# Generator Parameter Estimation by Q-Learning Based on PMU Measurements

Seyyed Rashid Khazeiynasab  Electrical and Computer Engineering
University of Central Florida
Orlando, FL 32816 USA
<EMAIL_ADDRESS>Junjian Qi  Electrical and Computer Engineering
Stevens Institute of Technology
Hoboken, NJ 07030 USA
<EMAIL_ADDRESS>Issa Batarseh  Electrical and Computer Engineering
University of Central Florida
Orlando, FL 32816 USA
<EMAIL_ADDRESS>
###### Abstract

In this paper, a novel Q-learning based approach is proposed for estimating
the parameters of synchronous generators using PMU measurements. Event
playback is used to generate model outputs under different parameters for
training the agent in Q-learning. We assume that the exact values of some
parameters in the model are not known by the agent in Q-learning. Then, an
optimal history-dependent policy for the exploration-exploitation trade-off is
planned. With given prior knowledge, the parameter vector can be viewed as
states with a specific reward, which is a function of the fitting error
compared with the measurements. The agent takes an action (either increasing
or decreasing the parameter) and the estimated parameter will move to a new
state. Based on the reward function, the optimal action policy will move the
parameter set to a state with the highest reward. If multiple events are
available, they will be used sequentially so that the updated
$\mathbfcal{Q}$-value can be utilized to improve the computational efficiency.
The effectiveness of the proposed approach is validated through estimating the
parameters of the dynamic model of a synchronous generator.

###### Index Terms:

Generator model, parameter estimation, phasor measurement unit (PMU),
Q-learning, sensitivity analysis.

##  I Introduction

In power systems, monitoring, protection, and control are usually model-based,
an accurate dynamic model for either synchronous generators [1, 2, 3] or
inverters [4] is thus essential. The inaccuracy of the power system model has
been witnessed in the blackout occurred in Western U.S. in 1996 [1], in which
model simulations showed a stable response while the system became unstable
[5, 6]. The synchronous generator is one of the most critical components in
power systems and its accurate modeling is important for studying the dynamics
of the system. This is no trivial task because: 1) The models may not be
available for all components; 2) Even if the models are available, the
parameters of the models may not be available; and 3) Even if the models and
the parameters are available, the parameters may have changed over time.

Offline methods have been proposed for estimating generator parameters, for
which the synchronous machine should be out of service. Besides, these methods
do not consider the changes of parameters due to aging [6]. With the
increasing installation of phasor measurement units (PMUs) that provide high-
quality online data to monitor the system status, there is an increasing
interest in estimating synchronous generator parameters using synchrophasor
data. The black-box optimization based method in [7] has a good estimation
accuracy, but for high-dimensional cases the estimation error will increase.
Kalman filter (KF) based methods are another popular methods that have been
applied to estimate the generator parameters. The extended Kalman filter (EKF)
used in [8] applies a linear approximation of models and its accuracy is thus
reduced [9]. The Ensemble Kalman filter (EnKF) [6] and Unscented Kalman filter
(UKF) [10] are further proposed to improve the accuracy of the estimation.
However, the KF based methods usually suffer from slow convergence rate [11].

In addition, Bayesian inference methods have been proposed for generator
parameter estimation [11, 12]. However, these methods need a likelihood
function for implementation which may lead to computational intractability for
high-dimensional cases. The accuracy of the method in [11] degrades under
large parameter errors. The performance of the method in [12] may deteriorate
for high-dimensional parameters.

Machine learning has been applied for parameter calibration. The method in
[13] generates extensive simulation data to train a multi-output convolutional
neural network (CNN) model and predict the generator parameters. However, it
requires a very large volume of data and expensive GPUs. Deep Q-learning based
methods are developed in [14, 15], in which multiple events are simultaneously
used for calibrating five parameters. Although deep Q-learning improves the
scalability of Q-learning and addresses its limitation of only working for
discrete and finite state and action spaces, it is very sensitive to hyper
parameters and requires a lot of tuning in order to converge. Its
implementation is also much more complicated since a lot of efforts are needed
to make the neural network function approximation actually work and the deep
Q-learning be stabilized. For generator parameter calibration problem in which
usually only a few critical parameters need to be calibrated, the benefit of
deep Q-learning is not obvious.

In this paper, different from [14, 15], we directly adopt Q-learning, a model-
free adaptive dynamic programming algorithm, to learn the optimal policy of
estimating parameters. The advantages and differences of our proposed approach
compared with [14, 15] include: 1) The implementation is much more
straightforward and reliable without the many complications in deep
Q-learning; 2) Q-learning works well for the parameter calibration problem
with a small number of parameters to be calibrated; 3) Instead of
simultaneously using multiple events we propose to use them sequentially to
improve the computational efficiency; and 4) We define a different reward
function based on the discrepancy between the PMU measurements and the outputs
of the model.

The remainder of this paper is organized as follows. Section II presents the
dynamic model of the synchronous generator and the event playback procedure in
generator model validation and parameter estimation. Section III introduces
the sensitivity approach for identifying critical parameters. In Section IV,
an overview is provided for reinforcement learning (RL), especially
Q-learning, and a Q-learning based method is developed for generator parameter
estimation. Section V presents case studies to validate the proposed method.
Finally concluding remarks are given in Section VI.

##  II Generator Dynamic Model

The dynamical model of a synchronous generator can be written in a general
form as:

$\displaystyle\dot{\bm{x}}=\bm{f}(\bm{x},\bm{u},\bm{\alpha})$ (1a)
$\displaystyle{\bm{y}}=\bm{h}(\bm{x},\bm{u},\bm{\alpha}),$ (1b)

where $\bm{f}(\cdot)$ and $\bm{h}(\cdot)$ are state transition and output
functions, $\bm{u}\in\mathbb{R}^{v}$ is the injected measured signals (voltage
magnitude, phase angle, and frequency), $\bm{y}\in\mathbb{R}^{o}$ is the
output variables (including active and reactive power of the generator),
$\bm{\alpha}\in\mathbb{R}^{z}$ is the parameter vector, and
$\bm{x}\in\mathbb{R}^{n}$ is the state vector that could include rotor angle,
rotor speed, transient or sub-transient voltages, and controller states.

The model includes a synchronous machine with GENTPJ model, an exciter with
ESST1A model, and a governor with IEEEG3 model. The d-axis GENTPJ model
without swing equations is shown in Fig. 1(a), where $X_{\mathrm{d}}$ and
$X_{\mathrm{q}}$ are d- and q-axis synchronous reactance,
$X^{\prime}_{\mathrm{d}}$ and $X^{\prime}_{\mathrm{q}}$ are d- and q-axis
transient synchronous reactance, $X^{\prime\prime}_{\mathrm{d}}$ and
$X^{\prime\prime}_{\mathrm{q}}$ are d- and q-axis sub-transient synchronous
reactance, $E^{\prime}_{\mathrm{d}}$ and $E^{\prime}_{\mathrm{q}}$ are d- and
q-axis transient voltages, $E^{\prime\prime}_{\mathrm{d}}$ and
$E^{\prime\prime}_{\mathrm{q}}$ are d- and q-axis sub-transient voltages.
$T^{\prime}_{\mathrm{do}}$ is d-axis transient open-circuit time constant,
$T^{\prime\prime}_{\mathrm{do}}$ is sub-transient open-circuit time constant,
$I_{\mathrm{d}}$ is d-axis current of the generator, and $E_{\mathrm{fd}}$ is
the field voltage.

The ESST1A exciter model is shown in Fig. 1(b), where $K_{\mathrm{A}}$ and
$K_{\mathrm{f}}$ are the main and feedback gains of the exciter,
$T_{\mathrm{C}}$, $T_{\mathrm{B}}$, $T_{\mathrm{C1}}$, $T_{\mathrm{B1}}$, and
$T_{\mathrm{A}}$ are time constants, $I_{\mathrm{LR}}$ is the maximum field
current, and $K_{\mathrm{LR}}$ is the gain on field current of the exciter.
The governor model is shown in Fig. 1(c), where $P_{\mathrm{m}}$ is the
mechanical power, $a_{23}$, $a_{21}$, $a_{11}$, and $a_{13}$ are turbine
coefficients, and the other parameters can be found in [16].

(a)

(b)

(c)

Figure 1: Generator, exciter, and governor models [16]: (a) d-axis GENTPJ
generator model; (b) ESST1A exciter model; (c) IEEEG3 governor model.

The “event playback” first uses a PMU to record the bus voltage magnitude,
phase angle, frequency, and active/reactive power at the point of common
coupling (PCC) during the events. Then by using the measurement signals it can
show the mismatches between real data and the model’s output [6, 8]. At time
step $k$, the output of the model is [6]:

$\displaystyle P_{\mathrm{model},k}$
$\displaystyle=\frac{E^{\prime\prime}_{k}V_{k}}{X^{\prime}_{\mathrm{d}}+X_{\mathrm{tr}}}\sin(\delta_{k}-\theta_{k})$
(2) $\displaystyle Q_{\mathrm{model},k}$
$\displaystyle=\frac{V^{2}_{k}-E^{\prime\prime}_{k}V_{k}\cos(\delta_{k}-\theta_{k})}{X^{\prime}_{\mathrm{d}}+X_{\mathrm{tr}}},$
(3)

where $E^{\prime\prime}_{k}$ and $\delta_{k}$ are, respectively, the generator
sub-transient voltage and rotor angle, $X_{\mathrm{tr}}$ is the reactance of
the step-up transformer, and $X^{\prime}_{\mathrm{d}}$ is the d-axis transient
synchronous reactance. The real and reactive power vectors from the model for
all time steps, denoted by $P_{\mathrm{model}}$ and $Q_{\mathrm{model}}$, are
compared with the real and reactive power measurement vectors for all time
steps, denoted by $P_{\mathrm{meas}}$ and $Q_{\mathrm{meas}}$. Let
$\bm{z}^{*}=[P_{\mathrm{meas}}^{\top}\;Q_{\mathrm{meas}}^{\top}]^{\top}$ be
the measurement from PMU and
$\bm{z}=[P_{\mathrm{model}}^{\top}\;Q_{\mathrm{model}}^{\top}]^{\top}$ be the
output of the model.

##  III Identifying Critical Parameters

After a model deficiency is revealed, the next step is to identify the
problematic parameters. A generator with its control can have many parameters.
Calibrating all of them could be computationally challenging and also not
every parameter is identifiable. Trajectory sensitivity has been used to
identify the most critical parameters [8]. The sensitivity of the output with
regard to parameter $\alpha_{i}$ can be calculated as [6]:

$\displaystyle S(\alpha_{i})$
$\displaystyle=\frac{1}{2K}\sum_{k=1}^{2K}{\frac{\alpha_{i}|z_{k}(\alpha_{i}^{+})-z_{k}(\alpha_{i}^{-})|}{\alpha_{i}^{+}-\alpha_{i}^{-}}},$

where $K$ is the number of time steps,
$\alpha_{i}^{+}=\alpha_{i}+\Delta\alpha_{i}$ and
$\alpha_{i}^{-}=\alpha_{i}-\Delta\alpha_{i}$, and $\Delta\alpha_{i}$ is a
small perturbation of $\alpha_{i}$. After sensitivity analysis, the parameters
selected to be estimated are those with large sensitivities [8, 6].

##  IV Q-Learning for Generator Parameter Estimation

Q-learning is a model-free RL algorithm with the goal of learning a policy to
tell an agent what action to take under what circumstances. In Q-learning, an
agent takes sequential actions at a series of states based on a state-action
value matrix, Q-table, until reaching an ultimate goal [17]. Let $\mathcal{A}$
and $\mathcal{S}$ be the action space and state space respectively. At each
episode $t$, the agent observes a state $s_{t}\in\mathcal{S}$ and chooses an
action $a_{t}\in\mathcal{A}$ based on policy $\pi$, which is a function that
maps states into actions. As a consequence of taking action $a_{t}$, the agent
receives a reward $R_{t}$ defined as $R_{t}=R(s_{t},a_{t},s_{t+1})$ and
observes the next state $s_{t+1}$ of the environment. The RL framework
considers the Markov decision process assumption, i.e $s_{t+1}$ is only
conditioned by $s_{t}$ and $a_{t}$ and is sampled according to the transition
probability $p(s_{t+1}|s_{t},a_{t})$. The above process is continued until the
agent reaches the last episode, called the terminal state [18].

The goal of the agent is to take actions so as to maximize the expected return
for a given state $s_{t}$. The expected return for selecting action $a$ in
state $s$, action-value function following a policy $\pi$, is defined as
$\mathbfcal{Q}^{\pi}(s,a)=\mathbb{E}[R_{t}|s=s_{t},a=a_{t},\pi]$ [18]. The
expected return at episode $t$ when the action $\pi^{*}$ maximizes the
$\mathbfcal{Q}$ function is given as:

$\displaystyle\mathbfcal{Q}^{\pi}(s_{t},a_{t})=\mathbb{E}\big{[}R_{t}+\gamma\,\mathrm{max}\underset{a_{t+1}\in\mathcal{A}}{\mathbfcal{Q}^{*}}(s_{t+1},a_{t+1})\big{]},$
(4)

where $\gamma\in[0,1]$ is the discount factor that weights the future rewards.
The state-action value functions are updated by [19]:

$\displaystyle\mathbfcal{Q}_{t+1}(s_{t+1},a_{t+1})=(1-\lambda)\mathbfcal{Q}_{t}(s_{t},a_{t})$
$\displaystyle+\lambda\big{[}R(s_{t},a_{t},s_{t+1})+\gamma\,\mathrm{max}\underset{a_{t+1}\in
A}{\mathbfcal{Q}_{t}}(s_{t+1},a_{t+1})\big{]},$ (5)

where $\lambda$ is the learning factor that controls the aggressiveness of the
learning. The balance of exploration and exploitation in Q-learning is
maintained by adopting a decaying $\varepsilon_{\mathrm{t}}$-greedy method
[18, 19], by which the agent takes random actions at the beginning while
reducing the randomness during the learning process [19]. The optimal policy
at episode $t$ is represented by a greedy strategy as:

$\displaystyle a_{t}=\underset{a\in A}{\arg\max}\;\mathbfcal{Q}_{t}(s,a).$ (6)

The values of $\mathbfcal{Q}$ are initialized to be zero and are updated
repeatedly by (IV) based on the action reward in the current state and the
maximum reward in the next state. The algorithm will converge to the optimal
policy, $\mathbfcal{Q}^{*}$, after $N$ episodes.

###  IV-A Design of State and Action

The state is the coordinate information of the spots that the agent moves in
the parameter space:

$\displaystyle\mathcal{S}=\\{s_{1},s_{2},\cdots,s_{N_{\mathrm{s}}}\\},$ (7)

where $N_{\mathrm{s}}$ is the number of states, which equals the number of
select-able states in the parameter space. For each element $1\leq j\leq L$ in
the parameter vector
$\bm{\alpha}_{\mathrm{c}}=[\alpha_{\mathrm{c},1},\cdots,\alpha_{\mathrm{c},L}]^{\top}$,
if the upper bounds and lower bounds of their priors are
$\bm{\alpha}_{\mathrm{c}}^{\mathrm{u}}=[\alpha_{\mathrm{c},1}^{\mathrm{u}},\cdots,\alpha_{\mathrm{c},L}^{\mathrm{u}}]^{\top}$
and
$\bm{\alpha}_{\mathrm{c}}^{\mathrm{l}}=[\alpha_{\mathrm{c},1}^{\mathrm{l}},\cdots,\alpha_{\mathrm{c},L}^{\mathrm{l}}]^{\top}$respectively.
We define a maximum acceptable error as $\tau$, then we discrete the domain
for parameter $j$ by $\Delta s_{j}$ given as:

$\displaystyle\Delta
s_{j}=2\tau\big{(}\alpha_{\mathrm{c},j}^{\mathrm{u}}-\alpha_{\mathrm{c},j}^{\mathrm{l}}\big{)}.$
(8)

For a parameter with initial value 5 and acceptable error as $1\%$, if the
prior is uniform as $\mathcal{U}(0,10)$, then $\Delta s=0.2$ and the number of
states for this parameter is equal to 50.

The action space $\mathcal{A}$ is composed of the select-able movement in the
parameter space:

$\displaystyle\mathcal{A}=\\{a_{1},a_{2},\cdots,a_{N_{\mathrm{a}}}\\},$ (9)

where $N_{\mathrm{a}}$ is the number of actions. The actions are increasing or
decreasing the parameter $j=1,\ldots,L$ by $\Delta s_{j}$ in the states. For
example, when there are two parameters to be calibrated, we have four
different actions: up, down, right, and left. The new state is updated
according to the chosen action.

###  IV-B Design of Dynamic Reward Function

For some RL problems, the reinforcement signal may not appear immediately
after the action that triggers the rewards [18]. In this type of problems, RL
faces delayed rewards. Handling delayed rewards is one of the major challenges
in RL. The agent must interact with the environment to adapt to it, and may
need to spend a lot of time in attaining the optimal behavior. Reward
functions describe how the agent “ought” to behave in this process based on
which the agent learns how to move in the environment [18]. To accelerate this
learning process, the reward function should describe the agent’s state in a
timely and accurate manner. If the reward function is better behaved, the
agent will learn better. Therefore, the design of the reward function is
critical for RL.

True parameters are assumed to be unknown. The reward function is based on the
discrepancy between the simulated outputs and the measurements,
$\epsilon_{\mathrm{s}}(\bm{z},\bm{z}^{*})$, is defined below:

$\displaystyle\epsilon_{\mathrm{s}}(\bm{z},\bm{z}^{*})=\frac{1}{2K}\left\lVert\bm{z}-\bm{z}^{*}\right\rVert_{1},$
(10)

where $\left\lVert\cdot\right\rVert_{1}$ is the 1-norm of a vector. Based on
the predefined thresholds $\bar{\epsilon}$ and $\underline{\epsilon}$, the
reward function is defined as:

$\displaystyle R(s,a)$
$\displaystyle=\begin{cases}\frac{10}{\epsilon_{\mathrm{s}}+0.01},&\text{if
$\,\epsilon_{\mathrm{s}}<\underline{\epsilon}$}\\\ 0,&\text{if
$\,\underline{\epsilon}\leq\epsilon_{\mathrm{s}}\leq\bar{\epsilon}$}\\\
-10(\epsilon_{\mathrm{s}}-\bar{\epsilon}),&\text{if
$\,\epsilon_{\mathrm{s}}>\bar{\epsilon}$}.\end{cases}$ (11)

If $\epsilon_{\mathrm{s}}<\underline{\epsilon}$, the agent gets positive
rewards and the states with smaller $\epsilon_{\mathrm{s}}$ get larger
rewards; if $\underline{\epsilon}\leq\epsilon_{\mathrm{s}}\leq\bar{\epsilon}$,
the agent does not get any rewards; if $\epsilon_{\mathrm{s}}>\bar{\epsilon}$,
the agent gets negative rewards.

###  IV-C Proposed Algorithm for Generator Parameter Estimation

We define a matrix $\mathcal{O}$ to show the status of every state during the
exploration and exploitation. Each state has a ‘searched’ or ‘un-searched’
status, and when the agent visits a state, the status of that state is set to
be ‘searched’. This matrix is useful for guaranteeing to run the model in each
state only once. If a state is explored by the agent for the first time, we
calculate $\epsilon_{\mathrm{s}}(\bm{z},\bm{z}^{*})$ and assign a reward for
it based on (11). The proposed Q-learning based algorithm for generator
parameter estimation is presented in Algorithm 1 [17]. In this algorithm
$\mathrm{Model}(\bm{\alpha}_{\mathrm{c}}^{*})$ generates the model outputs
$\bm{z}$ under parameter $\bm{\alpha}_{\mathrm{c}}^{*}$ using the generator
model and the event playback procedure in Section II. If multiple events are
available, we will use them sequentially so that the updated
$\mathbfcal{Q}$-value obtained from the training for the previous events can
be utilized to significantly improve the computational efficiency.

1:Set hyper parameters $\lambda,\gamma,\varepsilon,N$

2:Initialize experience pool $\mathbfcal{O}$ as an empty set

3:Discretize the parameter space based on (8)

4:$\mathbfcal{Q}\leftarrow{\bm{0}}$

5:for $1\leq t\leq N$ do

6: Start with an un-searched state with parameter
$\bm{\alpha}_{\mathrm{c}}^{*}$

7: Generate data $\bm{z}$ from $\bm{\alpha}_{\mathrm{c}}^{*}$:
$\bm{z}\sim\mathrm{Model}(\bm{\alpha}_{\mathrm{c}}^{*})$

8: Calculate discrepancy $\epsilon_{\mathrm{s}}(\bm{z},\bm{z}^{*})$ based on
(10)

9: Assign a reward to the state based on (11) or $\mathbfcal{O}$

10: With probability $\varepsilon$, select a random action $a_{t}$; otherwise
select $a_{t}=\underset{a\in A}{\arg\max}\;\mathbfcal{Q}_{t}(s,a)$

11: Update $\mathbfcal{Q}$-value matrix based on (IV)

12: Update $\mathbfcal{O}$-value matrix based on (11)

13:end for

Algorithm 1 Q-Learning based method for estimating the parameter
$\bm{\alpha}_{\mathrm{c}}$

##  V Simulation Results

We implement our method based on PSS/E and Python 2.7 and test it on the same
system used in [20]. All tests are carried out on a PC with Intel(R) Core(TM)
i7-8700 and 8 GB RAM. A PMU is installed at the 230-kV level of the
substation. The sampling rate of the PMU is 30 sample/s.

###  V-A Hyperparameter Setting

The discount factor $0<\gamma<1$, which makes a trade-off between the
immediate and long-term reward. In this paper, we consider $\gamma=0.9$. The
learning rate $\lambda$ determines the learning rate of the agent when
updating $\mathbfcal{Q}$-value of each state-action pair. We choose this
parameter to be $0.3$. The number of episodes $N$ is chosen as 2000. The
$\varepsilon_{\mathrm{t}}$ that determines the balance of exploration and
exploitation in Q-learning is chosen as $0.2$ [18]. We choose
$\underline{\epsilon}=0.001$ and $\bar{\epsilon}=2$ in (11).

###  V-B Critical Parameter Identification

For sensitivity analysis, a small perturbation
$\Delta\alpha_{i}=5\%|\alpha_{i}|$ is applied to each parameter. The top four
parameters and their sensitivities are listed in Table I. These are the gain
of the exciter $K_{\mathrm{A}}$, the time constant of the exciter
($T_{\mathrm{b}}$), the turbine coefficient ($a_{23}$), and the d-axis
transient rotor time constant ($T_{\mathrm{pdo}}$). They are identified as
critical parameters.

TABLE I:
Sensitivity Analysis of Parameters Parameter | Sensitivity
---|---
$K_{\mathrm{A}}$ | 1.75
$T_{\mathrm{B}}$ | 1.35
$a_{23}$ | 1.13
$T_{\mathrm{pdo}}$ | 1.11

###  V-C Estimation for Two Parameters

In practical implementation, the parameters provided by the manufacturer may
change due to aging. To consider this uncertainty, we assume that the
available parameters are deviated from the true values. We estimate the same
two parameters as in [12], which are the moment of inertia $H$ for synchronous
generator and the amplifier gain $K_{\mathrm{A}}$ for the exciter. Their true
values are $H_{\mathrm{True}}=5.4$ and $K_{\mathrm{A}_{\mathrm{True}}}=125$.
We assume that the mean values of these parameters are 10% greater than their
true values to account for parameter uncertainties. We choose the lower/upper
bounds of the uniform prior distributions as 50% or 70% less/greater than the
mean values, respectively. The results of parameter estimation under different
prior distributions are shown in Table II. It is seen that the proposed method
can provide accurate estimation of the parameters.

TABLE II:
Parameter Calibration Under Different Prior Distributions |  $H$ ($H_{\mathrm{True}}=5.40$)
---
|  $K_{\mathrm{A}}$ ($K_{\mathrm{A}_{\mathrm{True}}}=125$)
---
Prior |  | Estimated
---
(Error (%))
Prior |  | Estimated
---
(Error (%))
$\mathcal{U}$(2.9, 8.9) |  | 5.39
---
(0.1)
$\mathcal{U}$(68.8, 206.3) |  | 125.8
---
(0.6)
$\mathcal{U}$(1.8, 10.1) |  | 5.37
---
(0.6)
$\mathcal{U}$(41.2, 233.8) |  | 123.8
---
(0.9)

Fig. 2(a) shows the cumulative rewards for the case in the last row of Table
II. It is seen that the training converges after 421 iterations, which takes 5
hours.

###  V-D Estimation for Four Parameters

We also demonstrate the performance of the proposed method for estimating the
four critical parameters identified in Table I. We assumed that the mean
values of these parameters are 10% greater than their true values. We choose
the lower/upper bound of the uniform prior distributions as 50% less/greater
than the mean values. Table III lists the estimated values and the percentage
errors. It is seen that the estimate is very close to the true values. Fig.
2(b) shows the corresponding cumulative rewards. The training converges after
1000 iterations which takes 8 hours. Fig. 3 shows the results for real and
reactive power under the estimated parameters and the parameters before the
calibration. We consider the parameters before the calibration as 10% greater
than the true values. Before calibration there is obvious discrepancy between
the model outputs and the PMU measurements while with the estimated parameters
the model outputs match the PMU measurements very well.

TABLE III:
Calibration of Four Critical Parameters Parameter | True value | Estimated value | Error (%)
---|---|---|---
$K_{\mathrm{A}}$ | 125 | 123.8 | 0.9
$T_{\mathrm{b}}$ | 3.86 | 3.82 | 1
$a_{23}$ | 1.102 | 1.10 | 0.1
$T_{\mathrm{pdo}}$ | 5.4 | 5.34 | 1

(a)

(b)

Figure 2: Cumulative rewards for two and four-parameter case: (a) Two
parameter case; (b) Four parameter case.

(a)

(b)

Figure 3: Model performance before and after parameter calibration: (a) Real
power; (b) Reactive power. — PMU measurements; . model before calibration; \-
- model after calibration.

###  V-E Offline Training and Online Application

Parameter estimation is also conducted for another event and for the case in
which the true parameters are different from those in the case in Section V-C.
For these cases, we use the updated $\mathbfcal{Q}$-value in Section V-C which
can be considered as offline training. The cumulative reward for the second
event is shown in Fig. 4(a). It can be seen that the agent learns to find the
true parameters in 110 iterations which is much less than that needed for the
first event. For the case in which the true parameters are different, we use
the outputs of the model under $H=4.4$ and $K_{\mathrm{A}}=100$ as PMU
measurements and use the same prior distributions as in Section V-C. As can be
seen from Fig. 4(b), the agent learns to find the true parameter after 50
iterations which is less than the number of iterations in the case in Section
V-C. The estimations using updated $\mathbfcal{Q}$-value take around 5
minutes. The results here show that using the updated $\mathbfcal{Q}$-value
obtained from offline training can significantly improve the computational
efficiency of the online parameter estimation. This also indicates that the
computational efficiency can be greatly improved if multiple events are used
sequentially so that the learning for a later event can leverage the updated
$\mathbfcal{Q}$-value obtained from the learning for a previous event.

(a)

(b)

Figure 4: Cumulative rewards for (a) another event and (b) a case with
different true parameters.

##  VI Conclusion

This paper proposes a Q-learning based method for generator model parameter
estimation using PMU measurements. The simulation results for a decentralized
generator show that the proposed method can accurately estimate the system
parameters and the updated $\mathbfcal{Q}$-value obtained from offline
training can significantly improve the computational efficiency of the online
parameter estimation. Improving the scalability of Q-learning and addressing
curse of dimensionality while maintaining its relatively simple implementation
is a very challenging problem and will be studied in our future research.

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# Single production of vectorlike $B$ quarks at the CLIC

Jin-Zhong Han111E-mail<EMAIL_ADDRESS>Jing Yang, Shuai Xu, Han-Kui
Wang  School of Physics and Telecommunications Engineering, Zhoukou Normal
University, Zhoukou 466001, China

###### Abstract

The vector-like quarks are predicted in many new physics scenarios beyond the
Standard Model (SM) and could be seen potential signatures of new physics at
the TeV energy scale. In this work, we study single production of exotic
singlet and doublet vectorlike bottom quarks (VLQ-$B$) at future Compact
Linear Collider (CLIC) via the process $e^{+}e^{-}\to B\bar{b}$ with the decay
channel $B\to bZ$ and two types of modes: $Z\to\ell^{+}\ell^{-}$ and
$Z\to\nu\bar{\nu}$. We calculate the cross sections of signal and relevant SM
backgrounds. After a fast simulation of the signal and background events, the
exclusion limit at 95% confidence level and $5\sigma$ discovery prospects on
the parameters (the coupling strength $\kappa_{B}$ and the VLQ-$B$ mass) have
been, respectively, presented at the future CLIC with centre of mass energy
$\sqrt{s}=3$ TeV and integrated luminosity of 5 ab-1.

##  I Introduction

In order to solve the gauge hierarchy problem, the vector-like quarks (VLQs)
are predicted to regulate the Higgs boson mass-squared divergence
DeSimone:2012fs  in several extensions of the Standard Model (SM), such as
little Higgs models ArkaniHamed:2002qy , extra dimensions Agashe:2006wa ,
composite Higgs models Agashe:2004rs , and other TeV Scale new physics (NP)
models He:1999vp ; Wang:2013jwa ; He:2001fz ; He:2014ora . The left- and
right-handed components of these new VLQs are transformed in the same
properties under the SM electroweak symmetry group Aguilar-Saavedra:2013qpa .
The VLQs are therefore not excluded by present searches, unlike a fourth
generation of SM quarks that is ruled out by electroweak precision
measurements Kribs:2007nz ; Banerjee:2013hxa . Based on the electric charges
of $+2/3e$ ($T$ quark), $-1/3e$ ($B$ quark), $+5/3e$ ($X$ quark) or $-4/3e$
($Y$ quark), the VLQs could be grouped in multiplets, such as electroweak
singlet [$T$, $B$], electroweak doublets [ $\left(X,T\right),\left(T,B\right)$
or $\left(B,Y\right)$], or electroweak triplets [$\left(X,T,B\right)$ or
$\left(T,B,Y\right)$], and they could generate characteristic signatures at
the current and future high-energy colliders, see e.g. Atre:2011ae ;
Buchkremer:2013bha ; Barducci:2017xtw ; Cacciapaglia:2018qep ; Fuks:2016ftf ;
Yang:2014usa ; Liu:2018hum ; Cacciapaglia:2018lld ; Aguilar-Saavedra:2019ghg ;
Wang:2020ips ; Zhang:2017nsn ; Han:2017cvu ; Liu:2017rjw ; Liu:2017sdg ;
Liu:2019jgp ; Tian:2021oey ; Moretti:2016gkr ; Moretti:2017qby ;
Carvalho:2018jkq ; Roy:2020fqf ; Buckley:2020wzk ; Deandrea:2021vje ;
King:2021iah . Here we focus on the singlet or $(B,Y)$ doublet VLQ-$B$ quark,
which only couples to third-generation SM quarks.

Using the Run 2 data, the direct searches for VLQ-$B$ have been performed and
the constraints on VLQ-$B$ have been obtained at 95% confidence level (CL) by
the ATLAS and CMS Collaborations Aaboud:2018wxv ; Aaboud:2018xpj ;
Aaboud:2018uek ; Aaboud:2018ifs ; Sirunyan:2018qau ; Sirunyan:2019sza ;
Sirunyan:2018omb ; Sirunyan:2020qvb ; Aaboud:2018pii . For instance, an
analysis from CMS including single-lepton, dilepton, and multilepton final
states probed all decay modes of the VLQ-$B$, and excluded $B$ quark masses in
the range $910-1240$ GeV Sirunyan:2018omb . Recently, the CMS Collaboration
presented a search for VLQ-$B$ pair production in the fully hadronic final
state Sirunyan:2020qvb , and excluded the $B$ masses up to 1570, 1390, and
1450 GeV for 100% $B\to bh$, 100% $B\to tZ$, and $BY$ doublet cases,
respectively. The combination of searches utilizing various final states were
performed by the ATLAS Collaboration Aaboud:2018pii , and excluded values of
the $B$ mass up to 1220, 1370, and 1140 GeV for the singlet, the $(T,B)$
doublet and $(B,Y)$ doublet cases, respectively.

Up to now, many phenomenological analysis about the VLQ-$B$ have been
performed at the LHC and LHeC Nutter:2012an ; Gong:2019zws ; Gong:2020ouh .
Compared to the complicated QCD background at the hadron colliders, the future
linear $e^{+}e^{-}$ collider has a particularly clear background environment,
i.e., the final stage of Compact Linear Collider (CLIC) are operating at an
energy of 3 TeV CLIC1 ; CLIC2 ; CLIC3 ; Dannheim:2013ypa . Thus, the high-
energy linear collider is a precision machine that can accurately measure the
characteristics of the new VLQs Kitano:2002ss ; Kong:2007uu ; Senol:2011nm ;
Harigaya:2011yg ; Guo:2014piv ; Liu:2014pts ; Qin:2021cxl ; Han:2021kcr . In
this work, we focus on the observability of the single VLQ-$B$ production at
the CLIC via the process $e^{+}e^{-}\to B\bar{b}~{}(b\bar{B})$ combined with
the $B\to bZ$ and the subsequent decay channels $Z\to\ell^{+}\ell^{-}$ and
$Z\to\nu\bar{\nu}$, respectively. The advantage is that it has a higher
potential than paired production due to less phase space suppression. In
addition, this process can reveal the electroweak nature of the interaction
between the VLQ-$B$ and $Z$ boson. Therefore, we expect that once the VLQ-$B$
is discovered and its mass is determined, such work could serve as a
complementary option for future high-energy linear colliders.

The paper is organized as follows. In section II, we give a brief description
of the simplified model including the VLQ-$B$ with electrical charge $-1/3$,
and discuss its single production at the CLIC. In section III we investigate
the signal and discovery potential of the VLQ-$B$ in the $Zb$ decay channel at
the CLIC. Finally, we conclude in section IV.

##  II Vector-like bottom quark in the simplified model

The generic parametrization of an effective Lagrangian of VLQ-$B$ can be
expressed as (showing only the couplings relevant for our analysis):

$\displaystyle\mathcal{L}=$
$\displaystyle\kappa_{B}\bigg{\\{}\sqrt{\frac{\zeta_{i}\xi^{B}_{W}}{\Gamma^{0}_{W}}}\frac{g}{\sqrt{2}}[\bar{B}_{L/R}W^{+}_{\mu}\gamma^{\mu}u^{\,i}_{L/R}]+\sqrt{\frac{\zeta_{i}\xi^{B}_{Z}}{\Gamma^{0}_{Z}}}\frac{g}{2c_{W}}[\bar{B}_{L/R}Z_{\mu}\gamma^{\mu}d^{i}_{L/R}]$
(1)
$\displaystyle-\sqrt{\frac{\zeta_{i}\xi^{B}_{H}}{\Gamma^{0}_{H}}}\frac{M}{v}[\bar{B}_{R/L}Hd^{i}_{L/R}]\bigg{\\}}+\text{h.c.},$

where $g$ is the $SU(2)_{L}$ gauge coupling constant, $c_{W}$ is the usual
cosine of the weak mixing angle, $v\simeq 246$ GeV and $\xi^{V}$ parameters
controlling the relative strengths of the $V$ couplings to top partners, and
$\zeta_{i}$ parameters governing the mix of SM quark generations $i$ in each
coupling. $\xi$ and $\zeta$ are defined as $\sum_{V}\xi^{V}=1$ and
$\sum_{i}\zeta_{i}=1$, meaning $\zeta_{i}\xi^{V}=\textrm{BR}(B\rightarrow
Vq_{i})$ (for a detailed review, see Buchkremer:2013bha ).

Although couplings of VLQs to first- and second-generation SM quarks are not
excluded Atre:2008iu ; Atre:2011ae , much of the experimental and theoretical
attention is on VLQs that couple to third-generation SM quarks, as it is this
generation that requires fine-tuning in the SM. Here we focus on VLQ-$B$ that
couples exclusively to third-generation SM quarks. Our results are interpreted
assuming that the B quark belongs to a singlet or doublet representation and
that it decays exclusively to SM particles. In this case, the singlet $B$
quark has three different decay channels into SM particles: $tW$, $bZ$ and
$bH$. Using the equivalence theorem ET-hjh ; He:1992nga ; He:1993yd ;
He:1994br ; He:1996rb ; He:1996cm  the branching fractions for these three
decay modes are 0.5, 0.25 and 0.25, respectively. The $B$ doublet can decay to
$bZ$ or $bH$, each with a branching fraction of 0.5. Thus there are only two
free parameters: the $B$ quark mass $m_{B}$ and the coupling strength
$\kappa_{B}$.

##  III Event generation and discovery potentiality

In Fig. 1, we show the leading order Feynman diagram of the process
$e^{+}e^{-}\to B\bar{b}$ with the decay mode $B\to Zb$.

Figure 1: Representative Feynman diagrams of the process $e^{+}e^{-}\to B(\to
Zb)\bar{b}$.

In order to make a prediction for the signal, we calculate the cross section
for the process $e^{+}e^{-}\to B\bar{b}~{}(b\bar{B})$ times the branching
ratio of $B\to bZ$ at leading order (LO) by using MadGraph5-aMC$@$NLO mg5 .
The numerical values of the input parameters are taken from pdg .

Figure 2: Total cross sections as a function of $m_{B}$ with $\kappa_{B}=0.2$
and two cases.

In Fig. 2, we show the dependence of the cross sections $\sigma(e^{+}e^{-}\to
B\bar{b}+b\bar{B})\times Br(B\to bZ)$ on the $B$ quark mass $m_{B}$ at a 3 TeV
CLIC for $\kappa_{B}=0.2$. As the $B$ quark mass grows, the cross section of
single production decreases slowly due to a larger phase space. For
$\kappa_{B}=0.2$ and $m_{B}=1.5~{}(2)$ TeV, the cross section can reach about
0.05 (0.3) fb for the singlet case and 0.2 (0.12) fb for the doublet case,
respectively. Obviously, the cross section of single $B-$quark production is
proportional to the square of the coupling strength $\kappa_{B}$ for a given
$B$ quark mass.

In next section, we will perform the Monte Carlo simulation and explore the
discovery potentiality of VLQ-$B$ through the subsequent leptonic decay
channel $Z\to\ell^{+}\ell^{-}$ and the invisible decay channel
$Z\to\nu\bar{\nu}$, respectively.

Monte Carlo event simulations for signal and SM background are interfaced to
Pythia 8.20 pythia8  for fragmentation and showering. All event samples are
fed into the Delphes 3.4.2 deFavereau:2013fsa  with the CLIC detector card
designed for 3 TeV Leogrande:2019qbe . In our analysis, jets are clustered
with the Valencia Linear Collider (VLC) algorithm Boronat:2014hva ;
Boronat:2016tgd  in exclusive mode with a fixed number of jet ($N=2$ where $N$
corresponds to the number of partons expected in the final state) and fixed
one size parameter $R=0.7$. The $b$-tagging efficiency is taken as the loose
working points with 90% b-tagging efficiency in order not to excessively
reduce the signal efficiency. The misidentification rates are given as a
function of energy and pseudorapidity, i.e., in a bit where $E>500$ GeV and
$1.53<|\eta|\leq 2.09$, misidentification rates are $5\times 10^{-2}$.
Finally, event analysis is performed by using MadAnalysis5 ma5 .

###  III.1 The decay channel $Z\to\ell^{+}\ell^{-}$

In this subsection, we analyze the signal and background events at the 3 TeV
CLIC through the $Z\to\ell^{+}\ell^{-}$ ($\ell=e,\mu$) decay channel.

$\displaystyle e^{+}e^{-}\to B\bar{b}\to
Zb\bar{b}\to\ell^{+}\ell^{-}b\bar{b}.$ (2)

For this channel, the typical signal is two $b$-jet and two Opposite-Sign
Same-Flavor (OSSF) leptons. The dominant SM backgrounds come from the SM
processes $e^{+}e^{-}\to b\bar{b}\ell^{+}\ell^{-}$, and $e^{+}e^{-}\to
jj\ell^{+}\ell^{-}$ with the cross sections of 2.74 fb, and 8.39 fb,
respectively. Note that the contribution from the processes $e^{+}e^{-}\to
H(\to b\bar{b})Z$, $e^{+}e^{-}\to Z(\to b\bar{b})Z$, $e^{+}e^{-}\to b\bar{b}Z$
and $e^{+}e^{-}\to q\bar{q}Z$ are also included with the decay mode
$Z\to\ell^{+}\ell^{-}$.

To identify objects, we choose the basic cuts at parton level for the signals
and SM backgrounds as follows:

$\displaystyle p_{T}^{\ell}>~{}20~{}\text{GeV},\quad
p_{T}^{j/b}>~{}30~{}\text{GeV},\quad|\eta_{\ell/b/j}|<~{}2$ (3)

where $p_{T}^{\ell,b,j}$ are the transverse momentum of leptons, $b$-jets, and
light jets, respectively.

Figure 3: Normalized distributions for the signals (with $m_{B}$=1300, 1500
and 2000 GeV) and SM backgrounds at the CLIC.

For the signal, the leptons $\ell_{1}$ and $\ell_{2}$ are two OSSF leptons
that are assumed to be the product of the $Z$-boson decay, and at least two
$b$-tagged jet are present. In Fig. 3, we plot some differential distributions
for signals and SM backgrounds at the CLIC, such as the transverse momentum
distributions of the leading and subleading $b$-jets
($p_{T}^{b_{1}},p_{T}^{b_{2}}$), the separations $\Delta R_{b_{1},b_{2}}$, the
transverse momentum distributions of the leading and subleading leptons
($p_{T}^{\ell_{1}\ell_{2}}$), the separations $\Delta R_{\ell_{1},\ell_{2}}$,
and the invariant mass distribution for the $Z$ boson $M_{\ell_{1}\ell_{2}}$.
Due to the larger mass of VLQ-$B$, the decay products of VLQ-$B$ are highly
boosted. Therefore, the $p_{T}^{l,b}$ peaks of the signals are larger than
those of the SM backgrounds, and the lepton pairs of the signal are much
closer. Based on these kinematical distributions, we can impose the following
set of cuts:

  * •

Cut-1: There are exactly two isolated leptons and two $b$-tagged jets.

  * •

Cut-2: The transverse momenta of the leading and sub-leading $b$-jet are
required $p_{T}^{b_{1}}>500\rm~{}GeV$ and $p_{T}^{b_{2}}>250\rm~{}GeV$ with
$\Delta R_{b_{1},b_{2}}>1.2$.

  * •

Cut-3: The transverse momenta of the leading and sub-leading leptons are
required $p_{T}^{l_{1}l_{2}}>300\rm~{}GeV$, the invariant mass of the $Z$
boson is required to have $|M_{\ell_{1}\ell_{2}}-m_{Z}|<10\rm~{}GeV$ with
$\Delta R_{\ell_{1},\ell_{2}}<1$.

Table 1: Cut flow of the cross sections (in $10^{-3}$ fb) for the signals and SM backgrounds at the CLIC with $\kappa_{B}=0.2$ and three typical $B$ quark masses for the singlet case and doublet case (in the bracket).  Cuts | Signals | Backgrounds
---|---|---
|  1300 GeV |  1500 GeV |  2000 GeV |  $\ell^{+}\ell^{-}b\bar{b}$ |  $\ell^{+}\ell^{-}jj$ |  Total
Basic |  3.4 (15.2) |  3.3 (13.2) |  1.9 (7.6) |  1869 |  1179 |  3048
Cut 1 |  2.37 (9.5) |  2.1 (8.4) |  1.2 (4.8) |  689 |  103 |  792
Cut 2 |  2.25 (9.0) |  1.95 (7.8) |  1.1 (4.4) |  18.2 |  7.7 |  25.9
Cut3 |  1.62 (6.5) |  1.58 (6.3) |  0.95 (3.8) |  5.34 |  2.02 |  7.36

We present the cross sections of three typical signal ($m_{B}=1300,1500,2000$
GeV) and the relevant backgrounds after imposing the cuts in Table 1. One can
see that all the SM backgrounds are suppressed very efficiently with the cross
section of about 0.01 fb, while the signals still have a relatively good
efficiency at the end of the cut flow. The total SM background comes from the
$e^{+}e^{-}\to b\bar{b}\ell^{+}\ell^{-}$ process, with a total cross section
of $5.34\times 10^{-3}$ fb.

###  III.2 The decay channel $Z\to\nu\bar{\nu}$

In this subsection, we analyze the signal and background events through the
invisible decays $Z\to\nu\bar{\nu}$ decay channel.

$\displaystyle e^{+}e^{-}\to B\bar{b}\to Zb\bar{b}\to b\bar{b}+\not{E}_{T}.$
(4)

For this channel, the main SM backgrounds come from the processes
$e^{+}e^{-}\to\nu\bar{\nu}b\bar{b}$ and $e^{+}e^{-}\to\nu\bar{\nu}jj$ with the
cross sections of 0.26 fb and 0.61 fb, respectively. Note that the
contribution from the processes $e^{+}e^{-}\to HZ$,
$e^{+}e^{-}\to\nu_{e}\bar{\nu}_{e}H$, $e^{+}e^{-}\to\nu_{e}\bar{\nu}_{e}Z$,
and $e^{+}e^{-}\to Zb\bar{b}$ are also included with the decay mode
$Z\to\nu\bar{\nu}$ and $H\to b\bar{b}$.

In our simulations, we apply the following basic cuts on the signal and
background events at parton level:

$\displaystyle
p_{T}^{b/j}>~{}30~{}\text{GeV},\quad|\eta_{b/j}|<~{}2,\quad\not{E}_{T}>100~{}\text{GeV}.$
(5)

Figure 4: Normalized distributions for the signals and SM backgrounds for
$Z\to\nu\bar{\nu}$ decay channel at the CLIC.

Obviously, the signal events should contain large missing transverse energy
$\not{E}_{T}$ from the boosted $Z$ boson. In order to get some hints of
further cuts for reducing the SM backgrounds, we analyzed the normalized
distributions of $p_{T}^{b_{1},b_{2}}$, $p_{T}^{b_{2}}$, $\Delta
R_{b_{1},b_{2}}$, and $\not{E}_{T}$ for signals and SM backgrounds as shown in
Fig. 4. Based on these kinematical distributions, a set of further cuts are
given as:

  * •

Cut-1: There are at least two $b$-tagged jets and remove any electrons and
muons.

  * •

Cut-2: The transverse momenta of the leading and sub-leading $b$-jet are
required $p_{T}^{b_{1}}>300\rm~{}GeV$ and $p_{T}^{b_{2}}>200\rm~{}GeV$ with
$\Delta R_{b_{1},b_{2}}>1.5$.

  * •

Cut-3: The transverse missing energy is required $\not{E}_{T}>300\rm~{}GeV$.

Table 2: Cut flow of the cross sections (in $10^{-3}$ fb) for the signals and SM backgrounds at the CLIC with $\kappa_{B}=0.2$ and three typical $B$ quark masses for the singlet case and doublet case (in the bracket).  Cuts | Signals | Backgrounds
---|---|---
|  1300 GeV |  1500 GeV |  2000 GeV |  $\nu\bar{\nu}b\bar{b}$ |  $\nu\bar{\nu}jj$ |  Total
Basic |  11 (44) |  8.5 (38) |  5.5 (22) |  223590 |  50120 |  273710
Cut 1 |  8 (32) |  7 (28) |  4.1 (16.4) |  128500 |  5359 |  133859
Cut 2 |  7.5 (30) |  6.5 (26) |  3.6 (14.4) |  206 |  25 |  231
Cut3 |  6.1 (24.4) |  5.6 (22.4) |  3.3 (13.2) |  92 |  12 |  104

We summarize the cross sections of three typical signal
($m_{B}=1300,1500,2000$ GeV) and the relevant backgrounds after imposing the
cuts in Table 2. One can see that the total SM backgrounds are suppressed very
efficiently, with a cross section of about 0.1 fb.

###  III.3 Discovery and exclusion significance

In order to see whether the signatures of VLQ-$B$ can be detected at the CLIC,
we use the median significance to estimate the expected discovery and
exclusion significance Cowan:2010js :

$\displaystyle\mathcal{Z}_{\text{disc}}$
$\displaystyle=\sqrt{2\left[(s+b)\ln\left(\frac{(s+b)(1+\delta^{2}b)}{b+\delta^{2}b(s+b)}\right)-\frac{1}{\delta^{2}}\ln\left(1+\delta^{2}\frac{s}{1+\delta^{2}b}\right)\right]}$
(6) $\displaystyle\mathcal{Z}_{\text{excl}}$
$\displaystyle=\sqrt{2\left[s-b\ln\left(\frac{b+s+x}{2b}\right)-\frac{1}{\delta^{2}}\ln\left(\frac{b-s+x}{2b}\right)\right]-\left(b+s-x\right)\left(1+\frac{1}{\delta^{2}b}\right)},$

with

$\displaystyle x=\sqrt{(s+b)^{2}-4\delta^{2}sb^{2}/(1+\delta^{2}b)}.$ (7)

Here, the values of $s$ and $b$ were obtained by multiplying the total signal
and SM background cross sections, respectively, by the integrated luminosity.
$\delta$ is the percentage systematic error on the SM background estimate. In
the limit of $\delta\to 0$, these expressions can be simplified as

$\displaystyle\mathcal{Z}_{\text{disc}}$
$\displaystyle=\sqrt{2[(s+b)\ln(1+s/b)-s]},$ (8)
$\displaystyle\mathcal{Z}_{\text{excl}}$
$\displaystyle=\sqrt{2[s-b\ln(1+s/b)]}.$

In this work we choose two cases: no systematics ($\delta=0$) and a systematic
uncertainty of $\delta=10\%$.

Figure 5:  The exclusion limit (at $95\%$ CL) and discovery prospects (at
$5\sigma$) contour plots for the signal in $\kappa_{B}-m_{B}$ planes at 3 TeV
CLIC with integral luminosity 5 ab-1 for $Z\to\ell^{+}\ell^{-}$ decay channel.

Figure 6: Same as Fig. 5 but for $Z\to\nu\bar{\nu}$ decay channel.

In Figs. 5-6, we plot the exclusion limit at 95% confidence level (CL) and
$5\sigma$ sensitivity reaches for the coupling strength $\kappa_{B}$ as a
function of $m_{B}$ at 3 TeV CLIC with integral luminosity 5 ab-1,
respectively, for two decay channels without considering the effect of the
systematic error. One finds that, for the $Z\to\ell^{+}\ell^{-}$ decay
channel, the singlet (doublet) VLQ-$B$ quarks can be excluded in the region of
$\kappa_{B}\in[0.23,0.62]~{}([0.11,0.31])$ and $m_{B}\in$ [1200 GeV, 2500 GeV]
at the 3 TeV CLIC with the integrated luminosity of 5 ab-1, while the discover
region can reach $\kappa_{B}\in[0.4,0.53]~{}([0.2,0.55])$ and $m_{B}\in$ [1200
GeV, 2000 GeV] ([1200 GeV, 2500 GeV]). Similarly, for $Z\to\nu\bar{\nu}$ decay
channel, the singlet (doublet) VLQ-$B$ quarks can be excluded in the region of
$\kappa_{B}\in[0.22,0.5]~{}([0.11,0.3])$ and $m_{B}\in$ [1200 GeV, 2400 GeV]
([1200 GeV, 2500 GeV]), the discover region can reach
$\kappa_{B}\in[0.39,0.5]~{}([0.2,0.44])$ and $m_{B}\in$ [1200 GeV, 1900 GeV]
([1200 GeV, 2400 GeV]).

Figure 7: Combined exclusion limit (at 95% CL) and discovery prospects (at
$5\sigma$) contour plots for the signal in $\kappa_{B}-m_{B}$ planes at 3 TeV
CLIC with an integral luminosity of 5 ab-1 for the singlet (left) and doublet
(right) cases.

Certainly, the sensitivities with some systematic errors will be weaker than
those without any systematic error. Next, we combine the significance with
$\mathcal{Z}_{\text{comb}}=\sqrt{\mathcal{Z}^{2}_{\ell\bar{\ell}}+\mathcal{Z}^{2}_{\nu\bar{\nu}}}$
by using the results from above two decay channels with the aforementioned two
systematic error cases of $\delta=0$ and $\delta=10\%$.

In Fig. 7, the combined 95% CL exclusion limit and $5\sigma$ discovery
prospect lines are drawn in $\kappa_{B}-m_{B}$ planes at the 3 TeV CLIC. One
can see that, with a realistic 10% systematic error, the sensitivities are
slightly weaker than those without any systematic error. For the singlet and
doublet cases, the discovery region can, respectively, reach
$\kappa_{B}\in[0.42,0.52]$ with the VLQ-$B$ mass range $m_{B}\in$ [1200 GeV,
1900 GeV], and $\kappa_{B}\in[0.21,0.47]$ with the VLQ-$B$ mass range
$m_{B}\in$ [1200 GeV, 2400 GeV]. Otherwise, in the region of $m_{B}\in$ [1200
GeV, 2400 GeV], the 95% CL excluded region for the coupling parameter
$\kappa_{B}$ is [0.23, 0.5] for the singlet case and [0.11, 0.25] for the
doublet case, respectively, at the 3 TeV CLIC with an integrated luminosity of
5 ab-1.

Figure 8: The contour of 95% CL exclusion limit on the plane of $m_{B}$ versus
$\kappa_{B}$ in the (B, Y) doublet scenarios with a realistic 10% systematic
error. The solid line indicates the current expected limits from the LHC
ATLAS:2021gfv .

The ATLAS Collaboration has studied single production of a vector-like $B$
quark decaying into $bH$ with $H\to\gamma\gamma$ decay channel, assuming a
generalized coupling $\kappa_{B}=0.5$ and doublet branching ratios of 50% for
$B\to hb$ and $B\to Zb$, $B$ quark with masses less than 1210 GeV are excluded
at the 95% CL ATLAS:2018qxs . Very recently, the ATLAS Collaboration has
presented the limit from the single production of a VLQ-$B$ occurring as part
of a $(B,Y)$ weak isospin doublet via $B\to bH(H\to b\bar{b})$ decay channel
at 13 TeV LHC with 139 fb-1 luminosity, excluding
$1.0\rm~{}TeV<m_{B}<1.28\rm~{}TeV$ and $1.46\rm~{}TeV<m_{B}<2.0\rm~{}TeV$ for
$\kappa_{B}=0.25$, and $1.0\rm~{}TeV<m_{B}<2.0\rm~{}TeV$ for $\kappa_{B}=0.3$
ATLAS:2021gfv . In Fig. 8, we give the current results at 13 TeV LHC and
future reach at 3 TeV CLIC on the plane of $m_{B}$ versus $\kappa_{B}$ in the
$(B,Y)$ doublet scenarios. We can see that the future CLIC with
$\sqrt{s}=3\rm~{}TeV$ and integrated luminosity of 5 ab-1 could provide better
sensitivity to detect the $BbZ$ couplings than the current experimental
results obtained from the current 13 TeV LHC. For high energy hadron
colliders, the QCD backgrounds will be enhanced due to the pileup effect. By
comparison, the properties of the VLQ-$B$ can be measured accurately once it
is discovered due to the clean SM backgrounds at the future leptonic
colliders.

##  IV Conclusion

We have studied single production of VLQ-$B$ at the future 3 TeV CLIC via the
process $e^{+}e^{-}\to B\bar{b}\to Zb\bar{b}$ in a model-independent way. We
performed a full simulation for the signals and the relevant SM backgrounds
based on two types of decay channels $Z\to\ell^{+}\ell^{-}$ $(\ell=e,\mu)$ and
$Z\to\nu\bar{\nu}$. The $5\sigma$ discovery prospects and 95% CL exclusion
limits in the parameter plane of the two variables $m_{B}$ and $\kappa_{B}$
were, respectively, obtained at 3 TeV CLIC with an integral luminosity of 5
ab-1. Our numerical results show that, with the systematic error case of
$\delta=10\%$, in the region of $m_{B}\in$ [1200 GeV, 2300 GeV], the discovery
region can reach $\kappa_{B}\in[0.42,0.52]$ with the VLQ-$B$ mass range
$m_{B}\in$ [1200 GeV, 1900 GeV] for the singlet case, and
$\kappa_{B}\in[0.21,0.47]$ with the VLQ-$B$ mass range $m_{B}\in$ [1200 GeV,
2400 GeV] for the doublet case, respectively. Otherwise, in the region of
$m_{B}\in$ [1200 GeV, 2400 GeV], the excluded region for the coupling
parameter $\kappa_{B}$ is [0.23, 0.5] for the singlet case and [0.11, 0.25]
for the doublet case, respectively, at the 3 TeV CLIC with an integrated
luminosity of 5 ab-1.

###### Acknowledgements.

The work is supported by the Foundation of the Henan Science and Technology
Research Project:202102210223; Key Research Projects in Universities of
Henan:21A140031; Henan Colleges and Universities Youth Teacher Training
Project:2020GGJS212.

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