# A Distribution System State Estimator Based on an Extended Kalman Filter Enhanced with a Prior Evaluation of Power Injections at Unmonitored Buses

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## Abstract

**:**

## 1. Introduction

- The algorithm conceived to estimate the inputs to the prediction step of the filter. Such a sequence of data is obtained by disaggregating the active and reactive power injection variations measured at the substation;
- A deeper analysis of the relative impact of an increasing number of PMUs and SMs on state estimation performance when such measurements are used in the update step of the EKF.

## 2. Related Work

## 3. Models Description

#### 3.1. System Model

- The intrinsic voltage phase and RMS amplitude fluctuations at the slack bus in normal operating conditions (first two entries);
- The errors resulting from the linearization of the power injection equations (remaining $2N-2$ entries).

#### 3.2. Measurement Model

- $\mathbf{P}\left({\mathbf{x}}_{k}\right)$ and $\mathbf{Q}\left({\mathbf{x}}_{k}\right)$ are the $N-$long vectors including the active and reactive power injections as functions of the state. In practice, the i-th elements of such functions are given by (2).
- ${\mathbf{V}}_{k}^{\mathrm{PMU}}={[{V}^{1},{V}_{k}^{{m}_{1}},\dots ,{V}_{k}^{{m}_{M}}]}^{\mathrm{T}}$ comprises both the slack bus RMS voltage and the RMS voltage values measured by M PMUs deployed at buses ${m}_{1},\dots ,{m}_{M}$. Similarly, ${\theta}_{k}^{\mathrm{PMU}}={[{\theta}_{k}^{{m}_{1}},\dots ,{\theta}_{k}^{{m}_{M}}]}^{\mathrm{T}}$ is the vector of voltage angles measured directly by the same PMUs.
- Finally, vectors ${\mathbf{I}}_{k}^{\mathrm{PMU}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{[{I}_{k}^{{m}_{1}},\dots ,{I}_{k}^{{m}_{M}}]}^{\mathrm{T}}$ and ${\phi}_{k}^{\mathrm{PMU}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{[{\phi}_{k}^{{m}_{1}},\dots ,{\phi}_{k}^{{m}_{M}}]}^{\mathrm{T}}$ comprise the RMS values and the angles, respectively, of the phasors of the current flowing through one of the lines connected to the M buses where the PMUs are installed.

## 4. State Estimation Algorithm

#### 4.1. Prior Estimation of Power Injection Variations

- K be the number of $\Delta {\mathbf{u}}_{k}^{\mathrm{a}}$ samples collected in the observation interval $[{t}_{k-K+1},{t}_{k}]$ and arranged into vector $\Delta {\tilde{\mathbf{u}}}_{k}^{\mathrm{a}}={[\Delta {{\mathbf{u}}^{\mathrm{a}}}_{k}^{\mathrm{T}},\dots ,\Delta {{\mathbf{u}}^{\mathrm{a}}}_{k-K+1}^{\mathrm{T}}]}^{\mathrm{T}}$;
- ${\tilde{\delta}}_{k}={[{\delta}_{k}^{\mathrm{T}},\dots ,{\delta}_{k-K+1}^{\mathrm{T}}]}^{\mathrm{T}}$ be the column vector of the corresponding uncertainty contributions;
- $\Delta {\tilde{\mathbf{u}}}_{k}={[\Delta {\mathbf{u}}_{k}^{\mathrm{T}},\dots ,\Delta {\mathbf{u}}_{k-K+1}^{\mathrm{T}}]}^{\mathrm{T}}$ be the $2(N-1)K$-long column vector obtained by stacking elements $\Delta {u}_{{n}_{(i-1)}}$ and $\Delta {u}_{{n}_{(i+N-2)}}$ for $i=2,\dots ,N$ and $n=k-K+1,...,k$ (namely the active and reactive power injection variations at all the buses different from the slack bus in the observation interval $[{t}_{k-K+1},{t}_{k}]$).

- Linear constraint (9) with$${A}_{1}=\left[\begin{array}{cccc}-1& 0& \dots & 0\\ 0& -1& \dots & 0\\ \vdots & \dots & \vdots & \vdots \\ 0& 0& \dots & -1\\ -1& 0& \dots & 0\\ 0& -1& \dots & 0\\ \vdots & \dots & \vdots & \vdots \\ 0& 0& \dots & -1\end{array}\right]\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{A}_{2}=\begin{array}{c}\phantom{\rule{1.em}{0ex}}\underset{\ufe37}{\phantom{\rule{0.5.em}{0ex}}N-1\phantom{\rule{0.5.em}{0ex}}}\\ \left[\begin{array}{cccccc}1& \dots & 1& 0& \dots & 0\\ 0& 1& \dots & 1& \dots & 0\\ \vdots & \vdots & \dots & \vdots & \dots & \vdots \\ 0& 0& \dots & 1& \dots & 1\\ 1& \dots & 1& 0& \dots & 0\\ 0& 1& \dots & 1& \dots & 0\\ \vdots & \vdots & \dots & \vdots & \dots & \vdots \\ 0& 0& \dots & 1& \dots & 1\end{array}\right]\}2K\end{array}$$
- Constraint (10) relies on the assumption that the mean value of variables ${\delta}_{k}={[{\delta}_{{1}_{k}},{\delta}_{{2}_{k}}]}^{\mathrm{T}}$ is zero and provides a lower bound to the variances of the uncertainty contributions affecting the aggregate active and reactive power variations, respectively. Therefore, it provides a lower bound to the objective function as well. Indeed, the variance values estimated over the observation interval $[{t}_{k-K+1},{t}_{k}]$ cannot be smaller than the square of the combined standard uncertainties (denoted with ${\sigma}_{{\delta}_{P}}^{2}$ and ${\sigma}_{{\delta}_{Q}}^{2}$) of the power injection variations measured at the slack bus.
- Finally, the equality constraint (11) forces the sample variances of the unknown active and reactive bus power injections variations to be equal to given values denoted as ${\sigma}_{\Delta {P}^{i}}^{2}$ and ${\sigma}_{\Delta {Q}^{i}}^{2}$, respectively, for $i=2,\dots ,N$. Such values can be found a priori, for example, from past pseudo-measurements. Again, (11) relies on the realistic assumption that the mean values of the power injection variations over time are null.

#### 4.2. EKF Implementation

## 5. Case Study Description

## 6. State Estimation Results

#### 6.1. QP Settings for EKF Input Evaluation

#### 6.2. Process Noise (EKF Only)

#### 6.3. Power Injection Measurements

- The first scenario refers to the classic case in which just historical a priori information on power injections (i.e., pseudo-measurements only) is available. In this case, the unknown delays and the time misalignment between the collected data strongly affect measurement uncertainty, which consequently is as large as load inherent variability (scenario 1).
- The second scenario relies on the assumption that power injection measurements rely not only on pseudo-measurements, but also on the data collected by the SMs every 15 minutes. Such data are assumed to be aligned in time with a delay of a few seconds before being aggregated (scenario 2). In this case, we assume that the weight of SMs measurements on the total relative variance of power injection measurement data is about 33% at all buses (scenario 2).
- The third scenario is similar to the second one, but a deeper penetration of SMs is envisioned. Therefore, the weight of SMs measurements on the total relative variance of power injection data reaches about 67% (scenario 3).
- Finally, the fourth scenario relies on the assumption that all power measurements are collected from customers’ SMs and are properly aligned in time before being aggregated. Of course, in this case, the traditional pseudo-measurements are no longer used (scenario 4).

#### 6.4. PMU-Based Measurements

#### 6.5. Performance Analysis

- The initial EKF transient is quite short (just a few samples). This is most probably due to the good initialization of the EKF;
- The estimated state tracks the pseudo-periodic voltage pattern resulting from the time-varying load conditions shown in Figure 3 very well;
- State estimation accuracy is clearly high, since the differences between estimated and actual values at a given time are very small.

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AMI | Advanced Metering Infrastructure |

DSO | Distribution System Operator |

DSSE | Distribution System State Estimation |

EKF | Extended Kalman Filter |

GMM | Gaussian Mixture Model |

QP | Quadratic Programming |

PMU | Phasor Measurement Unit |

PSO | Particle Swarm Optimization |

RMS | Root Mean Square |

SE | State Estimation |

SM | Smart Meter |

TVE | Total Vector Error |

UTC | Coordinated Universal Time |

WLS | Weighted Least Squares |

## Appendix A. Load Aggregation Algorithm

- The sum of all the average load profiles included in the i-th subset is approximately equal to the nominal power injection at the corresponding bus, i.e., ${\sum}_{l\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathcal{L}}^{n}}\phantom{\rule{3.33333pt}{0ex}}{\overline{p}}_{l}\phantom{\rule{3.33333pt}{0ex}}\approx \phantom{\rule{3.33333pt}{0ex}}{P}_{{L}_{\mathrm{nom}}}^{i}$.
- The relative sample variance of the power at the i-th bus is approximately equal to a given fraction ${\gamma}^{2}$ of ${P}_{{L}_{\mathrm{nom}}}^{{i}^{2}}$, i.e., ${\sum}_{l\in {\mathcal{L}}^{i}}{s}_{{P}_{l}}^{2}\phantom{\rule{3.33333pt}{0ex}}\approx \phantom{\rule{3.33333pt}{0ex}}{\gamma}^{2}{P}_{{L}_{\mathrm{nom}}}^{{i}^{2}}$.

## Appendix B. PMU Placement Algorithm

- For each PMU configuration $m\in \mathcal{M}$, the state of the system under given load conditions (i.e., with known maximum variability) is estimated by the EKF or the WLS algorithm ${N}_{t}$ times, with ${N}_{t}$ large enough to ensure a reasonably accurate estimate of the state covariance matrix, i.e., in the order of a few hundred.
- If the sequences of estimation errors are realizations of ergodic processes (as it should be when either the EKF or the WLS estimator is used), the state covariance matrix ${\overline{C}}_{m}$ associated with the m-th PMU configuration can be estimated directly from the steady-state estimation error sequences. In the EKF case, ${\overline{C}}_{m}$ tends also to coincide with the solution of the linearized matrix Riccati equation.
- Once all covariance matrices ${\overline{C}}_{m}$ for $m\in \mathcal{M}$ are computed, the PMU configuration is selected as follows, i.e.,$${m}^{*}=arg\underset{m\in \mathcal{M}}{min}\sqrt{max\mathrm{Eig}\left({\overline{C}}_{m}\right)},$$
- Afterward, configuration ${m}^{*}$ is removed from set $\mathcal{M}$, i.e., $\mathcal{M}\leftarrow \mathcal{M}\backslash \left\{{m}^{*}\right\}$, and the algorithm starts over.

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**Figure 1.**Block diagram of the proposed DSSE algorithm based on an EKF with prior estimation of power injection variations.

**Figure 3.**Example of yearly load profile (with 1-week zoom in the inset) and corresponding histogram at bus no. 2, when the load variations are bounded within $\pm 60\%$. The dashed line fitting the histogram is the probability density function of a Gaussian Mixture Model (GMM) consisting of three components with mean values: 80.9, 93, 130.5 kW; standard deviations: 8.0, 56.5 and 123.4 kW and mixing factors: 0.32, 0.40 and 0.28, respectively.

**Figure 4.**RMS voltage (

**top picture**) and angles (

**bottom picture**) state estimates at different buses obtained with the proposed enhanced EKF algorithm during the first week of January. In this example, load variability is set to $\pm 60\%$ of the nominal values and no smart meter (SM) or Phasor Measurement Unit (PMU) data are used. The diamond markers represent the true values of the corresponding state variables and are shown for reference.

**Figure 5.**99-th percentiles of (

**a**) the bus RMS voltage relative estimation errors and (

**b**) the phase estimation errors obtained with the enhanced EKF and the WLS algorithm, respectively, under different load conditions and considering traditional pseudo-measurements only. The maximum TVE of all PMUs deployed is 1%.

**Figure 6.**99-th percentiles of (

**a**) the bus RMS voltage relative estimation errors and (

**b**) the phase estimation errors obtained with the enhanced EKF and the WLS algorithm when only pseudo-measurements (scenario 1) and an increasing number of PMUs with a different maximum TVE is used. The percentiles are computed over all buses and over the whole simulated year. In this case, the load profiles are included within $\pm 60\%$ of the respective nominal values.

**Figure 7.**99-th percentiles of the relative vector estimation errors returned by the WLS algorithm (on the left) and the EKF (on the right), when the weight of real-time SMs on the measurement variance of power injections increases from 0% (scenario 1) to 100% (scenario 4). In (

**a**,

**b**), the maximum TVE of the deployed PMUs is set to 0.1% and 1%, respectively. In both (

**a**) and (

**b**) from 1 to 9 PMUs are used and the load fluctuations lie within $\pm 60\%$, of the nominal values.

**Table 1.**Nominal active and reactive loads as well as line parameters of the IEEE 33-bus radial distribution system.

Bus | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mathbf{P}}_{{\mathit{L}}_{\mathbf{nom}}}^{\mathit{i}}/\mathbf{kW}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mathbf{Q}}_{{\mathit{L}}_{\mathbf{nom}}}^{\mathit{i}}/\mathbf{kvar}$ | Line | From | To | $\mathit{r}/\mathbf{\Omega}$ | $\mathit{x}/\mathbf{\Omega}$ |
---|---|---|---|---|---|---|---|

1 | – | – | 1 | 1 | 2 | 0.092 | 0.047 |

2 | 100 | 60 | 2 | 2 | 3 | 0.493 | 0.251 |

3 | 90 | 40 | 3 | 3 | 4 | 0.366 | 0.186 |

4 | 120 | 80 | 4 | 4 | 5 | 0.381 | 0.194 |

5 | 60 | 30 | 5 | 5 | 6 | 0.819 | 0.707 |

6 | 60 | 20 | 6 | 6 | 7 | 0.187 | 0.619 |

7 | 200 | 100 | 7 | 7 | 8 | 0.711 | 0.235 |

8 | 200 | 100 | 8 | 8 | 9 | 1.030 | 0.740 |

9 | 60 | 20 | 9 | 9 | 10 | 1.044 | 0.740 |

10 | 60 | 20 | 10 | 10 | 11 | 0.197 | 0.065 |

11 | 45 | 30 | 11 | 11 | 12 | 0.374 | 0.130 |

12 | 60 | 35 | 12 | 12 | 13 | 1.468 | 1.16 |

13 | 60 | 35 | 13 | 13 | 14 | 0.542 | 0.713 |

14 | 120 | 80 | 14 | 14 | 15 | 0.591 | 0.526 |

15 | 60 | 10 | 15 | 15 | 16 | 0.746 | 0.545 |

16 | 60 | 20 | 16 | 16 | 17 | 1.289 | 1.721 |

17 | 60 | 20 | 17 | 17 | 18 | 0.732 | 0.574 |

18 | 90 | 40 | 18 | 2 | 19 | 0.164 | 0.157 |

19 | 90 | 40 | 19 | 19 | 20 | 1.504 | 1.355 |

20 | 90 | 40 | 20 | 20 | 21 | 0.410 | 0.478 |

21 | 90 | 40 | 21 | 21 | 22 | 0.709 | 0.937 |

22 | 90 | 40 | 22 | 3 | 23 | 0.451 | 0.308 |

23 | 90 | 50 | 23 | 23 | 24 | 0.898 | 0.709 |

24 | 420 | 200 | 24 | 24 | 25 | 0.896 | 0.701 |

25 | 420 | 200 | 25 | 6 | 26 | 0.203 | 0.103 |

26 | 60 | 25 | 26 | 26 | 27 | 0.284 | 0.145 |

27 | 60 | 25 | 27 | 27 | 28 | 1.059 | 0.934 |

28 | 60 | 20 | 28 | 28 | 29 | 0.804 | 0.701 |

29 | 120 | 70 | 29 | 29 | 30 | 0.508 | 0.259 |

30 | 200 | 600 | 30 | 30 | 31 | 0.974 | 0.963 |

31 | 150 | 70 | 31 | 31 | 32 | 0.311 | 0.362 |

32 | 210 | 100 | 32 | 32 | 33 | 0.341 | 0.530 |

33 | 60 | 40 | – | – | – | – | – |

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**MDPI and ACS Style**

Macii, D.; Fontanelli, D.; Barchi, G.
A Distribution System State Estimator Based on an Extended Kalman Filter Enhanced with a Prior Evaluation of Power Injections at Unmonitored Buses. *Energies* **2020**, *13*, 6054.
https://doi.org/10.3390/en13226054

**AMA Style**

Macii D, Fontanelli D, Barchi G.
A Distribution System State Estimator Based on an Extended Kalman Filter Enhanced with a Prior Evaluation of Power Injections at Unmonitored Buses. *Energies*. 2020; 13(22):6054.
https://doi.org/10.3390/en13226054

**Chicago/Turabian Style**

Macii, David, Daniele Fontanelli, and Grazia Barchi.
2020. "A Distribution System State Estimator Based on an Extended Kalman Filter Enhanced with a Prior Evaluation of Power Injections at Unmonitored Buses" *Energies* 13, no. 22: 6054.
https://doi.org/10.3390/en13226054