I Introduction
Modern power grid is one of the most complex engineering systems in existence; the North American power grid is recognized as the supreme engineering achievement in the 20th century [1]. The complexity of grids is ever increasing: 1) the evolution of grid networks, especially the expansion in size; 2) the penetration of renewable/distributed resources, flexible/controllable electronic components, or even prosumers with dual loadgenerator behavior [2]; 3) the revolution of operation mechanisms, e.g., demandside management; and 4) the mechatronic disciplines (mechanics, electric and electronics) are realized in a more integrated way, and their interfaces will be more intertwined. All these driving forces lead to a nonlinear, diversified, hierarchical, and distributed power grid, which is hard to model and to analyse.
Digital twin (DT) is one of the most promising enabling technologies for realizing smart grids, especially for the construction of Ubiquitous SGeIoT (Electric Internet of Things proposed by State Grid Corporation of China [3]). A DT refers to the digital representation of a realworld entity or system. These DTs are linked to their realworld counterpart and are used to understand the state of the thing or system, respond to changes, improve operation and add value.
DT is characterized by datadriven mode, realtime interaction, and closedloop feedback. Automation and numerous stateoftheart technologies such as internet of things (IoT), 5G, drones, robots, edge computing, big data, and artificial intelligence (AI) serve as a basis for DT. Organizations often get a quick start of DT, i.e., build simple DTs and put them into operation at the very beginning, and then evolve them over time in an active and selfadaptive way—evolve DTs with the accumulation of operation data, practice feedbacks, and subjective experience. It is quite different from the procedure that we build a physical model, in which a global designing, a deliberate start, and some assumptions and simplifications, are required in advance. This quick start makes DT much more accessible than conventional simulations, e.g., Matpower, to engineering in practice.
It has been almost 16 years since the concept of DT was initially proposed in 2003 [4]. To date, many DT applications have been successfully implemented in different industries and DT becomes an emerging market. Tao et al. suggested 14 potential DT applications, such as product design, assembly, in a workshop [5]. Gartner identified DT as one of the Top 10 Strategic Technology Trends of 2018 [6]
. The DT market is estimated to grow from USD 3.8 billion in 2019 to USD 35.8 billion by 2025, at a CAGR of 37.8%. Major factors surging the demand for DT include increasing adoption of emerging technologies such as IoT and cloud
[7].This paper, from both science paradigms and engineering practice, outlines the backgrounds, challenges, framework, tools, and possible directions of DT for power systems (PSDT). To our best knowledge, it is also the first exploration on DT in the context of power systems. Our preliminary exploration is expected to contribute some novel discoveries, as well as some highdimensional analytics, to the engineering community. Besides, the connection of DT with big data analytics and AI may has deep impact on data science.
Ii Background and Framework of PSDT
Iia From Twins to Digital Twins
The concept of “twins” dates back to NASA’s Apollo Program. In the program, at least two identical space vehicles were built, allowing engineers to mirror conditions of the space vehicle during missions (analogous to Task 1, realtime situation awareness), and the vehicle remaining on earth is called the twin. The twin was also used extensively for training during flight preparations (analogous to Task 2, ultratime virtual test). During a flight mission it was used to simulate alternatives on Earthbased model, where available flight data were used to mirror flight conditions as well as possible, and thus assist astronauts in orbit in critical situations.
Another well known example of a “hardward” twin is the Iron Bird, a groundbased engineering tool used in aircraft industries to incorporate, optimize and validate vital aircraft systems [8]. Due to the development of simulation technologies, the hardware parts in the Iron Bird are replaced by virtual models. This allows system designers to use the concept of an Iron Bird in earlier development cycles, even when some physical components are not yet available. Extending this idea further along all phases of the life cycle leads to a complete digital model of the physical system, the Digital Twin (DT).
The term DT was brought to the public for the first time in NASA’s integrated technology roadmap [9]: A Digital Twin is an integrated multiphysics, multiscale simulation of a vehicle or system that uses the best available physical models, sensor updates, fleet history, etc., to mirror the life of its corresponding flying twin. The Digital Twin is ultrarealistic and may consider one or more important and interdependent vehicle systems, including propulsion/energy storage, avionics, life support, vehicle structure, thermal management/TPS, etc. Manufacturing anomalies that may affect the vehicle may also be explicitly considered. Reference [10] gives a graphical representation of some attributes of a DT as Fig. 1, and each of the nine subfigures presents a narrative.
Our work explores PSDT. Rather than life cycle phases or manufacturing anomalies, PSDT is more concerned with two major tasks in the field of power systems: 1) realtime situation awareness (SA), and 2) ultratime virtual test.
IiB Conventional SA and its Limitation
SA is defined, according to [11], as the perception of the elements in an environment, the comprehension of their meaning, and the projection of their status in the near future. Timely and accurate SA is essential for power system security. Inadequate SA is identified as one of the root causes for the largest blackout in history—the 14 August 2003 Blackout in the United States and Canada [12]. For a modern grid as stated in Sec. I, SA is in urgent need of a new prominence.
Conventional modelbased SA for power systems needs to be revisited. We would like to refer to Fig. 2 in book [13]
as an illustration. Under the 2nd and 3rd paradigms (from last few hundred years to last few decades), our insight into the world is mainly based on physical models. For a power grid, we use equations, formulas, or simulations to describe the operation regulations and interaction mechanisms of each units. This modelbased mode cannot make full use of massive data due to its own limitations—it aims at a deterministic solution which is always in lowdimensional space (low dimension is not well compatible with highdimensional data). Moreover, the assumptions and simplifications of system units (often small in size but large in number
[14]), and the increasing penetration of distributed energy resources (often susceptible to climate changing and usage lifetime [15]), will inevitably cause error; the error accumulations can hardly be addressed (described or analysed) with physical models or in lowdimensional space.IiC Science Paradigms and Datadriven SA in High Dimension
As shown in Fig. 2, we are now entering the age of the 4thparadigm—dataintensive scientific discovery. In this age, datadriven is an alternative paradigm, and even becomes a trend as data become more and more accessible. For smart grids, datadriven approaches become natural and stressing topics [16]. Datadriven approaches are also characterized by modelfree—we no longer heavily rely on physical models, and can handle the scenarios where the system topologies and network parameters are unreliable or even totally unavailable. Moreover, comparing datadriven results to modelbased ones, we can obtain some insights for further analysis. This phenomenon has some connection with Task 2, virtual test, and will be demonstrated by the case studies in Sec. IVC4.
High in dimensionality, rather than large in number, is the keystone and difficulty of datadriven SA designing. High dimension means that the datasets are represented in terms of large random matrices. These data matrices can be viewed as data points in highdimensional vector space of mathematics—each vector is very long. Traditional data transformations, however, are often in the form of low dimension, such as onedimensional Fourier Transformation (time domain to frequency domain), and threedimensional Park Transformation (ABC to dq0). In lowdimensional space, only two typical data matrices in the form of
are at our disposal: 1) are small, and 2) is small, is very large (compared with ). Low dimension is not well compatible with highdimensional data—lowdimensional tools are inadequate to a complex problem such as optimized operation or integrated planning for a modern grid, in which the behaviors and discipline of system units are closely intertwined.IiD Big Data Analytics and AI for eScience Data
The limitation of modelbased and lowdimensional statistical algorithms to eScience data has greatly spurred the development of big data analytics and artificial intelligence (AI). Modern grid operation is always accompanied with a temporalspatial dataset—the dataset is in highdimensional space, and in the form of time series. Temporal variations ( sampling instants) are simultaneously observed together with spatial variations ( signals). Only in highdimensional space do those statistical properties and benefits come out, and it will be revealed by a virtual test in our case studies.
The extraction of statistical information from this highdimensional space is a challenge that does not meet the prerequisites of most mathematical tools. Big data analytics and AI are employed to handle this challenge, and accordingly we select two tools: 1) random matrix theory (RMT), which is good at big data analytics, and 2) deep learning, which does well in massive data modeling. Both of the tools have already made huge impacts on many engineering fields. Compared to its modelbased counterpart, DT is more compatible with or even more naturally connected to these tools. PSDT realizes SA mainly
by means of mining information from temporalspatial dataset. This attribute of DT is marked as datadriven.IiE Datadriven Tools: RMT and Deep Learning
RMT has an advantage of transparency—unifying time and space through their ratio , RMT deals with temporalspatial data mathematically rigorously. The goal of RMT is to understand the joint eigenvalue distribution
as the statistic analytics from big data in the asymptotic regime. In particular, highdimensional analysis and visualization are treated as the functionals of the eigenvalue distributions. For instance, Linear eigenvalue statistics (LESs)
[17], built from data matrices, follow Gaussian distributions for very general conditions, and other statistical variables are studied due to the latest breakthroughs in probability on the central limit theorems of those LESs
[18]. The statistical properties of these variables are mostly derivable and provable. Moreover, RMT performs well with moderatesize data.Deep learning is the stateoftheart algorithm in data science. Deep learning does learn some nonhandcrafted features, so called deep features, from the massive labeled dataset without much prior knowledge, so that it can be generalized to different cases without making significant modifications. Moreover, the performance of the deep network model on fitting task and generalization task could be quantitative evaluated with test set, so as to ensure desired effects.
IiF Virtual Test and Framework of DT
Besides datadriven SA, DT also allows for simulations of new ideas that can be tested virtually to determine environmental impact before implementation in the real world. Software for automation can also be tested in advance using the virtual representation of the real system (i.e. ultratime virtual test).
Then we build the framework of PSDT, as shown in Fig. 3. The relationship among the PSDT, the physical grids, and the operators are similar to the relationship, in the aforementioned Apollo Program, among the vehicle remaining on earth, the space vehicle during missions, and the astronauts in orbit. PSDT helps the operators to understand the physical grid and to make a reliable decision in critical situations.
The arrow lines represent the data/information flow, which connects the physical object and its DT. Our two major tasks are deployed along these arrow lines: DT uses sampling spatialtemporal data from communication for SA, and uses virtual data/commands from decision makers for virtual test. These lines also form a closed loop, which means feedbacks are available for the tasks. Characterized by realtime data/information flow and closedloop feedbacks, the effectiveness of our tasks can be guaranteed. The involvement of humanmachine interfaces provides an access to some artificial inputs, e.g., hyper parameters setting for a virtual test; it makes DT more flexible and intelligent in the virtual space.
Iii Power Flow Operation Formulation and Conventional Analysis
Iiia Background of Power Grid Operation
Power flow (PF) analysis is a fundamental and most frequently used tool for many tasks in a power system, such as fault diagnosis, state estimation, security assessment, and optimal power dispatch. For each node in a grid network, choosing the reference direction as shown in Fig 4, Kirchhoff’s current law and ohm’s law say that:
(1) 
where is the admittance in Cartesian form^{1}^{1}1 is the conductance, is the susceptance, and is the imaginary unit., and and are node voltage and node current, respectively, in polar form^{2}^{2}2.
And thus, for each node in a power grid, Node for instance, taking account of the nodetoground admittance , its active power and reactive power are expressed as:
(2) 
Abstractly, a physical power system obeying Eq. (2) can be viewed as an analog engine—it takes bus voltage magnitude and bus voltage phaser as inputs, conductance and susceptance as given parameters, and “computes” active power injection and reactive power injection as outputs.
IiiB Modelbased Power Flow Analysis
Before exploring PSDT for power flow SA, we revisit the conventional PF calculation. Firstly we give the technical roadmaps of both, as shown in Fig. 5.
IiiB1 Classical PF Formulation
Fig. (a)a depicts a conventional PF calculation. Its solution is model and assumptionbased. That is to say, the information of topological parameters, i.e., the admittance , are prerequisite for the calculation, and the input (output) variables need to be preset as one of the following three categories:

and ( and ) for voltage controlled bus, bus;

and ( and ) for load bus, bus;

and ( and ) for reference bus, slack bus.
Conventional PF calculation deals mainly with the calculation of status variables, i.e., voltage magnitude and phase , for each network bus, for a given set of variables such as load demands, i.e., active power and reactive power , under certain assumptions such as in a balanced steadystate system operation [19]. With PF analysis, system operation conditions, e.g., power flows and power losses of each line in the grid network, and reactive power outputs of the generators, can be determined [20].
Consider a grid with buses, among which there are buses, buses, and slack bus . Starting with Eq. (2), PF analysis is formulated as Eq. (3), which solves a set of equations with an equal number of unknowns.
(3) 
where is the assignment symbol in computer science.
Eq. (3) builds a differentiable mapping , which consists of equations, from the state variables, and , to the power injections, and , i.e., .
IiiB2 Jacobian Matrix Estimation
Jacobian matrix is a matrix of all firstorder partial derivatives of a vectorvalued function. For PF analysis, it is a sparse matrix that results from a sensitivity analysis of PF equations. Tying together Eq.(2) and Eq.(3), we define the entries of , i.e., , as the partial derivatives of the outputs, and , with respect to the inputs, and . All in all, consists of four parts as follows:
(4) 
where .
And then
inherently contains the information about the most uptodate network topology, which can be taken care of by the topology processing in the Energy Management System. In practice, however, may not be accurately obtained due to following reasons: 1) topology error has long been cited as a major cause of inaccurate estimation results [21]: network topologies may be outofdata due to erroneous record, delay telemetry, or unexpected operation, especially for a distribution network, and line impedance parameters may be susceptible to climate changing and usage lifetime; 2) uncertainty and individuality of customer units, which are small in size but large in amount [15]; 3) ubiquitous noises, e.g., load/DG fluctuations, and 4) inevitable measurement errors, such as data missing, abnormal, and out of sync.
IiiB3 PF Analysis based on Jacobian Matrix Estimation
Under the modelbased mode, these two problems, i.e., PF calculation and estimation, are closely intertwined—the determination of is an essential part for PF calculation [20]. This phenomena introduces an additional uncertainty for the PF analysis, and also sets a high access threshold for the analysis. Numerical iteration algorithms and sparse factorization techniques, mainly based on NewtonRaphson and fastdecoupled methods, are used to approximate the nonlinear PF equations by linearized [22].
To formulate the linear approximation process that the system operation point shifts from to , the iteration is set as follows:
(5) 
The iteration, given in Eq. (5), depicts how to update the state variables from to . and are known variables which are measurable or calculable. , according to Eq. (3), is our desired on buses and desired on buses^{3}^{3}3For buses, neither nor are fixed; they are state variables that need to be estimated. For buses, is fixed, and needs to be estimated.. Then we focus on for the iteration. Traditionally, is computed via Eq. (4). The modelbased approach, however, is not ideal in practice, since the uptodate network topology and relevant parameters (admittance ), and the operation points are required at to be of high resolution and high precision; these requirements, unfortunately, are often unrealistic as aforementiond.
Iv DT for Realtime Power Flow Analysis
Iva Background of the Cases
Cases are built upon the simulation tool Matpower [23]. Specifying the power injection on each node ( in Eq. (3)), we solve the PF equation (Eq. (2)) with the known network model (given parameters and ), and obtain the voltage magnitude and angle ( in Eq. (3)).
For a standard IEEE 9bus system, Node 1 is the slack bus, Node 2, 3 are the PV buses, Node 5, 7, 9 are the PQ buses with actual load injection, and Node 4, 6, 8 are the PQ buses without load or generator injection, also seemed as tie line.
Suppose the sampling dataset is with 9600 points^{4}^{4}4The sampling rate for PMU, according to IEC61850 standard, can reach up to 4800 Hz [24], and for microPMU (PMU) can reach 120 Hz [25].. Considering the i.i.d. Gaussian power fluctuations according to reference [23], the normalization value of active power is obtained as shown in Fig. 6. The reactive power has similar trend. Note that the raw values of and on Node 5, 7, 9 (with actual load injection) are much larger than that on Node 4, 6, 8 (tie line). Thus, the standardization processes, consisting of an addition of very small artificial noise and the normalization of the matrix, do cause much larger amplitude vibrations on tie line Node 4, 6, 8.
IvB DT for Realtime Power Flow Monitoring
This section explores DT based on Artificial Neural Network (ANN), a typical AI algorithm as stated in Sec.
IID. ANN involves a network of simple processing elements (artificial neurons) which can exhibit complex global behavior, determined by connections between the processing elements and the element parameters. In most cases ANN is an adaptive system that changes its structure based on external or internal information that flows through the network.
Following the roadmaps given in Fig. (b)b, we build a 5 layers ANN to map the nonlinear relationship between the outputs ( in Eq. (3), ) and the inputs ( in Eq. (3), ), i.e., The neural number for each layers is set as
, and tanh is chosen as the activate function.
We use the data during 1:8400 for training (seen as a regression problem, and the label is ), and during 8401:9600 for testing. Taking the active power from the PQ buses with actual load injection (i.e., Node 5, 7 and 9) as the regression targets, Fig. (a)a shows the result—the regression value are very close to the truthvalue .
This PF Monitoring DT is datadriven and realtime. A quite well performance is achieved with a very simple start—only operation data are needed. PSDT is capable of handling the scenarios where system topologies and network parameters are unreliable or even totally unavailable, and thus, the physical models and admittance are no longer required information. Moreover, this PF monitoring DT is independent from Jacobian matrix .
IvC DT for Realtime Jacobian Matrix Estimation
IvC1 Background and Benchmark for Estimation
Under fairly general conditions, Jacobian matrix , according to Eq. (4), keeps nearly constant within a short time, called , due to the stability of the system, or concretely, of variables . The truthvalue of is calculated via Eq. (4) in a modelbased way. During the 9600 observations (Fig. 6), keeps nearly constant at around (Fig. (a)a
), and with a small standard deviation
(Fig. (b)b). is set as the benchmark during the whole period.IvC2 Sa— Estimation with ANN
Naturally, we try to continue our estimation task based on our welltrained ANN during our last realtime PF monitoring task. The nonlinear ANN is modeled as
Following Eq. (6), we obtain the result (Fig. (b)b), and find the task fail. It can deduce that the direct use of ANN may be unsuitable for handling derivative signal analysis, although a quite good result could be obtained for the regression. The derivative signal may have some connection with a residual network, and this topic will be discussed elsewhere.
IvC3 Sa— Estimation with LeastSquare Estimation
estimation is an inverse PF (IPF) problem [26]. During some period, for instance, considering observations at time instants , , operation points are obtained in the form of Eq. (3). Defining and , from Eq. (5) we deduce that . Since is nearly constant, the matrix form is formulated as
(7) 
where (), , and . Thus, we turn the estimation of into a standard regression problem, and the leastsquare estimation (LSE) is the first and most obvious choice as the solution[22]. Rewriting Eq. (7) as
(8) 
where and Since is overdetermined, i.e., , the LSE says that
(9) 
Due to the ubiquitous power fluctuations and measurement errors, the matrix would not be illconditioned under the aforementioned normal operation conditions. This property guarantees the performance of LSE estimation. Moreover, some small artificial noise can be added, according to our previous work [23], to prevent the (normalized) matrix from being illconditioned.
We use the proposed LSE to handle the large dataset and small dataset, respectively, and then compare the results with the benchmark (Fig. (a)a). Fig. 9 gives the estimation bias.
It is observed that LSE has good performances on the estimation task with a large dataset. With a small dataset, the performances become worse. Fig 9 reveals that for the proposed datadriven estimation algorithm, increasing data collection will improve the performance; it is not true, however, for modelbased one. Besides, datadriven estimation no longer needs the admittance . Conversely, the result of estimation inherently contains information about the most uptodate network topologies and corresponding parameters.
IvC4 Virtual Test—Closedloop Feedback on Estimation
Datadriven estimation can be seen as an application of Task 2, virtual test. Take IEEE 118bus system as a background. Running ‘case118.m’ in Matpower with the raw code, we calculate the benchmark (Fig. (a)a). Via a similar process to Sec. IVA, the estimation bias is observed. In Fig. (b)b
, we find some outliers exist such as Point (21, 29). According to Eq. (
3), Point (21, 29) represents , i.e., . With this clue, we check the description file (Fig. (e)e) and find that the description parameters for the branch connecting Node 49 and Node 66 are given twice, which may represents two lines exist in practice. We make correspond modification and a better result is acquired (Fig. (c)c). Step by step, we obtain a much better result () as Fig. (d)d.This example reveals that a DT, with closedloop feedbacks, is able to interact with the real physical system.
IvC5 RMTbased Analytics for Residues
We can conduct big data analytics to our DT. As what DT science paradigm tells in Sec. IID and IIE—Only in highdimensional space do those statistical properties and benefits exist. Factor models, often used for dimension reduction [27], are employed to analysis the mismatches (bias) between the modelbased and the DT results, based on an intuition that the estimation bias cannot always be regarded as pure noise—the bias does contain some statistical information especially when it is caused by poor assumptions or improper simplifications during the estimation.
From the view of factor models, the spectrum of a covariance matrix typically consists of two parts: A few spikes and the bulk. The former represents factors that mainly drive the features and the latter arises from idiosyncratic noise. Motivated by these two parts, we consider a minimum distance between two spectrum densities—one from a covariance structure model and the other from residues.
Regarding empirical data, factor models are formulated as
(10) 
where is empirical data, represents factors, represents factor loadings, is the number of factors, and represents residues.
Eq. (10) provides us a way to decompose the realworld data into systematic information and idiosyncratic noise. Conducting RMT analysis according to [28], we obtain the analytics as shown in Fig. (f)f and (g)g. ‘Estimation Bias 3’, which gets a much better performance than ‘Estimation Bias 1’, has a similar statistical trend curve but much fewer outliers. And from Fig. (e)e, we knows that for ‘Estimation 1’ there exist some duplications in the branch description file. This phenomenon indicates that the estimation result is sensitive to uptodate topology parameters.
V Conclusion
This paper conduct a preliminary exploration on digital twin for power systems (PSDT). More than a simulation tool or cyberphysical system, PSDT is characterized by seamless and active—datadriven, realtime, and closedloop—integration between digital and physical spaces. Around these attributes, we build a framework of PSDT, and then from both science paradigms and engineering practice, outlines the backgrounds, challenges, functions, technologies, and possible directions. Characterized by realtime data/information flow and closedloop feedbacks, the performance of PSDT can be guaranteed, and an example is given to reveals the interaction between the DT and the real physical system via closedloop feedbacks.
DT, mainly driven by data, has some advantages in data utilization. DT enable us to set a quick start only with observed data, which makes DT much more accessible to engineering in practice—we no longer heavily rely on physical models. Moreover, with the employment of RMT and Deep learning, DT is compatible with spatialtemporal data and thus can conduct big data analytics in highdimensional space. It means that PSDT is friendly to uncertainties and bias which are ubiquitous in a modern grid, such as Jacobian matrix estimation bias in our case studies. Besides, DT can be evolved, in an active and selfadaptive way, with data and feedbacks accumulation. Increasing data collection will improve the performance of DT; it is not true, however, for modelbased one.
High dimension is also discussed. Only in highdimensional space do some statistical properties and benefits come out. For instance, we separate Jacobian matrix estimation from PF analysis; these two tasks are closely intertwined under modelbased mode or in lowdimensional space.
DT is emerging and promising enabling technologies for realizing smart grids. In this work we discuss some steadystate applications; the realtime property is much more fit for the quasisteadystate or transientstate analysis. For instance, the NSFC’s funding (20202022) “Research on the Intelligent Fault Diagnosis with High Dimensional Criteria for Distribution Networks Based on the Merger of Random Matrix Theory and Deep Learning”. Beside, along the virtual test task, there are some directions. For instance, operational strategy optimization for all parties in a VPP (virtual power plant) [29], and MAS (multiagent system) [30] can be employed to enhance the interaction of each party. PSDT can also help with the operation, dispatch, management, and electricity market in the context of power systems. In addition, our PSDT is a good reference for other industries.
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