Setting-less transformer protection for ensuring security and dependability

Abstract

The “setting-less” protection approach is an innovative protection scheme with increased dependability and security. This method is based on dynamic state estimation, which continuously extracts the operating condition of a protection zone from real-time measurements and the protection zone dynamic model. The dynamic state estimation computes the probability that the measurements fit the dynamic model of the protection zone within the measurement accuracy via the Chi-square test. The probability represents the goodness of fit of measurements to the dynamic model of the protection zone. We refer to it as the confidence level that the protection zone is healthy. High confidence levels imply that the protection zone operates normally, while low confidence levels indicate that internal abnormalities/faults may have occurred. Note that the proposed method does not require coordination with other protection schemes which sometimes compromise security and/or dependability. In this paper, the setting-less protection approach is described and applied to transformer protection. The feasibility of the setting-less transformer protection is verified with numerical experiments of several case studies (e.g., transformer energization, external faults, internal faults, and high impedance faults). The method has been verified in the laboratory and there are plans for field testing.

Appendix

The dynamic state estimation of the setting-less protection is based on the time-domain dynamic model of the protection zone. Specifically, the component dynamic model is a set of algebraic and differential equations that describe the dynamics of the component as follows:

$$\begin{aligned}&0 = g(i(t),x(t),t),\end{aligned}$$

(21)

$$\begin{aligned}&\frac{\mathrm{d}x(t)}{\mathrm{d}t} = f(i(t),x(t),t),\end{aligned}$$

(22)

$$\begin{aligned}&x(t) = [{\begin{array}{ll} {v(t)}&{y(t)}\end{array}}]^\mathrm{T}, \end{aligned}$$

(23)

where x(t) is the vector of state variables, i(t) is the vector of terminal currents, v(t) is the vector of terminal voltages, y(t) is the vector of internal states, and t is the current time. These equations are first quadratized, i.e., in case of the existence of nonlinearities of degree larger than two, additional variables are introduced so that the resulting differential and algebraic equations do not include any nonlinearities greater than degree two [10, 11]. Subsequently, the quadratic integration method is used to integrate (21), (22), and (23) with a time step determined by three consecutive measured samples [12, 13], (i.e., by the sampling rate of the data acquisition system). As a result, a set of algebraic and differential equations are converted to a set of linear and quadratic algebraic equations, which are the component dynamic model expressed as (1), (2) and (3) in Sect. 3. This procedure is shown here for a three-phase transformer bank.

Figure 11 describes the equivalent circuit of the single-phase transformer model, which contains two resistances $r_{1}$ and $r_{2}$, two inductances $L_{1}$ and $L_{2}$, a shunt core resistance $r_{c}$, a shunt inductance $L_{m}$, and a magnetizing current $i_{m}(t)$. The currents $i_{1L}(t)$ and $i_{3L}(t)$ represent the terminal currents $i_{1}(t)$ and $i_{3}(t)$, respectively.

Fig. 11

The equivalent circuit diagram of a single-phase transformer

Fig. 12

Index mapping between the single-phase transformer model and the three-phase transformer model. Numbers in parenthesis mean the indices of state variables of the intermediate time $t_{m}$

The model equations for the single-phase transformer can be expressed as follows:

$$\begin{aligned}&i_1 (t)=i_{1L} (t),\end{aligned}$$

(24)

$$\begin{aligned}&i_1 (t)+i_2 (t)=0,\end{aligned}$$

(25)

$$\begin{aligned}&i_3 (t)=i_{3L} (t),\end{aligned}$$

(26)

$$\begin{aligned}&i_3 (t)+i_4 (t)=0,\end{aligned}$$

(27)

$$\begin{aligned}&r_c i_1 (t)+r_c Ni_3 (t)=r_c i_m (t)+e(t),\end{aligned}$$

(28)

$$\begin{aligned}&0=i_m (t)-z(t),\end{aligned}$$

(29)

$$\begin{aligned}&0=v_1 (t)-v_2 (t)-e(t)-r_1 i_{1L} (t)-L_1 \frac{\mathrm{d}}{\mathrm{d}t}i_{1L} (t),\end{aligned}$$

(30)

$$\begin{aligned}&0=v_3 (t)-v_4 (t)-Ne(t)-r_2 i_{3L} (t)-L_2 \frac{\mathrm{d}}{\mathrm{d}t}i_{3L} (t),\end{aligned}$$

(31)

$$\begin{aligned}&0=e(t)-\frac{\mathrm{d}}{\mathrm{d}t}\lambda (t),\end{aligned}$$

(32)

$$\begin{aligned}&0=y_1 (t)-\left( {\frac{\lambda (t)}{\lambda _0}} \right) ^{2},\end{aligned}$$

(33)

$$\begin{aligned}&0=y_2 (t)-y_1 (t)^{2},\end{aligned}$$

(34)

$$\begin{aligned}&0=y_3 (t)-y_2 (t)^{2},\end{aligned}$$

(35)

$$\begin{aligned}&0=y_4 (t)-y_1 (t)y_3 (t),\end{aligned}$$

(36)

$$\begin{aligned}&0=y_5 (t)-y_4 (t)\frac{\lambda (t)}{\lambda _0},\end{aligned}$$

(37)

$$\begin{aligned}&0=-i_0 y_5 (t)+z(t), \end{aligned}$$

(38)

where i(t) is the currents, v(t) is the voltages, N is the turn ratio of the transformer, y(t) and z(t) are the internal states, $i_{0}$ and $\lambda _{0}$ are the constants, and t is the current time. By applying the quadratic integration and model quadratization into equations, the single-phase transformer dynamic model is obtained as follows:

$$\begin{aligned} \left[ {{\begin{array}{c} {i_{1\phi } (t)}\\ 0\\ {i_{1\phi } (t_m)}\\ 0\end{array}}} \right]= & {} K_{1\phi } +L_{1\phi } \left[ {{\begin{array}{c} {v_{1\phi } (t)}\\ {y_{1\phi } (t)}\\ {v_{1\phi } (t_m)}\\ {y_{1\phi } (t_m)}\end{array}}}\right] \nonumber \\&+\left[ {{\begin{array}{c} {f_{1\phi } (t)}\\ {f_{1\phi } (t_m)}\end{array}}} \right] -\left[ {b_{1\phi } (t-h)} \right] , \end{aligned}$$

(39)

$$\begin{aligned} \left[ {{\begin{array}{c} {f_{1\phi } (t)}\\ {f_{1\phi } (t_m)}\end{array}}} \right]= & {} \left[ {{\begin{array}{c} \vdots \\ {{\left[ {{\begin{array}{cc} {v_{1\phi } (t)}\\ {y_{1\phi } (t)}\\ {v_{1\phi } (t_m)}\\ {y_{1\phi } (t_m)} \end{array}}} \right] ^{T}} {Q_i \left[ {{\begin{array}{c} {v_{1\phi } (t)}\\ {y_{1\phi } (t)}\\ {v_{1\phi } (t_m)}\\ {y_{1\phi } (t_m)} \end{array}}} \right] }}\\ \vdots \end{array}}} \right] , \end{aligned}$$

(40)

where the subscript $1\phi$ represents the single phase.

$$\begin{aligned} \left[ {b_{1\phi } (t-h)} \right] =M_{1\phi } \left[ {{\begin{array}{c} {i_{1\phi } (t-h)}\\ 0\end{array}}} \right] +N_{1\phi } \left[ {{\begin{array}{c} {v_{1\phi } (t-h)}\\ {y_{1\phi } (t-h)}\end{array}}} \right] . \end{aligned}$$

(41)

Finally, three single-phase transformer models are merged into a three-phase transformer dynamic model as illustrated in Fig. 12. The indices of the single-phase transformer models are re-assigned to those of the three-phase transformer model.

Based on the index mapping, the matrices and vectors (i.e., K, L, M, and N) of the single-phase transformer model in (39), (40), and (41) are integrated into one, finally forming the following three-phase transformer model:

$$\begin{aligned} \left[ {{\begin{array}{c} {i_{3\phi } (t)}\\ 0\\ {i_{3\phi } (t_m)}\\ 0\end{array}}} \right]= & {} K_{3\phi } +L_{3\phi } \left[ {{\begin{array}{c} {v_{3\phi } (t)}\\ {y_{3\phi } (t)}\\ {v_{3\phi } (t_m)}\\ {y_{3\phi } (t_m)} \end{array}}} \right] \nonumber \\&+\left[ {{\begin{array}{c} {f_{3\phi } (t)}\\ {f_{3\phi } (t_m)}\end{array}}} \right] -\left[ {b_{3\phi } (t-h)} \right] , \end{aligned}$$

(42)

$$\begin{aligned} \left[ {{\begin{array}{c} {f_{3\phi } (t)}\\ {f_{3\phi } (t_m)}\end{array}}} \right]= & {} \left[ {{\begin{array}{c} \vdots \\ {{\left[ {{\begin{array}{cc} {v_{3\phi } (t)}\\ {y_{3\phi } (t)}\\ {v_{3\phi } (t_m)}\\ {y_{3\phi } (t_m)} \end{array}}} \right] ^{T}} {Q_i \left[ {{\begin{array}{c} {v_{3\phi } (t)}\\ {y_{3\phi } (t)}\\ {v_{3\phi } (t_m)}\\ {y_{3\phi } (t_m)} \end{array}}} \right] }}\\ \vdots \end{array}}} \right] , \end{aligned}$$

(43)

$$\begin{aligned} \left[ {b_{3\phi } (t-h)} \right]= & {} M_{3\phi } \left[ {{\begin{array}{c} {i_{3\phi } (t-h)}\\ 0\end{array}}} \right] +N_{3\phi } \left[ {{\begin{array}{c} {v_{3\phi } (t-h)}\\ {y_{3\phi } (t-h)}\end{array}}} \right] ,\nonumber \\ \end{aligned}$$

(44)

where the subscript $3\phi$ indicates the three phase. Note that the above model is in the same form (syntax) as the model expressed with (1), (2), and (3). The model corresponds to a transformer bank. A similar model can be developed for shell-type or core-type three-phase transformers.