Conventional BBA method

The purpose of BBA is to find a reference orbit in which the beam passes the centers of all quadrupoles in a storage ring using beam position monitors (BPMs) and orbit corrector magnets (OCMs). The dipole fields seen by an off-axis particle in a quadrupole is given by $B_x=B_0{\rho }_0{K}_0{y}_0\mathrm{,}$ (1) $B_y=B_0{\rho }_0{K}_0{x}_0\mathrm{,}$ (2) where $B_0{\rho }_0$ is the magnetic rigidity, K₀ is the normalized quadrupole strength, and x₀ and y₀ are the beam offsets relative to the quadrupole center in the horizontal and vertical planes, respectively. Therefore, changing the quadrupole strength by Δ K causes a dipole field variation by $\text{Δ}{B}_x=B_0{\rho }_0\text{Δ}K{y}_0\mathrm{,}$ (3) $\text{Δ}{B}_y=B_0{\rho }_0\text{Δ}K{x}_0\mathrm{,}$ (4) resulting in a kick that leads to an orbital change at the observation point s by [17] $\begin{array}{c}\text{Δ}u(s)=\text{Δ}Ku(s_0)\left(\frac{1}{1-K_0\frac{L_0\beta \left(s_0\right)}{2\tan (\pi \nu )}}\right)\\ \times \frac{\sqrt{\beta (s)\beta \left(s_0\right)}}{2\sin (\pi \nu )}\cos \left(|\varphi (s)-\varphi \left(s_0\right)|-\pi \nu \right)\mathrm{,}\end{array}$ (5) where L₀ is the length of the quadrupole, ν is the betatron tune, and β(s₀) and β(s) are the beta functions at the locations of the quadrupole and observation points, respectively. ϕ(s₀) and ϕ(s) are the phase advances at the locations of the quadrupole and observation points, respectively, and u represents the beam positions in the horizontal and vertical planes. This equation shows that the beam orbit can be affected by the quadrupole strength variation and beam positions in the quadrupoles. To avoid this effect, the reference orbit of the orbit feedback system is typically set to the centers of all quadrupoles with u=0. The reference orbit is determined using the BBA technique.

The quadrupole center is measured using the nearest BPM. Suppose that when the beam passes through the quadrupole center, the related reading of this BPM is 𝒗₀. According to Eq. (5), by changing the quadrupole strength Δ K, the beam orbit change is given by $\begin{array}{l}\text{Δ}u=\text{Δ}K\mathcal{F}\left(v-v_0\right)\mathrm{,}\hfill \end{array}$ (6) where 𝒗 is the reading of the target BPM before the quadrupole strength change, and $\mathcal{F}$ is the coefficient that can be easily obtained from Eq. (5). To measure the quadrupole center, the beam is set to several different positions at its related BPM. For each position, the quadrupole strength is varied with the same Δ K, and the corresponding orbit changes are recorded. By applying a linear fit to Eq. (6), the quadrupole center 𝒗₀ is obtained. The conventional BBA always determines the horizontal and vertical offsets separately [18]. The above analysis implies that the coefficient $\mathcal{F}$ is treated as a constant, which implies that the beam optics remain unchanged during the BBA process. In fact, the change in quadrupole strength and closed orbit distortion can affect the beam optics. At the beginning of commissioning, the beam orbit and beam optics are possibly significantly different from the ideal model, which induces strong nonlinearity and coupling. In this case, several iterations are required for the conventional BBA method to eliminate the nonlinear effects and obtain more accurate quadrupole centers. A neural network with multiple layers that address nonlinear problems can be adopted for the BBA process [19, 20].

BBA using a neural network

BBA is based on the principle that the off-axis beam is affected by a quadrupole strength change. The golden orbit can then be evaluated using the relationship between the orbit changes and the initial beam orbits before varying the quadrupole strength. This relationship can be explored by training a neural network using orbital changes as the input data and initial orbits as the output data. By setting the orbital change to zero, the corresponding initial beam orbit becomes the predicted golden orbit. The main concept behind this proposed BBA method is illustrated in Fig. 1. To obtain data for training the NN model, a simulation or online experiment was performed as follows:

Fig. 1Full-screen View

Schematic of the neural network-based BBA method. Different orbits are generated by randomly adjusting the orbit correctors. On each orbit, all quadrupoles are changed with the same Δ K simultaneously to generate orbit changes. The orbit changes are used as the input data of the neural network, and the corresponding initial orbits are used as the output data for the training

• Randomly exciting all corrector magnets to form an initial closed orbit;

• recording all BPM readings;

• changing all quadrupoles by the same amount to form a new closed orbit.

• recording the changes in all BPM readings;

• resuming the quadrupole and corrector strengths to the original values;

• repeating the above procedures.

A typical dense neural network has one input layer, several hidden layers (also called middle layers), and one output layer, as illustrated in Fig. 2 [21]. The nodes where the data are transferred are called neurons. The nodes between adjacent layers are connected to each other by an arrow, which shows the flow of data. Each arrow represents a linear transformation combined with an activation function used to introduce nonlinearity if necessary [11]. A loss function is used to describe the performance of the neural network. An NN also requires an optimizer function to optimize the parameters used for data transmission. The optimization is performed by minimizing the loss value.

Fig. 2Full-screen View

Diagram of a typical dense neural network, which consists of one input layer, several hidden layers, and one output layer. Here, the hyperbolic tangent (tanh) is adopted as the activation function

Conventional BBA method

In the conventional BBA measurement for one quadrupole, the beam is moved to three different positions with the help of the corrector magnets [27]. At each position, the change in the beam orbit from all BPM readings was recorded after varying the strength of the target quadrupole by a certain Δ K. The orbital changes can be fitted linearly as a function of the beam position in the target quadrupole. An immobile point can be found by setting the position at which the BPM changes vanish. The quadrupole center was then obtained by adding all the immobile points from each BPM. The entire BBA routine repeats this process for all quadrupoles in both the horizontal and vertical planes in the storage ring. To increase the BBA accuracy, the measurement was repeated after moving the beam to the orbit obtained from the previous BBA experiment. This scheme is typically required in the machine commissioning stage. Figure 5 shows the simulated measurements of the horizontal and vertical centers of the quadrupole magnet in the HLS-II storage ring. At least three conventional BBA iterations are required to decrease the standard deviation of the fitted Gaussian function of the quadrupole center to several micrometers, which is of the same order as BPM measurement resolution [28].

Fig. 5Full-screen View

Simulated BBA measurement in the HLS-II storage ring. The horizontal and vertical measurements are applied, respectively. (a) Horizontal quadrupole center measurement. (b) Vertical qaudrupole center measurement. The plots show the orbit change observed from all BPMs by varying the target quadrupole strength with a certain Δ K when the beam is at three different positions. For each BPM, its changes can be fitted to find a center for the quadrupole. All the found centers are then fitted using the Gaussian function. The red line shows the fitted centers using all BPMs, which represents the BBA center of this quadrupole

BBA using an NN model

In the simulation, the correctors were set randomly within a certain range of kicks to move the beam orbit. The kick angle variations were generated using a normal distribution with a standard deviation of 0.05 mrad, and a truncation at three standard deviations was applied. For each random orbit, all quadrupoles were simultaneously changed by the same amount of Δ K (-0.02 m^-2). The corresponding initial beam orbit and orbit changes were recorded for all BPMs.

The entire simulation generates 10,000 samples. In each sample, there were 64 initial orbits and 64 orbit change data points, with 32 in the horizontal plane and 32 in the vertical plane. These samples were adopted to train the neural network. Figure 6 shows the random initial beam orbits within the range (-5,5) mm. Figure 7 shows the orbit change after quadrupole adjustment. The range of the orbit change was within (-1.5,1.3) mm and (-0.8,0.8) mm in the horizontal and vertical planes, respectively.

Fig. 6Full-screen View

Distribution of the initial orbits generated by randomly adjusting the orbit correctors within a certain range. (a) Initial orbits in the horizontal plane. (b) Initial orbits in the vertical plane

Fig. 7Full-screen View

Distribution of orbit change caused by varying the quadrupole strength for each random orbit in the simulation. (a) Horizontal orbit changes. (b) Vertical orbit changes

An NN model is trained using these data to obtain the golden orbit. The 64 sets of orbit change data were used as the input to the model, and the 64 sets of the corresponding initial orbit data were used as the output. Eighty percent of the data were used to train the model, and the rest were used to test the performance of the model. There were 128, 256, and 128 neurons in the three hidden layers, respectively. The tanh is considered the activation function that provides nonlinearity. The NN model was trained using the Adam optimizer [29]. The loss function is the mean squared error (MSE) between the measured data and model-predicted results, which is $\text{loss}=\text{mean}\left({\left({\text{r}}_{\text{model}}-{\text{r}}_{\text{real}}\right)}^2\right)\mathrm{.}$ (7) Figure 8 compares the golden orbits obtained from the conventional BBA and BBA using a neural network. The consistency between these two BBA methods demonstrates the validity and effectiveness of the new BBA technique. Subsequently, an online experiment was conducted in the HLS-II storage ring.

Fig. 8Full-screen View

The quadrupole centers obtained from the conventional BBA and the NN-based BBA. The simulation shows good consistency between these two methods

Training data acquisition

Similar to the simulation, training data can be obtained from a real storage ring. Before the experiment, the magnet strengths were set according to the results of the early commissioning of the storage ring. In this case, the beam is not on the golden orbit that connects the centers of the quadrupoles. An online experiment to obtain the training data is presented in this subsection.

During the online experiment, the orbit feedback system was turned off, and the correctors were randomly set to generate different orbits. As a compromise between beam stability and data diversity, the adjustment range of all correctors was set to ±0.8 A relative to the starting point. This corrector adjustment range ensured no beam loss during the experiment by controlling the orbit change within a distinguishable range, as shown in Fig. 10. The horizontal tune in the HLS-II storage ring was approximately 4.44, whereas the vertical tune was 2.36, which is further from the half-integer. When the quadrupole strength is simultaneously increased, the horizontal tune increases accordingly which may reach the half-integer resonance and thus cause beam loss. Therefore, all quadrupole strengths were adjusted in the decreasing direction by -0.02 m^-2 (normalized focusing strength). After the orbital change was recorded, all quadrupole strengths were restored to their original values. For the HLS-II storage ring, the time constant for the orbit corrector power supply is approximately 15 ms [30]. The time constant for the quadrupole power supply is approximately 30 ms. This implies that one complete loop of this measurement could be performed within 1 second. To ensure the data acquisition accuracy, the measurement time for each loop was set to 2 s.

Fig. 10Full-screen View

Distribution of the orbits generated by randomly adjusting the orbit correctors within a certain range. (a) Horizontal BPM readings. (b) Vertical BPM readings

An online experiment was conducted during the machine study time [31]. The entire measurement process generated 21,000 samples. These samples were used to train the neural network. Figure 10 shows the randomly generated initial beam orbits before the quadrupole strength is varied. The distribution shows that orbits were generated within a range of approximately (-10,10) mm, and the densest distribution was approximately 0. The orbit change after the quadrupole adjustment was also analyzed, and the distribution of the orbit differences is plotted in Fig. 11. The range of the orbit change was within (-3,2) mm in the horizontal plane and within (-1.5,1.5) mm in the vertical plane.

Fig. 11Full-screen View

Distribution of orbit change after varying the quadrupoles for each random orbit. (a) Horizontal orbit change. (b) Vertical orbit change

NN model training using online data

In this subsection, the relationship between the initial orbit and the orbit change after quadrupole adjustment is explored using a dense neural network. Similar to the simulation, 64 sets of orbit change data were set as the input to the model, and 64 sets of the corresponding initial orbit data were set as the output. To determine the data-size requirements, the two models were trained using different numbers of samples. In Model I, all 21000 samples were adopted, 5/6 of the samples were used for training, and 1/6 were the validation data set. For comparison, Model II was trained with only 3000 samples in the training set and 600 samples in the validation set; 3600 samples were adopted in total. The Adam optimizer and MSE loss function were used to train the models.

The trained NN models were evaluated by calculating the mean absolute error (MAE) between the measured and model-predicted values of the validation samples for each BPM: $\text{MAE}=\text{mean}\left(|{\text{r}}_{\text{measured}}-{\text{r}}_{\text{predicted}}|\right)\mathrm{.}$ (8) The absolute errors for both models are shown in Fig. 12, which shows that the errors in model I are smaller than those in model II. In the horizontal plane, the overall average absolute error was approximately 99 μm for Model I and 125 μm for Model II. In the vertical plane, the overall average absolute error was approximately 62 μm for Model I and 71 μm for Model II. The results show that increasing the number of samples for the NN model training can improve the model’s accuracy.

Fig. 12Full-screen View

The mean absolute error (MAE) between the measured beam orbits and predicted values of the validation samples for all BPMs. (a) Horizontal plane. (b) Vertical plane

Golden orbit from the NN model

In the NN training, the orbit changes caused by varying quadrupoles were used as the input data. The corresponding initial orbits were used as the output data. The beam on the golden orbit should have the least orbital distortion (ideally zero) owing to the change in quadrupole strength. Therefore, the input can be set to zero for the NN model, and the corresponding output is the golden orbit.

To estimate its accuracy, this golden orbit was compared with that obtained using the conventional BBA method, and the results are shown in Fig. 13. The sub-figures in Fig. 13 illustrate the differences between the novel and conventional BBA. The results show that this golden orbit is consistent with that obtained from the conventional BBA. In the horizontal plane, the average difference between the conventional BBA and the model prediction was approximately 46 μm for Model I and 53 μm for Model II. In the vertical plane, the average difference between the conventional BBA and the model prediction is approximately 42 μm for Model I and 39 μm for Model II.

Fig. 13Full-screen View

Comparison of the golden orbit obtained using the NN model and the conventional BBA method. The difference between the two golden orbits is shown in the subfigure. (a) Horizontal golden orbit. (b) Vertical golden orbit

Although the training error of Model I was smaller than that of Model II, the difference in the predicted golden orbits from these two models did not exhibit a large deviation [32]. In the HLS-II storage ring, the typical experimental period for the conventional BBA process is approximately 5 h. In the machine commissioning phase, this BBA process is needed to be repeated several times to obtain precise results. Model II used only 3600 samples, which resulted in a shorter online measurement time (approximately 2 h). As discussed previously, the online measurement time for this new method is irrelevant to the total quadrupole number. This differs from the conventional BBA, where the larger storage ring requires more time. In contrast, the NN-trained golden orbit can be set as the starting point for the conventional BBA. This helps reduce the iterative process of the BBA starting from the initial commissioning orbit and, hence, the experimental time.