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Preliminary Exploration on Digital Twin for PowerSystems: Challenges, Framework, and Applications

Xing He, Member, IEEE, Qian Ai, Senior Member, IEEE, Robert C. Qiu, Fellow, IEEE, Dongxia Zhang

Abstract—Digital twin (DT) is one of the most promisingenabling technologies for realizing smart grids. Characterizedby seamless and active—data-driven, real-time, and closed-loop—integration between digital and physical spaces, a DT is muchmore than a blueprint, simulation tool, or cyber-physical system(CPS). Numerous state-of-the-art technologies such as internetof things (IoT), 5G, big data, and artificial intelligence (AI)serve as a basis for DT. DT for power systems aims at situationawareness and virtual test to assist the decision-making on powergrid operation and management under normal or urgent condi-tions. This paper, from both science paradigms and engineeringpractice, outlines the backgrounds, challenges, framework, tools,and possible directions of DT as a preliminary exploration. To ourbest knowledge, it is also the first exploration on DT in the contextof power systems. Starting from the fundamental and mostfrequently used power flow (PF) analysis, some typical applicationscenarios are presented. Our work is expected to contribute somenovel discoveries, as well as some high-dimensional analytics, tothe engineering community. Besides, the connection of DT withbig data analytics and AI may has deep impact on data science.

Index Terms—Digital twin, data-driven, real-time, closed-loop,situation awareness, big data analytics, modeling

I. INTRODUCTION

MODERN power grid is one of the most complex en-gineering systems in existence; the North American

power grid is recognized as the supreme engineering achieve-ment in the 20th century [1]. The complexity of grids is everincreasing: 1) the evolution of grid networks, especially theexpansion in size; 2) the penetration of renewable/distributedresources, flexible/controllable electronic components, or evenprosumers with dual load-generator behavior [2]; 3) the revolu-tion of operation mechanisms, e.g., demand-side management;and 4) the mechatronic disciplines (mechanics, electric andelectronics) are realized in a more integrated way, and theirinterfaces will be more intertwined. All these driving forceslead to a non-linear, diversified, hierarchical, and distributedpower grid, which is hard to model and to analyse.

Digital twin (DT) is one of the most promising enablingtechnologies for realizing smart grids, especially for the con-struction of Ubiquitous SG-eIoT (Electric Internet of Thingsproposed by State Grid Corporation of China [3]). A DT refersto the digital representation of a real-world entity or system.These DTs are linked to their real-world counterpart and areused to understand the state of the thing or system, respondto changes, improve operation and add value.

DT is characterized by data-driven mode, real-time inter-action, and closed-loop feedback. Automation and numerousstate-of-the-art technologies such as internet of things (IoT),

5G, drones, robots, edge computing, big data, and artificialintelligence (AI) serve as a basis for DT. Organizations oftenget a quick start of DT, i.e., build simple DTs and put theminto operation at the very beginning, and then evolve themover time in an active and self-adaptive way—evolve DTs withthe accumulation of operation data, practice feedbacks, andsubjective experience. It is quite different from the procedurethat 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 moreaccessible than conventional simulations, e.g., Matpower, toengineering in practice.

It has been almost 16 years since the concept of DT wasinitially proposed in 2003 [4]. To date, many DT applicationshave been successfully implemented in different industries andDT becomes an emerging market. Tao et al. suggested 14potential DT applications, such as product design, assembly,in a workshop [5]. Gartner identified DT as one of the Top 10Strategic Technology Trends of 2018 [6]. The DT market isestimated to grow from USD 3.8 billion in 2019 to USD 35.8billion by 2025, at a CAGR of 37.8%. Major factors surgingthe demand for DT include increasing adoption of emergingtechnologies such as IoT and cloud [7].

This paper, from both science paradigms and engineeringpractice, outlines the backgrounds, challenges, framework,tools, and possible directions of DT for power systems(PSDT). To our best knowledge, it is also the first explorationon DT in the context of power systems. Our preliminaryexploration is expected to contribute some novel discoveries,as well as some high-dimensional analytics, to the engineeringcommunity. Besides, the connection of DT with big dataanalytics and AI may has deep impact on data science.

II. BACKGROUND AND FRAMEWORK OF PSDTA. From Twins to Digital Twins

The concept of “twins” dates back to NASA’s ApolloProgram. In the program, at least two identical space vehicleswere built, allowing engineers to mirror conditions of thespace vehicle during missions (analogous to Task 1, real-timesituation awareness), and the vehicle remaining on earth iscalled the twin. The twin was also used extensively for trainingduring flight preparations (analogous to Task 2, ultra-timevirtual test). During a flight mission it was used to simulatealternatives on Earth-based model, where available flight datawere used to mirror flight conditions as well as possible, andthus assist astronauts in orbit in critical situations.

Another well known example of a “hardward” twin is theIron Bird, a ground-based engineering tool used in aircraft

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industries to incorporate, optimize and validate vital aircraftsystems [8]. Due to the development of simulation technolo-gies, the hardware parts in the Iron Bird are replaced by virtualmodels. This allows system designers to use the concept ofan Iron Bird in earlier development cycles, even when somephysical components are not yet available. Extending this ideafurther along all phases of the life cycle leads to a completedigital model of the physical system, the Digital Twin (DT).

The term DT was brought to the public for the first time inNASA’s integrated technology roadmap [9]: A Digital Twinis an integrated multiphysics, multiscale simulation of a vehicleor system that uses the best available physical models, sensorupdates, fleet history, etc., to mirror the life of its correspondingflying twin. The Digital Twin is ultra-realistic and may considerone or more important and interdependent vehicle systems, includingpropulsion/energy storage, avionics, life support, vehicle structure,thermal management/TPS, etc. Manufacturing anomalies that mayaffect the vehicle may also be explicitly considered. Reference [10]gives a graphical representation of some attributes of a DT asFig. 1, and each of the nine subfigures presents a narrative.

Fig. 1: Graphical Representation of DT Paradigm [10]

Our work explores PSDT. Rather than life cycle phases ormanufacturing anomalies, PSDT is more concerned with twomajor tasks in the field of power systems: 1) real-time situationawareness (SA), and 2) ultra-time virtual test.

B. Conventional SA and its Limitation

SA is defined, according to [11], as the perception ofthe elements in an environment, the comprehension of theirmeaning, 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 thelargest blackout in history—the 14 August 2003 Blackout inthe United States and Canada [12]. For a modern grid as statedin Sec. I, SA is in urgent need of a new prominence.

Conventional model-based SA for power systems needs tobe revisited. We would like to refer to Fig. 2 in book [13] as anillustration. Under the 2nd and 3rd paradigms (from last fewhundred years to last few decades), our insight into the worldis mainly based on physical models. For a power grid, we useequations, formulas, or simulations to describe the operation

regulations and interaction mechanisms of each units. Thismodel-based mode cannot make full use of massive data due toits own limitations—it aims at a deterministic solution whichis always in low-dimensional space (low dimension is notwell compatible with high-dimensional data). Moreover, theassumptions and simplifications of system units (often small insize but large in number [14]), and the increasing penetrationof distributed energy resources (often susceptible to climatechanging and usage lifetime [15]), will inevitably cause error;the error accumulations can hardly be addressed (described oranalysed) with physical models or in low-dimensional space.

xviii

escience: WhaT is iT?

eScience is where “IT meets scientists.” Researchers are using many different meth-ods to collect or generate data—from sensors and CCDs to supercomputers and particle colliders. When the data finally shows up in your computer, what do you do with all this information that is now in your digital shoebox? People are continually seeking me out and saying, “Help! I’ve got all this data. What am I supposed to do with it? My Excel spreadsheets are getting out of hand!” So what comes next? What happens when you have 10,000 Excel spreadsheets, each with 50 workbooks in them? Okay, so I have been systematically naming them, but now what do I do?

science Paradigms

I show this slide [Figure 1] every time I talk. I think it is fair to say that this insight dawned on me in a CSTB study of computing futures. We said, “Look, computa-tional science is a third leg.” Originally, there was just experimental science, and then there was theoretical science, with Kepler’s Laws, Newton’s Laws of Motion, Maxwell’s equations, and so on. Then, for many problems, the theoretical mod-els grew too complicated to solve analytically, and people had to start simulating. These simulations have carried us through much of the last half of the last millen-nium. At this point, these simulations are generating a whole lot of data, along with

FIGURE 1

4πGp3 K c2

a2=aa

2.

• Thousand years ago:science was empirical

describing natural phenomena

• Last few hundred years:theoretical branch

using models, generalizations

• Last few decades:a computational branch

simulating complex phenomena

• Today: data exploration (eScience) unify theory, experiment, and simulation

– Data captured by instrumentsor generated by simulator

– Processed by software– Information/knowledge stored in computer– Scientist analyzes database / files

using data management and statistics

Science Paradigms

Jim graY oN eSCiENCE

Fig. 2: Science Paradigms [13]

C. Science Paradigms and Data-driven SA in High Dimension

As shown in Fig. 2, we are now entering the age of the4th-paradigm—data-intensive scientific discovery. In this age,data-driven is an alternative paradigm, and even becomes atrend as data become more and more accessible. For smartgrids, data-driven approaches become natural and stressingtopics [16]. Data-driven approaches are also characterized bymodel-free—we no longer heavily rely on physical models,and can handle the scenarios where the system topologies andnetwork parameters are unreliable or even totally unavailable.Moreover, comparing data-driven results to model-based ones,we can obtain some insights for further analysis. This phe-nomenon has some connection with Task 2, virtual test, andwill be demonstrated by the case studies in Sec. IV-C4.

High in dimensionality, rather than large in number, is thekeystone and difficulty of data-driven SA designing. Highdimension means that the datasets are represented in terms oflarge random matrices. These data matrices can be viewed asdata points in high-dimensional vector space of mathematics—each vector is very long. Traditional data transformations,however, are often in the form of low dimension, such asone-dimensional Fourier Transformation (time domain to fre-quency domain), and three-dimensional Park Transformation(ABC to dq0). In low-dimensional space, only two typicaldata matrices in the form of X∈RN×T are at our disposal: 1)N,T are small, and 2) N is small, T is very large (comparedwith N ). Low dimension is not well compatible with high-dimensional data—low-dimensional tools are inadequate to a

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complex problem such as optimized operation or integratedplanning for a modern grid, in which the behaviors anddiscipline of system units are closely intertwined.

D. Big Data Analytics and AI for eScience Data

The limitation of model-based and low-dimensional sta-tistical algorithms to eScience data has greatly spurred thedevelopment of big data analytics and artificial intelligence(AI). Modern grid operation is always accompanied with atemporal-spatial dataset—the dataset is in high-dimensionalspace, and in the form of time series. Temporal variations (Tsampling instants) are simultaneously observed together withspatial variations (N signals). Only in high-dimensional spacedo those statistical properties and benefits come out, and itwill be revealed by a virtual test in our case studies.

The extraction of statistical information from this high-dimensional space is a challenge that does not meet theprerequisites of most mathematical tools. Big data analyticsand AI are employed to handle this challenge, and accordinglywe select two tools: 1) random matrix theory (RMT), whichis good at big data analytics, and 2) deep learning, which doeswell in massive data modeling. Both of the tools have alreadymade huge impacts on many engineering fields. Compared toits model-based counterpart, DT is more compatible with oreven more naturally connected to these tools. PSDT realizesSA mainly by means of mining information from temporal-spatial dataset. This attribute of DT is marked as data-driven.

E. Data-driven Tools: RMT and Deep Learning

RMT has an advantage of transparency—unifying time andspace through their ratio c=T/N , RMT deals with temporal-spatial data mathematically rigorously. The goal of RMT isto understand the joint eigenvalue distribution as the statisticanalytics from big data in the asymptotic regime. In particular,high-dimensional analysis and visualization are treated as thefunctionals of the eigenvalue distributions. For instance, Lineareigenvalue statistics (LESs) [17], built from data matrices,follow Gaussian distributions for very general conditions,and other statistical variables are studied due to the latestbreakthroughs in probability on the central limit theorems ofthose LESs [18]. The statistical properties of these variablesare mostly derivable and provable. Moreover, RMT performswell with moderate-size data.

Deep learning is the state-of-the-art algorithm in data sci-ence. Deep learning does learn some non-handcrafted features,so called deep features, from the massive labeled datasetwithout much prior knowledge, so that it can be generalizedto different cases without making significant modifications.Moreover, the performance of the deep network model onfitting task and generalization task could be quantitative eval-uated with test set, so as to ensure desired effects.

F. Virtual Test and Framework of DT

Besides data-driven SA, DT also allows for simulations ofnew ideas that can be tested virtually to determine environmen-tal impact before implementation in the real world. Software

for automation can also be tested in advance using the virtualrepresentation of the real system (i.e. ultra-time virtual test).

Then we build the framework of PSDT, as shown in Fig. 3.The relationship among the PSDT, the physical grids, and theoperators are similar to the relationship, in the aforementionedApollo Program, among the vehicle remaining on earth, thespace vehicle during missions, and the astronauts in orbit.PSDT helps the operators to understand the physical grid andto make a reliable decision in critical situations.

Physical Object

Digital Twin

SensorsActuators

①Data-driven Tools③Empirical

Tools

Models, Formulas, Simulations, … ClassicalStatistics

SubjectiveExperience

Monitoring, Dispatching

Management, Control

Decision Makers

Big Data, AI, 5G, Cloud, IoT, …

I. RMT II. Deep Learning

Commu-nication

Reliable Decision

Virtual Data/Commands

Field Data

Human-machine

Interfaces

Task 2) Virtual Test

Task 1) SA

②Model-based Tools

Matpower, PSCAD, PSSE, …

a) Data-driven

b) Real-time

c) Closed-loop

Major Tasks

Basic Tools

Main Properties

Decisions

Iteration in

Virtual Space

Seamless & Active Integration

Fig. 3: Framework for DT of Power Systems

The arrow lines represent the data/information flow, whichconnects the physical object and its DT. Our two majortasks are deployed along these arrow lines: DT uses samplingspatial-temporal data from communication for SA, and usesvirtual data/commands from decision makers for virtual test.These lines also form a closed loop, which means feed-backs are available for the tasks. Characterized by real-timedata/information flow and closed-loop feedbacks, the effec-tiveness of our tasks can be guaranteed. The involvement ofhuman-machine interfaces provides an access to some artificialinputs, e.g., hyper parameters setting for a virtual test; it makesDT more flexible and intelligent in the virtual space.

III. POWER FLOW OPERATION FORMULATION ANDCONVENTIONAL ANALYSIS

A. Background of Power Grid Operation

Power flow (PF) analysis is a fundamental and most fre-quently used tool for many tasks in a power system, such asfault diagnosis, state estimation, N−1 security assessment, andoptimal power dispatch. For each node i in a grid network,choosing the reference direction as shown in Fig 4, Kirchhoff’scurrent law and ohm’s law say that:

Ii =

n∑j=1j 6=i

Ij =∑j 6=i

Yij ·(Uj − Ui

). (1)

where Yij=Gij+j ·Bij is the admittance in Cartesian form1,and Ui=

∣∣∣Ui∣∣∣∠θi=Vi∠θi=Viejθi and Ii =∣∣∣Ii∣∣∣∠φi are node

voltage and node current, respectively, in polar form2.

1G is the conductance, B is the susceptance, and j is the imaginary unit.2Vi∠θi = Viejθi = Vi(cos θi + j · sin θi).

4

GiI

iI

iU

1U

2UjU

NU

1iI

2iI

* * * *

j

j i ij

i ji i j i ij

i j ij i i

j

j

i

ij

U U Y

U I U U U Y

V V G B

I

V

*

*

G

ji i i i i

i ji i

j i

S U I P Q

U I I

NiI

Fig. 4: Schematic Diagram for Grid Network Operation

And thus, for each node in a power grid, Node i for instance,taking account of the node-to-ground admittance yi, its activepower P and reactive power Q are expressed as:

Pi=Vi∑k 6=i

Vk (Gikcosθik+Biksinθik)−Vi2∑k 6=i

Gik−Vi2gi

Qi=Vi∑k 6=i

Vk (Giksinθik−Bikcosθik)+Vi2∑k 6=i

Bik+Vi2bi

(2)Abstractly, a physical power system obeying Eq. (2) can be

viewed as an analog engine—it takes bus voltage magnitudeV and bus voltage phaser θ as inputs, conductance G andsusceptance B as given parameters, and “computes” activepower injection P and reactive power injection Q as outputs.

B. Model-based Power Flow Analysis

Before exploring PSDT for power flow SA, we revisit theconventional PF calculation. Firstly we give the technicalroadmaps of both, as shown in Fig. 5.

Power Flow AnalysisGrid Topologies Admittance Matrices Y

Preset Physical Nodes as 3 Types

(n): PV (m), PQ (l), Vθ (1)

Matpower

Give an initial point P(t0)

(m+2l) Functions

Newton-Raphson

Iteration

m+2l Status Variables

θPV (m), θPQ (l), VPQ (l)

Derived Parameters

e.g., Line Losses, Total Cost

(a) Conventional Model-based PF Analysis

1 2( ) Ω ΩfΩ1 Ω2

(b) Data-driven PF Analysis

Fig. 5: Model-based PF and Data-driven PF

1) Classical PF Formulation:Fig. 5a depicts a conventional PF calculation. Its solution is

model- and assumption-based. That is to say, the informationof topological parameters, i.e., the admittance Y , are prereq-uisite for the calculation, and the input (output) variables needto be preset as one of the following three categories:• P and V (Q and θ) for voltage controlled bus, PV bus;• P and Q (V and θ) for load bus, PQ bus;• V and θ (P and Q) for reference bus, slack bus.

Conventional PF calculation deals mainly with the calcula-tion of status variables, i.e., voltage magnitude V and phase θ,for each network bus, for a given set of variables such as loaddemands, i.e., active power P and reactive power Q, undercertain assumptions such as in a balanced steady-state systemoperation [19]. With PF analysis, system operation conditions,e.g., power flows and power losses of each line in the gridnetwork, and reactive power outputs of the generators, can bedetermined [20].

Consider a grid with n buses, among which there are m PVbuses, l PQ buses, and 1 slack bus (n=m+l+1). Starting withEq. (2), PF analysis is formulated as Eq. (3), which solves a setof equations with an equal number (p=m+2l) of unknowns.

y :=

P1

...Pn−1

Qm+1

...Qn−1

=f

θ1

...θn−1

Vm+1

...Vn−1

=:f (x) J=

∂y1∂x1

· · · ∂y1∂xp

.... . .

...∂yp∂x1

· · · ∂yp∂xp

(3)

where := is the assignment symbol in computer science.Eq. (3) builds a differentiable mapping f , which consists of

p equations, from the state variables, θ and V , to the powerinjections, P and Q, i.e., f :x∈Rp→y∈Rp.

2) Jacobian Matrix Estimation:Jacobian matrix J is a matrix of all first-order partial

derivatives of a vector-valued function. For PF analysis, itis a sparse matrix that results from a sensitivity analysis ofPF equations. Tying together Eq.(2) and Eq.(3), we define theentries of J, i.e., [J ]ij , as the partial derivatives of the outputs,P and Q, with respect to the inputs, V and θ. All in all, Jconsists of four parts H,N,K,L as follows:

Hij= ViVj (Gij sin θij−Bij cos θij)−δij ·Qi+δij ·V 2i bi

Nij= ViVj (Gij cos θij+Bij sin θij)+δij ·Pi−δij ·V 2i gi

Kij= −ViVj (Gij cos θij+Bij sin θij)+δij ·Pi+δij ·V 2i gi

Lij= ViVj (Gij sin θij−Bij cos θij)+δij ·Qi+δij ·V 2i bi

(4)

where Hij= ∂Pi

∂θj, Nij= ∂Pi

∂VjVj ,Kij= ∂Qi

∂θj, Lij= ∂Qi

∂VjVj .

And then

J =

[[H]n−1,n−1 [N]n−1,l

[K]l,n−1 [L]l,l

]J inherently contains the information about the most up-

to-date network topology, which can be taken care of by thetopology processing in the Energy Management System. Inpractice, however, J may not be accurately obtained due tofollowing reasons: 1) topology error has long been cited asa major cause of inaccurate estimation results [21]: networktopologies may be out-of-data due to erroneous record, delaytelemetry, or unexpected operation, especially for a distribu-tion network, and line impedance parameters may be suscep-tible to climate changing and usage lifetime; 2) uncertaintyand individuality of customer units, which are small in sizebut large in amount [15]; 3) ubiquitous noises, e.g., load/DGfluctuations, and 4) inevitable measurement errors, such as datamissing, abnormal, and out of sync.

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3) PF Analysis based on Jacobian Matrix Estimation:Under the model-based mode, these two problems, i.e.,

PF calculation and J estimation, are closely intertwined—the determination of J is an essential part for PF calculation[20]. This phenomena introduces an additional uncertaintyfor the PF analysis, and also sets a high access thresholdfor the analysis. Numerical iteration algorithms and sparsefactorization techniques, mainly based on Newton-Raphsonand fast-decoupled methods, are used to approximate thenonlinear PF equations by linearized J [22].

To formulate the linear approximation process that the sys-tem operation point shifts from (x(k),y(k)) to (x(k+1),y(k+1)),the iteration is set as follows:

x(k+1) := x(k) + J−1(x(k)

)(y(k+1) − y(k)

)(5)

The iteration, given in Eq. (5), depicts how to updatethe state variables from x(k) to x(k+1). y(k) and x(k) areknown variables which are measurable or calculable. y(k+1),according to Eq. (3), is our desired P,Q on PQ buses anddesired P on PV buses3. Then we focus on J for the iteration.Traditionally, J is computed via Eq. (4). The model-basedapproach, however, is not ideal in practice, since the up-to-date network topology and relevant parameters (admittanceY ), and the operation points (x(k),y(k)) are required at tobe of high resolution and high precision; these requirements,unfortunately, are often unrealistic as aforementiond.

IV. DT FOR REAL-TIME POWER FLOW ANALYSIS

A. Background of the Cases

Cases are built upon the simulation tool Matpower [23].Specifying the power injection on each node (y in Eq. (3)),we solve the PF equation (Eq. (2)) with the known networkmodel (given parameters G and B), and obtain the voltagemagnitude and angle (x in Eq. (3)).

For a standard IEEE 9-bus system, Node 1 is the slack bus,Node 2, 3 are the PV buses, Node 5, 7, 9 are the PQ buseswith actual load injection, and Node 4, 6, 8 are the PQ buseswithout load or generator injection, also seemed as tie line.

Suppose the sampling dataset is with 9600 points4. Con-sidering the i.i.d. Gaussian power fluctuations according toreference [23], the normalization value of active power P isobtained as shown in Fig. 6. The reactive power Q has similartrend. Note that the raw values of P and Q on Node 5, 7,9 (with actual load injection) are much larger than that onNode 4, 6, 8 (tie line). Thus, the standardization processes,consisting of an addition of very small artificial noise and thenormalization of the matrix, do cause much larger amplitudevibrations on tie line Node 4, 6, 8.

B. DT for Real-time Power Flow Monitoring

This section explores DT based on Artificial Neural Net-work (ANN), a typical AI algorithm as stated in Sec. II-D.

3For PQ buses, neither V nor θ are fixed; they are state variables that needto be estimated. For PV buses, V is fixed, and θ needs to be estimated.

4The sampling rate for PMU, according to IEC-61850 standard, can reachup to 4800 Hz [24], and for micro-PMU (µPMU) can reach 120 Hz [25].

0 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 24:00-1

0

1

-1

0

1

P5

P9

P7

P4, P6, P8 P2, P3

Fig. 6: Power Consumption for IEEE 9-Bus System

ANN involves a network of simple processing elements (ar-tificial neurons) which can exhibit complex global behavior,determined by connections between the processing elementsand the element parameters. In most cases ANN is an adaptivesystem that changes its structure based on external or internalinformation that flows through the network.

Following the roadmaps given in Fig. 5b, we build a 5 layersANN to map the non-linear relationship f between the outputs(y in Eq. (3), P,Q) and the inputs (x in Eq. (3), θ, V ), i.e.,P = f (V,θ) . The neural number for each layers is set as[14, 50, 50, 50, 14], and tanh is chosen as the activate function.

We use the data during 1:8400 for training (seen as aregression problem, and the label is y), and during 8401:9600for testing. Taking the active power P from the PQ buseswith actual load injection (i.e., Node 5, 7 and 9) as theregression targets, Fig. 7a shows the result—the regressionvalue P ∗5 , P

∗7 , P

∗9 are very close to the truth-value P5, P7, P9.

8200 8400 8600 8800 9000 9200 9400 9600

0.4

0.5

0.6

0.7

0.8

0.9

P5

P5*

P9

P9*

P7*P7

(a) Prediction of P on load nodes

JErr

16 0.25 46 4.1 6.5 2.1 1.2 0.23

.046 17 7.7 1.3 4 2.2 0.62 .023

.01 11 .017 1.8 3.1 .073 0.1

.024 22 23 2.2 1.1 1.7 0.39

1.6 0.18 14 .049 1.7

.019 17 0.57 4.3 13 .075 1.8

.075 0.58 33 25 100 31 20 12

0.18 .034 15 2.6 14 1.4 0.88 12

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

810

20

30

40

50

60

70

80

90

100

(b) J estimation via Eq. (6)

Fig. 7: Power Prediction and J Estimation Using ANN

This PF Monitoring DT is data-driven and real-time. Aquite well performance is achieved with a very simple start—only operation data are needed. PSDT is capable of handlingthe scenarios where system topologies and network parametersare unreliable or even totally unavailable, and thus, thephysical models and admittance Y are no longer requiredinformation. Moreover, this PF monitoring DT is independentfrom Jacobian matrix J.

C. DT for Real-time Jacobian Matrix Estimation

1) Background and Benchmark for J Estimation:Under fairly general conditions, Jacobian matrix J, accord-

ing to Eq. (4), keeps nearly constant within a short time,

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called ∆t, due to the stability of the system, or concretely,of variables V, θ, Y . The truth-value of J is calculated viaEq. (4) in a model-based way. During the 9600 observations(Fig. 6), J keeps nearly constant at around JMean (Fig. 8a),and with a small standard deviation JStd (Fig. 8b). JMean isset as the benchmark during the whole period.

JMean

-16 16 -1.6

-17 17 -.84

-39 10 11 -3.3 2.2 1.7

11 -16 5.5 1.7 -2.8 2

17 5.8 -33 9.9 0.6 -2.5 1

9.9 -24 14 1.3 -2 2.3

16 14 -36 6.1 0.97 -2.8 0.25

11 5.7 -17 1 2.1 -1.9

3.3 -2.2 -1.7 -39 10 12

-1.6 3.6 -2 11 -16 5.4

-.85 -.6 2.5 -1 5.8 -33 9.9

-1.3 3.6 -2.3 9.8 -23 14

-1.6 -.97 2.8 -.25 14 -36 6.1

-1 -2.1 3.1 11 5.6 -17

2 4 6 8 10 12 14

2

4

6

8

10

12

14-35

-30

-25

-20

-15

-10

-5

0

5

10

15P2P3

P9Q4

Q9

V4 V9θ2 θ3 θ9

(a) Truth-value of J0

JStd

.035 .035

.032 .032

0.19 .068 .09 0.15 0.17

0.12 0.17 .046 0.13 0.17 .025

.032 .05 0.12 .042 .02 .021

.044 0.11 .065 .018 .048 .018

.035 .063 0.15 .057 .022 .02

0.13 .054 0.18 0.16 .024 0.2

.016 0.16 0.18 .097 .011

0.13 0.1 .028 .094 0.16 .036

.02 .023 .012 .061 .018

.018 .034 .017 .027 .07 .037

.024 .012 .02 .029 .076 .013

0.15 .027 0.13 0.1 .043 0.2

2 4 6 8 10 12 14

2

4

6

8

10

12

140

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

(b) Standard deviation of J0

Fig. 8: Basic Statistical Information of J from 9600 Samplings

2) SA—J Estimation with ANN:Naturally, we try to continue our J estimation task based on

our well-trained ANN during our last real-time PF monitoringtask. The non-linear ANN is modeled as

y=f (x),fL(WL· · ·f2

(W2f1 (W1x+b1)+b2

)· · ·+bL

)With repeatedly use of the Chain Rule, J is solved as

J =∂y

∂xT =∂a(L)

∂xT ==∂f (L)

(z(L)

)∂z(L)T

∂(WL−1aL−1 + bL−1

)∂xT

=diag(fL

′

z=zL

)WL−1 ∂aL−1

∂xT ,ΓLWL−1 ∂aL−1

∂xT

= ΓLWL−1ΓL−1WL−2 · · ·Γ2W1

(6)where Γl=diag

(f l

′

z=zl

), l=2, · · · , L

Following Eq. (6), we obtain the result (Fig. 7b), and findthe task fail. It can deduce that the direct use of ANN may beunsuitable for handling derivative signal analysis, although aquite good result could be obtained for the regression. Thederivative signal may have some connection with a residualnetwork, and this topic will be discussed elsewhere.

3) SA—J Estimation with Least-Square Estimation:J estimation is an inverse PF (IPF) problem [26]. During

some period, t∆ for instance, considering T observations attime instants ti, (i = 1, 2, · · · , T, tT − t1 = t∆), operationpoints (x(i),y(i)) are obtained in the form of Eq. (3). Defining∆x(k),x(k+1)−x(k) and ∆y(k),y(k+1)−y(k), from Eq. (5)we deduce that ∆y(k)≈J(k)∆x(k). Since J is nearly constant,the matrix form is formulated as

B≈JA (7)

where J∈RN×N (N=p=m+2l), B=[∆y(1), · · · ,∆y(T )

]∈

RN×T , and A =[∆x(1), · · · ,∆x(T )

]∈RN×T . Thus, we turn

the estimation of J into a standard regression problem, andthe least-square estimation (LSE) is the first and most obviouschoice as the solution[22]. Rewriting Eq. (7) as

Θ ≈ ΛJT, (8)

JErr

:= LSE; nSample

:= 5000

.013 .018 .1 .02 .036 .038 .038 0.16 .06 .032 .052 .067 .07

.029 .035 .051 .017 .06 .033 .011 .013 .02 .028

.013 .012

.01 .013 .025 .017 .032 .013

.012 .012 .019 .011

.041 .015 0.16 .04 .029 .026 .087 .058 0.2 .064 .075 .084 0.11 .089

.033 0.2 .011 .091 .058 .075 0.34 .076 .043 .081 0.11 0.14

.015 .026 .03 .014 .017 .078 .022 .021 .028

.012 .011 .02 .018 .014

.013 .014 .016

.01 .011 .011 .019 .012 .011

.035 0.11 .02 .026 .017 .076 .045 0.17 .041 .039 .051 .08 .075

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0.05

0.1

0.15

0.2

0.25

0.3

(a) Bias using 4800 samplings

JErr

:= LSE; nSample

:= 0240

.013

0.41 0.49 0.48 0.34 0.58 0.15 0.39 0.86

0.28 .031 .028 .025 .079 0.1 .034 0.35

0.57 .039 0.76 0.15 0.86 0.39 0.76 0.37

0.22 0.4 0.19 0.21 0.4 0.3 0.32 1.1

.065 .029 .026 .031 .021 .06 .095

0.2 .027 0.45 .029 0.67 0.59 0.64 0.84

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

(b) Bias using 240 samplings

Fig. 9: Estimation Bias of J with Large and Small Dataset

where Θ := BT ∈ RT×N and Λ := AT ∈ RT×N . Since Λ isover-determined, i.e., T >N , the LSE says that

JT =(ΛTΛ

)−1ΛTΘ =

(AAT)−1

ABT (9)

Due to the ubiquitous power fluctuations and measurementerrors, the matrix Λ would not be ill-conditioned under theaforementioned normal operation conditions. This propertyguarantees the performance of LSE estimation. Moreover,some small artificial noise can be added, according to ourprevious work [23], to prevent the (normalized) matrix Λ frombeing ill-conditioned.

We use the proposed LSE to handle the large dataset andsmall dataset, respectively, and then compare the results withthe benchmark (Fig. 8a). Fig. 9 gives the estimation bias.

It is observed that LSE has good performances on the Jestimation task with a large dataset. With a small dataset,the performances become worse. Fig 9 reveals that for theproposed data-driven J estimation algorithm, increasing datacollection will improve the performance; it is not true, how-ever, for model-based one. Besides, data-driven J estimationno longer needs the admittance Y. Conversely, the result ofJ estimation inherently contains information about the mostup-to-date network topologies and corresponding parameters.

4) Virtual Test—Closed-loop Feedback on J Estimation:Data-driven J estimation can be seen as an application of

Task 2, virtual test. Take IEEE 118-bus system as a back-ground. Running ‘case118.m’ in Matpower with the raw code,we calculate the benchmark (Fig. 10a). Via a similar processto Sec. IV-A, the estimation bias is observed. In Fig. 10b, wefind some outliers exist such as Point (21, 29). According toEq. (3), Point (21, 29) represents ∂y29/∂x21, i.e., ∂p66/∂θ49.With this clue, we check the description file (Fig. 10e) andfind that the description parameters for the branch connectingNode 49 and Node 66 are given twice, which may representstwo lines exist in practice. We make correspond modificationand a better result is acquired (Fig. 10c). Step by step, weobtain a much better result (10−4) as Fig. 10d.

This example reveals that a DT, with closed-loop feedbacks,is able to interact with the real physical system.

5) RMT-based Analytics for Residues:We can conduct big data analytics to our DT. As what DT

science paradigm tells in Sec. II-D and II-E—Only in high-dimensional space do those statistical properties and benefits

7

Model-Based: J0

-28 8.520 2.76.6-133 12213 273.4-62 1745 4.79.7-92 3834 20 -3.46.8 1-34 34 -1.7-122 145.527451311 6.64.21.77.1143.72.8 1.4-5622 3.54.520 6.9 1.11.47.1 2.1-3618 18 5.22218-51 3.5 7.5 1.8-26 2.44.7 19 5.1-47286.1 1328-40 12 -1.25.6-3911 10 12 2.1 2.6-408.6 5.6 25 1.88.6118.8-46 4.3 7.5 14 3 3.23.5-14433 965.4 271.432-122 90 20-414.8 4.81418 2.455.14.7-121.3 6.5 2-19 6.87.34.9 1.72.51.71.5-681.6 5.4 4.91519126.6 1.34.66.13.81.71.5-11112883.8 7.1 1.512-715448854-1661.6 8.48.4 3.43.244.21.8-486.1 6.325 1.91.66.3-14425 7137 1325-554.5 178.2 3.62.1-13329 10 3363 -1.3 1.155.74.729-49 10 1.62.4 -486.6 266.2 6.324.7 -9.7 5 1.4-21 21 4.16.5-27 21 7-215.7 15 4.86-1116.1 4.57510 1.1223.86.3-80 4.9 1429 5.5119.5 2.5 1.91.7-415.5 6.2127.49.1 1.35.51.92.4-4.7 4.75.7-273.34.6 14 1.62.9-131011-186.84.37.1-483.3 115.8 17 2.71.33.34.8 -17123.212-71184.7 165.57.64.1 5.11.21.918-345.85.55.34.55.6-33235.323-614.7 1612 4.24.24.6-9.1 4.5 1.65-411213 11 4.312-1213-134.3 -34 30 9.2-255 255 228.514 -23 -6.9205.5 -348.4 -9.52.51221838 8.6-20013 1.3-363.34527 -72 -173533 -68 -5.31346 13-8412 4.7-253.43.5 12-15 4-4.4134.5 -17 -5.111 -165 -3.91.52018 5.6 30 5.1-100 26 1.1-23 2.37.5 -1810 -3.82.610-199 2-4.22.59.2-155.7 1.6-3.21.819127.9 5.9-45 -1110 -209.3 -4.62.225 9.2-35 2.6-1120 11 26 -7619 -2.4-4.46.9 -136.2 -3.81.990 -11118 -244.5965.2 6.219-166268.4 1.83.9-372.81.89.7 1826-54 2.2-2.8-2.514 8-22 3.2-6.7186.5 -24 -75.4 -93.6 -2.11.13.7-1410 -3.32.96.64.6 10-21 2.2-5.77.515 -26 -7.4518 -23 -7.712 -186.5 -6.426.2 -321412 -114.8414-195.2 5.3-6.21.27.1 5.2-12 1.5-2.78.6 6.3-15 2.6-5.38.4 11-20 4.4-7.26.57017 -94 -1825 -7449 -4.25.83733 49-122 2.6-7.28.39.9 -18 -3.763 255 -39951 -314.7265.121 -52 -126.2214.4 -56 18 -17 5.274 -11139 -319.114 40-54 8.5-1129 51-80 4-4.410 -5124 17 -168.2 5.25.9 24-365.8 6.9-123.212 6.1-18 2.6-8.67.34.7 -12 -2.59.313 -22 -3.910 -2312 -6.93.35.516 12-642110 4.2-206.13.120-3716 6.6-115.15.4 17 1016-6011 5.23.45.2-162.211 11-22 2-49.3 5.5 -15 -2.97.7 -168.3 -3.32.217 8.4-25 1.5-5.53.8164.7 -24 -6.212 -4029 -151111 29-40 11-1514 -9984 -231912 84-96 19-226.5 -6.5 -1.315 17 -33 5.8 -9.9-2.5-4.6 7.1 -23-5.8-1.7 10-2.5 -358.4-25-3-3.4 -1.336-3.2 8.9-20014-10-6.7 17 -731.6-7.1 5.5 -65-4.6-13 -4.726-3.3 13-8412-1.1 -3.95 12-16-4-1.3 5.3 -17-2.8 4.3-1.4 -165.1-5.1-3.6-1.6 -9 -1.123 -2.3 5.2-111 25-1.5 4-2.5 -1911-1.94.3-2.4 11-209.3-1.53.3-1.8 9.6-155.7-5.2-3.9-1.3 11 6.1-45-2.6 4.8-2.2 -209.6-8.3 -2.511 9.6-36-2.4 -3.3 2.3 4.4 26 -7519-2.2 4.1-1.9 -136.3-20 24-4.4 -11119-25-1.2 -1.7-3.837-2.8-1.8 6.419-166268.7-3.1 -2.22.82.5 1826-54-3.8 -3.27 8.1-23-5.6-1.9 7.5 -25-1.3 2.4-1.1 -9.13.73.6-2.9 3.8-1410-2.3-2.2 -2.26.7 10-21-1.9-4.7 8.2 -26-1.4-6.8 8.3 -23-4.8 6.7-1.9 -186.8-2.9 11-4.6-3.9 -331512-5.16.3-1.2 14-205.5-1.6 -1.43 5.4-13-2.8 -2.65.4 6.3-15-2.9 -4.37.1 12-21-15-3.8 20 -941.6 4.2-5.8 -7549-4.8 -2.67.2 50-122-1.5-2.8 4.3 -18-5.8 -19 31-4.8 -39950-6.6-1.1-3.9 12 -52-1.8-5.9-1.6 18 -5 -58 19-23 32-9.1 -11139-3.8 -8.512 39-53-44.5 51-78-3.5 17-8 -5.2 -5124 17-2.2 -6.812-3.1 25-375.9-6.2 -2.58.7 6.2-18-2.3 2.9 -12-1.2-3.6 4.9 -23-3.7 7-3.3 -2312-2.2-4.7 -4.120-5.9-3.1 12-652110-6.512-5.1 21-3716-1.2 -5.1-3.3-5.117-2.2 17 1017-6011-2.4 -24.4 11-21-2.4 -1.2 3.6 -14-1.5 3.7-2.2 -168.4-4.1 -1.55.6 8.5-25-1.6-4.1-1.1 6.8 -25-4.6 15-10 -4230-3.8 -1114 30-41-3.3 22-19 -10088-2.8 -1922 88-100-1.7 1.7 -6.5-4.5 -5.6 10 18 -3420 40 60 80 100 120 140 160 180

20

40

60

80

100

120

140

160

180

-300

-200

-100

0

100

200

(a) Truth-value of J0

JErr

20 40 60 80 100 120 140 160 180

20

40

60

80

100

120

140

160

180 0

2

4

6

8

10

12

14

16

[X,Y]: [21 29]Index: 5.745[R,G,B]: [0 0.9373 1]

↑

(b) Bias of Estimation 1

JErr

20 40 60 80 100 120 140 160 180

20

40

60

80

100

120

140

160

180 0

2

4

6

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10

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16

(c) Bias of Estimation 2

JErr

20 40 60 80 100 120 140 160 180

20

40

60

80

100

120

140

160

180 0

1

2

3

4

5

10-4

(d) Bias of Estimation 3

(e) Description File for Branch

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 Probability Density Function for Bias 1 M-P LawP = 0P = 1P = 2P = 3P = 4P = 5P = 6

Outliers

(f) Factor Model for Bias 10 1 2 3 4 5 6 7 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Probability Density Function for Bias 3

M-P LawP = 0P = 1

(g) Factor Model for Bias 3

Fig. 10: Estimation Result for IEEE 118-bus System

exist. Factor models, often used for dimension reduction [27],are employed to analysis the mismatches (bias) between themodel-based J0 and the DT results, based on an intuition thatthe estimation bias cannot always be regarded as pure noise—the bias does contain some statistical information especiallywhen it is caused by poor assumptions or improper simplifi-cations during the estimation.

From the view of factor models, the spectrum of a co-variance matrix typically consists of two parts: A few spikesand the bulk. The former represents factors that mainly drivethe features and the latter arises from idiosyncratic noise.Motivated by these two parts, we consider a minimum dis-tance between two spectrum densities—one from a covariancestructure model and the other from residues.

Regarding empirical data, factor models are formulated as

X = L(p)F(p) + R. (10)

where X ∈ RN×T is empirical data, F ∈ Rp×T representsfactors, L ∈ RN×p represents factor loadings, p is the number

of factors, and R ∈ RN×T represents residues.Eq. (10) provides us a way to decompose the real-world data

into systematic information and idiosyncratic noise. Conduct-ing RMT analysis according to [28], we obtain the analyticsas shown in Fig. 10f and 10g. ‘Estimation Bias 3’, whichgets a much better performance than ‘Estimation Bias 1’,has a similar statistical trend curve but much fewer outliers.And from Fig. 10e, we knows that for ‘Estimation 1’ thereexist some duplications in the branch description file. Thisphenomenon indicates that the estimation result is sensitive toup-to-date topology parameters.

V. CONCLUSION

This paper conduct a preliminary exploration on digital twinfor power systems (PSDT). More than a simulation tool orcyber-physical system, PSDT is characterized by seamless andactive—data-driven, real-time, and closed-loop—integrationbetween digital and physical spaces. Around these attributes,we build a framework of PSDT, and then from both scienceparadigms and engineering practice, outlines the backgrounds,challenges, functions, technologies, and possible directions.Characterized by real-time data/information flow and closed-loop feedbacks, the performance of PSDT can be guaranteed,and an example is given to reveals the interaction between theDT and the real physical system via closed-loop feedbacks.

DT, mainly driven by data, has some advantages in datautilization. DT enable us to set a quick start only with observeddata, which makes DT much more accessible to engineering inpractice—we no longer heavily rely on physical models. More-over, with the employment of RMT and Deep learning, DT iscompatible with spatial-temporal data and thus can conductbig data analytics in high-dimensional space. It means thatPSDT is friendly to uncertainties and bias which are ubiquitousin a modern grid, such as Jacobian matrix estimation bias inour case studies. Besides, DT can be evolved, in an activeand self-adaptive way, with data and feedbacks accumulation.Increasing data collection will improve the performance of DT;it is not true, however, for model-based one.

High dimension is also discussed. Only in high-dimensionalspace do some statistical properties and benefits come out.For instance, we separate Jacobian matrix estimation from PFanalysis; these two tasks are closely intertwined under model-based mode or in low-dimensional space.

DT is emerging and promising enabling technologies forrealizing smart grids. In this work we discuss some steady-state applications; the real-time property is much more fit forthe quasi-steady-state or transient-state analysis. For instance,the NSFC’s funding (2020-2022) “Research on the IntelligentFault Diagnosis with High Dimensional Criteria for Distri-bution Networks Based on the Merger of Random MatrixTheory and Deep Learning”. Beside, along the virtual test task,there are some directions. For instance, operational strategyoptimization for all parties in a VPP (virtual power plant)[29], and MAS (multi-agent system) [30] can be employed toenhance the interaction of each party. PSDT can also help withthe operation, dispatch, management, and electricity market inthe context of power systems. In addition, our PSDT is a goodreference for other industries.

8

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