Constant temperature control of tundish induction heating power supply for metallurgical manufacturing

Abstract

The tundish induction heating power supply (TIHPS) is one of the most important equipment in the continuous casting process for metallurgical manufacturing. Specially, the constant temperature control is greatly significant for metallurgical manufacturing. In terms of the relationship between TIH load temperature and output power of TIHPS, the constant temperature control can be realized by power control. In this paper, a TIHPS structure with three-phase PWM rectifiers and full-bridge cascaded inverter is proposed. Besides, an input harmonic current blocking strategy and a load voltage feedforward control are also proposed to realize constant temperature control. To meet the requirement of the system, controller parameters are designed properly. Experiments are conducted to validate the feasibility and effectiveness of the proposed TIHPS topology and the control methods.

Keywords

tundish induction heating power supply (TIHPS) / constant temperature control / input harmonic current blocking / load voltage feedforward

Introduction

The tundish induction heating provides an important technology with a superior performance for metallurgical manufacturing industrial applications featuring its advantages of fast heating performance, reliability, cleanness, and safety [¹]. The tundish induction heating power supply (TIHPS) is the most important equipment to implement the heating system [²,³]. During the heating process, one of the challenges to be solved for TIHPS is to keep the temperature constant and ensure low superheat of the conductive material [⁴]. It is a challenge to control the TIH load since its impedance is highly nonlinear. In addition, the load power fluctuation may emulate harmonics in grid currents, reducing power quality. Motivated by the problem, it is meaningful to investigate a kind of topology and strategy for TIHPS.

One of the key issues of AC/AC TIHPS control locates at the constant temperature control of TIH load [⁵–⁸]. In Ref. [⁵], the model and control of temperature dynamics of induction furnace for TIH is investigated. Three PI controllers are discussed and the proper controller is selected for melting metal. From another view that the load power reference can be obtained from the temperature control [⁶], the object can be realized by directly controlling load power for TIHPS. Some studies have been conducted to address direct power control (DPC). In Ref. [⁹], a model predictive DPC is proposed for power compensation in a single-phase quasi-Z-source inverter to fulfill zero tracking error between the APF power and the load power. In Ref. [¹⁰], an instantaneous reactive power control is proposed to eliminate the distortion in grid current and degrade the total harmonic distortion (THD). A model-based predictive power control method, which reduces the possibility of low-frequency oscillation and suppresses current harmonic, is presented for traction line-side converter in Ref. [¹¹]. In addition, a power compensation scheme is introduced for doubly fed induction generators to achieve excellent steady-state stability and dynamic performance and improve the power quality [¹²]. However, the fluctuation of load power will bring forth harmonics into grid current, deteriorating power quality of grid.

On the other hand, since TIHPS is a kind of AC-AC converter, some different converters have been presented depending on the required output frequency and power [¹³–¹⁶]. In Ref. [¹³], a power supply topology with a magnetic energy recovery switch is proposed to realize the current control of frequency variation and amplitude variation. In Refs. [¹⁴,¹⁵], a voltage-fed high-frequency series load resonant inverter topology and an efficiency-improved zero voltage soft-switching high-frequency resonant (HF-R) inverter are presented for induction heating. Besides, the multilevel inverter proposed in Ref. [¹⁶] can synthesize quantized approximations of arbitrary AC waveforms.

In the present paper, a topology consisted of a former three-phase PWM rectifier and a later cascaded inverter composed of full-bridge submodules (FBSMs) is proposed for TIHPS. The former rectifier provides a stable DC source for the later inverter. The cascaded structure power supply for the TIH load can reduce the THD of the load current and voltage stress of switches. An input harmonics blocking strategy is presented in the rectifier to improve power grid and to suppress voltage ripples of DC-link voltage. Load voltage and power feedforward are employed to improve the dynamics of closed-loop control and compensate the power loss.

Topology and operation principle of TIHPS

The proposed topology of TIHPS is depicted in Fig. 1. In Fig. 1(a), the TIHPS is comprised of a three-phase PWM rectifier and a cascaded inverter (N = 6). In addition, u_sx (x= a, b, c) and i_sx represent the three phase voltages and currents of secondary side of transformer respectively. The indispensable multi-winding transformer with a ratio of 10 kV/0.4 kV is connected in series between the grid and the TIHPS. The phase lead of the transformer is 30° when N = 6. The electromagnetic heater load and its equivalent circuit are shown in Fig. 1(b). The molten steel can be regarded as a static load due to its very slow speed. The operation principle of the induction heater can be approximately seen as a single-phase transformer [¹⁴]. R₁, L₁ and R₂, L₂ denote the equivalent impedance of the coil and molten steel respectively, and C_com is the series compensating capacitor. The equivalent resistance and inductance are R_o = R₁ + R₂, L_o = L₁ + L₂ + 1/ω_oC_com. The circuit equation of load is given as

(1)u_{\mathrm{inv}}=L_{\mathrm{o}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{o}}}{\mathrm{d}t}}+R_{\mathrm{o}}{i}_{\mathrm{o}}+u_{\mathrm{o}},

where u_inv is the output voltage of the cascaded inverter. Defining T as the switching period, the discrete mathematical model can be obtained as

(2)u_{\mathrm{inv}}\left(k\right)={\displaystyle \frac{L_{\mathrm{o}}}{T}}\left[i_{\mathrm{o}}^{*}\right(k)-i_{\mathrm{o}}(k\left)\right]+R_{\mathrm{o}}{i}_{\mathrm{o}}+u_{\mathrm{o}}\left(k\right).

Hence, the operation principle of TIHPS is shown as follows: The AC power from the grid can be converted into the desired AC load power. According to the positive directions of voltages and currents in Fig. 1(a), the equations of fore-stage PWM rectifier in abc rotating coordinate can be obtained as

(3)\left[\begin{array}{l}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{s}a}}{\mathrm{d}t}}\hfill \\ {\displaystyle \frac{\mathrm{d}{i}_{\mathrm{s}b}}{\mathrm{d}t}}\hfill \\ {\displaystyle \frac{\mathrm{d}{i}_{\mathrm{s}c}}{\mathrm{d}t}}\hfill \end{array}\right]={\displaystyle \frac{\mathrm{1}}{L}}\left[\begin{array}{l}{u}_{\mathrm{s}a}\hfill \\ u_{\mathrm{s}b}\hfill \\ u_{\mathrm{s}c}\hfill \end{array}\right]-{\displaystyle \frac{U_{\mathrm{dc}}}{L}}\left[\begin{array}{l}{s}_{ca}\hfill \\ s_{cb}\hfill \\ s_{cc}\hfill \end{array}\right].

The switching signal s_x can be defined as

(4)\{\begin{array}{l}{s}_{ca}=s_a-{\displaystyle \frac{s_a+s_b+s_c}{\mathrm{3}}}\hfill \\ s_{cb}=s_b-{\displaystyle \frac{s_a+s_b+s_c}{\mathrm{3}}}\hfill \\ s_{cc}=s_c-{\displaystyle \frac{s_a+s_b+s_c}{\mathrm{3}}}\hfill \end{array}.

The discrete model can be derived as

(5)\left[\begin{array}{l}{u}_{ca}\left(k\right)\hfill \\ u_{cb}\left(k\right)\hfill \\ u_{cc}\left(k\right)\hfill \end{array}\right]=U_{\mathrm{dc}}\left[\begin{array}{l}{s}_{ca}\left(k\right)\hfill \\ s_{cb}\left(k\right)\hfill \\ s_{cc}\left(k\right)\hfill \end{array}\right]=\left[\begin{array}{l}{u}_{\mathrm{s}a}\left(k\right)\hfill \\ u_{\mathrm{s}b}\left(k\right)\hfill \\ u_{\mathrm{s}c}\left(k\right)\hfill \end{array}\right]-{\displaystyle \frac{L}{T}}\left[\begin{array}{l}{i}_{\mathrm{s}a}^{*}\left(k\right)-i_{\mathrm{s}a}\left(k\right)\hfill \\ i_{\mathrm{s}b}^{*}\left(k\right)-i_{\mathrm{s}b}\left(k\right)\hfill \\ i_{\mathrm{s}c}^{*}\left(k\right)-i_{\mathrm{s}c}\left(k\right)\hfill \end{array}\right].

Proposed control strategies of TIHPS

Temperature closed-loop control of TIH load

To realize the constant temperature, a closed-loop temperature control strategy is proposed, as demonstrated in Fig. 2. The temperature can be controlled to generate power reference for the TIHPS by employing a PI controller PI_t [⁵,⁶]. T and T* represent the actual and reference temperature.

Three phase PWM rectifier control strategy

Derivation of 2nd ripple of DC-link voltage

The sum of three-phase input power P_in only contains DC component {\overline{P}}_{\mathrm{in}} without harmonics because the 2nd harmonic component can be offset in terms of Eq. (6). The DC-link voltage u_dc will not be affected by the input power.

(6)P_{\mathrm{in}}=P_a+P_b+P_c={\overline{P}}_{\mathrm{in}}.

Owing to the fact that the load operates at ω_o supplied by TIHPS, the output voltage and current are defined as u_o = U_omsin(ω_ot) and i_o = I_omsin(ω_ot + ϕ_o). The instantaneous load power p_o and its AC fluctuation component p_o and {\tilde{p}}_o can be respectively expressed as

(7)\{\begin{array}{l}{p}_{\mathrm{o}}=u_{\mathrm{o}}\times i_{\mathrm{o}}\hfill \\ =U_{\mathrm{om}}{I}_{\mathrm{om}}·[\sin {\omega }_{\mathrm{o}}t·\sin ({\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{o}}\left)\right]\hfill \\ ={\displaystyle \frac{U_{\mathrm{om}}{I}_{\mathrm{om}}}{\mathrm{2}}}·[\cos {\varphi }_{\mathrm{o}}-\cos (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{o}}\left)\right]\hfill \\ {\tilde{p}}_{\mathrm{o}}={\displaystyle \frac{U_{\mathrm{om}}{I}_{\mathrm{om}}}{\mathrm{2}}}·\cos (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{o}})\hfill \end{array},

where u_dc contains the 2nd load frequency (2ω_o) fluctuation, expressed as

(8)\mathrm{\Delta }u=\delta \sin (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}{\omega }_{\mathrm{o}}}),

where δ and ϕ_2ωo are the amplitude and initial phase of Du respectively. Considering u_dc = U_dc + Du, the DC-link power p_dc can be obtained as

(9)p_{\mathrm{dc}}=u_{\mathrm{dc}}·C{\displaystyle \frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}}=C[U_{\mathrm{dc}}+\delta \sin (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}\omega }\left)\right]·\left[\mathrm{2}{\omega }_{\mathrm{o}}\delta \cos \right(\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}\omega }\left)\right]=\mathrm{2}{\omega }_{\mathrm{o}}C{U}_{\mathrm{dc}}\delta \cos (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}\omega })+{\omega }_{\mathrm{o}}C{\delta }^{\mathrm{2}}\sin (\mathrm{4}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}\omega }).

Ignoring the 4th ripples, Du can be derived as

(10)\mathrm{\Delta }u=\delta \sin (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}{\omega }_{\mathrm{o}}})=-{\displaystyle \frac{U_{\mathrm{om}}{I}_{\mathrm{om}}}{\mathrm{4}{\omega }_{\mathrm{o}}C{U}_{\mathrm{dc}}}}\sin (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{o}}).

As expressed in Eq. (10), Du is proportional to U_omI_om, and inversely proportional to CU_dc.

Input harmonic current blocking strategy

The control diagram of the former PWM rectifier is displayed in Fig. 3. Since Du is introduced, the load power could generate input harmonic components. When the outer voltage loop uses the PI controller, the input harmonic current reference i_{\mathrm{sh}}^{*} can be calculated as

(11)i_{\mathrm{sh}}^{*}=K\mathrm{\Delta }u·\sin \left(\omega t\right)=K\delta ·\sin (\mathrm{2}{\omega }_{\mathrm{o}}t+{\varphi }_{\mathrm{2}{\omega }_{\mathrm{o}}})·\sin \left(\omega t\right)={\displaystyle \frac{K\delta }{\mathrm{2}}}\left[\cos \right((\mathrm{2}{\omega }_{\mathrm{o}}-\omega )t+{\varphi }_{\mathrm{2}{\omega }_{\mathrm{o}}})-\cos ((\mathrm{2}{\omega }_{\mathrm{o}}+\omega )t+{\varphi }_{\mathrm{2}{\omega }_{\mathrm{o}}}\left)\right],

where ω is the grid frequency, and K is the proportional coefficient of the PI controller.

In Eq. (11), i_{\mathrm{sh}}^{*} contains 2ω_o±ω components with an amplitude of (Kδ)/2. To suppress the input current ripples, an input harmonics blocking method is proposed. The error of u_dc can be obtained by subtracting the 2nd ripple component calculated using Eq. (10) from the difference between u_dc and u_{\mathrm{dc}}^{*}. Then, the error can be sent to the PI controller to generate the regulation signal I_dc which is used to multiply with the synchronous angular of the grid to produce the input current reference i_{\mathrm{s}x}^{*}.

Additionally, because the inverter provides power for TIH load supplied from dc-link voltage, the load power feedforward should be compensated and the feedforward current amplitude I_p can be obtained as

(12)I_{\mathrm{p}}={\displaystyle \frac{\mathrm{2}{P}_{\mathrm{o}}^{*}}{\mathrm{3}{U}_{\mathrm{s}x\mathrm{(rms)}}}},

where U_sx(rms) is the root mean square (RMS) of u_sx, P_{\mathrm{o}}^{*} is the reference of P_o, and I_p can be added to I_dc.

Cascaded inverter control strategy

The control diagram of the cascaded inverter is presented in Fig. 4 and the reference i_{\mathrm{o}}^{*} can be obtained as

(13)I_{\mathrm{o}\left(\mathrm{rms}\right)}^{*}=\sqrt{{\displaystyle \frac{\mathrm{2}{P}_{\mathrm{o}}^{*}}{Z_{\mathrm{o}}}}},

where I_{\mathrm{o(rms)}}^{*} is the RMS amplitude of i_{\mathrm{o}}^{*}, Z_o = \sqrt{R_{\mathrm{o}}^{\mathrm{2}}+X_{\mathrm{o}}^{\mathrm{2}}} is the equivalent impedance of load, and I_{\mathrm{o(rms)}}^{*} is multiplied with the synchronous angular signal u_o to produce the final reference signal i_{\mathrm{o}}^{*}.

Similarly, the closed-loop system with a PR controller can ensure the precise load current tracking. Furthermore, to improve the anti-disturbance ability of the TIH load, the load voltage feedforward can be employed.

Dynamic analysis of controllers and parameters design

According to Figs. 3–4, the block diagram of the closed-loop control for DC-link voltage u_dc, input current i_sx, and load current i_o using different PR controllers are given in Fig. 5 respectively. Owing to the fact that the temperature control speed is obviously lower than that of the converter, the rectifier, the inverter and the TIH load can be regarded as the 1st order inertia, as shown in Fig. 5(a), and G_pwm(s) can be obtained from Eq. (14), K_pwm = 1.

(14)G_{\mathrm{pwm}}\left(s\right)=K_{\mathrm{pwm}}{\displaystyle \frac{\mathrm{1}-\mathrm{0.5}Ts}{\mathrm{1}+\mathrm{0.5}Ts}}.

Considering tracking i_{\mathrm{s}x}^{\mathrm{ref}} and i_{\mathrm{o}}^{\mathrm{ref}} at f and f_o respectively, the PR₁ and PR₂ controller can be given as

(15)\{\begin{array}{l}{G}_{{\mathrm{PR}}_1}\left(s\right)=K_{\mathrm{p}\mathrm{1}}+K_{\mathrm{r}\mathrm{1}}{\displaystyle \frac{s}{s^{\mathrm{2}}+{\omega }_{}^{\mathrm{2}}}}\hfill \\ G_{{\mathrm{PR}}_2}\left(s\right)=K_{\mathrm{p}\mathrm{2}}+K_{\mathrm{r}\mathrm{2}}{\displaystyle \frac{s}{s^{\mathrm{2}}+{\omega }_{\mathrm{o}}^{\mathrm{2}}}}\hfill \end{array}.

The PI controller can be expressed as

(16)G_{\mathrm{PI}}\left(s\right)={\displaystyle \frac{k_{\mathrm{p}}(T_{\mathrm{i}}s+\mathrm{1})}{T_{\mathrm{i}}s}}.

Temperature closed-loop system

In Fig. 5(a), G_rec(s), G_inv(s), and G_load(s) indicate the equivalent transfer functions of the three parts of the TIHPS. The transfer functions can be defined as

(17)\{\begin{array}{l}{G}_{\mathrm{rec}}\left(s\right)={\displaystyle \frac{\mathrm{1}-\mathrm{0.5}{T}_{\mathrm{1}}s}{\mathrm{1}+\mathrm{0.5}{T}_{\mathrm{1}}s}}\hfill \\ G_{\mathrm{inv}}\left(s\right)={\displaystyle \frac{\mathrm{1}-\mathrm{0.5}{T}_{\mathrm{2}}s}{\mathrm{1}+\mathrm{0.5}{T}_{\mathrm{2}}s}}\hfill \\ G_{\mathrm{load}}\left(s\right)={\displaystyle \frac{\mathrm{1}-\mathrm{0.5}{T}_{\mathrm{3}}s}{\mathrm{1}+\mathrm{0.5}{T}_{\mathrm{3}}s}}\hfill \end{array}.

The PI controller used in the temperature closed-loop can be expressed as G_PIt(s) = k_pt(T_its + 1)/T_its. The open-loop and closed-loop transfer function can be obtained from Eq. (18). The characteristic equation is shown in Eq. (19). According to the Routh stability criterion, the sufficient conditions should meet the requirement of nonnegative coefficient in Eq. (19). The relationship between k_pt, T_it and Y₁, Y₂, Y₃ is shown in Fig. 6. In Figs. 6(a) and 6(c), Y₁ and Y₃ can always remain positive, regardless of the values obtained for k_pt and T_it. In Fig. 6(b), when k_pt is fixed, a smaller T_it could lead to a negative Y₂. When k_pt and T_it are valued in the area of Y₂>0, the system can be ensured to be stable.

(18)\{\begin{array}{l}{G}_{\mathrm{o\backslash vskip\; 2pt-\backslash vskip\; -2ptT}}\left(s\right)=G_{\mathrm{PIt}}\left(s\right)·G_{\mathrm{rec}}\left(s\right)·G_{\mathrm{inv}}\left(s\right)·G_{\mathrm{load}}\left(s\right)\hfill \\ G_{\mathrm{c}\mathrm{\backslash vskip\; 2pt-\backslash vskip\; -2ptT}}\left(s\right)={\displaystyle \frac{G_{\mathrm{PIt}}\left(s\right)·G_{\mathrm{rec}}\left(s\right)·G_{\mathrm{inv}}\left(s\right)·G_{\mathrm{load}}\left(s\right)}{\mathrm{1}+G_{\mathrm{PIt}}\left(s\right)·G_{\mathrm{rec}}\left(s\right)·G_{\mathrm{inv}}\left(s\right)·G_{\mathrm{load}}\left(s\right)}}\hfill \end{array},

(19)\{\begin{array}{l}{b}_{\mathrm{0}}{s}^{\mathrm{4}}+b_{\mathrm{1}}{s}^{\mathrm{3}}+b_{\mathrm{2}}{s}^{\mathrm{2}}+b_{\mathrm{3}}{s}^{\mathrm{1}}+b_{\mathrm{4}}=\mathrm{0}\hfill \\ b_{\mathrm{4}}>\mathrm{0}\hfill \\ Y_{\mathrm{1}}=b_{\mathrm{1}}>\mathrm{0}\hfill \\ Y_{\mathrm{2}}=(b_{\mathrm{1}}{b}_{\mathrm{2}}-b_{\mathrm{3}}{b}_{\mathrm{0}})/b_{\mathrm{1}}>\mathrm{0}\hfill \\ Y_{\mathrm{3}}=b_{\mathrm{3}}-b_{\mathrm{1}}^{\mathrm{2}}{b}_{\mathrm{4}}/(b_{\mathrm{1}}{b}_{\mathrm{2}}-b_{\mathrm{3}}{b}_{\mathrm{0}})>\mathrm{0}\hfill \end{array}.

Input current and voltage closed-loop system

For current control, PR controllers are adopted owing to the fact that they have infinite gain at the resonant frequency and very small gain at other frequencies [¹³]. In Fig. 5(b), the open-loop transfer function can be expressed as

(20)G_{\mathrm{o\backslash vskip\; 2pt-\backslash vskip\; -2ptis}}\left(s\right)=G_{{\mathrm{PR}}_1}\left(s\right)·G_{\mathrm{pwm}}\left(s\right)·(\mathrm{1}/(L_{\mathrm{s}}s+R_{\mathrm{s}}\left)\right).

K_p1 and K_r1 directly affect the dynamics of the input current control system. As shown in Fig. 7(a), when K_r1 remains constant, a higher K_p1 brings forth a higher gain, accelerating the response speed to a low frequency range and a high frequency range.

In light of Fig. 5(b), for analyzing the outer voltage closed-loop system, the dual-loop control can be simplified as presented in Fig. 5(c). The closed-loop transfer function of the inner input current G_{c_is}(s) is approximately equal to 1 on account of a much faster current response speed. Then, the open-loop and closed-loop transfer function can be expressed as

(21)\{\begin{array}{l}{G}_{\mathrm{o\backslash vskip\; 2pt-\backslash vskip\; -2ptudc}}\left(s\right)=G_{\mathrm{PI}}\left(s\right)·G_{\mathrm{c\backslash vskip\; 2pt-\backslash vskip\; -2ptis}}\left(s\right)·(d/sC)\hfill \\ G_{\mathrm{c\backslash vskip\; 2pt-\backslash vskip\; -2ptudc}}\left(s\right)={\displaystyle \frac{G_{\mathrm{PI}}\left(s\right)·G_{\mathrm{c\backslash vskip\; 2pt-\backslash vskip\; -2ptis}}\left(s\right)·(d/sC)}{\mathrm{1}+G_{\mathrm{PI}}\left(s\right)·G_{\mathrm{c\backslash vskip\; 2pt-\backslash vskip\; -2ptis}}\left(s\right)·(d/sC)}}\hfill \end{array},

(22)T_{\mathrm{i}}C{s}^{\mathrm{2}}+T_{\mathrm{i}}{k}_{\mathrm{p}}ds+k_{\mathrm{p}}d=\mathrm{0.}

In Eq. (22), the sufficient condition meet CT_i>0, dk_pT_i>0, and dk_p>0. In Figs. 7(b) and 7(c), a higher k_p brings forth a larger control gain over a broad frequency range, but results in a lower phase margin and a higher crossover frequency under the circumstance of T_i = 0.5. The bode plots in Fig. 7(b) indicate that the maximum phase margins exist in the case of appropriate k_p. For the same k_p in Fig. 7(c), the phase margin will be increased with the increasing T_i. Synthesizing the requirements of phase margin or amplitude margin for the system, the parameters of the PI and the PR controller can also be designed properly.

Load current closed-loop system

In terms of Fig. 5(d), the load model can be expressed as Z_o = L_os + R_o. The load voltage feedforward can be introduced into the load current closed-loop control. Therefore, the open-loop transfer function G_{o_io}(s) and G_{o_fed}(s) and the transfer function of the tracking error G_{e_io}(s) and G_{e_fed}(s), without and with feedforward control in Fig. 5(d) can be expressed in Eqs. (23) and (24) respectively.

(23)\{\begin{array}{l}{G}_{\mathrm{o\backslash vskip\; 2pt-\backslash vskip\; -2ptio}}\left(s\right)=G_{{\mathrm{PR}}_{\mathrm{2}}}\left(s\right)·G_{\mathrm{pwm}}\left(s\right)·(\mathrm{1}/Z_{\mathrm{o}})\hfill \\ G_{\mathrm{o\backslash vskip\; 2pt-\backslash vskip\; -2ptfed}}\left(s\right)=\left(G_{{\mathrm{PR}}_{\mathrm{2}}}\right(s)+Z_{\mathrm{o}})·G_{\mathrm{pwm}}\left(s\right)·(\mathrm{1}/Z_{\mathrm{o}})\hfill \end{array},

(24)\{\begin{array}{l}{G}_{\mathrm{e\backslash vskip\; 2pt-\backslash vskip\; -2ptio}}\left(s\right)={\displaystyle \frac{e_{\mathrm{i}}\left(s\right)}{i_{\mathrm{o}}^{\mathrm{ref}}\left(s\right)}}={\displaystyle \frac{Z_{\mathrm{o}}}{Z_{\mathrm{o}}+G_{{\mathrm{PR}}_2}\left(s\right)·G_{\mathrm{pwm}}\left(s\right)}}\hfill \\ G_{\mathrm{e\backslash vskip\; 2pt-\backslash vskip\; -2ptfed}}\left(s\right)=={\displaystyle \frac{e_{\mathrm{i}}\left(s\right)}{i_{\mathrm{o}}^{\mathrm{ref}}\left(s\right)}}={\displaystyle \frac{Z_{\mathrm{o}}-Z_{\mathrm{o}}·G_{\mathrm{pwm}}\left(s\right)}{Z_{\mathrm{o}}+G_{{\mathrm{PR}}_2}\left(s\right)·G_{\mathrm{pwm}}\left(s\right)}}\hfill \end{array}.

Hence, in Fig. 8, the bode plot comparisons of the load current control system can be depicted.

In Fig. 8(a), compared with the conventional method, the open-loop gain in a high frequency range is approximately 1 when introducing feedforward control, improving the dynamic response to high frequency input signals while the open-loop gain with conventional method is attenuated in high frequency range, demonstrating that the system has a poor sensitivity to a high frequency signal. In Fig. 8(b), the error gain with the feedforward control is attenuated compared to the conventional one, revealing that the feedforward of load voltage can reduce the tracking error.

Experimental results

To validate the proposed topology of TIHPS and the proposed control method, an experimental system is established in the industrial manufacturing according to Fig. 1. The parameters of the system are listed in Table 1.

Input harmonic current blocking strategy validation

The experimental results of three phase grid currents without and with the proposed input harmonics blocking strategy are depicted in Figs. 9(a) and 9(b), respectively. In Fig. 9(a), without the proposed input harmonics blocking, the harmonic components at 350 Hz are i_sa (58.76 A), i_sb (34.71 A) and i_sc (53.58 A), and i_sa (21.76 A), i_sb (15.96 A) and i_sc (33.45 A) for 450 Hz. When employing the proposed strategy, it should be pointed out that the harmonic components of input currents can be reduced greatly. The input harmonic current at 350 Hz are i_sa (6.26 A), i_sb (11.36 A) and i_sc (8.62 A), and i_sa (9.19 A), i_sb (6.41 A) and i_sc (8.58 A) for 450 Hz, as shown in Fig. 9(b).

In terms of Eq. (10), the ripple Du_{dc_cal} can be calculated. The comparison of Du_{dc_cal} and the measured ripple Du_{dc_real} are shown in Fig. 9(c). Combining with the error Du_{dc_err} between Du_{dc_cal} and Du_{dc_real}, Du_{dc_cal} can basically track Du_{dc_real} comparatively precisely, verifying the feasibility of the proposed input harmonic current blocking strategy and the correctness of the derivation process.

Load control combining feedback and feedforward loop validation

The transient case is set as: before t₁, TIHPS operates with P_{\mathrm{o}}^{*}; at t₁, there occurs a disturbance of 50 kW on P_{\mathrm{o}}^{*}, remaining for 0.05 s. At (t₁ + 0.05), disturbance disappears.

Figures 10(a) and 10(b) show the response curves of load current and its reference under the occurrence of load power disturbance. Here, i_{o_wo} and i_{o_fed} represent the load current only with closed-loop control and with the additional feedforward control respectively. It is observed that i_{o_fed} has a better dynamics and faster response speed to the disturbance of P_{\mathrm{o}}^{*} compared to those of i_{o_wo}. To further validate the effectiveness, the corresponding current error Di_{o_wo} and Di_{o_fed} are shown in Fig. 10(c). Obviously, the maximum of Di_{o_wo} is 44.3 A, while only 36.5 A for Di_{o_fed}. Besides, when P_{\mathrm{o}}^{*} decreases 50 kW at t₁, Di_{o_wo} increases apparently and Di_{o_fed} still basically keeps the constant value, in consistent with the theoretical analysis.

TIH load temperature closed-loop control validation

Figure 11 shows the dynamics of TIH load temperature control. The objective temperature T* is set at 1000°C. During 0–t₁, T* remains 1000°C; during t₁–t₂ and t₃–t₄, there occurs a temperature interruption as 50°C in T*. During t₂–t₃ and after t₄, T* recovers to 1000°C. Figure 11(a) shows the initial rise of temperature. The process can be finished less than 60 ms. When the temperature disturbance DT^ref occurs at t₁ and t₃ in Fig. 11(b), T can respond to the transient interruption quickly within 10 ms and return to a steady-state. The tracking error (T*–T) maintains within 3°C–5°C under normal operation condition, validating the good dynamic performance and control precision of the closed-loop temperature control strategy.

Conclusions

To realize the constant temperature control of the TIH load, a TIHPS structure constructed by a former PWM rectifier and a latter cascaded inverter for tundish induction heating is proposed and the corresponding control methods are investigated in this paper. The contributions are described as follows:

(1) To ensure the constant temperature operation of the TIH load, a temperature closed-loop control strategy is proposed to improve the control precision and anti-disturbance performance. Experiments prove that TIHPS operates with only 3°C temperature error.

(2) For the PWM rectifier, an input harmonic current blocking strategy is proposed to suppress the input harmonic currents caused by load power ripples, improving the power quality of power grid.

(3) For the cascaded inverter, a load voltage feedforward loop is introduced into the load current closed-loop control system. Compared with the conventional method only with current closed-loop control, the proposed comprehensive control combining feedback loop and feedforward loop can accelerate the dynamic response speed and improve the tracking precision of load current.

(4) The parameters of the controller are designed properly and experiments are conducted to verify the feasibility and effectiveness of the proposed structure and strategies for the TIHPS.

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