Temperature control in induction heating forging applications

Abstract

Temperature is a key parameter in the industrial forging process. Induction heating is a fast, efficient and easily controllable system to have a uniform and controlled thermal profile before forging. The possibility to have a finer control over the temperature management leads to the improvement in process, a better product quality level, and lower maintenance cost. In this paper, the pyrometer noise and unstable signal is filtered and corrected using a Kalman filter based on a numerical electro-thermal model to obtain a reliable signal for the process management.

Keywords

Industrial process control Kalman filter induction heating induction forge numerical model Dyna Forge Zone

1. Introduction

The forging process requires a controlled temperature profile to keep the products quality [1,2]. The advantages are both in the product quality and process management: better material stress distribution, less scale, longer die life, possibility to easily manage production standby keeping the material ready for rapid restart with a reduction of rejected parts.

The problem with the temperature control, both in continuous and batch heating system, is to have an accurate temperature feedback to control the power generator. Figure 1 shows a real data acquisition from a static induction heating of a steel cylinder. The pyrometer signal can be influenced by a wide range of factor like the presence of scales, surface defects, smoke, uneven coating, etc. giving sudden temperature rises and drops or instabilities.

Fig. 1.

Real pyrometer data from induction heating forging process.

An averaging filter and a control system (PID, feedforward, …) could be used for continuous process but are not effective to manage partial or batch heating, frequent stops and rapid restart required in the industrial production.

2. Solution proposed

The idea is the implementation of a Kalman Filter (KF) in the system to remove measure inaccuracies and noise giving a better feedback for the system temperature controller using a numerical model for the induction heating.

KF is a prediction algorithm ideal for systems which are continuously changing. KFs have the advantage that they are light on memory (they don’t need to keep any history other than the previous state), and they are very fast, making them well suited for real time problems and embedded systems as in Fig. 2.

Fig. 2.

Control system description: the induction feed u (voltage or current) are adjusted according to the temperatures (measured, modelled and target) using a KF.

The predictive model implemented for the KF control is based on a 1D thermal and electromagnetic numerical model. The power distribution and the temperature variation from the generator data can easily evaluated using the embedded model that can be used for the production optimization or recipe changes.

2.1. Numerical model description

The physics and the geometry involved in forging induction heating process can be accurately estimated using a 1D electro-thermal numerical model. The original algorithm, developed and coded for SAET by Lavers [3,4] in the 1985, has been rewritten in modern C # language and updated to take advantage of the current computation capabilities.

The model is a mix of finite-difference and analytical method as in [5]: the coupled electromagnetic and thermal problems are iteratively solved along a radius for an infinite cylindrical configuration. The EM module computes the power density distribution w [W/mm³] that is used into the heat transfer module to solve for temperatures distribution; the material properties are then updated according the temperature rise for the next EM step. Given the input data as in Fig. 3(a), the SOLVER computes the time evolution of electrical and thermal variables Fig. 3(b). Results of the simulation has been proven to be accurate in more than 30 years of industrial coils design.

Fig. 3.

(a) Model: solver input and output data and (b) example of computed temperature transient for a set of two inductors.

2.2. Kalman filter

KF idea is to reduce measures uncertainty in a dynamic system blending the raw data with values predicted with a mathematical model [6,7]. If the current state of the system x_t can be evaluated from the previous state x_t−1 then an iterative algorithm can be implemented to predict each next step knowing the evolution laws and the excitation components. In the KF loop, at each step the predicted value $\hat{x}_{t}$ is compared with the measured value z_t: the estimated value x_t is a weighted average between the two as shown in Fig. 4a. The “weight” K_t is named Kalman Gain and it’s updated at each step considering the uncertainty connected to the measure and the model: the information with lower variance (uncertainty) has a higher weight.

Fig. 4.

(a) KF “prediction” and “update” for a single variable and (b) visual representation of a 2D PDFs multiplications and estimated solution position.

The system evolution from the state x_k−1 to the new state x_k is described in (1) [8]: $\begin{eqnarray}\displaystyle x_{k}=\boldsymbol{A}\cdot x_{k-1}+\boldsymbol{B}\cdot u_{k}+w_{k} & & \displaystyle\end{eqnarray}$ (1) where A is the state transition matrix, B is the input-to-state matrix, u_k is the external input excitation vector, and w_k is the process noise. $\begin{eqnarray}\displaystyle z_{k}=\mathbf{H}\cdot x_{k}+v_{k}. & & \displaystyle\end{eqnarray}$ (2)

The measurement process is described in (2): the measured values z_k are a linear combination of the state x_k with the additional measure noise term v_k.

The KF loop consists in two steps: prediction and update. The prediction is the projection of the previous state information into the next step: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hat{x}_{k}=\boldsymbol{A}\cdot \hat{x}_{k-1}+\boldsymbol{B}\cdot u_{k}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{P}_{k}=\mathbf{A}\cdot \mathbf{P}_{k-1}\cdot \mathbf{A}^{T}+\mathbf{Q}_{k}\end{eqnarray}$ (4) where P_k is the state covariance matrix describing the uncertainty of the state variables, and Q_k is the process perturbation covariance matrix, i.e. the additional uncertainty introduced by the process evolution. The update is the correction step where the measures and the predicted state are weighted to guess the current state: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle y_{k}=z_{k}-\mathbf{H}\cdot \hat{x}_{k}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{K}_{k}=\mathbf{P}_{k}\cdot \mathbf{H}_{k}^{T}\cdot (\mathbf{H}_{k}\cdot \mathbf{P}_{k}\cdot \mathbf{H}_{k}^{T}+\mathbf{R}_{k})^{-1}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \hat{x}_{k}:=\hat{x}_{k}+\mathbf{K}_{k}\cdot y_{k}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{P}_{k}:=(\mathbf{I}-\mathbf{K}_{k}\cdot \mathbf{H}_{k})\cdot \mathbf{P}_{k}\end{eqnarray}$ (8) where y_k is the estimated error, R_k is the measure covariance matrix, and I is the identity matrix. The theoretical analysis of the KF can be found in different papers [6,8]: the key concept is that the estimated state $\hat{x}_{k}$ is the point with maximum likelihood given the model and measure uncertainties as shown in Fig. 4b.

2.3. Model-filter coupling example

The model evolution described with (3) expresses the state variables changes as a linear combination of the previous step state. As it usually happens in real cases, the temperature system response to voltage variation, that can be effortlessly evaluated for any given configuration with the numerical model, is not linear as shown in Fig. 5a. Extended Kalman Filter (EKF), continuous model parameters update or Unscented Kalman Filter (UKF) can be used to have better results, but the complexity and computational effort increase [9,10]. The simplest problem approach is to linearize the system response around the working point as in the example in Fig. 3b: in this continuous forging line, the surface temperature of the steel bars after the second inductor can be approximated with T = m ⋅ V + q in the range 900–1100 °C as shown in Fig. 5a.

Fig. 5.

(a) Correlation between voltage and max final surface temperature for a given system and (b) the results from the filtering data for the same system.

Thus, using the temperature/voltage coefficient m evaluated from the numerical model, a first simple evolution law can be represented as: $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}T\\ {\rm\Delta}V\end{array}\right)_{t}=\left[\begin{array}{@{}cc@{}}1 & m\\ 0 & 1\end{array}\right]\cdot \left(\begin{array}{@{}c@{}}T\\ {\rm\Delta}V\end{array}\right)_{t-1}+\left[\begin{array}{@{}c@{}}0\\ 1\end{array}\right]\cdot V_{a}(t) & & \displaystyle\end{eqnarray}$ (9) where the temperature evolution and voltage variation ΔV are updated and adjusted using the V_a parameter that is the external Voltage control. For this model, the KF parameters are defined as follows: $\begin{eqnarray}\displaystyle \boldsymbol{A}=\left[\begin{array}{@{}cc@{}}1 & m\\ 0 & 1\end{array}\right]\quad \boldsymbol{B}=\left[\begin{array}{@{}c@{}}0\\ 1\end{array}\right]\quad \mathbf{H}=\left[\begin{array}{@{}cc@{}}1 & 0\end{array}\right]\quad \mathbf{R}=[{\sigma}_{\mathit{pyrom}}^{2}]\quad \boldsymbol{Q}=\left[\begin{array}{@{}c@{}}{\sigma}_{T}^{2}\\[3.0pt] {\sigma}_{V}^{2}\end{array}\right] & & \displaystyle\end{eqnarray}$ (10) where ${\sigma}_{\mathit{pyrom}}^{2}$ is the variance of the pyrometer’s measures, and ${\sigma}_{T}^{2}$ and ${\sigma}_{V}^{2}$ are process variance parameters. An example of initial values to describe the system can be the following set: $\begin{eqnarray}\displaystyle x_{0}=\left(\begin{array}{@{}c@{}}1100∼^{\circ }\text{C}\\ 0∼\text{V}\end{array}\right)\quad u_{0}=0∼\text{V}\quad \mathbf{P}_{0}=\left[\begin{array}{@{}cc@{}}{\sigma}_{\mathit{pyrom}}^{2} & 0\\ 0 & {\sigma}_{volt}^{2}\end{array}\right]. & & \displaystyle\end{eqnarray}$ (11)

The estimated temperature at each iteration is used to adjust the inductor voltage to keep the required temperature setpoint as in Fig. 5b and Fig. 6. Tuning the R, Q , and the feedback control on the voltage, the system response can be optimized to fit the system inertia.

Fig. 6.

Filtered noisy temperature data keeping the setpoint temperature (1100 °C) with a change in the system behavior at 90 s; response time can be tuned playing with the settings.

Different models can be implemented to include, for example, the inlet part temperature in a multi-coils system.

3. Further development and conclusions

An interesting feature of the KF for a future development is the damage detection as in Fig. 6: if the implemented model is accurate, a constant deviation of the measure from the prevision can represent a change in the model behavior [8]. Different metrics can be implemented to track the system evolution. In a forging inductor, the refractory can be damaged or worn increasing the heat exchange and thus reducing the global efficiency of the system that can rise an automatic warning to the user.

The pyrometer measures are affected by two different error types: the instrument class and the non-gaussian error connected to the surface scales emissivity. This second error always gives a negative contribute. The KF theory requires a normal distribution but some authors [11,12] implemented modified versions of the filter to consider this effect. In [11], a Robust Sequential Estimator (RSE) is used to validate the data before the KF, removing the outlier values.

To sum up, the implemented numerical model gives the possibility to predict the behavior of a set of forging inductors: the temperature response for each heating sections of a inductors forging line can be correlated with voltage and the previous temperature. A linearized model of the system evolution can hence be implemented in the prediction step of a Kalman Filter to virtually increase the control pyrometer class, giving a finer control over system temperature.

References

Snape

R.G.

Clift

S.E.

and Bramley

A.N.

, Sensitivity of finite element analysis of forging to input parameters, Journal of Materials Processing Technology82(1-3) (1998) 10.21–26, doi:10.1016/S0924-0136(98)00005-3.

Rudnev

and Totten

G.E.

, ASM Handbook Volume 4C: Induction Heating and Heat Treatment, ASM International, Ohio, 2014.

Lavers

, Numerical solution methods for electroheat problems, IEEE Transactions on Magnetics19(6) (1983) 11.2566–2572, doi:10.1109/TMAG.1983.1062899.

Lavers

J.D.

, State of the art of numerical modeling for induction processes, COMPEL - The international journal for computation and mathematics in electrical and electronic engineering27(2) (2008) 37.335–349, doi:10.1108/03321640810847625.

Lupi

Forzan

and Aliferov

, Induction and Direct Resistance Heating: Theory and Numerical Modeling, Springer International Publishing, 2015: 370 pp.978-3-319-03478-2.

Kalman

R.E.

, A new approach to linear filtering and prediction problems, J. Basic Eng.82(1) (1960), 35–45.

Grewal

M.S.

and Andrews

A.P.

, Kalman Filtering: Theory and Practice with MATLAB, 4th edn, Wiley-IEEE Press, Hoboken, New Jersey, 2014.

Bulut

, Applied Kalman filter theory, PhD Thesis, Northeastern University, 2011.

Julier

S.J.

and Uhlmann

J.K.

, Unscented Filtering and Nonlinear Estimation, Proceedings of the IEEE92(3) (2004) 3.401–422, doi:10.1109/JPROC.2003.823141.

10.

Sastry

, Decomposition of the extended Kalman filter, IEEE Transactions on Automatic Control16(3) (1971) 6.260–261, doi:10.1109/TAC.1971.1099709.

11.

Mirza

M.J.

, A modified Kalman filter for non-Gaussian measurement noise, in: Communication Systems and Information Technology, Springer, Berlin, Heidelberg, 2011, pp. 401–409.978-3-642-21761-6.

12.

Naveau

Philippe

Genton

Marc G.

and Shen

Xilin

, A skewed Kalman filter, Journal of Multivariate Analysis94(2) (2005) 6.382–400, doi:10.1016/j.jmva.2004.06.002.