Sensitivity analysis of volt-VAR optimization to data changes in distribution networks with distributed energy resources
Introduction
The Volt-VAR optimization (VVO) is one of the crucial functions in advanced distribution management systems (ADMSs), employed for efficient managing and controlling of active distribution networks with distributed energy resources (DERs) (e.g., solar photovoltaic (PV) system, energy storage system (ESS), and electric vehicle), advanced metering infrastructure (AMI), and demand response programs [1]. In active distribution networks, VVO computes the optimal voltage magnitude for any node in an optimization problem. Thus, the total voltage deviation from the nominal voltage is minimized while considering all operational constraints of distribution systems. Operational constraints in the VVO are formulated using many different types of data—voltage limits, distribution line parameters, operation parameters of PV/ESS, the predicted PV generation output and load demand, demand reduction and load exponents in exponential load models. A change in these heterogeneous data may result in a severe malfunction of the VVO. This study attempts to provide an analytical framework for the quantification of the impact of changes in VVO data on the performance of VVO.
The primary goal of the VVO is to determine optimal voltage levels along the distribution feeder under all loading conditions through the coordination of conventional voltage regulators such as on-load tap changers (OLTCs) at substations, step-type voltage regulators (SVRs), and capacitor banks (CBs) [2]. Recently, advanced information and communication technology (ICT) such as AMI with smart meters provides voltage regulators with real-time loading and voltage measurements, thereby adjusting the voltage profile more accurately [3]. Furthermore, the smart inverters of DERs can be employed as voltage regulators, through the injection or absorption of their real and reactive power, to maintain the desired local voltage profile based on the designed droop control curve when voltage violations occur rapidly owing to intermittent PV generation [4], [5]. A mixed-integer second-order cone programming (MISOCP) model was formulated to determine the optimal parameters for the droop control curves of DERs [6]. More recently, with an increasing number of EVs connected to distribution networks, a new multi-agent based VVO method was developed where fast vehicle-to-grid reactive power dispatch, charging coordination and VVO for EVs are carried out in a distributed and fast manner [7]. The impact of different EV penetration levels on an AMI-based quasi real-time VVO was investigated [8].
As the coupling between the VVO and the ICT system with DERs is further strengthened for efficient voltage regulation of an entire active distribution network, a large number of heterogeneous data including sensor measurements and operating parameters for networks and DERs are used for the VVO process as input data. With the aforementioned circumstances, the change in these data due to a natural error or man-made attack may significantly distort the optimal solution of VVO. Recent studies have shown that an adversary can maliciously change the distribution feeder voltage profile by stealthily malfunctioning the voltage regulators through the injection of false data into sensors [9], [10], [11].
This study investigates the sensitivity of the VVO with respect to changes in various types of data that are used for the VVO process. We specifically aim to evaluate the impact of these data on the VVO optimal solutions (e.g, nodal voltage and real/reactive power flow at the substation) and the optimal total voltage variation. A large body of literature has been accumulated on the subject of sensitivity analysis regarding data changes for power system applications at the transmission level: (i) locational marginal price (LMP) in the DC and AC optimal power flow models [12], [13], (ii) state estimation [14], [15], and iii) look-ahead security constrained economic dispatch (SCED) [16]. Recently, we conducted data perturbation-based sensitivity analysis of power system applications at the distribution level, including the home energy management system (HEMS) [17] and distribution system state estimation (DSSE) integrated with HEMS [18]. Moreover, voltage sensitivity analysis was performed in distribution systems, including development of the expression for voltage sensitivity that illustrates the relationship between real power and voltage magnitude in a DC distribution system [19], calculation of voltage sensitivity using historical smart meter data without low voltage (LV) grid topology information [20] and of decentralized voltage sensitivity with local and neighborhood measurements only [21], and derivation of the nodal sensitivity of voltage and real/reactive power at nodes with distribution phasor measurement unit (DPMU) [22].
Although much work has been conducted on the subject of sensitivity analysis for power system applications at both the transmission and distribution system levels, to the best of our knowledge, no analytical sensitivity framework has been developed to directly assess the impact of various types of data on the VVO. Previous studies [19], [20], [21], [22] have focused on a sensitivity analysis of nodal voltage magnitude considering the change in a few variables such as real and reactive power without explicitly considering the VVO process. Our study is motivated by a desire to investigate the impact of change in all input data applied in VVO on the optimal solution of VVO. In this study, we assume that data change refers to both natural noise and man-made attacks. Our proposed sensitivity framework is based on a perturbation approach using Karush-Kuhn-Tucker (KKT) conditions for general nonlinear optimization problems [23], which have been verified and illustrated in [13], [14].
The main contributions of this study are summarized as follows:
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We present a closed-form analytical framework to evaluate the sensitivity of the optimal VVO solution with respect to changes in various types of data used for VVO.
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To the best of our knowledge, data sensitivity analysis of mixed integer nonlinear programming (MINLP)-based VVO via a perturbation approach based on its corresponding KKT equations has not been studied yet; therefore the proposed sensitivity analysis framework will provide system operators with an online analysis tool to quickly assess the impact of data on the performance of the VVO.
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The key part of the proposed approach is to develop a closed-form linear sensitivity matrix based on the perturbed KKT equations from the VVO formulation in the following three steps: (i) VVO formulation using MINLP, (ii) reformulation of the MINLP-based VVO into nonlinear programming (NLP)-based VVO with an integer optimal solution from the MINLP-based VVO, and (iii) perturbation of KKT equations from NLP-based VVO.
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Through the performance evaluation of the developed sensitivity matrix in the IEEE 33-node and 123-node systems, we show that the proposed approach can successfully generate the accurate sensitivities within the allowable VVO execution time.
The remainder of this paper is organized as follows. Section 2 introduces a system model and mathematical optimization problem for VVO. In Section 3, we develop a mathematical framework based on the perturbed KKT conditions of the VVO formulation to quantify the sensitivity of VVO to change in data. Section 4 presents numerical examples that illustrate the impact of various types of data on VVO in the IEEE 33-node and 123-node test feeders, along with validation of the accuracy and computation time of the proposed sensitivity framework. We make concluding remarks and suggest future work in Section 5.
Section snippets
Preliminary
The main notations used throughout this paper are summarized in Table 1. Bold symbols represent vectors or matrices. Hat symbols represent predictions of true parameter values. The other undefined symbols in the nomenclature section are explained in the text.
Proposed sensitivity framework for VVO
This section introduces the proposed analysis framework used to quantify the sensitivity of the VVO with respect to changes in the data. In Section 3.1, the sensitivity framework for a general NLP optimization problem is presented, which can be obtained by perturbing the decision variables and data in the KKT equations from the NLP optimization problem. In Section 3.2, the proposed sensitivity matrix for the NLP-based VVO is proposed along with the classification of both the optimal solution
Numerical results
In this section, the developed sensitivity matrices and in Section 3 are tested and validated to assess the impact of changes in various types of data on VVO. The impact assessment is illustrated in the modified IEEE 33-node [25] and 123-node test system [29] as shown in Fig. 3. A simulation environment for the sensitivity analysis is described in Section 4.1, which is followed by the subsequent five subsections that include the sensitivity results for different data groups in Table 2:
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Conclusions
The Volt-VAR optimization is formulated with many different types of input data that involve distribution line parameters, PV and load forecasts with demand reduction, voltage-dependent load models, and operating conditions for PV system/ESS and voltage regulators. Since data change may take place due to a natural error or a man-made attack, maintaining data quality in Volt-VAR optimization is of vital importance for reliable distribution system operations. In this context, sensitivity analysis
Declaration of Competing Interest
None.
Acknowledgments
This work was supported in part by the Korea Electric Power Corporation under Grant R17XA05-75, and in part by the Korea Government (MSIP) through the National Research Foundation of Korea (NRF) under Grant 2018R1C1B6000965.
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