalties with device reserve costs may be modeled
as a linear program (LP) for fixed power levels and commitment decisions $p$,
$q$, and $u$. While these constraints are not complicated individually, the
sheer number of constraints and variables required to encode the full reserve
model (five headroom constraints and ten reserve variables per device) adds
significant complexity to the AC-UC problem.

###  II-C Contingency constraints

Security constraints are an important feature of electric power markets, and
may be incorporated into Unit Commitment problems [12] or Optimal Power Flow
problems [13]. The GOC AC-UC problem considers security constraints that
penalize branch flow thermal limit violations in a defined set of
contingencies, where each contingency is defined as the loss of a single
specified branch in the network. While solutions computed in this work are
evaluated in the context of these contingency constraints, the algorithms
presented do not account for them as they did not contribute to significant
penalties in the objective function value.

###  II-D Problem scale

Early approaches to AC Unit Commitment applied Lagrangian relaxation [14] and
Benders decomposition [15] to problems defined on the IEEE 118-bus network. Fu
et al. have considered a security-constrained version of this problem on the
same network [16] using an augmented Lagrangian and Benders decomposition
approach. Castillo et al. apply an outer approximation method to solve AC-UC
problems on networks of up to 118 buses and compare with a method that employs
DC unit commitment followed by an AC feasibility solve [4], while Tejada-
Arango et al. benchmark a direct mixed integer nonlinear programming (MINLP)
formulation, an approach based on sequential linear programming, and a second-
order conic programming formulation [17]. These previous works on the AC-UC
problem have applied exact local MINLP algorithms to this problem and have
only demonstrated scalability up to power networks of on the order of 100
buses and 24 time points, which is far from the size desired by industrial
practitioners. Section IV-A investigates the scalability of the Knitro [18]
MINLP solver on AC-UC problems and confirms that large-scale instances are out
of reach. This work considers AC-UC problems on the Grid Optimization
Competition ranging from 73 buses to more than 8,000 buses with 48 time
points, and approaches solving such large problems by decomposing the full AC-
UC problem into several subroutines and parallelizing these tasks across
multiple cores.

##  III Methods

The decomposition algorithms that are developed in this work rely on several
subroutines, each focusing on different aspects of the complete AC-UC problem.
This section introduces these subroutines then describes how they are combined
into four distinct heuristic solution approaches.

###  III-A Subroutines

####  III-A1 Copper-plate scheduling

The unit commitment approach taken is to model the full-horizon scheduling
problem as a MIP. For tractability, the scheduling problem considers only
copper-plate real and reactive power balances. That is, no network model is
considered. Only total demand and generation must match, as if all devices are
connected to a single conductive plate. The full zonal reserve requirement
model is considered in this subproblem as well. The outputs of this subproblem
are discrete commitment decisions $u$, real and reactive power estimates $p$
and $q$, and real and reactive reserve commitments $p^{\rm RES}$ and $q^{\rm
RES}$.

####  III-A2 AC optimal power flow (AC-OPF)

This work achieves AC feasibility by solving an AC optimal power flow (AC-OPF)
problem at individual time periods $t$. Each subproblem has the discrete
commitment decisions fixed and only considers devices that are online in time
period $t$. Device costs are convex piecewise-linear cost functions modeled
with the $\Delta$ formulation described by [19]. In this work, shunts and
transformer taps are fixed to their operating conditions at the initial point.
To keep AC-OPF problem sizes manageable, reserve constraints are not
considered by these subproblems. Incorporating these constraints would add ten
additional variables and five additional inequality constraints per
dispatchable device and several additional variables and inequality
constraints per reserve zone, nearly doubling the size of the model.

####  III-A3 Reserve allocation

While reserve products and zonal reserve requirements are incorporated in the
unit commitment model, they are not explicitly considered by the AC-OPF model.
For this reason, care is taken to reallocate feasible reserves after the
commitment schedule and AC-feasible power dispatch is generated. The
algorithms discussed in Section III-B differ primarily in how they allocate
reserve products based on the methods discussed here.

Greedy reserve allocation, the first method considered for allocating
reserves, is a simple greedy procedure that, for every dispatchable device,
attempts to allocate the headroom to the device’s upper and lower bounds to
the appropriate reserve products, starting with the highest value products. In
this approach, the cost of providing a reserve and the zonal reserve
requirements are not considered. That is, reserve is assigned for every device
possible even if doing so is not necessary to satisfy the requirements. This
is motivated by the observation that, in most cases, the penalty for failing
to meet a reserve requirement is much larger than the cost of providing the
reserve, which is often zero.

Tighten device bounds, the second method considered, fixes real reserve
products computed by the copper-plate unit commitment schedule on all devices
where doing so does not lead to a guaranteed local power balance violation on
the device’s bus. A local power balance violation occurs if, for example, a
lone producing device on a bus is scheduled to provide more regulation down
reserve than the sum of upper bounds on real power that can be transmitted
from adjacent lines. For devices where this type of violation does not occur,
bounds are tightened on device real power variables $p$ to guarantee that the
reserves committed by the unit commitment problem are feasible after the AC-
OPF subproblems. If a large number of devices provide reserves, this method
can become overly restrictive for the AC-OPF subproblems and can lead to power
balance violations. For this reason, a variation of this method is considered
where only some fraction $\gamma$ of devices providing real power reserve have
their bounds tightened. When $\gamma<1$, devices are sorted by the value of
reserve that they provide and the top $\gamma$ of them are constrained in the
AC-OPF subproblems. This method can be also be used in conjunction with post-
AC-OPF reserve allocation.

Reserve re-dispatch via linear programs, the final method, computes reserve
products post-AC-OPF by fixing commitment decisions $u$ and dispatched real
and reactive power levels $p$ and $q$ by solving the reserve model (zonal
reserve balances, headroom constraints, costs, and penalties) as a linear
program (LP). These LPs are independent at each time period and are solved in
parallel.

###  III-B Decomposition algorithms

The subroutines described in Section III-A are combined into the follow four
algorithms for solving the full AC-UC problem.

Algorithm 1 is a simple decomposition designed to mimic a basic version of
current industrial practice. A commitment schedule $u$, along with nominal
real power $p$, reactive power $q$, and real reserve products $p^{\rm RES}$,
is computed via the copper-plate unit commitment problem described in Section
III-A1. Co-optimization of commitment decisions with real power reserves is
chosen to mimic a joint reserve scheduling market [20]. Real reserve products
are then fixed, imposing tightened bounds on real power. AC-OPF subproblems
are solved sequentially, with a bound-tightening step before each solve to
ensure feasibility of ramping constraints. Reactive power reserve products are
allocated after solving AC-OPF subproblems by the greedy strategy described in
Section III-A3.

1: Initialize: $p,q,u\leftarrow$ initial status

2: $p,q,u,p^{\rm RES}\leftarrow$ Copper-plate unit commitment

3: Tighten bounds on $p$ by fixing $p^{\rm RES}$

4: for $t=1\dots 48$ do

5: Tighten bounds on $p_{t}$ using ramping constraints

6: $p_{t},q_{t}\leftarrow$ AC-OPF at $t$

7: end for

8: $q^{\rm RES}\leftarrow$ Greedy reserve allocation

9: return $p,q,u,p^{\rm RES},q^{\rm RES}$

Algorithm 1 Reserve-preserving decomposition

Algorithm 2 is designed to prioritize unit commitment-feasible, AC-feasible
solutions. It solves a copper-plate unit commitment problem to obtain a
commitment schedule and nominal real and reactive powers, then solves AC-OPF
subproblems sequentially. After solving AC-OPF subproblems, real and reactive
powers are fixed and both real and reactive reserve products are allocated
using the greedy approach.

1: Initialize: $p,q,u\leftarrow$ initial status

2: $p,q,u\leftarrow$ Copper-plate unit commitment

3: for $t=1\dots 48$ do

4: Tighten bounds on $p_{t}$ using ramping constraints

5: $p_{t},q_{t}\leftarrow$ AC-OPF at $t$

6: end for

7: $p^{\rm RES},q^{\rm RES}\leftarrow$ Greedy reserve allocation

8: return $p,q,u,p^{\rm RES},q^{\rm RES}$

Algorithm 2 Simple greedy decomposition

Algorithm 3 uses the same basic decomposition, but is refined by the
observation that tightening bounds on a large number of devices can lead to AC
infeasibility. It employs the same bound tightening strategy as Algorithm 1
but only for the top 5% of devices providing reserve value, as described in
Section III-A3. After AC-OPF subproblems, reserves are re-computed via linear
programs that balance costs with zonal reserve penalties subject to fixed real
power, reactive power, and on status variables previously computed.

1: Parameter $\gamma=5$

2: Initialize: $p,q,u\leftarrow$ initial status

3: $p,q,u,p^{\rm RES},q^{\rm RES}\leftarrow$ Copper-plate unit commitment

4: Tighten $p$ bounds by fixing $p^{\rm RES}$ for top $\gamma\%$ of devices

5: for $t=1\dots 48$ do

6: Tighten bounds on $p_{t}$ using ramping constraints

7: $p_{t},q_{t}\leftarrow$ AC-OPF at $t$

8: end for

9: $p^{\rm RES},q^{\rm RES}\leftarrow$ Reserve re-dispatch via linear programs

10: return $p,q,u,p^{\rm RES},q^{\rm RES}$

Algorithm 3 Reserve/AC-balancing heuristic

Algorithm 4 is the balancing heuristic algorithm with a parallel decomposition
of the AC-OPF subproblems at individual time periods. This is followed by a
sequential projection of real power into the bounds implied by ramping
constraints at the previous time point. The resulting solutions satisfy the
unit commitment and ramping constraints, have modest AC power balance
violations due to ramping constraint projections, and have reserves that are
balanced between the pre-OPF copper-plate reserve schedule and the post-OPF
reserve re-dispatch. This algorithm variant is comparable to the solution
method that was used for the “ARPA-E Benchmark” algorithm in Event 4 of GOC
Challenge 3.

1: Parameter $\gamma=5$

2: Initialize: $p,q,u\leftarrow$ initial status

3: $p,q,u,p^{\rm RES},q^{\rm RES}\leftarrow$ Copper-plate unit commitment

4: Tighten $p$ bounds by fixing $p^{\rm RES}$ for top $\gamma\%$ of devices

5: for $t=1\dots 48$ in parallel do

6: $p_{t},q_{t}\leftarrow$ AC-OPF at $t$

7: end for

8: for $t=1\dots 48$ do

9: Project $p_{t}$ to satisfy ramping constraints

10: end for

11: $p^{\rm RES},q^{\rm RES}\leftarrow$ Reserve re-dispatch via linear
programs

12: return $p,q,u,p^{\rm RES},q^{\rm RES}$

Algorithm 4 Parallel decomposition heuristic

These decomposition algorithms consider all hard constraints of the GOC
problem formulation [5]. That is, omitted features such as contingency
constraints are modeled as soft constraints. As such, solutions produced by
these algorithms are guaranteed to be feasible for the GOC problem
formulation. However, as all of these algorithms are heuristics, they provide
no particular guarantees on solution speed or quality. To address this, future
work may focus on using these decompositions in a Benders decomposition or
outer approximation framework to develop approaches that, given enough time,
are guaranteed to converge to a locally optimal solution. The next section
conducts detailed experiments on a variety of realistic network instances to
explore the strengths and weaknesses of these algorithms.

##  IV Results and discussion

###  IV-A Full AC-UC problem evaluation

To motivate the use of heuristic decomposition algorithms for large scale
instances of the AC-UC problem, we first evaluate the performance of Knitro
14.0 on small instances of the full AC-UC problem, using a heuristic branch-
and-bound algorithm [21] with nonlinear programming (NLP) subproblems solved
by an interior point method [22]. We use small networks and construct an AC-UC
problem containing only unit commitment and AC network constraints (i.e.
omitting reserve products). The test datasets have 3, 14, 37, 73, and 617
buses, and come either from GOC Event 4 or the “sandbox” dataset, indicated by
“S0” in the Case ID. Solve times and objective values for these small
instances of the simplified problem are shown in Table I. The results indicate
that, even for this simplified problem, Knitro is unable to solve a 617-bus
instance of the full AC-UC problem as an MINLP with 627,768 variables and
705,544 constraints. The largest instance considered in this work, with 8,316
buses, has 8,260,224 variables and 8,947,330 constraints in this simplified
version of the AC-UC problem. This is well beyond the scope of off-the-shelf
MINLP solvers, and motivates the use of heuristic decomposition methods to
produce high-quality solutions for industry-scale instances of this problem.

TABLE I: Results of solving full AC-UC problem with Knitro Case ID | N. Var. | N. Con. | Objective ($) | Solve time (s)
---|---|---|---|---
S0N00003-003 | 2260 | 3850 | 9.07e+05 | 71
S0N00014-003 | 22080 | 26183 | 2.23e+06 | 33
S0N00037-003 | 43584 | 47494 | 1.07e+07 | 20
N00073-333 | 159072 | 195415 | 2.30e+08 | 84
N00617-002 | 627768 | 705544 | N/A |  $>$10000

###  IV-B Test data

The algorithms presented in Section III-B are evaluated in terms of solution
quality and solve time on 28 day-ahead AC-UC cases provided by Event 4 of the
Grid Optimization Competition, Challenge 3. Networks with 73, 617, 2,000,
4,224, 6,049, 6,717, and 8,316 buses are considered. Four representative
scenarios are considered for each network. Problem data for each network are
given in Table II. Except for the 73-bus network, all scenarios for a given
network have the same numbers of dispatchable devices, AC lines, and reserve
zones. Each case is assigned a unique identifier (“Case ID”) using the network
size and the scenario number from GOC Event 4. The objective values of
solutions are computed by an independent evaluation program provided by the
GOC that considers the full problem formulation [23].

TABLE II: Statistics for the Event 4 networks considered Buses | Devices | AC lines |  $p$ res. zones |  $q$ res. zones
---|---|---|---|---
73 (scenario 991) | 208 | 127 | 1 | 1
73 (other scenarios) | 205 | 105 | 1 | 1
617 | 499 | 723 | 10 | 10
2000 | 1894 | 2345 | 4 | 10
4224 | 2151 | 2605 | 2 | 2
6049 | 3774 | 4920 | 6 | 6
6717 | 5826 | 7173 | 9 | 12
8316 | 5585 | 7723 | 7 | 7

###  IV-C Computational Setting

All results in this section are computed using HPE ProLiant XL170r servers
with two Intel 2.10 GHz CPUs and 128 GB of memory using Julia v1.6 [24]. These
machine have 36 physical cores and up to 72 simultaneous threads. Optimization
models are constructed with JuMP v1.14 [25]. The unit commitment MIP models
are solved with Gurobi v10.0 and the AC-OPF subproblems are solved with Ipopt
v3.14 [26] using the MA27 linear solver [27] and a symbolic automatic
differentiation approach [28] that is similar to the approach of Gravity [29].
To reduce the runtime of ill-conditioned instances that require many interior-
point iterations to converge, solves are terminated via callback if they have
reached a $10^{-3}$ unscaled primal residual and a 1.0 unscaled dual residual.
The reserve assignment LPs are solved using HiGHS v1.5 [30].

Implementations of the models and subroutines used in this work can be found
at https://github.com/lanl-ansi/GOC3Benchmark.jl.

###  IV-D Solution quality

Solution quality results for Algorithm 1 are shown in Table III. In Tables
III-VI, “Objective” is the market surplus value of the solution produced by
presented algorithm, while “Best-known solution” is the value of the best
solution produced by any team in the GOC, according to the leaderboards [31].
Each of these best solutions was obtained within a two hour time limit on the
standardized hardware used by the competition. “Gap” is defined as the percent
difference with respect to the best-known solution. “Res. penalty” is the
total penalty incurred by zonal reserve shortfalls, while “$p$-penalty” and
“$q$-penalty are total real and reactive power balance violation penalties.
The penalty factor is $10^{6}$ US dollars per power unit, so achieving a
$10^{2}$ power balance penalty is equivalent to a total power balance
violation magnitude of $10^{-4}$. “Solve time” is the time elapsed between the
start of Julia code execution and the point at which the solution file is
written.

While the algorithm produces acceptable solutions for many cases, 13 cases
have gaps to the best-known solution of over 30%. In 12 of these 13 cases (and
several others), the dominant penalties are real and reactive power balance
violation penalties. In Table III, rows corresponding to solution gaps of
larger than 30% are highlighted. This suggests that fixing reserve products
for many devices is too restrictive to achieve AC feasibility.

Algorithm 2 prioritizes AC feasibility over reserve feasibility. It includes
reserves in its copper-plate unit commitment model, but does not fix any
reserve products before the AC-OPF subproblems. Table IV shows solution
quality results for Algorithm 2, which produces surprisingly high-quality
solutions for a simple greedy approach. Except for seven cases, these
solutions are computed within the 2 hour time limit. Of the 28 cases, 12 have
a gap of less than 10% to the best-known solution. In 9 of the remaining 16
cases, the dominant penalty is the reserve shortfall penalty. This suggests
that post-AC-OPF greedy reserve allocation is not sufficient to minimize
reserve penalty, and that solutions could be improved by including information
from the reserves computed by the unit commitment subproblem.

Algorithm 3 attempts to improve reserve shortfall penalties by tightening
bounds on $\gamma=5\%$ of devices providing reserve value. In addition,
reserve products are re-dispatched after the AC-OPF subproblems via linear
programs with zonal balance models. The solution quality results for this
algorithm are shown in Table V. The results show that this algorithm produces
high-quality results in most cases, with the gap to the best-known solution
less than 10% for 19 of 28 cases. Reserve penalty dominates in only one of the
remaining nine cases (N02000-031). The total market surplus across all cases
is 1.74$\times$1010$ for Algorithm 3, compared to 1.37$\times$1010$ for
Algorithm 2, a 20% improvement with a value of over 3 billion US dollars. Fig.
1 compares the penalties incurred by Algorithms 1-3 on cases 4,000 buses or
larger. The comparison shows large power balance violation penalties for
Algorithm 1 and large reserve shortfall penalties for Algorithm 2. In the few
cases where Algorithm 3 incurs large penalties, they are dominated by moderate
power balance violations.

The results demonstrate that surprisingly good solutions can be obtained by a
simple algorithm that does not consider contingency constraints, line
switching, shunt control, or transformer control. In 16 of 28 cases
considered, Algorithm 3 obtains solutions with objectives within 2% of the
best-known. In case N08316-131, however, the algorithm takes longer than the 2
hour time limit to produce a solution.

Figure 1: Summary of objective penalties incurred for Algorithms 1-3 on the
larger networks considered in this work. Algorithm 3 reliably balances AC and
reserve feasibility requirements, finding solutions with low penalties.

###  IV-E Solve time

Fig. 2 shows a breakdown of the solve times of Algorithm 3 for the 28 cases
presented. The results show that solving the sequential AC-OPF subproblems is
the dominant computational expense, although for the 6,717-bus cases the unit
commitment solve time also contributes significantly.

Figure 2: A breakdown of solve times with Algorithm 3, where the scenarios for
each network have the same order as in Table V. These results highlight how
AC-OPF solve times are a dominant factor in the runtime of the algorithm.

Algorithm 4 solves individual AC-OPF subproblems in parallel to attempt to
reduce the dominant computational burden and provide solutions within the two
hour time limit. It does so at some expense to solution quality, however, as
the post-processing projection step that recovers ramping feasibility can also
introduce power balance violations. Solution quality results for Algorithm 4
are shown in Table VI. While the parallel solution strategy does produce high-
quality solutions for most cases, it does incur a penalty compared to
Algorithm 3 on cases N04224-005, N04224-022, and N04224-038 due to power
balance violations.

Fig. 3 shows the solve times and speed-up factors as functions of the number
of threads allocated to the Julia process running Algorithm 4 on one scenario
from each of the five largest network sizes considered. While Algorithm 3
exceeds the time limit on case N08316-131, Algorithm 4 solves within the time
limit when using only a single thread. With only four threads, Algorithm 4
solves within the time limit by a margin of at least 2,000 s for all cases.
Additional runtime gains can be achieved using additional cores.

Figure 3: Scaling of Algorithm 4’s solve time with number of threads for five
of the large cases considered in this work. The results show that 32 parallel
processes are sufficient for satisfying a two hour runtime limit.

According to the results presented in Fig. 2, Algorithm 3 spends between 36%
and 90% of its time in the sequential AC-OPF portion of the algorithm for
networks 2,000 buses and larger. As only this portion of the algorithm is
parallelized, linear speedup with the number of processors is not expected. A
theoretical upper bound on speedup for the most OPF-intensive instance is
10$\times$, while an upper bound on speedup for the average instance is
approximately 4$\times$. Fig. 3 shows speedups between 1.4$\times$ and
2.6$\times$ with 64 threads for the most challenging instances, indicating the
presence of some overhead. Thread scheduling is done by Julia’s built-in
Threads package. While parallelizing subproblems across time periods is
sufficient to solve the presented cases within the time limit, additional
gains may be obtained by decomposing in space as well. Overall, these results
suggest that parallelization can be an effective approach to balancing
solution quality and runtime requirements in industrial-scale AC Unit
Commitment.

##  V Conclusion

This work provides a simple decomposition framework for solving large-scale
instances of a challenging AC Unit Commitment problem posed by Challenge 3 of
the Grid Optimization Competition. Although the competition formulation
includes several features that this work does not consider, such as
contingency constraints and line switching, the proposed algorithms are
successful in producing high-quality solutions to the majority of problem
instances considered. A variant of the proposed method using a parallel
decomposition of AC-OPF subproblems is capable of producing solutions to all
instances considered within the two hour time limit, with some reduction in
solution quality compared to a fully sequential computational approach. The
success of this algorithm and those from other participants in the grid
optimization competition suggest that industry-scale AC Unit Commitment is
within reach of current numerical optimization methods.

## Acknowledgements

This work was funded by the U.S. Department of Energy Advanced Research
Projects Agency-Energy (ARPA-E), as part of the Grid Optimization Competition.
LA-UR-23-31000.

TABLE III: Solution Quality Results for Algorithm 1 Case ID | Best-known solution ($) | Objective ($) | Gap | Res. penalty ($) |  $p$-penalty ($) |  $q$-penalty ($) | Solve time (s)
---|---|---|---|---|---|---|---
N00073-303 | +1.48e+08 | +1.48e+08 | 0.03% | -4.90e+04 | -2.42e+00 | -3.18e+00 | 181
N00073-333 | +2.31e+08 | +2.30e+08 | 0.02% | -7.36e+04 | -1.44e+01 | -2.70e+01 | 180
N00073-373 | +2.35e+08 | +2.34e+08 | 0.48% | -3.43e+05 | -7.28e+01 | -3.91e+02 | 187
N00073-991 | +5.90e+07 | +7.95e+06 | 86.51% | -1.16e+05 | -6.50e+01 | -4.92e+07 | 193
N00617-002 | +1.64e+08 | +1.64e+08 | 0.17% | -3.58e-12 | -2.68e+02 | -1.45e+03 | 257
N00617-015 | +2.65e+08 | +2.65e+08 | 0.11% | -2.55e+04 | -3.58e+02 | -2.21e+03 | 304
N00617-039 | +1.64e+08 | +1.64e+08 | 0.18% | -2.72e+04 | -2.79e+02 | -1.49e+03 | 299
N00617-069 | +2.66e+08 | +2.65e+08 | 0.31% | -8.14e+03 | -6.91e+02 | -5.04e+03 | 295
N02000-022 | +7.59e+08 | +7.31e+08 | 3.78% | -2.45e+07 | -1.27e+02 | -1.08e+03 | 984
N02000-031 | +8.26e+08 | +7.84e+08 | 5.12% | -1.83e+07 | -1.64e+07 | -1.73e+03 | 2621
N02000-074 | +7.58e+08 | +5.13e+08 | 32.36% | -2.05e+06 | -2.44e+08 | -1.97e+03 | 608
N02000-080 | +8.30e+08 | +5.51e+08 | 33.62% | -5.83e+05 | -2.79e+08 | -6.09e+03 | 641
N04224-005 | +4.96e+08 | +2.18e+08 | 55.99% | -5.01e+07 | -5.56e+07 | -1.52e+08 | 1813
N04224-022 | +4.96e+08 | -5.32e+07 | 110.71% | -2.59e+07 | -2.01e+08 | -3.04e+08 | 2013
N04224-038 | +4.97e+08 | +2.24e+08 | 54.82% | -3.11e+07 | -9.72e+07 | -1.26e+08 | 1729
N04224-054 | +7.22e+08 | +4.92e+08 | 31.90% | -2.26e+08 | -3.72e+05 | -1.22e+05 | 1074
N06049-003 | +6.09e+08 | +5.71e+08 | 6.30% | -5.64e+05 | -5.93e+05 | -1.24e+07 | 5173
N06049-031 | +6.78e+08 | +4.09e+08 | 39.69% | -1.18e+06 | -2.54e+05 | -2.63e+07 | 4360
N06049-057 | +6.09e+08 | +2.83e+08 | 53.58% | -3.47e+05 | -3.19e+05 | -9.33e+07 | 3887
N06049-069 | +8.27e+08 | +3.82e+08 | 53.74% | -3.22e+05 | -2.65e+06 | -9.78e+06 | 5095
N06717-044 | +9.05e+08 | +8.14e+08 | 10.05% | -2.02e+07 | -3.15e+07 | -1.65e+04 | 3870
N06717-050 | +1.32e+09 | +1.23e+09 | 6.79% | -1.80e+07 | -1.97e+07 | -3.34e+04 | 3748
N06717-062 | +9.11e+08 | +2.99e+08 | 67.20% | -1.33e+07 | -5.50e+08 | -4.41e+06 | 3407
N06717-068 | +1.33e+09 | +3.11e+08 | 76.57% | -2.05e+07 | -9.31e+08 | -1.53e+06 | 4242
N08316-109 | +1.43e+09 | +1.36e+09 | 4.55% | -1.94e+06 | -5.22e+07 | -4.07e+06 | 4661
N08316-121 | +1.16e+09 | +1.03e+09 | 11.28% | -3.37e+06 | -2.32e+07 | -9.62e+07 | 5723
N08316-131 | +1.43e+09 | +1.41e+09 | 1.85% | -6.96e+06 | -5.66e+06 | -5.01e+06 | 4928
N08316-157 | +1.18e+09 | +6.44e+08 | 45.54% | -4.01e+06 | -2.59e+08 | -2.66e+08 | 6957
TABLE IV: Solution Quality Results for Algorithm 2 Case ID | Best-known solution ($) | Objective ($) | Gap | Res. penalty ($) |  $p$-penalty ($) |  $q$-penalty ($) | Solve time (s)
---|---|---|---|---|---|---|---
N00073-303 | +1.48e+08 | +1.48e+08 | 0.11% | -9.39e+02 | -2.42e+00 | -3.18e+00 | 163
N00073-333 | +2.31e+08 | +2.30e+08 | 0.11% | -1.94e+04 | -1.44e+01 | -2.70e+01 | 165
N00073-373 | +2.35e+08 | +2.34e+08 | 0.48% | -2.00e+05 | -7.28e+01 | -3.91e+02 | 171
N00073-991 | +5.90e+07 | +7.96e+06 | 86.50% | -9.12e+04 | -6.50e+01 | -4.92e+07 | 188
N00617-002 | +1.64e+08 | +1.64e+08 | 0.01% | -0.00e+00 | -5.39e+01 | -2.69e+02 | 253
N00617-015 | +2.65e+08 | +2.65e+08 | 0.12% | -2.70e+05 | -2.29e+02 | -1.22e+03 | 271
N00617-039 | +1.64e+08 | +1.64e+08 | 0.17% | -2.70e+05 | -7.80e+01 | -4.14e+02 | 262
N00617-069 | +2.66e+08 | +2.66e+08 | 0.10% | -2.70e+05 | -2.46e+02 | -1.30e+03 | 280
N02000-022 | +7.59e+08 | -2.01e+09 | 364.29% | -2.77e+09 | -6.88e+01 | -3.14e+02 | 1345
N02000-031 | +8.26e+08 | -4.40e+09 | 632.79% | -5.23e+09 | -8.40e+01 | -4.31e+02 | 2886
N02000-074 | +7.58e+08 | +1.62e+08 | 78.64% | -5.96e+08 | -5.18e+01 | -2.89e+02 | 926
N02000-080 | +8.30e+08 | -1.87e+07 | 102.25% | -8.49e+08 | -6.19e+01 | -2.93e+02 | 943
N04224-005 | +4.96e+08 | +4.40e+08 | 11.34% | -2.78e+05 | -2.08e+07 | -3.01e+07 | 2240
N04224-022 | +4.96e+08 | +2.92e+08 | 41.26% | -0.00e+00 | -8.21e+07 | -1.17e+08 | 2246
N04224-038 | +4.97e+08 | +2.57e+08 | 48.20% | -0.00e+00 | -9.04e+07 | -1.41e+08 | 2068
N04224-054 | +7.22e+08 | +6.05e+08 | 16.22% | -1.15e+08 | -1.82e+03 | -9.52e+03 | 2471
N06049-003 | +6.09e+08 | +5.95e+08 | 2.27% | -6.95e+05 | -8.79e+04 | -1.00e+07 | 5951
N06049-031 | +6.78e+08 | +5.54e+08 | 18.32% | -1.13e+07 | -1.08e+05 | -1.61e+07 | 5283
N06049-057 | +6.09e+08 | +3.20e+08 | 47.57% | -1.00e+06 | -2.98e+05 | -7.96e+07 | 6050
N06049-069 | +8.27e+08 | +4.40e+08 | 46.72% | -1.26e+06 | -1.29e+06 | -5.07e+06 | 7597
N06717-044 | +9.05e+08 | -4.23e+08 | 146.70% | -1.33e+09 | -1.42e+02 | -3.82e+02 | 6385
N06717-050 | +1.32e+09 | -1.29e+08 | 109.81% | -1.45e+09 | -2.92e+02 | -1.65e+03 | 7302
N06717-062 | +9.11e+08 | -5.57e+08 | 161.19% | -1.47e+09 | -1.55e+02 | -9.73e+02 | 6585
N06717-068 | +1.33e+09 | -4.71e+08 | 135.40% | -1.80e+09 | -4.77e+02 | -3.40e+03 | 7614
N08316-109 | +1.43e+09 | +1.42e+09 | 0.76% | -7.80e+05 | -1.13e+06 | -8.53e+05 | 8866
N08316-121 | +1.16e+09 | +1.06e+09 | 9.09% | -3.81e+06 | -1.10e+07 | -7.68e+07 | 9318
N08316-131 | +1.43e+09 | +1.42e+09 | 1.23% | -7.64e+06 | -9.54e+05 | -4.55e+05 | 8838
N08316-157 | +1.18e+09 | +1.07e+09 | 9.57% | -3.23e+06 | -1.06e+07 | -8.26e+07 | 10000
TABLE V: Solution Quality Results for Algorithm 3 Case ID | Best-known solution ($) | Objective ($) | Gap | Res. penalty ($) |  $p$-penalty ($) |  $q$-penalty ($) | Solve time (s)
---|---|---|---|---|---|---|---
N00073-303 | +1.48e+08 | +1.48e+08 | 0.09% | -2.11e+03 | -6.09e+01 | -3.73e+02 | 217
N00073-333 | +2.31e+08 | +2.30e+08 | 0.09% | -1.65e+04 | -4.70e+01 | -2.87e+02 | 212
N00073-373 | +2.35e+08 | +2.34e+08 | 0.39% | -2.69e+05 | -1.32e+02 | -7.99e+02 | 212
N00073-991 | +5.90e+07 | +7.36e+06 | 87.52% | -1.12e+05 | -3.51e+02 | -5.05e+07 | 239
N00617-002 | +1.64e+08 | +1.64e+08 | 0.09% | -3.64e-03 | -6.19e+01 | -2.64e+02 | 342
N00617-015 | +2.65e+08 | +2.65e+08 | 0.12% | -1.32e+03 | -4.25e+02 | -8.20e+02 | 368
N00617-039 | +1.64e+08 | +1.64e+08 | 0.10% | -1.26e+04 | -5.03e+01 | -2.24e+02 | 337
N00617-069 | +2.66e+08 | +2.66e+08 | 0.09% | -1.39e+03 | -1.89e+02 | -3.91e+02 | 380
N02000-022 | +7.59e+08 | +7.47e+08 | 1.61% | -1.21e+07 | -1.57e+03 | -5.73e+03 | 1614
N02000-031 | +8.26e+08 | +7.29e+08 | 11.73% | -1.00e+08 | -2.13e+03 | -8.05e+03 | 3154
N02000-074 | +7.58e+08 | +7.57e+08 | 0.08% | -1.62e+01 | -6.37e+02 | -2.80e+03 | 1831
N02000-080 | +8.30e+08 | +8.29e+08 | 0.08% | -1.50e+01 | -4.47e+02 | -2.45e+03 | 1763
N04224-005 | +4.96e+08 | +4.88e+08 | 1.57% | -1.67e+00 | -9.62e+03 | -2.38e+06 | 1462
N04224-022 | +4.96e+08 | +3.85e+08 | 22.47% | -1.19e+00 | -4.38e+07 | -6.12e+07 | 1454
N04224-038 | +4.97e+08 | +3.19e+08 | 35.77% | -6.00e-09 | -6.66e+07 | -1.05e+08 | 1464
N04224-054 | +7.22e+08 | +7.19e+08 | 0.43% | -4.63e-02 | -2.60e+03 | -2.59e+04 | 902
N06049-003 | +6.09e+08 | +5.75e+08 | 5.57% | -2.00e+05 | -1.69e+05 | -1.36e+07 | 4872
N06049-031 | +6.78e+08 | +5.38e+08 | 20.65% | -1.89e+05 | -1.56e+05 | -1.78e+07 | 4581
N06049-057 | +6.09e+08 | +2.92e+08 | 52.17% | -1.37e+03 | -3.49e+05 | -8.46e+07 | 3937
N06049-069 | +8.27e+08 | +3.84e+08 | 53.58% | -1.40e-10 | -1.16e+06 | -8.18e+06 | 4532
N06717-044 | +9.05e+08 | +8.94e+08 | 1.26% | -1.28e+07 | -8.45e+02 | -3.59e+03 | 4799
N06717-050 | +1.32e+09 | +1.32e+09 | 0.22% | -2.41e+06 | -8.21e+02 | -5.60e+03 | 5008
N06717-062 | +9.11e+08 | +8.31e+08 | 8.79% | -1.34e+07 | -6.33e+07 | -2.45e+05 | 4503
N06717-068 | +1.33e+09 | +1.23e+09 | 7.20% | -9.18e+06 | -8.07e+07 | -7.49e+04 | 5313
N08316-109 | +1.43e+09 | +1.41e+09 | 1.29% | -3.04e+05 | -5.99e+06 | -5.25e+06 | 6938
N08316-121 | +1.16e+09 | +1.03e+09 | 11.02% | -2.09e+06 | -1.97e+07 | -9.93e+07 | 6469
N08316-131 | +1.43e+09 | +1.40e+09 | 1.96% | -9.57e+06 | -5.87e+06 | -5.61e+06 | 8061
N08316-157 | +1.18e+09 | +1.06e+09 | 10.60% | -1.26e+06 | -9.71e+06 | -1.10e+08 | 6747
TABLE VI: Solution Quality Results for Algorithm 4 Case ID | Best-known solution ($) | Objective ($) | Gap | Res. penalty ($) |  $p$-penalty ($) |  $q$-penalty ($) | Solve time (s)
---|---|---|---|---|---|---|---
N00073-303 | +1.48e+08 | +1.48e+08 | 0.09% | -2.11e+03 | -6.09e+01 | -3.73e+02 | 215
N00073-333 | +2.31e+08 | +2.30e+08 | 0.09% | -1.65e+04 | -4.70e+01 | -2.87e+02 | 207
N00073-373 | +2.35e+08 | +2.34e+08 | 0.39% | -2.69e+05 | -1.32e+02 | -7.99e+02 | 209
N00073-991 | +5.90e+07 | +7.36e+06 | 87.52% | -1.12e+05 | -3.51e+02 | -5.05e+07 | 234
N00617-002 | +1.64e+08 | +1.64e+08 | 0.09% | -3.64e-03 | -6.19e+01 | -2.64e+02 | 249
N00617-015 | +2.65e+08 | +2.65e+08 | 0.12% | -1.32e+03 | -4.25e+02 | -8.20e+02 | 273
N00617-039 | +1.64e+08 | +1.64e+08 | 0.10% | -1.26e+04 | -5.03e+01 | -2.24e+02 | 265
N00617-069 | +2.66e+08 | +2.66e+08 | 0.09% | -1.39e+03 | -1.89e+02 | -3.91e+02 | 296
N02000-022 | +7.59e+08 | +7.47e+08 | 1.61% | -1.21e+07 | -1.58e+03 | -5.90e+03 | 755
N02000-031 | +8.26e+08 | +7.29e+08 | 11.73% | -1.00e+08 | -1.98e+03 | -7.70e+03 | 2390
N02000-074 | +7.58e+08 | +7.57e+08 | 0.08% | -1.59e+01 | -4.11e+02 | -2.30e+03 | 738
N02000-080 | +8.30e+08 | +8.29e+08 | 0.08% | -1.55e+01 | -3.61e+02 | -2.05e+03 | 652
N04224-005 | +4.96e+08 | +1.80e+08 | 63.61% | -6.81e-01 | -2.19e+08 | -9.09e+07 | 827
N04224-022 | +4.96e+08 | +2.74e+08 | 44.84% | -9.18e-01 | -1.61e+08 | -5.56e+07 | 775
N04224-038 | +4.97e+08 | +2.06e+08 | 58.53% | -4.14e-09 | -2.01e+08 | -8.42e+07 | 921
N04224-054 | +7.22e+08 | +7.19e+08 | 0.47% | -4.63e-02 | -3.32e+05 | -2.58e+04 | 720
N06049-003 | +6.09e+08 | +5.64e+08 | 7.39% | -2.03e+05 | -1.12e+07 | -1.32e+07 | 1934
N06049-031 | +6.78e+08 | +5.38e+08 | 20.68% | -1.88e+05 | -4.29e+05 | -1.76e+07 | 2059
N06049-057 | +6.09e+08 | +2.92e+08 | 52.06% | -1.59e+03 | -3.47e+05 | -8.36e+07 | 1537
N06049-069 | +8.27e+08 | +3.80e+08 | 53.97% | -1.57e-10 | -2.16e+06 | -1.01e+07 | 2636
N06717-044 | +9.05e+08 | +8.94e+08 | 1.26% | -1.28e+07 | -8.09e+02 | -3.50e+03 | 3868
N06717-050 | +1.32e+09 | +1.32e+09 | 0.22% | -2.41e+06 | -8.65e+02 | -5.65e+03 | 3977
N06717-062 | +9.11e+08 | +8.31e+08 | 8.79% | -1.34e+07 | -6.33e+07 | -2.44e+05 | 3217
N06717-068 | +1.33e+09 | +1.23e+09 | 7.20% | -9.18e+06 | -8.07e+07 | -7.49e+04 | 4379
N08316-109 | +1.43e+09 | +1.38e+09 | 3.14% | -3.05e+05 | -3.23e+07 | -5.18e+06 | 4114
N08316-121 | +1.16e+09 | +1.01e+09 | 13.00% | -2.10e+06 | -3.78e+07 | -1.05e+08 | 3046
N08316-131 | +1.43e+09 | +1.39e+09 | 3.24% | -9.74e+06 | -2.41e+07 | -5.58e+06 | 3995
N08316-157 | +1.18e+09 | +1.02e+09 | 13.59% | -1.20e+06 | -4.37e+07 | -1.12e+08 | 3438

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††institutetext: Department of Physics, Beijing Normal University, Beijing
100875, China

#  $SL(2,R)\times U(1)$ symmetry and quasinormal modes in the self-dual warped
AdS black hole

Yuan Chen,  Wei Guo  Kai Shi  Hongbao Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The algebraic approach to the spectrum of quasinormal modes has been made as
simple as possible for the BTZ black hole by the strategy developed in Zhang .
By working with the self-dual warped AdS black hole, we demonstrate in an
explicit way that such a strategy can be well adapted to those warped AdS
balck holes with the $SL(2,R)\times U(1)$ isometry. To this end, we first
introduce two associated tensor fields with the quadratic Casimir of
$SL(2,R)\times U(1)$ Lie algebra in the self-dual warped AdS black hole and
show that they correspond essentially to the metric and volume element up to a
constant prefactor, respectively. Then without appealing to any concrete
coordinate system, we can further show that the solutions to the equations of
motion for the scalar, vector, spinor fields all fall into the representations
of the $SL(2,R)\times U(1)$ Lie algebra by a purely abstract tensor and spinor
analysis. Accordingly, the corresponding spectrum of quasinormal modes for
each fixed azimuthal quantum number can be derived algebraically as the
infinite tower of descendants of the highest weight mode of the $SL(2,R)$ Lie
subalgebra.

##  1 Introduction

As we know, not only does the photon ring outside of the black hole play a
vital role in predicting the intricate patterns of the black hole image, but
also controls the photon sphere quasinormal ringdowns of massless fields such
as the gravitational waves. Very recently, it has been found in HKLS  that the
photon sphere quasinormal modes exhibit an emergent $SL(2,R)$ symmetry for the
Kerr black hole, namely the spectrum of the photon sphere quasinormal modes
forms the highest weight representation of the emergent $SL(2,R)$ symmetry.
Later on, to gain a deeper understanding of this emergent $SL(2,R)$ symmetry,
the authors in KLS  initiate the exploration of the self-dual warped AdS black
hole, which arises as an approximation to the near-extremal Kerr black hole
with the $SL(2,R)$ isometry. In particular, the resultant spectrum of
quasinormal modes by explicit calculation can be identified as the highest
weight representation of the $SL(2,R)$ isometry. It is further shown in CHH
that the aforementioned emergent $SL(2,R)$ symmetry associated with the
spectrum of the photon sphere quasinormal modes, if redefined appropriately by
virtue of the residual degrees of freedom, can be regarded precisely as the
eikonal limit of the $SL(2,R)$ isometry.

As in the context of AdS/CFT correspondence, the identification of such a
symmetry is suspected to offer us a guiding principle to search for the
holographic dual field theory description of the black hole under
consideration through Kerr/CFT correspondence and warped AdS/CFT
correspondence, respectivelyGHSS ; Compere ; CZZZ ; SX . In particular, the
$SL(2,R)$ symmetry dictated quasinormal modes correspond to the poles of the
retarded Green function of the putative dual CFT. Such a specific
correspondence has been confirmed in AdS3/CFT2, where analytic results can be
readily obtained for both sides. Speaking specifically, the analytic result
for the quasinormal modes of the scalar, vector, and spinor perturbations in
the BTZ black hole is first obtained by directly solving the equations of
motionCL ; BSS . Later on, inspired by MS  and BKL , the authors in SS
demonstrate that the quasinormal modes can be constructed as the left and
right chiral highest weight representation of the $SL(2,R)$ isometry for the
scalar and metric perturbations. This algebraic construction has been further
generalized to other three dimensional black holes with the vector
perturbation includedCLC ; CZ . However, the involved calculation is somewhat
complicated, which makes the $SL(2,R)$ isometry obscure albeit recovered in
the final result as it should be the case. Such a technical deficiency is
rescued in Zhang  by resorting to the two intrinsic tensor fields associated
with the $SL(2,R)$ isometry of the BTZ black hole as well as the covariant
derivative rather than the ordinary derivative in the course of analysis.
Among others, one very advantage of this strategy is that the spinor
perturbation can be incorporated and treated in a uniform manner as other
tensor fields.

As alluded to in the very beginning, the $SL(2,R)$ isometry of the self-dual
warped AdS black hole is supposed to play a similar role in controlling the
quasinormal modes algebraically, which may lead to some intriguing
observational signatures of holographic gravity. However, such a suspicion has
so far been confirmed solely for the (massless) scalar perturbation due to its
obvious simplicity. So it is tempting to check this suspicion for other
perturbations explicitly by utilizing the strategy developed in Zhang . The
purpose of this paper is to show this is also the case for both vector and
spinor perturbations. To make the whole analysis as simple as possible, we are
required to introduce the two intrinsic tensor fields associated with the
Casimir of the full $SL(2,R)\times U(1)$ isometry of the self-dual warped AdS
black hole. This is the main difference from the BTZ black hole, where the
full isometry is given by $SL(2,R)_{L}\times SL(2,R)_{R}$ but the two
intrinsic tensor fields are associated with the Casimir of each $SL(2,R)$
sector. It is noteworthy that the warp factor makes the additional $U(1)$
isometry also display itself in Riemann curvature, which turns out to have an
extra effect on the conformal weight of the spinor perturbation. Except for
these nuances, the whole procedure devised in Zhang  proves to be applicable
to the self-dual warped AdS black hole with the $SL(2,R)\times U(1)$ symmetry
manifest throughout the whole analysis.

The rest of paper is organized as follows. In the subsequent section, we
provide a brief review of the self-dual warped AdS black hole as a solution to
the topological massive gravity, where the $SL(2,R)\times U(1)$ isometry is
identified and relevant geometric quantities are presented in an explicit way
for our later usage. In Section 3, with the quadratic Casimir of
$SL(2,R)\times U(1)$ Lie algebra, we introduce its Lie derivative
representation and two associated tensor fields, which turn out to be
proportional to the metric and the volume element, respectively. In Section 4,
we first derive how the solutions to the equations of motion fall in a uniform
manner into the representations of the $SL(2,R)\times U(1)$ Lie algebra for
the scalar, vector, and spinor field in the self-dual warped AdS black hole
and then present the algebraic approach to the quasinormal modes as the
highest weight representation of the $SL(2,R)$ Lie subalgebra for each fixed
azimuthal quantum number, where the main bulk of tensor and spinor analysis is
performed in a purely abstract way without resorting to any specific
coordinate system111To our best knowledge, the equation of motion for the
metric perturbation on top of the warped AdS is essentially third order in the
topological massive gravity, and has not been tamed into so well understood a
form as in AdS. So we leave it to future work. . We conclude our paper in the
last section.

Notation and conventions follow Wald  unless specified otherwise.

##  2 Self-dual warped AdS black hole, $SL(2,R)\times U(1)$ isometry and
relevant geometric quantities

Let us start with the self-dual warped AdS black hole

$ds^{2}=\ell^{2}\left[\frac{-dT^{2}+dx^{2}}{\sinh^{2}x}+\Lambda^{2}\left(d\phi+\frac{dT}{\tanh
x}\right)^{2}\right],$ (1)

where $\ell$ is the warped AdS radius and the warp factor $\Lambda$ is so
defined that the unwarped self-dual AdS black hole is given by $\Lambda=1$.
Whence the inverse metric can be obtained as

$\displaystyle g^{ab}=$
$\displaystyle\frac{1}{\ell^{2}}\left\\{{-\sinh^{2}x}(\frac{\partial}{\partial
T})^{a}(\frac{\partial}{\partial T})^{b}+\cosh x\sinh
x[(\frac{\partial}{\partial
T})^{a}(\frac{\partial}{\partial\phi})^{b}+(\frac{\partial}{\partial\phi})^{b}(\frac{\partial}{\partial
T})^{a}]\right.$ (2) $\displaystyle\left.+\sinh^{2}x(\frac{\partial}{\partial
x})^{a}(\frac{\partial}{\partial
x})^{a}+\frac{1-\Lambda^{2}\cosh^{2}{x}}{\Lambda^{2}}(\frac{\partial}{\partial\phi})^{a}(\frac{\partial}{\partial\phi})^{b}\right\\},$

and the associated volume element reads

$\epsilon=\Lambda\ell^{3}\ \text{csch}^{2}x\ dT\wedge dx\wedge d\phi.$ (3)

Such a black hole has the $SL(2,R)\times U(1)$ isometry with the corresponding
Killing fields defined as follows

$\displaystyle L_{0}^{a}$ $\displaystyle=$
$\displaystyle-(\frac{\partial}{\partial T})^{a},$ $\displaystyle L_{-1}^{a}$
$\displaystyle=$ $\displaystyle e^{-T}[-\cosh{x}(\frac{\partial}{\partial
T})^{a}+\sinh{x}(\frac{\partial}{\partial
x})^{a}+\sinh{x}(\frac{\partial}{\partial\phi})^{a}],$ $\displaystyle
L_{+1}^{a}$ $\displaystyle=$ $\displaystyle
e^{T}[-\cosh{x}(\frac{\partial}{\partial
T})^{a}-\sinh{x}(\frac{\partial}{\partial
x})^{a}+\sinh{x}(\frac{\partial}{\partial\phi})^{a}],$ $\displaystyle
W_{0}^{a}$ $\displaystyle=$
$\displaystyle(\frac{\partial}{\partial\phi})^{a},$ (4)

whereby the commutators satisfy the following $SL(2,R)\times U(1)$ Lie algebra

$[L_{0},L_{\pm 1}]=\mp L_{\pm
1},\quad[L_{+1},L_{-1}]=2L_{0},\quad[W_{0},L_{m}]=0.$ (5)

with $m=0,\pm 1$. Note that our self-dual warped AdS black hole is not locally
maximally symmetric. Instead, one can show that the corresponding Riemann
tensor, Ricci tensors, Ricci scalar, and Einstein tensor are given by

$\displaystyle R_{abcd}$
$\displaystyle=2(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})-Rg_{a[c}g_{d]b},$ (6)
$\displaystyle R_{ab}$
$\displaystyle=\frac{\Lambda^{2}-2}{2\ell^{2}}g_{ab}+\frac{1-\Lambda^{2}}{\Lambda^{2}\ell^{4}}W_{0a}W_{0b},$
$\displaystyle R$ $\displaystyle=\frac{\Lambda^{2}-4}{2\ell^{2}},$
$\displaystyle G_{ab}$
$\displaystyle=\frac{\Lambda^{2}}{4\ell^{2}}g_{ab}+\frac{1-\Lambda^{2}}{\Lambda^{2}\ell^{4}}{W_{0}}_{a}{W_{0}}_{b}.$

As one can see, the Riemann tensor receives an extra contribution from the
$U(1)$ generator. In particular, this $U(1)$ generator will also leave its
footprint in the following expression, i.e.,

$R_{abcd}\epsilon^{ab}{}_{e}\epsilon^{cd}{}_{f}=\frac{\Lambda^{2}}{\ell^{2}}g_{ef}+\frac{4(1-\Lambda^{2})}{\Lambda^{2}\ell^{4}}W_{e}W_{f},$
(7)

which, as mentioned before and shown later, will have an effect on the
conformal weight of the spinor perturbation. On the other hand, although the
Weyl tensor vanishes automatically, the self-dual warped AdS black hole is not
conformally flat because the Cotton tensor defined as
$C_{ab}=\epsilon_{a}{}^{cd}\nabla_{c}(R_{db}-\frac{1}{4}Rg_{db})$ is given by

$C_{ab}=\frac{\Lambda(1-\Lambda^{2})}{2\ell^{3}}g_{ab}+\frac{3(\Lambda^{2}-1)}{2\Lambda\ell^{5}}{W_{0}}_{a}{W_{0}}_{b},$
(8)

which does not vanish when $\Lambda\neq 1$. Accordingly, we have

$G_{ab}-\frac{4-\Lambda^{2}}{12\ell^{2}}g_{ab}+\frac{2\ell}{3\Lambda}C_{ab}=0,$
(9)

which means that our self-dual warped AdS black hole can be supported by the
gravitational Chern-Simons term as a solution to the topological massive
gravityKLS .

##  3 Quadratic Casimir, its Lie derivative representation and two associated
tensor fields

The quadratic Casimir operator of $SL(2,R)\times U(1)$ Lie algebra is defined
as

$C^{2}=L^{2}+(1-\frac{1}{\Lambda^{2}})W_{0}^{2}$ (10)

with

$L^{2}=L_{0}^{2}-\frac{1}{2}(L_{+1}L_{-1}+L_{-1}L_{+1})$ (11)

the Casimir operator of $SL(2,R)$ Lie subalgebra. Note that the Lie derivative
of tensor and spinor fields obeys
$[\mathcal{L}_{X},\mathcal{L}_{Y}]=\mathcal{L}_{[X,Y]}$ and
$\mathcal{L}_{\alpha X}=\alpha\mathcal{L}_{X}$ for the arbitrary Killing
vector fields $X$ and $Y$ with the arbitrary constant $\alpha$. Therefore the
above Lie algebra can be naturally represented by the Lie derivative. In
particular, the quadratic Casimir operators can be realized by the Lie
derivative as follows

$\mathcal{C}^{2}=\mathcal{L}^{2}-(1-\frac{1}{\Lambda^{2}})\mathcal{L}_{W_{0}}\mathcal{L}_{W_{0}}=\mathcal{L}_{L_{0}}\mathcal{L}_{L_{0}}-\frac{1}{2}(\mathcal{L}_{L_{+1}}\mathcal{L}_{L_{-1}}+\mathcal{L}_{L_{-1}}\mathcal{L}_{L_{+1}})-(1-\frac{1}{\Lambda^{2}})\mathcal{L}_{W_{0}}\mathcal{L}_{W_{0}}.$
(12)

Now inspired by the strategy developed in Zhang , we construct the following
two tensor fields associated with the quadratic Casimir of $SL(2,R)\times
U(1)$, i.e.,

$H^{ab}=L_{0}^{a}L_{0}^{b}-\frac{1}{2}(L_{+1}^{a}L_{-1}^{b}+L_{-1}^{a}L_{+1}^{b})-(1-\frac{1}{\Lambda^{2}})W_{0}^{a}W_{0}^{b}$
(13)

and

$\displaystyle Z_{abc}$ $\displaystyle=$ $\displaystyle
L_{0a}\nabla_{b}L_{0c}-\frac{1}{2}(L_{+1a}\nabla_{b}L_{-1c}+L_{-1a}\nabla_{b}L_{+1c})-(1-\frac{1}{\Lambda^{2}}){W_{0}}_{a}\nabla_{b}{W_{0}}_{c}.$

Obviously, $H$ is a symmetric tensor field. By a straightforward calculation,
one can show that $H$ is actually proportional to our metric, i.e.,

$H^{ab}=\ell^{2}g^{ab}.$ (14)

Regarding $Z$, we first notice that it is antisymmetric between the last two
indices due to the Killing equation $\nabla_{a}\xi_{b}=\nabla_{[a}\xi_{b]}$.
Second, $\nabla_{b}H_{ac}=0$ tells us that it is also antisymmetric between
the first and third indices. So $Z$ is virtually a $3$-form, which implies
that it should be proportional to the volume element. In addition, we have

$\displaystyle\nabla^{a}Z_{abc}$ $\displaystyle=$ $\displaystyle
L_{0a}\nabla^{a}\nabla_{b}L_{0c}-\frac{1}{2}(L_{+1a}\nabla^{a}\nabla_{b}L_{-1c}+L_{-1a}\nabla^{a}\nabla_{b}L_{+1c})-(1-\frac{1}{\Lambda^{2}}){W_{0}}_{a}\nabla^{a}\nabla_{b}{W_{0}}_{c}$
(15) $\displaystyle=$
$\displaystyle{R_{cb}}^{ad}[L_{0a}L_{0d}-\frac{1}{2}(L_{+1a}L_{-1d}+L_{-1a}L_{+1d})-(1-\frac{1}{\Lambda^{2}}){W_{0}}_{a}{W_{0}}_{c}]=0,$

where $R_{abcd}=R_{ab[cd]}$ is used in the last step while the Killing
equation and the identity

$\nabla_{a}\nabla_{b}\xi_{c}={R_{cba}}^{d}\xi_{d}$ (16)

for any Killing field $\xi$ are used in the first and second steps,
respectively. Therefore, the proportional coefficients in front of the volume
element should be constant. In particular, an explicit computation leads to

$Z_{abc}=-\frac{\Lambda\ell}{2}\epsilon_{abc}.$ (17)

It is noteworthy that the nice properties exhibited in Eq. (14) and Eq. (17)
respectively for $H$ and $Z$ fields will be lost if one naively constructs
them simply out of the quadratic Casimir of $SL(2,R)$ Lie subalgebra.

##  4 $SL(2,R)\times U(1)$ symmetry and quasinormal modes in the self-dual
warped AdS black hole

As a warm-up, let us rework with the scalar field $\phi$, whose equation of
motion is given by

$(\nabla_{a}\nabla^{a}-\mu^{2})\phi=0.$ (18)

The Lie derivative acting on the scalar field gives rise to

$\mathcal{L}_{X}\mathcal{L}_{Y}\phi=X^{a}\nabla_{a}(Y^{b}\nabla_{b}\phi)=(X^{a}\nabla_{a}Y^{b})\nabla_{b}\phi+X^{a}Y^{b}\nabla_{a}\nabla_{b}\phi,$
(19)

whereby we have

$\mathcal{C}^{2}\phi={{Z^{a}}_{a}}^{b}\nabla_{b}\phi+H^{ab}\nabla_{a}\nabla_{b}\phi=\ell^{2}g^{ab}\nabla_{a}\nabla_{b}\phi=(\mu\ell)^{2}\phi.$
(20)

Now let us proceed with the massive vector field $A$, whose equation of motion
is given by

${\epsilon_{a}}^{bc}\nabla_{b}A_{c}=-\mu A_{a}.$ (21)

Whence we can see that the Lorenz condition is satisfied automatically as
follows

$\nabla_{a}A^{a}=-\frac{1}{\mu}\epsilon^{abc}\nabla_{a}\nabla_{b}A_{c}=-\frac{1}{2\mu}\epsilon^{abc}{R_{abc}}^{d}A_{d}=-\frac{1}{2\mu}\epsilon^{abc}{R_{[abc]}}^{d}A_{d}=0$
(22)

due to the cyclic identity $R_{[abc]d}=0$. Furthermore, we have

$\displaystyle\mu^{2}A^{d}$ $\displaystyle=$
$\displaystyle-\mu\epsilon^{dea}\nabla_{e}A_{a}=\epsilon^{dea}\nabla_{e}({\epsilon_{a}}^{bc}\nabla_{b}A_{c})=\epsilon^{ade}{\epsilon_{a}}^{bc}\nabla_{e}\nabla_{b}A_{c}$
(23) $\displaystyle=$
$\displaystyle(g^{dc}g^{eb}-g^{db}g^{ec})\nabla_{e}\nabla_{b}A_{c}=\nabla_{a}\nabla^{a}A^{d}-\nabla_{a}\nabla^{d}A^{a}$
$\displaystyle=$
$\displaystyle\nabla_{a}\nabla^{a}A^{d}+\nabla^{d}\nabla_{a}A^{a}-\nabla_{a}\nabla^{d}A^{a}=\nabla_{a}\nabla^{a}A^{d}+R^{dabc}A_{c}g_{ab}$
$\displaystyle=$ $\displaystyle\nabla_{a}\nabla^{a}A^{d}-R^{dc}A_{c}.$

On the other hand, by the Lie derivative acting on this vector field, we have

$\displaystyle\mathcal{L}_{X}\mathcal{L}_{Y}A_{a}$ $\displaystyle=$
$\displaystyle
X^{b}\nabla_{b}(\mathcal{L}_{Y}A_{a})+\mathcal{L}_{Y}A_{b}\nabla_{a}X^{b}$
(24) $\displaystyle=$ $\displaystyle
X^{b}\nabla_{b}(Y^{c}\nabla_{c}A_{a}+A_{c}\nabla_{a}Y^{c})+(Y^{c}\nabla_{c}A_{b}+A_{c}\nabla_{b}Y^{c})\nabla_{a}X^{b}$
$\displaystyle=$
$\displaystyle(X^{b}\nabla_{b}Y^{c})\nabla_{c}A_{a}+X^{b}Y^{c}\nabla_{b}\nabla_{c}A_{a}+(X^{b}\nabla_{a}Y^{c})\nabla_{b}A_{c}+A_{c}X^{b}\nabla_{b}\nabla_{a}Y^{c}$
$\displaystyle+(Y^{c}\nabla_{a}X^{b})\nabla_{c}A_{b}+A_{c}\nabla_{b}(Y^{c}\nabla_{a}X^{b})-A_{c}Y^{c}\nabla_{b}\nabla_{a}X^{b}.$

Whence we can further obtain

$\displaystyle\mathcal{C}^{2}A_{a}$ $\displaystyle=$
$\displaystyle{{Z^{b}}_{b}}^{c}\nabla_{c}A_{a}+H^{bc}\nabla_{b}\nabla_{c}A_{a}+2{{Z^{c}}_{a}}^{b}\nabla_{c}A_{b}+A_{c}\nabla_{b}{{Z^{c}}_{a}}^{b}+A^{c}R_{cabd}H^{bd}-A_{c}R_{ad}H^{dc}$
(25) $\displaystyle=$
$\displaystyle\ell^{2}g^{bc}\nabla_{b}\nabla_{c}A_{a}+\ell\Lambda{\epsilon_{a}}^{cb}\nabla_{c}A_{b}-\ell^{2}R_{ac}A^{c}$
$\displaystyle=$ $\displaystyle[(\mu\ell)^{2}-\Lambda\mu\ell]A_{a},$

where the identity (16) is used in the first step.

With the above experience, let us manipulate the spinor field, which is more
involved. To this end, we start with the Dirac equation

$(\gamma^{a}\nabla_{a}+\mu)\psi=0.$ (26)

Here $\gamma^{a}=e_{I}^{a}\gamma^{I}$ and the covariant derivative acting on
the spinor field is given by
$\nabla_{a}=\partial_{a}+\frac{1}{4}\omega_{IJa}\gamma^{IJ}$, where
$e_{I}^{a}$ constitute a set of orthogonal normal vector bases, and Gamma
matrices satisfy $\\{\gamma^{I},\gamma^{J}\\}=2\eta^{IJ}$ with the spin
connection $\omega_{IJa}=e_{Ib}\nabla_{a}e_{J}^{b}$ and
$\gamma^{IJ}=\frac{1}{2}[\gamma^{I},\gamma^{J}]$. Next acting on the Dirac
equation with $\gamma^{b}\nabla_{b}-\mu$, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(\gamma^{b}\nabla_{b}-\mu)(\gamma^{a}\nabla_{a}+\mu)\psi=(\gamma^{a}\gamma^{b}\nabla_{a}\nabla_{b}-\mu^{2})\psi$
(27) $\displaystyle=$
$\displaystyle(g^{ab}\nabla_{a}\nabla_{b}+\gamma^{ab}\nabla_{a}\nabla_{b}-\mu^{2})\psi=(\nabla_{a}\nabla^{a}-\mu^{2})\psi+\gamma^{ab}\nabla_{[a}\nabla_{b]}\psi$
$\displaystyle=$
$\displaystyle(\nabla_{a}\nabla^{a}-m^{2}+\frac{1}{8}R_{abcd}\gamma^{ab}\gamma^{cd})\psi,$

where $\gamma^{ab}=e_{I}^{a}e_{J}^{b}\gamma^{IJ}$. On the other hand, the Lie
derivative of spinor fields with respect to Killing fields is given byCD

$\mathcal{L}_{X}\psi=X^{a}\nabla_{a}\psi-\frac{1}{4}\gamma^{ab}\psi\nabla_{b}X_{a},$
(28)

whereby we have

$\displaystyle\mathcal{L}_{X}\mathcal{L}_{Y}\psi$ $\displaystyle=$
$\displaystyle
X^{a}\nabla_{a}\mathcal{L}_{Y}\psi-\frac{1}{4}\gamma^{ab}\mathcal{L}_{Y}\psi\nabla_{b}X_{a}$
$\displaystyle=$ $\displaystyle
X^{a}\nabla_{a}(Y^{c}\nabla_{c}\psi-\frac{1}{4}\gamma^{cd}\psi\nabla_{d}Y_{c})-\frac{1}{4}\gamma^{ab}(Y^{c}\nabla_{c}\psi-\frac{1}{4}\gamma^{cd}\psi\nabla_{d}Y_{c})\nabla_{b}X_{a}$
$\displaystyle=$
$\displaystyle(X^{a}\nabla_{a}Y^{c})\nabla_{c}\psi+X^{a}Y^{c}\nabla_{a}\nabla_{c}\psi-\frac{1}{4}\gamma^{cd}\psi
X^{a}\nabla_{a}\nabla_{d}Y_{c}-\frac{1}{4}(X^{a}\nabla_{d}Y_{c})\gamma^{cd}\nabla_{a}\psi$
$\displaystyle-\frac{1}{4}(Y^{c}\nabla_{b}X_{a})\gamma^{ab}\nabla_{c}\psi+\frac{1}{16}\gamma^{ab}\gamma^{cd}\psi\nabla_{d}(Y_{c}\nabla_{b}X_{a})-\frac{1}{16}\gamma^{ab}\gamma^{cd}\psi
Y_{c}\nabla_{d}\nabla_{b}X_{a}.$

Then it is not hard to show

$\displaystyle\mathcal{C}^{2}\psi$ $\displaystyle=$
$\displaystyle{{Z^{a}}_{a}}^{c}\nabla_{c}\psi+H^{ac}\nabla_{a}\nabla_{c}\psi-\frac{1}{4}\gamma^{cd}\psi
R_{cdae}H^{ae}-\frac{1}{2}{Z^{a}}_{dc}\gamma^{cd}\nabla_{a}\psi$ (30)
$\displaystyle+\frac{1}{16}\gamma^{ab}\gamma^{cd}\psi\nabla_{d}Z_{cba}-\frac{1}{16}\gamma^{ab}\gamma^{cd}\psi
R_{abde}{H^{e}}_{c}$ $\displaystyle=$
$\displaystyle\ell^{2}\nabla_{a}\nabla^{a}\psi+\frac{\Lambda\ell}{4}{\epsilon^{a}}_{bc}\gamma^{cb}\nabla_{a}\psi+\frac{\ell^{2}}{16}R_{abcd}\gamma^{ab}\gamma^{cd}\psi$
$\displaystyle=$
$\displaystyle\ell^{2}(\mu^{2}-\frac{1}{16}R_{abcd}\gamma^{ab}\gamma^{cd})\psi-\frac{\Lambda\ell}{4}{\epsilon^{a}}_{bc}\gamma^{bc}\nabla_{a}\psi$
$\displaystyle=$
$\displaystyle\ell^{2}(\mu^{2}-\frac{1}{16}R_{abcd}\epsilon^{abe}\epsilon^{cdf}\gamma_{e}\gamma_{f})\psi+\frac{\Lambda\ell}{2}\gamma^{a}\nabla_{a}\psi$
$\displaystyle=$
$\displaystyle[(\mu\ell)^{2}-\frac{\Lambda\mu\ell}{2}]\psi-(\frac{\Lambda^{2}}{16}g_{ef}+\frac{1-\Lambda^{2}}{4\Lambda^{2}\ell^{2}}W_{e}W_{f})\gamma^{(e}\gamma^{f)}\psi$
$\displaystyle=$
$\displaystyle[(\mu\ell)^{2}-\frac{\Lambda\mu\ell}{2}]\psi-(\frac{3\Lambda^{2}}{16}+\frac{1-\Lambda^{2}}{4})\psi$
$\displaystyle=$
$\displaystyle[(\mu\ell)^{2}-\frac{\Lambda\mu\ell}{2}+\frac{\Lambda^{2}-4}{16}]\psi,$

where the identity $\gamma^{ab}=\epsilon^{abc}\gamma_{c}$ and Eq. (7) have
been used in the fourth and fifth steps, respectively.

So the upshot of the whole bulk of tensor and spinor analysis presented above
is that the solutions to the equations of motion for various fields all turn
out to fall into the representations of the $SL(2,R)\times U(1)$ Lie algebra
characterized by the value of the Casimir, i.e.,

$\mathcal{C}^{2}\Phi=\lambda\Phi$ (31)

with $\lambda=(u\ell)^{2}$ for the scalar field,
$\lambda=(u\ell)^{2}-\Lambda\mu\ell$ for the vector field, and
$\lambda=(u\ell)^{2}-\frac{\Lambda\mu\ell}{2}+\frac{\Lambda^{2}-4}{16}$ for
the spinor field.

With this in mind, we can construct the quasinormal modes by the standard
algebraic approach. Namely, we start from the highest weight mode of the
$SL(2,R)$ Lie subalgebra with a fixed azimuthal quantum number $m$ as follows

$\mathcal{L}_{W_{0}}\Phi^{(m0)}=im\Phi^{(m0)},\quad\mathcal{L}^{2}\Phi^{(m0)}=\lambda_{L}\Phi^{(m0)},\quad\mathcal{L}_{L_{+1}}\Phi^{(m0)}=0,\quad\mathcal{L}_{L_{0}}\Phi^{(m0)}=h\Phi^{(m0)},$
(32)

where the component of $\Phi^{m0}$ associated with the orthogonal normal basis
chosen in Appendix B can be written formally as

$\Phi^{(m0)}=e^{-hT+im\phi}\Psi^{(m0)}(x)$ (33)

with the conformal weight $h$ and the eigenvalue $\lambda_{L}$ of the
$SL(2,R)$ Casmir given respectively by

$h=\frac{1\pm\sqrt{1+4\lambda_{L}}}{2},\quad\lambda_{L}=\lambda-\frac{(\Lambda^{2}-1)m^{2}}{\Lambda^{2}}.$
(34)

Then the quasinormal modes can be obtained as the infinite tower of the
descendent modes, i.e.,

$\Phi^{(mn)}=\mathcal{L}_{L_{-1}}^{n}\Phi^{(m0)}$ (35)

with $n=0,1,2,\cdots$. Note that
$\mathcal{L}_{L_{0}}\Phi^{(mn)}=(h+n)\Phi^{(mn)}$, so we have the following
spectrum of quasinormal frequencies

$\omega_{mn}=-i(h+n)$ (36)

with the imaginary part of $\omega_{mn}$ required to be negative by
definition.

We conclude this section by mentioning that in the eikonal regime $|m|\gg 1$,
$\lambda$ in Eq. (34) can be neglected. This amounts to saying that the
effects from the mass and spin are both subleading in the eikonal limit, which
substantiates the claim made in KLS .

##  5 Conclusion

Partially motivated by the emergent $SL(2,R)$ symmetry in the photon sphere
quasinormal modes of the Kerr black hole, we have successfully obtained the
analytic expression for the spectrum of quasinormal modes of the scalar,
vector, and spinor fields in the exactly soluble self-dual warped AdS black
hole in a uniform manner by fully exploiting its $SL(2,R)\times U(1)$
isometry. To achieve this, we have introduced the two tensor fields associated
with the Casimir of the full $SL(2,R)\times U(1)$ Lie algebra and unveiled
their pleasing relations to the metric and volume element respectively. Then
we show that the solutions to the equations of motion of the scalar, vector,
and spinor fields all fall into the representations of the $SL(2,R)\times
U(1)$ Lie algebra by our tensor and spinor analysis, where no specific
coordinate system is used and the aforementioned two tensor fields make
$SL(2,R)\times U(1)$ symmetry transparent in the whole analysis. The resultant
spectrum of quasinormal modes can be further constructed as the highest weight
representation of the $SL(2,R)$ Lie subalgebra.

Although the self-dual warped AdS black hole is more involved than the
simplest BTZ black hole, our work demonstrates that the strategy previously
developed in Zhang  for the algebraic approach to the spectrum quasinormal
modes in the BTZ black hole turns out to be utterly applicable to the self-
dual warped AdS black hole. As we know, there are other warped AdS black hole
solutions with the $SL(2,R)\times U(1)$ isometryMCL ; BC ; ALPSS ; CMN  ,
where a variety of quasinormal modes have been calculated out mainly by
solving the equations of motion analyticallyCX1 ; CX2 ; CN ; CNX ; LR . With
our present work, we are convinced that the derivation of these quasinormal
modes can also be made as simple as possible by our algebraic approach.

## Acknowledgements

This work is partly supported by the National Key Research and Development
Program of China with Grant No. 2021YFC2203001 and National Natural Science
Foundation of China with Grant No. 12075026.

## Appendices

### A An explicit calculation of $Z$

By $\nabla_{a}\xi_{b}=\frac{1}{2}(d\xi)_{ab}$ for any Killing field $\xi$, we
have

$\displaystyle
Z_{abc}=Z_{[abc]}=\frac{1}{2}[L_{0[a}dL_{0bc]}-\frac{1}{2}(L_{+1[a}dL_{-1bc]}+L_{-1[a}dL_{+1bc]})-(1-\frac{1}{\Lambda^{2}}){W_{0}}_{[a}{dW_{0}}_{bc]}]$
(37) $\displaystyle=\frac{1}{6}[L_{0}\wedge dL_{0abc}-\frac{1}{2}(L_{+1}\wedge
dL_{-1abc}+L_{-1}\wedge
dL_{+1abc})-(1-\frac{1}{\Lambda^{2}}){W_{0}}\wedge{dW_{0}}_{abc}],$

where

$\displaystyle L_{0a}$ $\displaystyle=$ $\displaystyle
g_{ab}L_{0}^{b}=-\ell^{2}[(\Lambda^{2}\coth^{2}x-\text{csch}^{2}x)(dT)_{a}+\Lambda^{2}\coth
x(d\phi)_{a}]$ (38) $\displaystyle L_{-1a}$ $\displaystyle=$ $\displaystyle
g_{ab}L_{-1}^{b}=-e^{-T}\ell^{2}\text{csch}\ x[(\Lambda^{2}-1)\coth
x(dT)_{a}-(dx)_{a}+\Lambda^{2}(d\phi)_{a}],$ $\displaystyle L_{+1a}$
$\displaystyle=$ $\displaystyle g_{ab}L_{-1}^{b}=-e^{T}\ell^{2}\text{csch}\
x[(\Lambda^{2}-1)\coth x(dT)_{a}+(dx)_{a}+\Lambda^{2}(d\phi)_{a}],$

and

$\displaystyle(dL_{0})_{ab}$ $\displaystyle=$
$\displaystyle\ell^{2}\text{csch}^{2}x[\Lambda^{2}(dx)_{a}\wedge(d\phi)_{b}-2(\Lambda^{2}-1)\coth{x}(dT)_{a}\wedge(dx)_{b}]$
$\displaystyle(dL_{-1})_{ab}$ $\displaystyle=$ $\displaystyle
e^{-T}\ell^{2}\text{csch}\
x[(2-\Lambda^{2}-\Lambda^{2}\cosh^{2}{x})\text{csch}^{2}x(dT)_{a}\wedge(dx)_{b}-\Lambda^{2}(d\phi)_{a}\wedge(dT)_{b}$
$\displaystyle+\Lambda^{2}\coth{x}(dx)_{a}\wedge(d\phi)_{b}],$
$\displaystyle(dL_{+1})_{ab}$ $\displaystyle=$ $\displaystyle
e^{T}\ell^{2}\text{csch}\
x[(2-\Lambda^{2}-\Lambda^{2}\cosh^{2}{x})\text{csch}^{2}\
x(dT)_{a}\wedge(dx)_{b}+\Lambda^{2}(d\phi)_{a}\wedge(dT)_{b}$
$\displaystyle+\Lambda^{2}\coth{x}(dx)_{a}\wedge(d\phi)_{b}],$
$\displaystyle(dW_{0})_{ab}$ $\displaystyle=$
$\displaystyle\ell^{2}\Lambda^{2}\text{csch}^{2}x(dT)_{a}\wedge(dx)_{b}.$ (40)

With this, we can finally obtain

$Z=-\frac{\Lambda^{2}\ell^{4}\text{csch}^{2}x}{2}dT\wedge dx\wedge d\phi.$
(41)

### B A little bit of spinor analysis

Associated with the choice of the orthogonal normal bases as
$e_{0}^{a}=\frac{\sinh{x}}{\ell}(\frac{\partial}{\partial
T})^{a}-\frac{\cosh{x}}{\ell}(\frac{\partial}{\partial\phi})^{a}$,$e_{1}^{a}=\frac{\sinh{x}}{\ell}(\frac{\partial}{\partial
x})^{a}$, and
$e_{2}^{a}=\frac{1}{\Lambda\ell}(\frac{\partial}{\partial\phi})^{a}$, the non-
vanishing spin connections can be written as

$\displaystyle\omega_{01a}=-\omega_{10a}$
$\displaystyle=\frac{(2-\Lambda^{2})\coth{x}}{2}(dT)_{a}-\frac{\Lambda^{2}}{2}(d\phi)_{a},$
(42) $\displaystyle\omega_{02a}=-\omega_{20a}$
$\displaystyle=-\frac{\Lambda\text{csch}\ x}{2}(dx)_{a},$
$\displaystyle\omega_{12a}=-\omega_{21a}$
$\displaystyle=\frac{\Lambda\text{csch}\ x}{2}(dT)_{a}.$

Thus we have

$\displaystyle\mathcal{L}_{L_{0}}\Psi(x)$
$\displaystyle=L_{0}^{a}\nabla_{a}\Psi(x)-\frac{1}{4}\gamma^{ab}\Psi(x)\nabla_{b}L_{0a}$
(43)
$\displaystyle=L_{0}^{a}\partial_{a}\Psi(x)+\frac{1}{4}L^{a}_{0}\omega_{aIJ}\gamma^{IJ}\Psi(x)-\frac{1}{4}\gamma^{IJ}\Psi(x)e_{I}^{a}e_{J}^{b}\nabla_{b}L_{0a}$
$\displaystyle=\frac{1}{4}\gamma^{IJ}\Psi(x)(L_{0}^{a}\omega_{aIJ}-e_{I}^{a}e_{J}^{b}\nabla_{b}L_{0a})=0.$

Similarly, we can also obtain $\mathcal{L}_{W_{0}}\Psi(x)=0$.

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