Active distribution system state estimation incorporating photovoltaic generation system model

Electric Power Systems Research 182 (2020) 106247

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Active distribution system state estimation incorporating photovoltaic generation system model

T

Zhi Fanga, Yuzhang Linb,*, Shaojian Songa, Chunning Songa, Xiaofeng Lina, Gang Chenga a b

School of Electrical Engineering, Guangxi University, Nanning, Guangxi, 530004, China Department of Electrical and Computer Engineering, University of Massachusetts, Lowell, MA, 01854, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Active distribution system Photovoltaic power generation State estimation Per unit system Bad data processing

The inherently intermittent and varying power generation of distributed energy resources (DER) makes the situational awareness of active distribution systems (ADS) a challenging issue. Considering the detailed photovoltaic (PV) system model and environmental influencing factors including solar irradiance and temperature, this paper develops a state estimation model for the ADS with integrated PV power generation systems. Firstly, in order to achieve high computational efficiency, an explicit function fitting model of PV arrays is proposed and used to accurately approximate the conventional five-parameter model. Then, the state variables of a PV generation system are selected, and the measurement equations and the Jocobian matrix are derived. Finally, combined with the conventional distribution network model, state estimation and bad data processing of an ADS with PV systems are realized. A generalized per unit system is also developed for unified standardization of electrical and non-electrical quantities. Simulation results in the IEEE 33-bus system show that the proposed model significantly expands the scopes and capabilities of state estimation and bad data processing in ADSs, which facilitates comprehensive and reliable monitoring of ADSs.

1. Introduction With the integration of distributed energy resources (DER) with intermittent and varying nature, conventional power distribution systems are evolving into the so-called active distribution systems (ADS) [1]. It requires more flexible and accurate control and optimization methods for the energy management of DERs and active loads. State estimation is one of the basic tools for situational awareness of the ADS. It is important to improve the functionality of state estimation in every aspect in order to provide a solid foundation for the real-time monitoring and control of the system [2]. Since traditional distribution systems generally do not contain power sources, in the traditional state estimation methods, only the network part of the system is modeled and estimated. The power sources, if any, are simply modeled as power injections into the network [3–5]. When DERs of ineligible capacities are integrated into distribution networks, the real-time monitoring and estimation of the operating state of these power sources will be of great significance for the situational awareness and decision making in the ADS. Photovoltaic (PV) plants are the most common type of power sources in the ADS. In general, the Maximum Power Point Tracking (MPPT) mode is the commonly used operating mode of PV power generations. However, in

⁎

order to satisfy the needs of system operation, such as frequency and voltage controls, PV power plants may also operate away from the maximum power point [6,7]. Furthermore, under different solar irradiance and temperature, the maximum power points of PV systems vary drastically. Real-time estimation of the maximum power point can help estimate the potential maximum output of a PV plant, and the adjustable ranges of its active and reactive power, which provide useful information for the dispatching and operation of the ADS. Real-time monitoring of PV systems can also provide accurate model for PV plants, which facilitates online security assessment of the ADS. In addition, the fault detection, location, and classification of PV plants are also based on their real-time modeling and monitoring. Accurate estimation of real-time PV variables is also greatly beneficial for ultrashort-term renewable energy forecasting which needs to be estimated by the PV variables. In summary, the extension of the ADS state estimation model to the integrated PV plants is of great significance to situation awareness and operation of the entire system. At present, the existing research on grid-connected PV systems focuses on the modeling and analysis of the uncertainty of the injected power into the ADS. Few studies have been conducted on detailed state estimation modeling of PV plants in the ADS. In addition, for ADS state estimation, most existing methods only use traditional AC voltage and

Corresponding author. E-mail address: [email protected] (Y. Lin).

https://doi.org/10.1016/j.epsr.2020.106247 Received 21 March 2019; Received in revised form 23 January 2020; Accepted 23 January 2020 Available online 12 February 2020 0378-7796/ Published by Elsevier B.V.

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Nomenclature z x e R H G r rN Pr NS NP Ipv Vpv Ipvs Vpvs

measurement vector state vector measurement error vector error covariance matrix Jacobian matrix gain matrix estimated residual normalized residuals the covariance matrix of the residual the series number of a PV array the parallel number of a PV array the output current of the five-parameter array the output voltage of the five-parameter array the output current of the fitting function array the output voltage of the fitting function

G T Dpv Vpvdc Ipvdc Mpv θpv Epvo θpvo Xpv Kpv Ppvac Qpvac Vpvac θpvac Ppvi Qpvi ηpvdc ηpvac

model of a PV model of a PV model of a PV model of a PV

array solar irradiance temperature the duty cycle of the DC/DC converter the output voltage of the DC/DC converter the output current of the DC/DC converter the modulation ratio of the DC/AC inverter the modulation phase angle of the DC/AC inverter the output voltage amplitude of the DC/AC inverter the output voltage phase angle of the DC/AC inverter equivalent impedance the turn ratio of the transformer the active power flowing from the PV system to the PCC the reactive power flowing from the PV system to the PCC the voltage amplitude of the PCC the voltage phase angle of the PCC the active power injections at the PCC the reactive power injections at the PCC the overall efficiencies of the DC/DC converter the overall efficiencies of the DC/AC inverter

accurately approximate the conventional five-parameter model. A state estimation model of an ADS with PV systems and an extended per unit system which is for unified standardization of electrical and non-electrical quantities are developed in Section 4. Case study results are provided in Section 5, and Section 6 concludes the paper.

power measurements, but do not consider non-electrical measurements or measurements obtained from the DC part of the systems. Ref. [8] takes the outputs of DERs as pseudo-measurements. Similarly, Refs. [9–11]. uses Gaussian mixture model to express the different probability distribution functions of DERs and loads, and solves the state estimation problem based on the weighted least squares (WLS) estimation algorithm. Ref. [12] develops the membership function for active loads and different DER outputs using credibility measure, and subsequently solves a linear state estimation model using fuzzy analysis method. Ref. [13], a large number of the historical data of DERs and loads are trained efficiently by using the technology of fuzzy neural network to obtain more accurate pseudo-measurements of the node injection power. The placement and integration of micro-phasor measurement units (μPMU) and advanced measurement infrastructures (AMI) is studied in Refs. [14–16]. Ref. [17], a three-phase state estimation method with DG is proposed. The DER is regarded as a PQ node, but the specific physical model of DER is not developed. In view of the need of the real-time monitoring and control of the ADS and the limitations of the existing approaches, this paper presents a novel ADS state estimation model which accounts for the models and measurements of PV plants in detail. Compared with the existing literature, the innovations and contributions of this paper are highlighted as below:

2. Distribution network state estimation and bad data processing Generally, the measurement equations of a distribution system can be written as follows: (1)

z = h (x ) + e

where z is the (m × 1) measurement vector, x is the (n × 1) state vector, h(x) is the (m × 1) measurement function, e is the (m × 1) measurement error vector, m is the number of measurements, and n is the number of state variables. The entries of the measurement error vector e are usually assumed to be uncorrelated, and satisfies Gaussian distributions with zero mean values, i.e. E(e) = 0. In traditional state estimation of distribution networks, the entries of the state vector x are voltage amplitudes and phase angles of each nodes. The measurement vector z includes bus voltage amplitudes, branch active and reactive power flows, and bus active and reactive power injections. The specific expressions of measurement equations in distribution networks can be found in Ref. [23]. The WLS method is the most widely used method for state estimation. The formulation of the problem is given as follows [23]:

(1) An explicit function fitting model of PV arrays is proposed, which can approximate the five-parameter model of PV arrays with high precision, while greatly enhance the computational efficiency of state estimation. (2) A detailed model of the ADS with PV systems for state estimation is proposed, which expands the scope of ADS state estimation from the power network to the power sources. (3) The non-electrical measurements, such as solar irradiance and temperature, are considered in the proposed ADS state estimation model, which improves the information redundancy. (4) A unified per unit system for both electrical and non-electrical quantities is proposed for ADS state estimation. (5) The feasibility of performing state estimation and bad data identification and correction in a detailed PV system model is verified.

min J (x ) = (z − h (x ))T R−1 (z − h (x ))

(2)

where R = cov(e) = E(ee ) is the error covariance matrix. It is a diagonal matrix, and the diagonal entry Rii is equal to σi2, where σi is the standard deviation of the ith measurement. The first-order optimality condition can be written as: T

g (x ) =

∂J (x ) = −H T (x ) R−1 (z − h (x )) = 0 ∂x

(3)

where H(x) = ∂h(x)/∂x is the (m × n) Jacobian matrix. The nonlinear function g(x) is expanded into a Taylor series at xk:

g (x ) = g (x k ) + G (x k )(x − x k ) + ...=0

(4)

Ignoring the higher order terms, the Gauss-Newton method is used to solve the problem iteratively:

The rest of the paper will be organized as follows. Section 2 briefly reviews the WLS state estimation formulation and the standard implementation of the largest normalized residual test [18–22]. Section 3 proposes an explicit function fitting model of PV arrays which is used to

G (x k ) Δx k = H T (x k ) R−1Δz k

(5)

where k is the number of iterations, x is the state vector of the k-th k

2

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iteration, Δxk = xk+1-xk, Δzk = z-h(xk), G(xk) = HT(xk)R−1H(xk) is the (n × n) gain matrix. Substituting Δxk = xk+1-xk into (5) will yield:

x k + 1 = G (x k )−1H T (x k ) R−1Δz k + x k

PV cell, Ω; Rp is the equivalent parallel resistance of the PV cell, Ω; n is the ideal factor of the diode; Vt is the thermal voltage of the PV cell, V, Vt = kq/T, where k is the disposal of Boltzmann’s constant, k = 1.381 × 10−23 J/K, q is the unit charge, q = 1.602 × 10−19 C, T is the temperature of PV cells, K, N = n × Vt. According to the open-circuit voltage, short-circuit current, the voltage and current at the maximum power point, and the correction coefficient, the five parameters can be calculated under any conditions. The specific procedures can be found in Ref. [24]. The mathematical model of a PV array with series number NS and parallel number NP under any conditions is as follows:

(6)

Due to various reasons, the measurement or communication systems of the ADS may occasionally produce bad data, and these bad data will have a significant impact on the results of state estimation. Therefore, the identification and correction of bad data is a very important task of state estimation. In this paper, the well-known normalized residual method is used for bad data processing [18,19]. For each measurement, the corresponding normalized residual can be computed. The normalized residuals are defined as follows [23]:

r N = (diagΩ)−1/2r

Vpv + Ipv ⋅ Rs ⋅ Ns / Np N ⋅ Ns

Ipv = Np⋅Iph − Np⋅I0⋅(e Vpv⋅Np + Ipv⋅Rs⋅Ns − Rp⋅Ns

(7)

where r is the estimated residual, r = z-h(x), and Ω is the covariance matrix of the residual:

Ω = cov(r ) = R − H (x )(H (x )T R−1H (x ))−1H (x )T

− 1) (11)

where Ipv and Vpv are the output current and output voltage of the fiveparameter model of the PV array, respectively. The five-parameter model of PV arrays is a complex implicit transcendental equation. If it is used in the measurement equation in the state estimation model of PV arrays, it will be too complex and computationally costly. In this paper, in order to improve the overall computational efficiency of state estimation of the ADS with PV systems, an explicit fitting function is used to approximate the five-parameter model of PV arrays, as will be described below.

(8)

In the absence of errors, each entry of the normalized residual vector rN has a standard normal distribution, and with a great probability its value is less than a threshold value of 3.0. When a measurement is bad, the normalized residual corresponding to the measurement is the largest among all, and greater than the threshold of 3.0 with a high probability. Detailed principle and implementation of the normalized residual method can be found in Ref. [23]. Assuming that the measurement identified as bad data is zjbad , correction is carried out based on the following equation [23]:

3.2. Fitting function model of PV arrays

where Rjj , r jbad , and Ωjj are the variance of noise, residual, and variance of the residual associated with this measurement, respectively.

The design of the fitting function model of PV arrays needs to balance between computational efficiency and accuracy. By observing the five-parameter model expression of PV arrays under various conditions, a fitting function model for PV arrays under arbitrary conditions is proposed. The expression is as follows:

3. Modeling of PV arrays

Vpvs = k⋅ln(a⋅Ipvs + b⋅G + c⋅T + p⋅G⋅T + d ) + r⋅Ipvs + s⋅Ipvs⋅G + m

zj = zjbad −

Rjj Ωjj

r jbad

(9)

⋅G + n⋅T + t

The modeling of PV arrays is an important issue, which seriously affects the accuracy and computational efficiency of ADS state estimation. Compared with the four-parameter model of PV arrays, the five-parameter model can more accurately express the relationship between solar irradiance, temperature, output voltage, and output current of PV arrays, especially under the condition of low temperature and low irradiance. However, the relation expressed by the five-parameter model is a complex implicit function. When the state estimation is performed and the Jacobian matrix of the PV branch is evaluated, the computational burden is substantial. In order to achieve high computational efficiency while maintaining reasonably accuracy, the fiveparameter model of PV array is fitted and replaced by an explicit function based on the WLS method.

(12) where Ipvs and Vpvs are the output current and output voltage; a, b, c, d, k, p, r, s, m, n, and t are the parameters to be fitted; G and T are the solar irradiance and temperature. Based on the expression of the five-parameter model of PV arrays, k (k ≥ 5000 , and k ∈ N+) sets of data of the irradiance, temperature, the output voltage, and output current are randomly obtained across the practically feasible region. That is, in the data taken, the solar irradiance Gn, the temperature Tn, the PV array five-parameter model output voltage Vpvn, and the PV array output current Ipvn are respectively a k×1 column vector, and the corresponding elements in the vectors Gn, Tn, Vpvn, and Ipvn satisfy the relationship given by the five-

3.1. Five-parameter model of PV arrays The five-parameter equivalent circuit of PV cells is shown in Fig. 1 [24]. It is not only suitable for a single PV cell, but also suitable for PV cell modules and arrays. The mathematical relationship between terminal voltage V and current I is expressed as follows:

V + IRs Rp V + IRs V + IRs − Io [exp( ) − 1) − nVt Rp

I = Iph − ID − = Iph

(10)

where Iph is the photo-generated current of the PV cell, A; Io is the reverse saturation current and leakage current of the diode, A; ID is the current through the diode, A; Rs is the equivalent series resistance of the

Fig. 1. Equivalent circuit representing the five-parameter model of a PV cell. 3

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measurements, and methods in the papers [32–34] can be adopted to integrate them. If the sampling rate of the PV systems is slower than SCADA, the situation is similar to the integration of AMI measurements, and methods in the papers [35–37] can be adopted to integrate them. 5. In this paper, the measurements of PV systems are based on the existing measurement devices, such as sensors of the DC/DC converter and the DC/AC inverter, meteorological sensors in PV power plants or weather stations, etc., hence the proposed method typically does not require placement of new measurement devices.

parameter model of PV array. In other words, suppose Gn(i), Tn(i), Vpvn (i), and Ipvn(i) are the i-th (1 ≤ i ≤ k , and i ∈ N+) entries of the vectors Gn, Tn, Vpvn, and Ipvn, respectively, then Gn(i), Tn(i), Vpvn(i), and Ipvn(i) satisfy Eq. (11). Substituting vectors Gn, Tn, and Ipvn to obtain vector Vpvsn (k × 1), then the following WLS problem can be solved to find the optimal fitting parameters:

min D (x ) = (Vpvn − Vpvsn )T Wn (Vpvn − Vpvsn )

(13)

where Wn is the weight matrix (k × k). Considering the significant number of fitting parameters, the selection of their initial values may not be a trivial task. Optimization algorithms free from initial values selection can be used, such as the differential evolution algorithm [25,26]. The effectiveness of the proposed function fitting model will be verified in the simulation studies section.

4.1. State estimation model of PV systems As shown in Fig. 2, a PV system typically adopts a two levels structure with a DC/DC converter and a DC/AC inverter, and connects to the distribution network through a transformer. The PV arrays may operate in the MPPT mode or other modes based on the needs of the ADS. Given the solar irradiance G and temperature T, denote the output current as Ipvs, and the output voltage as Vpvs. The DC/DC converter operates in the constant voltage mode. With the duty cycle Dpv, the output voltage and the output current are denoted as Vpvdc and Ipvdc, respectively. The DC/AC inverter also operates in a constant voltage mode, using double loop control where the outer loop is the voltage loop and the inner loop is the current loop. With the modulation ratio Mpv and the modulation phase angle θpv, the output voltage amplitude and phase angle of the DC/AC inverter are denoted as Epvo and θpvo, respectively. The equivalent impedance is denoted as Xpv, and the turn ratio of the transformer is assumed to be 1: Kpv. The active power and reactive power flowing from the PV system to the Point of Common Coupling’s (PCC) PV are denoted as Ppvac and Qpvac, respectively. The voltage amplitude and phase angle of the PCC are denoted as Vpvac and θpvac, respectively. The active and reactive power injections at the PCC (excluding the power from the PV system) are denoted as Ppvi and Qpvi, respectively. The overall efficiencies of the DC/DC converter and DC/ AC inverter are denoted as ηpvdc and ηpvac, respectively. In the state estimation of a PV system, the state variables are selected as: xpv = [G, T, Ipvs, Vpvdc, Epvo, θpvo]T, and the available measurements are collected as: zpv = [G, T, Ipvs, Vpvs, Ipvdc, Vpvdc, Dpv, Mpv, θpv, Epvo, Ppvac, Qpvac]T. With the selected measurements, state variables, the topology of PV systems, and the fitting function model for PV arrays, the measurement equations in PV systems are derived as follows:

4. State estimation model of the ADS with PV systems In this section, a PV system state estimation model compatible with steady-state monitoring of the ADS will be developed. State variables are selected, available electrical and non-electrical measurements are fully considered, and relationships between the state variables and measurements are derived. The most widely used WLS method is applied to perform joint estimation of the state variables of grid-connected PV systems and the distribution network, and the normalized residual method is used to process the associated bad data. Before moving forward, the following remarks should be made: 1. The focus of this paper is on the detailed modeling of PV systems and their joint state estimation with the distribution network, rather than improving the state estimation algorithm itself. While in this paper, the traditional WLS and normalized residual method are used for state estimation and bad data identification, respectively, the proposed model is readily applicable to other methods, such as the robust state estimation methods [27–29]; 2. In this paper, only three-phase balanced distribution systems are considered. The three-phase state estimation methods [30,31] for unbalanced power distribution systems are not discussed, but owing to the generality of the proposed model, it can be readily considered as a follow-up extended study; 3. Because the focus and contributions of this paper are on the PV systems, only the measurement information of the PV systems are discussed in detail, and the measurements of the distribution networks are not specifically discussed. However, thanks to the generality of the model, measurement data from Phasor Measurement Units (PMU) [32–34], Remote Terminal Units (RTU), and AMI [35–37], and pseudo-measurement data from generation scheduling and load forecasting as needed, can all be integrated as measurements of the distribution network, which does not affect the objectives, methodologies, and conclusions of this paper. 4. It is the utility’s decision whether the measurements in the PV systems should be communicated via the legacy Supervisory Control and Data Acquisition (SCADA), or using a separate new system. If a separate system is used, the sampling rate of the PV systems may be different than SCADA. If the sampling rate of the PV systems is faster than SCADA, the situation is similar to the integration of PMU

G z = G + Ge

(14)

Tz = T + Te

(15)

z e Ipvs = Ipvs + Ipvs

(16)

z e Vpvs = h pv (G, T , Ipvs) + V pvs

(17)

e z Vpvdc = Vpvdc + V pvdc

(18)

z Ipvdc =

=

Ipvs⋅Vpvs⋅ηpvdc Vpvdc

e + Ipvdc

Ipvs⋅h pv (G, T , Ipvs)⋅ηpvdc Vpvdc

e + Ipvdc

(19)

Fig. 2. Topological structure of a PV system.

4

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z Dpv =1−

=1− Epvo

z Mpv =

Vpvdc

Vpvs Vpvdc

h pv (G, T , Ipvs) Vpvdc

e + Dpv

(20)

e + Mpv

(21)

z e Epvo = Epvo + Epvo z θpv

(22)

= θpvo − θpvac +

z Ppvac =

distribution network is relatively simple, and the distribution network can be readily standardized by the conventional standardization methods. However, the PV systems contain non-electrical quantities such as the solar irradiance and temperature, and the relationships between the output voltage, output current, and the non-electric quantities are nonlinear. It is not feasible for the PV systems to obtain the per unit values using the conventional standardization methods. In this section, an extended per unit system is developed for unified standardization of electrical and non-electrical quantities. For a system y = f(x1, x2, x3, …, xn), x1, x2, x3, …, xn are the inputs, y is the output, and the relationship between them may be linear or nonlinear. The values of x1, x2, x3, …, xn, and y are actual values. The base values of x1, x2, x3, …, xn, y are selected as xb1, xb2, xb3, …, xbn, yb. The system can be standardized as follows:

e + Dpv

e θpv

(23)

Vpvac⋅Epvo⋅sin(θpvo − θpvac) k pv⋅Xpv

e + Ppvac

(24)

Vpvac⋅Epvo⋅cos(θpvo − θpvac)

z Qpvac =

y x x x x ⋅y = f ⎛ 1 ⋅xb1, 2 ⋅xb2 , 3 ⋅xb3, ..., n ⋅xbn ⎞ xb2 xb3 xbn yb b ⎝ xb1 ⎠ ⎜

k pv⋅Xpv Vpvac⋅Vpvac

−

k pv⋅k pv⋅Xpv

+

e Qpvac

m⋅ypu = f (k1⋅x pu1, k2⋅x pu2 , k3⋅x pu3, ..., kn⋅x pun )

k pv⋅Xpv

ypu = e + l pv

(30)

where xpu1, xpu2, xpu3, …, xpun, and ypu are the per unit values of x1, x2, x3, …, xn, and yb, respectively. Eq. (30) can be rewritten as

z l pv = Ipvs⋅h pv (G, T , Ipvs)⋅ηpvdc⋅ηpvac

− 3⋅

(29)

Since xb1, xb2, xb3, …, xbn, and yb are constants, set (k1, k2, k3, …, kn) = (xb1, xb2, xb3, …, xbn), m = yb, then:

(25)

where quantities with superscript z are the measurements, quantities with superscript e are the errors of the corresponding measurements, and hpv is the fitting function (12). Based on the law of power conservation, a virtual measurement equation can be derived as follows:

Vpvac⋅Epvo⋅sin(θpvo − θpvac)

⎟

1 ⋅f (k1⋅x pu1, k2⋅x pu2 , k3⋅x pu3, ..., kn⋅x pun ) m

(31)

Which can further be denoted as follows:

(26)

ypu = fpu (x pu1, x pu2 , x pu3 , ..., x pun )

Assuming that the PCC is indexed as bus i of the connected distribution network, the measurement equations of the injected active power Ppvi and reactive power Qpvi into the PCC are given as follows:

(32)

(27)

The electrical and non-electrical quantities in the ADS can be standardized using the method presented above. When the base values of the electrical quantities are selected, the base values only need to satisfy Ib × Vb = Pb. When the non-electrical quantities base values are selected, the base values only need to satisfy Eq. (32). The standardization process for the PV system using the abovementioned standardization method is elaborated as follows. In the PV system, Eq. (12) shows that the relationship between G, T, Ipvs, and Vpvs is nonlinear. Using the methods in this section, the PV system can be standardized as follows:

(28)

Vpvs

where hPi and hQi are the measurement functions of power injections in the conventional power network [23]; Vi and θi are the voltage amplitude and phase angle of bus i of the distribution network. From the above analysis, it can be found that for each PV power plant, the amount of data that needs to be transmitted to the control center is modest. The total number of measurements per plant ranges from 10 to 15. Considering that the state estimation is performed at a frequency of tens of seconds (referring to the conventional execution frequency of the current transmission network), it is required to upload about 10–15 floating point numbers every tens of seconds, which can be readily implemented by any common types of communication infrastructure.

Vbpvs

z Ppv i = h Pi (V1, V2, …, Vi , …, θ1, θ2, …, θi , …)

− 3⋅

Vpvac⋅Epvo⋅sin(θpvo − θpvac) k pv⋅Xpv

e + Ppv i

z Qpv i = hQi (V1, V2, …, Vi , …, θ1, θ2, …, θi , ...)

− 3⋅( −

Vpvac⋅Epvo⋅cos(θpvo − θpvac)

Vpvac⋅Vpvac k pv⋅k pv⋅Xpv

k pv⋅Xpv e ) + Qpv i

h pv (G b. =

G , Gb

Tb.

Ipvs T , Ibpvs. I ) Tb bpvs

Vbpvs

(33)

where Gb, Tb, Ibpvs, and Vbpvs are the base values of G, T, Ipvs, and Vpvs, respectively. Set (a1, a2, a3, a4) = (Gb, Tb, Ibpvs, Vbpvs), then:

Vpupvs =

1 h pv (a1. Gpu, a2 . Tpu, a3. Ipupvs) a4

(34)

where Gpu, Tpu, Ipupvs, and Vpupvs are the per unit values of G, T, Ipvs, and Vpvs, respectively. When the base values of G, T, Ipvs, and Vpvs are selected, Gb, Tb, and Vbpvs can be arbitrarily selected. Generally, Gb and Tb can be selected as the irradiance and temperature under standard conditions, and Vbpvs can be selected as the base value of the voltage level where the PV system is located. Ibpvs cannot be selected arbitrarily, and must satisfy:

4.2. Per unit system for electrical and non-electrical quantities

Vbpvs. Ibpvs = Xb

In the per unit system of conventional distribution networks, the electrical quantities are in similar scales, and the per unit values are easy to be computed. It only needs to determine the base values of each voltage level, then the per unit values can be easily computed. In order that the PV systems can be integrated into the distribution networks, the PV systems and the distribution networks need to be uniformly standardized. The computational relationship between electrical quantities in the

(35)

where Xb is the three-phase power base value of the ADS. 5. Simulation results This section is divided into two parts. The first part of simulation is to verify the accuracy of the fitting function model of the PV arrays with respect to the five-parameter model. The second part of simulation is to 5

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verify the proposed state estimation model using the IEEE 33-bus test system.

182.5 V. The power factor at the PCC is 0.8. The equivalent impedance Xpv is 0.01 j. The turn ratio of the transformer is 1:37.

5.1. Verification of the PV array fitting function model

5.2.1. State estimation without random errors (ideal condition) In MATLAB, the proposed state estimation model is simulated on the modified IEEE 33-bus test system given above which contains 4 PV systems. When there assumes no random error in the measurements, the algorithm converges after 7 iterations, and the true and estimated values of the PV systems state variables (phase angles expressed in degrees, and other quantities expressed in per unit) are shown in Table 3. As is apparent, the state estimation model developed in this paper can effectively converge and accurately estimate the state variables of the IEEE 33-bus test system and the PV systems. Thus, the operating state of the entire ADS can be captured. The simulation case validates the proposed state estimation model and the extended per unit system.

In order to verify of the fitting function model of PV arrays proposed in this paper, the simulation is performed in MATLAB. In this paper, the PV array consists of TDB125 × 125-72-P PV cells in series and parallel. The specific parameters of TDB125 × 125-72-P PV cells under standard operating conditions (solar irradiance G = 1000 W/m2 and temperature T = 298 K) are shown in Table 1. The number of PV cells in series is NS = 10, and the number of PV cells in parallel is NP = 300. Based on the five-parameter model of PV arrays, 8000 sets of data of solar irradiances, temperatures, output voltages, and output currents are randomly generated across the practically feasible region. The irradiance Gn, and the temperature Tn are 8000 × 1 column vectors, with the interval of the entries being [0,1200] (W/m2) and [238,318] (K), respectively. The five-parameter model output voltage Vpvn and output current Ipvn of the PV arrays are 8000 × 1 column vectors with the interval of the entries being [0,450] (V) and [0,1600] (A), respectively. The parameters to be fitted are solved for according to the method described in Section 3. The condition number of the gain matrix in this WLS problem is 6.534×108, which is acceptable. The fitting function model of PV arrays is obtained as follows:

5.2.2. State estimation with random errors In this section, the state estimation results will be compared between the proposed framework with detailed PV system modeling and the conventional state estimation framework where PV systems are not modeled in detail and simply treated as power injections into the network. Random errors obeying the normal distribution are introduced into the measurement set of the two models.

z imeans = z itrue + rand⋅σi

Vpvs = 23.3 × ln(−723 × Ipvs + 996 × G + 0.0177 × T + 0.452 × G × T − 5.72) − 0.0114 × Ipvs + 2.10 × 10−7 × Ipvs × G − 0.0147 × G − 1.52 × T + 585

(37)

where zimeas is the ith measurement value, zitrue is the ith true measurement value provided by power flow analysis, rand is a random number obeying standard normal distribution N (0, 1), and σi is the ith standard deviation of the measurement error. In this paper, the error with σi = 0.001 p. u. is introduced into the measurement set of two state estimation models. After 200 repetitions, the mean square deviations of the estimated state variables associated with the distribution network obtained by the two models in the IEEE 33-bus test system is illustrated in Fig. 10. In order to further illustrate performance of the proposed model under different penetrations of PV generations, 2 additional PV systems (the capacity of each PV system is also 0.525 MW) are connected to the 22th and 30th buses of the IEEE 33-bus test system. The results are obtained under the condition of 4 PV systems and 6 PV systems, as shown in Fig. 10. As shown in Fig. 10, the estimated values of state variables of the proposed method are more accurate than those of the conventional model under the different penetration conditions of PV systems, especially at the 3–9th, 14–18th, 20–22th, 27–31th, and 32th buses. For example, the accuracy of estimated values of buses 2, 4, 5, 20, and 22 of the IEEE 33-bus test system with 4 PV systems is improved by 12.68%, 12.38%, 11.76%, 12.68%, and 11.74%, respectively. It demonstrates that the model proposed in this paper yields more accurate results than the conventional model. This improvement owes to the increased information redundancy. From Section 4, it can be found that PV systems typically have higher redundancy level than the distribution network, which, by the principles in estimation theories [38], improves state

(36)

The comparison between the output voltages of the proposed model and those of the standard five-parameter model under several sets of typical operating conditions are shown in Fig. 3–8 and Table 2. It can be observed that the PV arrays fitting function model proposed in this paper has a fairly high fitting accuracy in a wide range of irradiance and temperature. In almost all the given cases of irradiance, temperature, and current, the errors of the output voltages predicted by the fitting function model stay below 1%. When the temperature and irradiance are close to the standard condition (G = 1000 W/m2 and T = 298 K), the fitting accuracy is the highest. Therefore, in the state estimation of PV systems, this model can be used to replace the traditional fiveparameter model to achieve high efficiency of computing the Jacobian matrix. 5.2. Verification of the proposed ADS state estimation model In order to verify the proposed state estimation model, this paper uses the IEEE 33-bus test system for simulation. The topology of the IEEE 33-bus test system with a total of 32 lines is shown in Fig. 9. The three-phase power base value of the IEEE 33-bus test system is 10 MW, and the line voltage rms base value is 12.66 kV. Bus 1 is considered as the reference bus with the voltage phase angle equal to zero, and the convergence tolerance of the state estimation algorithm is set as 10−6. In this paper, the PV systems are connected to the 8th, 14th, 24th, and 26th buses, as shown in Fig. 9. The capacity of each PV system is 0.525 MW, the number of PV cells in series is NS = 10, and the number of PV cells in parallel is NP = 300. The efficiencies of the DC/DC converter and DC/AC inverter are assumed to be 95% and 96%, respectively. The simulations are performed under the standard operating condition (G = 1000 W/m2 and T = 298 K). The PV arrays operate in the MPPT mode, and the output voltage Vpvs is 360 V. The DC/DC converters operate in a constant voltage mode, and the output voltage Vpvdc is 600 V. The DC/AC inverters operate in a constant voltage mode, adopting double loop control where the outer loop is the voltage loop and the inner loop is the current loop, and the output voltage Epvo is

Table 1 Parameters of PV cells.

6

Parameters

Parameter Values

Types of solar cells Maximum power (Pmax/W) Maximum power point voltage (Vm/V) Maximum power point current (Im/A) Short-circuit current (Isc/A) Open-circuit voltage (Voc/V) Short-circuit current temperature coefficient (K1(/K)) Open circuit voltage temperature coefficient (K2(/K))

Monocrystalline silicon 175.0 36.00 4.870 5.210 44.40 0.002084 −0.001554

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Fig. 3. Relationship between (a) the PV array output current and voltage I–V, and (b) the output power and voltage P–V at G = 1000 W/m2 and T = 298 K. Fig. 5. Relationship between (a) the PV array output current and the voltage I–V, and (b) the output power and voltage P–V at G = 100 W/m2 and T = 298 K.

Fig. 4. Relationship between (a) the PV array output current and voltage I–V, and (b) the output power and voltage P–V at G = 500 W/m2 and T = 298 K.

estimation solutions. Regarding computational performances, in this paper, the experimental hardware environment is Intel Core i7 2.70 GHz CPU, and 4GB RAM. The software environment is MATLAB R2014a running on Windows 10 operating system. The average computing time of the conventional ADS state estimation model without the detailed PV

Fig. 6. Relationship between (a) the PV array output current and voltage I–V, and (b) the output power and voltage P–V at G = 1200 W/m2 and T = 278 K.

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model is 0.438 s. The average computing time of the ADS state estimation model in this paper with the detailed PV model is 0.446 s. The condition number of the gain matrix of the state estimation of ADS with PV systems is 1.273 × 106, which is acceptable. Therefore, the proposed model does not lead to significant additional computational burden compared to the conventional model. It should be pointed out that the advantages of the proposed method are not limited to improving the accuracy of the state estimates associated with the distribution network. More importantly, by modeling the PV systems and utilizing its associated measurements data, the coverage of the ADS state estimation is extended to the grid-connected PV systems. The operation states of the PV systems are effectively monitored, which lay the foundation for the comprehensive situation awareness of the ADS. 5.2.3. Bad data processing In this subsection, the capability of bad data processing of the proposed model is tested. In each case, five gross errors are introduced into the measurement set of the IEEE 33-bus test system with 4 PV systems. The WLS method and the normalized residual method are carried out iteratively until the largest normalized residual is lower than the set threshold of 3.0. Case 1: The measurement dataset of the distribution network is intentionally corrupted with 5 gross errors in P6, P15, P29, Q21, and V8, respectively. The simulation results are shown in Table 4. In each cycle of identification, the variables associated with the largest two normalized residuals are listed. In each cycle, the measurement corresponding to the largest normalized residual is corrected or deleted from the measurement dataset. As can be seen, the first five cycles successfully identify bad data in the distribution network. The normalized residuals of the sixth cycle are all lower than the detection threshold, indicating that there is no more bad data. Case 2: The measurement dataset of a single PV system is intentionally corrupted with 5 gross errors in G1, T1, Dpv1, Ppvac1, and lpv1. The simulation results are shown in Table 5. As shown in Table 5, five bad data in the PV system are identified. After correcting or deleting bad data, the measurements in the PV system become gross-error-free. Case 3: The measurement datasets of the multiple PV systems are intentionally corrupted with 5 gross errors in G1, G3, T2, T4, and Ppvac1. The simulation results are shown in Table 6. Clearly, five bad data in multiple PV systems are identified one at a time. After correcting or deleting bad data, the measurements in the system become gross-error-free. Case 4: The measurement data set of the multiple PV systems is intentionally corrupted with 4 gross errors in measurements G3, T2, lpv4, and Ppvac1. Meanwhile, the measurement dataset of the distribution network is intentionally corrupted with one gross error in P29. In order to show the situation of mixed SCADA measurements and pseudomeasurements, it is assumed that half of the measurements are SCADA measurements with high accuracy (standard deviations being set as 0.001 p.u.), and the other half of the measurements are pseudo-measurements with low accuracy (standard deviations being set as 0.01 p.u.). The simulation results are shown in Table 7. As shown in Table 7, five bad data in the PV systems and the distribution network are identified. This case shows that the proposed method in this paper is still effective when the measurement data set contain measurements with different accuracy levels. Case 5: This case is a comparison of the bad data processing methods of the traditional distribution network and the bad data processing methods of the ADS proposed in this paper. First of all, the measurements are configured reasonably to ensure that the system is observable. The measurement dataset of the distribution network is intentionally corrupted with a gross error in P24. Table 8 shows the simulation results of bad data processing using the traditional state estimation model which represents the PV systems as power injections

Fig. 7. Relationship between (a) the PV array output current and voltage I–V, and (b) the output power and voltage P–V at G = 800 W/m2 and T = 318 K.

Fig. 8. Relationship between (a) the PV array output current and voltage I–V, and (b) the output power and voltage P–V at G = 300 W/m2 and T = 278 K.

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Table 2 Comparison of output voltage and current values between the five-parameter model and fitting function model. G = 1000W/m2, T = 298 K

G = 800W/m2, T = 318 K

G = 500W/m2, T = 278 K

G = 100W/m2, T = 298 K

Ipv/Ipvs

Vpv

Vpvs

δ(%)

Ipv/Ipvs

Vpv

Vpvs

δ(%)

Ipv/Ipvs

Vpv

Vpvs

δ(%)

Ipv/Ipvs

Vpv

Vpvs

δ(%)

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

444.2 441.1 437.9 434.6 431.1 427.6 423.8 419.8 415.5 410.9 405.8 400.1 393.4 385.2 373.8 353.1

442.2 439.6 436.8 433.9 430.9 427.7 424.3 420.6 416.6 412.2 407.3 401.67 394.9 386.3 374.1 351.1

0.4528 0.3515 0.2521 0.1551 0.0612 0.0289 0.1140 0.1928 0.2631 0.3221 0.3653 0.3855 0.3696 0.2913 0.0827 0.5509

0 90 180 270 360 450 540 630 720 810 900 990 1080 1170 1200 1250

411.4 408.2 404.9 401.4 397.7 393.9 389.7 385.3 380.3 374.8 368.4 360.7 350.4 334.0 324.9 287.7

409.7 407.0 404.1 401.1 397.9 394.4 390.7 386.6 381.9 376.7 370.5 362.9 352.5 335.5 325.9 286.1

0.4025 0.2915 0.1813 0.0724 0.0348 0.1396 0.2410 0.3378 0.4278 0.5079 0.5725 0.6105 0.5964 0.4433 0.3015 0.5745

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 720

459.6 457.4 455.2 452.8 450.4 447.8 445.0 442.0 438.8 435.2 431.2 426.5 420.9 413.7 403.3 397.2

463.7 461.6 459.3 456.9 454.5 451.8 448.9 445.8 442.3 438.4 433.9 428.7 422.3 413.9 401.6 394.3

0.8927 0.9036 0.9101 0.9115 0.9067 0.8947 0.8738 0.8417 0.7956 0.7311 0.6415 0.5161 0.3355 0.0596 0.4180 0.7354

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 145

397.8 396.3 394.7 392.9 391.1 389.1 386.9 384.5 381.9 378.9 375.4 371.3 366.2 359.6 349.8 342.4

401.8 400.2 398.4 396.5 394.5 392.3 389.8 387.2 384.2 380.8 376.9 372.3 366.5 358.9 347.7 339.2

1.018 0.9859 0.9506 0.9109 0.8662 0.8155 0.7574 0.6904 0.6120 0.5187 0.4055 0.2640 0.0792 0.1791 0.5887 0.9220

into the distribution network only. The simulation results of bad data processing using the comprehensive ADS model containing the detailed models of the PV systems are shown in Table 9. As shown in Table 8, when the conventional state estimation model is used, the normalized residuals associated with P24-25 and P24 are almost identical, thus the bad data only be detected, but cannot be identified. The reason is that due to the lack of information redundancy, measurements P24-25 and P24 form a critical measurement pair [17], where the true source of error cannot be distinguished. In comparison, as shown in Table 9, when the proposed state estimation model is used, the critical measurement is broken thanks to increased measurement redundancy, and the bad data can be correctly identified. After correcting or deleting bad data, the measurements in the system become gross-error-free. The improvement in the capability of error identification brought by detailed modeling of the PV systems and consideration of their associated measurements is evident. In summary, the proposed state estimation model not only enables bad data processing for measurements associated with the grid- connected PV systems, but also improves the bad data processing capability for the measurements associated with the distribution network. These

Table 3 True and estimated values of the PV systems state variables. State variable

True value

Estimated value

State variable

True value

Estimated value

G1 T1 Ipvs1 Vpvdc1 Epvo1 δpvo1 G2 T2 Ipvs2 Vpvdc2 Epvo2 δpvo2

1.000 1.000 0.08830 1.000 1.000 2.016 1.000 1.000 0.08820 1.000 1.000 2.248

1.000 1.001 0.08831 1.000 1.007 2.015 1.002 1.000 0.08830 1.000 1.006 2.249

G3 T3 Ipvs3 Vpvdc3 Epvo3 δpvo3 G4 T4 Ipvs4 Vpvdc4 Epvo4 δpvo4

1.000 1.000 0.08830 1.000 1.000 2.225 1.000 1.000 0.08840 1.000 1.000 2.398

1.002 1.001 0.08831 1.001 1.021 2.226 1.001 1.000 0.08830 1.001 1.007 2.397

Note: The number in the character subscript indicates the number of the PV branch.

Fig. 9. Topology of the IEEE 33-bus test system with PV systems. 9

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Fig. 10. Mean square error between the estimated and true voltage magnitudes of the IEEE 33-bus system for both models.

Table 4 Simulation results of case 1.

Table 6 Simulation results of case 3.

Iterations

Parameters

rN

Bad data identified

Iterations

Parameters

rN

Bad data identified

1

P29 Q29 Q21 P21 P6 Q6 P15 Q15 V8 Q6 P21 Q23

46.32 46.31 19.81 19.80 30.71 30.68 16.59 16.58 9.731 1.667 0.1239 0.1228

P29

1 2

P6

3

P15

4

V8

5

None

6

580.5 160.9 117.8 117.7 69.32 69.31 10.19 10.18 10.18 10.18 0.1239 0.1228

Ppvac1

Q21

Ppvac1 lpv1 T4 G4 T2 G2 G1 T1 G3 T3 P21 Q23

2 3 4 5 6

Note: P and Q represent active and reactive power, respectively, and the number in the subscript indicates the number of the bus.

Bad data identified

Ppvac1 lpv4 lpv4 θpv4 P29 Q29 T2 G2 G3 T3 P21 Q23

193.8 133.8 133.8 119.2 106.5 100.2 69.23 69.21 20.13 20.12 0.1239 0.1228

Ppvac1

Bad data identified

1

1

Ppvac1 Dpv1 Dpv1 G1 lpv1 θpv1 T1 G1 G1 T1 P21 Q23

966.9 819.5 857.9 625.8 355.9 311.8 113.8 113.7 10.19 10.18 0.1239 0.1228

Ppvac1

2

Dpv1

3

lpv1

4

T1

5

G1

6

6

None

rN

rN

5

G3

Parameters

Parameters

4

G1

Iterations

Iterations

3

T2

Table 7 Simulation results of case 4.

Table 5 Simulation results of case 2.

2

T4

lpv4 P29 T2 G3 None

None

Table 8 Simulation results of conventional state estimation model.

features ensure the robustness of situational awareness provided by the proposed state estimation model.

Iterations

Parameters

rN

Bad data identified

1

P24-25 P24

18.39 18.38

P24-25

6. Conclusion are accordingly derived. Based on the proposed model, state estimation and bad data processing of the ADS are carried out by applying the WLS method and the normalized residual method, respectively. Simulation results show that the proposed fitting function model is capable of approximating the five-parameter PV array model with high accuracy, and the real-time operating state of the ADS, including the integrated PV systems, can be accurately and reliably estimated. The proposed work will facilitate comprehensive situation awareness and reliable

In this paper, an ADS state estimation model with explicit consideration of grid-connected PV systems is developed. In order to achieve high computational efficiency, an explicit fitting model of PV arrays is proposed to characterize the relationships between the solar irradiance, temperature, current, and voltage in replacement of the five-parameter model. Power electronic devices of PV systems are also effectively modeled. Subsequently, state variables are selected for the PV systems, and the equations associated with available measurements 10

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Table 9 Simulation results of proposed state estimation model. Iterations

Parameters

rN

Bad data identified

1

P24 P24-25 P21 Q23

20.12 16.99 0.1239 0.1228

P24

2

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None

operation of the ADS. Declaration of Competing Interests None. CRediT authorship contribution statement Zhi Fang: Methodology, Software, Validation, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization. Yuzhang Lin: Conceptualization, Methodology, Formal analysis, Writing - review & editing, Supervision. Shaojian Song: Methodology, Validation, Writing - review & editing, Project administration, Funding acquisition. Chunning Song: Methodology, Formal analysis, Validation, Resources, Data curation. Xiaofeng Lin: Conceptualization, Resources, Supervision, Project administration, Funding acquisition. Gang Cheng: Software, Resources, Data curation. References [1] O. Samuelsson, S. Repo, R. Jessler, et al., Active distribution network — demonstration project ADINE, Innovative Smart Grid Technologies Conference Europe, IEEE, 2010. [2] M. Sebastian, O. Devaux, O. Huet, Description and benefits of a situation awareness tool based on a distribution state estimator and adapted to smart grids, (2008). [3] A. Ranković, B.M. Maksimović, A.T. Sarić, et al., ANN-based correlation of measurements in micro-grid state estimation, Int. Trans. Electr. Energy Syst. 25 (10) (2015) 2181–2202. [4] A. Ranković, B.M. Maksimović, A.T. Sarić, A three-phase state estimation in active distribution networks, Int. J. Electr. Power Energy Syst. 54 (1) (2014) 154–162. [5] A.R. Abbasi, A.R. Seifi, A new coordinated approach to state estimation in integrated power systems, Int. J. Electr. Power Energy Syst. 45 (1) (2013) 152–158. [6] E.I. Batzalis, G.E. Kampitsis, S.A. Papathanassiou, Power reserves control for PV systems with real-time MPP estimation via curve fitting, IEEE Trans. Sustainable Energy 8 (3) (2017) 1269–1279. [7] M.J.Z. Zadeh, S.H. Fathi, A new approach for photovoltaic arrays modeling and maximum power point estimation in real operating conditions, IEEE Trans. Ind. Electr. 64 (12) (2017) 9334–9343. [8] J. Liu, J. Tang, F. Ponci, et al., Trade-Offs in PMU deployment for state estimation in active distribution grids, IEEE Trans. Smart Grid 3 (2) (2012) 915–924. [9] G. Valverde, A.T. Saric, V. Terzija, et al., Stochastic monitoring of distribution networks including correlated input variable, IEEE Trans. Power Syst. 28 (1) (2013) 246–255. [10] E. Manitsas, R. Singh, B.C. Pal, et al., Distribution system state estimation using an artificial neural network approach for pseudo measurement modeling, IEEE Trans. Power Syst. 27 (4) (2012) 1888–1896. [11] D.A. Haughton, G.T. Heydt, A linear state estimation formulation for smart distribution systems, IEEE Trans. Power Syst. 28 (2) (2013) 1187–1195.

11