Transformer Monitoring Using Kalman Filtering* Subramanian V. Shastri, Member, IEEE, Emma Stewart, Senior Member, IEEE, and Ciaran Roberts, Member IEEE Abstract— In this paper, we present a systems-theoretic approach to real-time in-situ monitoring of operating transformers. The most significant and novel result is the estimation of partial discharge buildup in transformers. In addition, it is capable of calculating secondary side power factor, and detecting voltage fluctuations, reactive buildup and core saturation. The paper discusses critical design considerations such as sampling time, model excitation, and system order. Concerns regarding power quality, reliability and resilience are increasing in the distribution grid with the injection of power from renewables. The algorithm presented here could help mitigate these by continuously monitoring transformer health and performance during operation.
I. INTRODUCTION New transformers start with efficiencies greater than 98%. But as their components age, units begin to operate at lower efficiencies, and with increased I2R losses and elevated thermal profiles. On an average, about 65% of transformers on the U.S. grid tend to operate at 125% to 150% of their capacity during peak summer hours. Operating for 4 hours at 125% capacity can cause a decrease in life expectancy of 1 week. Operating for 8 hours at 150% reduces it by ½ year! This implies that loading transformers above rated capacity can cause exponential aging of critical components such as insulation and coolant oil. This, in turn, further reduces efficiency, increases thermal gradients and leads to a Loss of Life (LoL) model similar to ([10]). Understanding the age and condition of insulation and coolant oil in transformers, therefore, is important for maintaining grid reliability ([3]). Transformers are also beginning to operate in regimes beyond their intended design and control capability. The aggressive growth in distributed energy resources (DER) forecast over the next 25 years, if not curtailed, will most likely amplify short and long-term voltage fluctuations in the grid ([7], [15]) and cause bi-lateral flow of power through transformers. The thermal gradients resulting from reactive buildup and discharge in transformers will accelerate aging in its critical components. With high DER injection and electric vehicle (EV) integration, very fast transients (VFTs) in the KHz band from solid-state inverters will further exacerbate aging in nearby transformers ([9] and [13]). Hence, in-situ monitoring of transformers during operation will be essential for ensuring power quality and reliability in a future when DER and EV become significant in the distribution grid ([19]).
* This work has been funded in part by the Office of Electricity Delivery and Energy Reliability, Distributed Energy Program of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. S.V. Shastri is Professor of Practice & Director of Industry Partnerships at the Shiley-Marcos School of Engineering, University of San Diego, CA (email:
[email protected]) and President of Transense, Inc.
In this paper, we refer to in-situ monitoring of operating transformers as inline monitoring. The fastest growing inline method in the industry is Dissolved Gas Analysis (DGA). It involves the analysis of gases generated in the cooling oil and insulation at elevated temperatures for the detection of fault conditions in transformers ([5], [11] and [18]). DGA has demonstrated some success in monitoring larger power transformers, despite suffering from sampling and classification errors, and lacking the ability to predict the onset of faults. Its applicability to distribution transformers is limited by the fact that it cannot be used on dry-type units, and it is not cost effective for smaller oil-cooled units. There are microphasor-based grid monitoring techniques that can be used to infer transformer state ([16]) by looking at circuit-level impedance ([2] and [6]). But they do not describe transformer condition. In this paper, we take a direct approach to the monitoring problem ([25]). At present, the only direct technique capable of in-situ monitoring is called Sweep Frequency Response Analysis (SFRA) (or a closely related technique called Dielectric Frequency Response Analysis, or DFRA). It is an input-output method used to empirically determine the presence of faults in unloaded transformers and is applied to a unit that has been taken offline for maintenance [27]. In contrast, our approach is capable of inline transformer monitoring. It relies on the development of a theoretical model of an operating transformer and a practical algorithm to estimate model parameters. Parameter estimates may then be used to determine the health and condition of transformers. The discrete-time state space theoretical model describes an extended equivalent circuit representation of a practical transformer under load, and includes the effects of aging and thermal gradients. Estimation of model parameters is performed using Kalman filtering and Akaike criterion. II. DIRECT ELECTRICAL MONITORING It is well known since the work of Steinmetz ([26]) that the input-output properties of an ideal transformer may be represented using the RL circuit in Figure 1. It assumes that power transfer from the primary to secondary windings is governed solely by the windings ratio. Circuit elements represent I2R, flux, hysteresis and eddy current losses, and the model has historically been the basis for describing how transformers work. Actual values of the circuit elements may be calculated using open- and short-circuit tests. The transfer function between input and output terminals is given by E. Stewart is Deputy Group Leader and Research Scientist, Lawrence Berkeley National Laboratory, Berkeley, CA (email:
[email protected]). C.M. Roberts is a Senior Scientific Engineering Associate, Lawrence Berkeley National Laboratory, Berkeley, CA (email:
[email protected])..
Equation 1, where (𝑉# 𝐼# ) , (𝑉& 𝐼& ) are the corresponding voltage-current pairs, and 𝑁( = 𝑁+ and 𝑁, = 𝑁-. .1 −1𝑅,3 0
𝑁, ⎡ + 𝑠 𝐿,3 79 ⎢𝑁= 1 ⎢0 ⎣
1
B 𝑠𝐿𝑝𝑐 +𝑅𝑝𝑐 – 𝑠𝐿 𝑅 𝑝𝑐 𝑝𝑐
Z[
B
ZH
0
1
0
Z\ N c 𝑠𝐿𝑝𝑐 +𝑅𝑝𝑐 Z[
–
𝑠𝐿𝑝𝑐 𝑅𝑝𝑐
− 1 𝑅𝑝𝑓 + 𝑠 𝐿𝑝𝑓 7 gF𝑠𝐿𝑝𝑐 +𝑅𝑝𝑐 Mh𝑅𝑝𝑓 +𝑠 𝐿𝑝𝑓 i+ 𝑠𝐿𝑝𝑐 𝑅𝑝𝑐 je .
𝑠𝐿𝑝𝑐 𝑅𝑝𝑐
𝑉𝑖 9 𝐼𝑖
(3)
0⎤ ⎥ 𝑁= ⎥ 𝑁> ⎦
Equation (3) describes the transfer function of a practical transformer. (Ri, Ci) represents the resistance and capacitance of the ith partial discharge, and (Rt, Ct), those of the coolant oil.
− 1𝑅(3 + 𝑠 𝐿(3 7
The theoretical basis for SFRA is also Equation 3. It is an input-output technique that uses Bode Plots of the transfer function to sinusoidal excitation signals to determine the condition of a transformer. SFRA is typically conducted offline on unloaded units. Variances between a healthy unit (blue) and a test unit (red) are used to detect potential faults in the latter. Since issues related to the core, windings, and taps and connections are predominantly manifest in distinct subchannels (Figure 3), fault assessment using visual inspection or simple correlation techniques is common ([27]).
𝑉𝑖 𝑉𝑜 F1,GHI JKHI 71KHL J, GHL 7J ,GHIKHI MN O Q = O Q 𝐼𝑖 𝐼𝑜
(1)
,GHI KHI
Figure 1: Steinmetz Circuit of Idealized Transformer
Figure 2: Equivalent Circuit of Practical Transformer Figure 1 does not take into account power transfer losses in the coolant oil and insulation. Thermal gradients in the former and degradation of the dielectric properties in the latter increase such losses. Furthermore, the cellulose dielectric material cracks or forms voids and build up electrical charge. This happens when the material loses moisture and hardens as it ages. The capacitive effect of bubbles created by the gases released from mineral oil at elevated temperature can also cause a charge buildup. In turn, buildups introduce partial discharge paths during operation. The work in [23] analyzes such phenomena in some detail and proposes an RC network to model their effect on power transfer. By integrating this into Figure 1, we derive an equivalent circuit of what is known as a practical transformer (Figure 2). Figure 2(b) describes the RC network. Parallel branches of resistances and capacitances placed in series are used to represent the impact of partial discharge paths ([22] discusses the use of such a model for insulation). The final parallel branch of resistor and capacitor represents electrical properties of the coolant oil. While the structure of the network in 2(b) remains constant, the number of branches as well as values of circuit elements change during operation. The RC network transfer function is: S−
Z[
1
KT JUT , KT UT ,
0 1 0 ZH UX , V S Y ∑ 1 − + (K U ,J+) 1V B 0 X X
0
Z\ N Z[
𝑉 𝑉 . #] 9 = . ^] 9 (2) 𝐼#] 𝐼^]
Note that the windings ratio is applied to (Viw, Iiw) pair. Integrating the circuit elements from (1) into (2), we have: 𝑉 . ^ 9 = .1 𝐼^ 0
1 −1𝑅,3 + 𝑠 𝐿,3 79 _ 𝑅` + 𝐶` 𝑠 − 1 𝑅` 𝐶` 𝑠
0
Y
1
𝐶# 𝑠 b 1 c− d (𝑅# 𝐶# 𝑠 + 1) +
0 1e
Figure 3: SFRA Example from ([14]) Limitations of SFRA are artifacts of an input-output (instead of state-space) approach and the transformers being off-line (as opposed to inline). In the process, a lot of dynamic information is lost. For example, the number of partial discharge RC branches in new units, 𝑛 = 0. During operation, there will be a reactive component build up which in turn will generate gases in the coolant oil and increase n. These changes will be transient and not be present when the oil cools to ambient temperature. Permanent changes in n will occur as a result of insulation degradation. Each new partial discharge path will add an RC branch and increase n. Changes in the (Rt, Ct) pair describes how electrical properties of the coolant oil varies with temperature. To capture dynamic information, we develop a state-space model of the transfer function in Equation (3) and include the effects of load. We argue that this will provide a more complete view of the inline electrical characteristics of a transformer during operation. The model thus derived describes the relationship between discrete-time sequences of primary and secondary voltages and currents. We allow both system order and model parameters to vary, and develop an algorithm to estimate them on-line. Estimation is performed using Kalman filtering and Akaike criterion. We then discuss how order and parameter estimates can be used to perform inline monitoring of transformer condition and performance, a capability beyond SFRA. Since the algorithm has the potential to derive estimates within a few seconds, real-time inline transformer monitoring is possible.
A. Transformer State Space Model Our development begins with the transformation of the transfer function in Equation (3) into a discrete-time state space model using a bilinear transform 𝑧 = 𝑒 ,n , where T is the sampling period ([21]). Equations (4) – (9) perform these operations on the corresponding terms from Equation (3): r
r
- FKpL qsGpL MJtFKpL JsGpL M
𝑅,3 + 𝑠𝐿,3 → n
(tJ+)
( (t)
u = (tJ+)
(4)
Note that 𝟎n is used to denote an n-vector or n x n matrix depending on the context. Equation (11) is denoted the discrete extended equivalent circuit (E2C) model of a practical transformer. Its parameters are circuit elements at the terminals and those representing insulation and coolant oil. Order of the model can be used to track partial discharge paths due to insulation aging and thermal effects, while parameter estimates reflect values of circuit elements during operation.
F1𝑠𝐿(z + 𝑅(z 71𝑅3 + 𝑠 𝐿(3 7 + 𝑠𝐿(z 𝑅(z M
B. ARX Model of Transformer In practice, Equation (11) will require the consideration of noise and interference that will likely be prevalent around transformers during operation. E2C can be re-stated with noise and interference as a standard autoregressive moving average exogenous (ARMAX) model [17]. A particular simplification, the ARX model, is developed here. The voltage side of Equation (11) can be re-written as:
𝑠𝐿(z 𝑅(z 2 2 2 2 𝑇 OF𝑅(z − 𝑇 𝐿(z M + 𝑧 F𝑅(z + 𝑇 𝐿(z MQ OF𝑅(3 − 𝑇 𝐿(3 M + 𝑧 F𝑅(3 + 𝑇 𝐿(3 MQ → (𝑧 − 1)(𝑧 + 1)𝑅(z 𝐿(z 2
𝑑+,+ 𝑉& (𝑘 + 𝑛 + 4) = S 𝑑+,YJ‡
2 2 𝑇 F𝑅` − 𝑇 𝐶` M + 𝑧 F𝑅` + 𝑇 𝐶` M
𝑅` + 𝑠𝐶` → 𝑠𝑅` 𝐶` 2
(𝑧 − 1)𝑅` 𝐶`
𝑅(3 + 𝑠𝐿(3 →
(tq+)(tJ+)KHI GHI
+ (tq+)(tJ+)K
HI GHI
1,GHI JKHI 7 ,KHI GHI Y
d +
→
2 𝑇 𝑝𝑓
=
𝑝- (𝑧) (5) (𝑧 − 1)
2 𝑇 𝑝𝑓
h𝑅𝑝𝑓 − 𝐿 i+𝑧h𝑅𝑝𝑓 + 𝐿 i (𝑧+1)
=
𝑝3 (𝑧) (𝑧+1)
( 1t r 7
{ = (tq+)(tJ+) r s
r s
(tq+)KHI GHI
( (t)
| = (tq+)
+ S
(8)
2𝐶 2𝐶 + − # +𝑧 # 𝐶# 𝑠 𝑇 𝑇 → d 2𝑅# 𝐶# 2𝑅 𝐶 (𝑅# 𝐶# 𝑠 + 1) M + 𝑧 F1 + # # M Y F1 − 𝑇 𝑇
∑Y#•+ F− =
𝐷(𝑧
0 − .
𝑉 0 9 . ^9 = 𝐷(𝑧 qYq‡ ) 𝐼^ (10)
k
where Di(z ) and Nij(z ) are polynomials in z of order k. With z-kf(z(t)) = f(t+k(Dt)), and sequences [Vi(k), ……, Vi(k+n+4)]T, [Ii(k), ……, Ii(k+n+4)]T, [Vo(k), ……, Vo(k+n+4)]T and [Io(k), ……, Io(k+n+4)]T, we transform (10) into the discrete-time voltage and current relationship:
.
𝑑+,+ , 𝑑+,-, … , 𝑑+,YJ‡ 𝟎YJ‡
𝑛+-,+ 𝑑+,YJ‡
…
𝑉• (𝑘) … 𝑉• (𝑘 + 𝑛 + 4) , 𝐼• (𝑘) … 𝐼• (𝑘 + 𝑛 + 4)] 𝑑+,YJŽ 𝑛++,+ 𝑑+,+ , , .. S , .. 𝑑 𝑑 𝑑+,YJ‡ +,YJ‡ +,YJ‡
𝑛+-,YJ‡ n 𝑑+,YJ‡ V
+ 𝑒(𝑘 + 𝑛 + 4) (13)
𝑉𝐼 (𝑘)
⎡ ⎤ ⋮ 𝑛+-,+ , 𝑛+-,-, … , 𝑛+-,YJ‡ ⎢𝑉𝐼 (𝑘 + 𝑛 + 4)⎥ ⎢ ⎥ 𝑛--,+, 𝑛--,- , … , 𝑛--,YJ‡Q ⎢ 𝐼𝐼 (𝑘) ⎥ ⋮ ⎢ ⎥ ⎣ 𝐼𝐼 (𝑘 + 𝑛 + 4) ⎦
𝒀𝑽 = 𝑿𝑽 𝜽𝑽 + 𝒆𝑽 𝒀𝑽𝒊
(14)
where = 𝑉^ (𝑘 + 𝑖(𝑛 + 4)) is the i version of Equation (13), and 𝒆𝑽 # is the corresponding disturbance. Similarly, we derive a model for currents as: 𝒀𝑰 = 𝑿𝑰 𝜽𝑰 + 𝒆𝑰
th
(15)
where 𝒀𝑰 , 𝑿𝑰 , 𝜽𝑰 , 𝒆𝑰 have interpretations identical to 𝒀𝑽 , 𝑿𝑽 , 𝜽𝑽 , 𝒆𝑽 respectively. Combining Equations (14) and (15) we derive an ARX model of a practical transformer as:
𝑉𝑜 (𝑘) ⎡ ⎤ ⋮ ⎢𝑉 (𝑘 + 𝑛 + 4)⎥ 𝟎YJ‡ ⎥= 9⎢ 𝑜 𝑑-,+, 𝑑-,- , … , 𝑑-,‡YJ‡ ⎢ 𝐼𝑜(𝑘) ⎥ ⋮ ⎢ ⎥ ⎣ 𝐼𝑜(𝑘 + 𝑛 + 4) ⎦
𝑛++,+ , 𝑛++,-, … , 𝑛++,YJ‡ O𝑛 , 𝑛 , … , 𝑛 -+,+ -+,-+,YJ‡
,
Stacking together the outputs 𝑉& 1𝑘 + 𝑘 “ (𝑛 + 4)7, 𝑘 “ ∈ [0, … , (3𝑛 + 11)] as a column vector, we have a voltage side ARX model in the form:
−𝑁++ (𝑧 qYq‡ ) 𝑁+- (𝑧 qYq‡ ) 𝑉# 9. 9 𝑁-+(𝑧 qYq‡ ) 𝑁--(𝑧 qYq‡ ) 𝐼# k
𝑛++,YJ‡ 𝑑+,YJ‡
𝑉& (𝑘 + 𝑛 + 4) = [𝑉^ (𝑘) … 𝑉^ (𝑘 + 𝑛 + 3)
2𝑅# 𝐶# 2𝑅 𝐶 M + 𝑧 F1 + # # Mj 𝑇 𝑇
where pi(zj) and qi(zj) are polynomials in z of order j. Equation (3) may then be expressed in the Z domain as: .
…
where eo(k) represents the measurement error resulting from the electromagnetic field surrounding the transformer windings in Vo(k). Equation (12) can be expressed in the form:
𝑝‚ (𝑧 Y ) = Y (9) ∏#•+(𝑎# + 𝑏# 𝑧)
qYq‡ )
𝑛++,+ 𝑑+,YJ‡
𝑉 𝐼 ( 𝑘) ⎡ ⎤ ⋮ ⎢ ⎥ ( ) 𝑛+-,YJ‡ ⎢𝑉𝐼 𝑘 + 𝑛 + 4 ⎥ , V⎢ ⎥ 𝑑+,YJ‡ 𝐼𝐼 (𝑘) ⎢ ⎥ ⋮ ⎢ ⎥ ⎣ 𝐼𝐼 (𝑘 + 𝑛 + 4) ⎦
+ 𝑒^ (𝑘 + 𝑛 + 4) (12)
2𝐶# 2𝐶 2𝑅 𝐶 2𝑅 𝐶 + 𝑧 # M ∏Y~•+,~•# gF1 − # # M + 𝑧 F1 + # # Mj 𝑇 𝑇 𝑇 𝑇 ∏Y#•+ gF1 −
𝑉^ (𝑘) 𝑑+,YJŽ V_ ⋮ b 𝑑+,YJ‡ 𝑉^ (𝑘 + 𝑛 + 3)
…
(7)
n FKHI q GHI MJtFKHI J GHI M -
(6)
(11)
𝑽 𝑽 𝟎 9 .𝜽𝑽 9 + .𝒆𝑽 9 or, 𝒀 = 𝑿𝜽 + 𝒆 (16) .𝒀 𝑰 9 = .𝑿 𝒀 𝟎 𝑿𝑰 𝜽𝑰 𝒆𝑰 C. Parameter Estimation Using Kalman Filtering The elements of 𝜽 in Equation (16) change during operation and need to be determined using input/output data. œ represent an estimate of the parameter vector. Then 𝒀 œ, Let 𝜽 œ , can be written as: an estimate of the output using 𝜽
œ œ = 𝑿𝜽 𝒀
(17)
Equation (17) does not have the measurement error term since it is a computational model. Using Equations (16) and (17), recursive estimates of 𝜽 can be derived with the following equations (based on Kalman Filter theory [12], [8]): œ(𝑖) = 𝜽 œ(𝑖 − 1) + 𝑲(𝑖)ž𝒀(𝑖) − 𝒀 œ (𝑖)Ÿ (18) 𝜽 œ (𝑖) is calculated where K is the Kalman Gain matrix and 𝒀 œ(𝑖 − 1). The assumption here is that 𝜽 is stationary using 𝜽 during the estimation. K can be recursively calculated as: n (𝑖)𝑷(𝑖
𝑲(𝑖) = 𝑷(𝑖 − 1)𝑿(𝑖 − 1)[𝑰 + 𝑿
10MHz. Since load characteristics are unlikely to contain artifacts in MHz band, it would be reasonable to assume a top frequency of 10MHz and propose a value of 𝑇 ≤ 50 nS. 2) Design of Excitation Signals (VI, II) Injected signals are voltages and currents at the input terminals. Their selection is made so that: -
𝑰 + 𝑿n (𝑖)𝑷(𝑖 − 1) 𝑿(𝑖) is invertible ∀ 𝑖 in the recursive estimation procedure.
-
Injected power is sufficiently high to overcome the noise and interference around the transformer.
-
Injected signals are rich in spectrum to capture the frequencies of interest.
-
Finally, the hardware used must be capable of injecting power at the input terminals.
− 1)𝑿(𝑖)] (19) q+
𝑷(𝑖) = 𝑷(𝑖 − 1) − 𝑷(𝑖 − 1)𝑿(𝑖)[𝑰 + 𝑿n (𝑖)𝑷(𝑖 − 1)𝑿(𝑖)]q+ 𝑿𝑻 (𝑖)𝑷(𝑖 − 1) (20)
A key assumption in the derivation of the recursive estimation equations is that measurements are affected by a Gaussian process. In a transformer, errors are introduced due to electromagnetic interference (EMI) in the form of radiation, and (capacitive, inductive or common ground) cross-coupling. Radiation effects can be minimized (and therefore neglected) by shielding components involved in the injection of signals, and read-out of data. Cross-coupling effects are traditionally modeled with success as Gaussian processes which validates the error model assumptions. Substituting Equation (17) into (18) and accounting for measurement errors, we get a recursive parameter estimate error equation: 𝜽𝑬 (𝑖) = 𝜽“𝑬 (𝑖) + 𝑲(𝑖)[𝑿𝜽“𝑬 + 𝒆(𝑖)] (21) œ(𝑖)M and 𝜽“¤ = F𝜽(𝑖) − 𝜽 œ(𝑖 − Where, 𝜽¤ = F𝜽(𝑖) − 𝜽 1)M. The covariance of the estimation error can be written as: 𝑪𝜽 (𝑖) = 𝑬1𝜽𝑬 𝜽𝑻𝑬 7
3) Determination of Error Term e Estimation of electrical parameters assumes knowledge of the disturbance term e which must be gathered from the inputoutput data set. Assuming errors are caused by a Gaussian process, we have a probability density function in the form: r
𝑔(𝒀|𝜇, 𝜎
+
∏ŽYJ++ 𝑒 = √-±² #•+
1´ µ ¶7 q³ X r ¸ r·
(24)
where 𝜇, 𝜎 are the mean and variance of the PDF respectively. Note that yi refers to the ith element of the vector Y. Determination of the error model, therefore, involves the estimation of its mean and variance. To do so, we make the assumption that the sampled data Y consists of independent, identically distributed (i.i.d) elements. Then using well understood Maximum Likelihood Estimation (MLE) methods (e.g., [20]) we derive the following: +
= [𝑰 – 𝑲(𝑖)] 𝑪𝜽 (𝑖 − 1) [𝑰 – 𝑲(𝑖)]
𝜇̂ = ŽYJ++ ∑ŽYJ++ # • + 𝑦#
n (𝑖)7 𝑲(𝑖) (22)
+𝑲(𝑖) 𝐸1𝒆(𝑖)𝒆
where the trace of the covariance matrix contains the mean squared errors. The optimal Kalman Gain Matrix K* is derived by setting the partial derivative of the trace w.r.t to K to zero: 𝑲∗ (𝑖) = 𝑪𝜽 (𝑖 − 1)ž𝑪𝜽 (𝑖 − 1) + 𝐸1𝒆(𝑖)𝒆n (𝑖)7Ÿ
-)
q+
(23)
Equation (17) through (23) provide the means to estimate the electrical parameters of the E2C model based on injection and measurements of voltages and currents. Note that considerations such forgetting factors can be considered to optimize both the speed and accuracy in estimation. As a rule, the initial value of P is set to something large. D. Considerations in Parameter Estimation Implementation of the proposed algorithm requires several considerations, most critical of which are described here. 1) Choice of Sampling Time T T must be sufficiently small to capture the frequencies of interest in E2C, but it can be difficult to determine analytically since the order n of the system is not likely to be known. So, we draw from the practical results available from SFRA of unloaded transformers, and note that their Bode Plots indicate that the frequency band of interest lies between 10Hz -
(25)
+
𝜎» - = ŽYJ++ ∑ŽYJ++ # • + (𝑦# − 𝜇̂ )
(26)
where 𝜇̂ , 𝜎» are maximum likelihood estimates of the mean and variance of the process involved in the measurement errors and represent parameter values that yield the best match with data. Note that there may be more than one such estimate. We argue that identical distribution is a reasonable assumption since the underlying probability distribution of measurement errors will likely not change in the time scale of the estimation process. Elements of Y being independent is a major assumption that needs validation through experimentation. E. Estimation of System Order Using Akaike Criterion The final design consideration is that of determining the value of n, the model order. To do this, we use the Akaike Information Criterion (AIC) ([1]), a measure that relates model order to estimation error. AIC for the E2C model is defined as: 𝐴𝐼𝐶(3𝑛 + 11) = 2(3𝑛 + 11) − 2𝑙𝑛1ℒ ∗ (𝜇, 𝜎| 𝒀)7
(27)
where ℒ ∗ , the maximum likelihood, assumes the form: +
ℒ ∗ = − - (3𝑛 + 11)𝑙𝑛(2𝜋𝜎» - ) − ∑ŽYJ++ #• + F
(ÀX q Á Â )r -² Âr
M
(28)
The value of n that minimizes AIC gives the system order. Without apriori knowledge of n, we need to consider the
possibility that our data set may not be adequate in size. To account for small sample sizes, AIC can be extended to: 𝐴𝐼𝐶U (𝑚) = 2(3𝑛 + 11) − 2𝑚(𝑚+1)
2𝑙𝑛1ℒ ∗ (𝜇, 𝜎| 𝒀)7 + 13𝑛+11− (𝑚−1)7
(29)
where the last term is added as a penalty factor when Ä < 40. In the analysis presented, we propose the use of ŽYJ++ Equations (27) and (29) for model selection depending on sample size. If there are multiple model orders for similar values of AIC or AICC, we propose the use of multimodal œ ([4]), with an average inference for the estimation of 𝜽 weighted using evidence ratios. E. Transformer Model Parameter Estimation Algorithm We now present an algorithm for transformer state space model parameter estimation based on the analysis conducted thus far. The minimum order of the ARX model is 11 and occurs in a new transformer (for 𝑛 = 0), which means that the smallest data set would correspond to 𝑚& = 11 × 40 = 440. Let the data collected accommodate a large enough number of partial discharges, e.g., 𝑛 = 500 (note that a justifiable upper bound on n will need experimental work). This leads to a data set 𝒮# for the ith estimate qi that corresponds to 𝑚# = 60,440. We assume the data is pre-processed. Step 1: Separate the data into 501 new data sets 𝒮#~ , 𝑗 ∈ [0, 1, 2, 3, … . . , (𝑛 − 1), 𝑛] where 𝒮#~ , the jth data set corresponds to 𝑚#~ = (3 × 𝑗 + 11) × 40. Step 2: For each 𝒮#~ calculate the corresponding 𝜇̂ ~ and 𝜎»~ , of the associated Gaussian. Step 3: Calculate AICj and (AICC)j using 𝜇̂ ~ and 𝜎»~ . ∗ Determine the optimal system order as 𝑚#~ = (3 × 𝑛#∗ + ∗ 11) × 40, where 𝑛# assumes the value of 𝑗 that minimizes the Akaike Criterion. Denote the corresponding data set as 𝒮#∗, and mean and variance as 𝜇#∗ and 𝜎#∗ respectively. Note that 𝑛#∗ directly provides an estimate of the number of partial discharge paths at sampling window i. Step 4: Apply 𝜇#∗ , 𝜎#∗ , 𝑛#∗ to construct the error vector ei. Step 5: Using Equation (24), calculate the matrix 𝑷# (𝑙), 𝑙 > 0. For l = 0, choose a 𝑷# matrix with large eigenvalues. Note that 𝑿# is constructed using (3𝑛#∗ + 11) × 40 data samples. œ𝒊 (0) = 𝟎, 𝑖 = 1 and 𝜽 œ𝒊 (0) = 𝜽∗#q+ , 𝑖 > 1 Step 6: Assume 𝜽 (from Step 9). Using Equation (23), calculate the 𝑲# (𝑙), 𝑙 > 0. œ# (𝑙). Note that 𝒀# Step 7: Using Equation (22) calculate 𝜽 is constructed using (3𝑛#∗ + 11) × 40 data samples. Step 8: Rewrite Equation (21) as 𝒀œË (𝑙 + 1) = œ𝒊 (𝑙), 𝑙 > 0, and calculate 𝒀# (𝑙 + 1). Assume 𝒀œË (0) = 𝟎. 𝑿# 𝜽 œ 𝒊 (𝑙) − 𝒀# 7n 1𝒀 œ 𝒊 (𝑙) − Step 9: Repeat Steps 5 – 8 until 1𝒀 ∗ 𝒀# 7 < 𝜀, where e is small. Let 𝜽# denote the estimated value œ 𝒊(𝑙) − 𝒀# 7. of the parameter vector that minimizes 1𝒀 Step 10: Estimate 𝑅Í(3 (𝑖), 𝐿Í(3 (𝑖), 𝑅Í(z (𝑖), 𝐿Í(z (𝑖), 𝑅Í,3 (𝑖), 𝐿Í,3 (𝑖), 𝑅Ín (𝑖), 𝐶În (𝑖), and 𝑅ÍÄ (𝑖), 𝐶ÎÄ (𝑖), 𝑚 ∈ [1, 𝑛#∗ ], the electrical parameters, from 𝜽∗# .
E. From Model Parameters to Transformer Condition The algorithm presented in the previous section estimates the parameters of the E2C model. We now describe the kind of transformer condition information available in the model. 1) Partial Discharge Paths Perhaps the most significant and novel result of the proposed method is a direct estimation of whether partial discharge paths are building up in the transformer unit. If 𝑛∗ (𝑖) > 0, then partial discharges are present. If 𝑛∗ (𝑖) ≠ 𝑛∗ (𝑗), 𝑖 ≠ 𝑗, then partial discharge paths are being created/destroyed over time. If there are changes present between nighttime 𝑛∗ values, there exists a high likelihood that insulation aging is affecting transformer operation. The trend of 𝑛∗ values between 8AM and 2PM describes the thermal loading present on the transformer. Drastic transient changes in 𝑛∗during these intervals point to significant load changes and the potential for transformer overload. In addition, as utilities increase injection levels of renewables and eventually discard curtailment, 𝑛∗ trends can be used to study the reactive buildup resulting from changes in power flow direction. 2) Voltage Fluctuations Parameter trends may be estimated from a data set Ξ = [𝜽∗Ñ , 𝜽∗ÑJ+ , … . , 𝜽∗ÑJÒq+ , 𝜽∗ÑJÒ ], 𝑎, 𝑏 ∈ 𝛪. In the event Ξ contains parameter vectors whose orders or element values change over time, voltage on the secondary side will tend to vary for a given voltage/current pair on the primary side. This implies that parameter trends may be utilized in the simulation environment to investigate whether voltage fluctuations greater than the allowable ± 5% can occur and conduct a sensitivity analysis. Note that fluctuations resulting from order changes allow us to understand the impact of partial discharge. 3) Power Factor With a given set of input voltages and currents, the simulation model also easily calculates the corresponding output voltages and currents. This implies that the phase angles of the voltages and currents, and the power factor, all on the secondary side of the transformer can be easily determined. Power factor is vital in the determination of reactive power buildup, which plays a key role in the thermal stresses present during power flow direction reversals. 4) Geomagnetically Induced Currents (GIC) Geomagnetically induced currents (GIC) that emanate from plasma discharge during solar storms have become an important consideration for grid resiliency. GIC typically produces core saturation, hot spots, reactive component buildup and resonance in transformers. Conceptually, saturation will be manifest in how the (Steinmetz) parameters 𝑅Í(3 (𝑖), 𝐿Í(3 (𝑖), 𝑅Í(z (𝑖), 𝐿Í(z (𝑖), 𝑅Í,3 (𝑖), 𝐿Í,3 (𝑖) change due to GIC. The impact of hot spots on insulation degradation, dissolved gases and partial discharge will be captured by how 𝑅Ín (𝑖), 𝐶În (𝑖), and 𝑅ÍÄ (𝑖), 𝐶ÎÄ (𝑖), 𝑚 ∈ [1, 𝑛#∗ ] vary. And finally, nonlinearity in model inductance and its relationship with resistance will provide information about the nature and scale of ferromagnetic resonance.
III. SUMMARY AND FUTURE WORK In this paper, we have described a systems theoretic approach to transformer monitoring and contrasted it with commonly used in-situ (DGA) and off-line (SFRA) methods. It relies on the development of a practical equivalent circuit of a transformer and transforming it into a discrete-time extended state space (E2C) model. It is shown that the model lends itself to parameter identification using standard approaches in estimation theory such as MLE, Akaike Criterion and Kalman estimation. The paper concludes with a description how model parameters may be utilized to assess transformer condition. Experimental validation of the algorithm will be dealt with in a forthcoming paper ([24]).
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ACKNOWLEDGMENT Authors wish to thank Professor Judy Franklin for her feedback on the monitoring approach presented in this paper.
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