# Two Decentralized Dynamic State Estimation Schemes for Multimachine Power Systems with Transmission Losses

Authors: Natanael Vieyra; Paul Maya-Ortiz; César Angeles-Camacho
Journal: International Transactions on Electrical Energy Systems
 (2023)

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## Abstract

This article addresses the problem of estimating the state of a multimachine power system (MPS). To work with power networks with lossy transmission lines, a variation of the classical third-order MPS is proposed by considering generators’ electrical power injected into the grid as a state variable. Based on a linear decentralized estimation model (tailored for a specific purpose), the state variables of generators (load angle, relative speed, and electrical power) together with terminal voltage magnitudes are estimated in a decentralized fashion through two new robustly convergent linear Luenberger estimators, one based on load angle measurement and the other based on relative speed measurement. The new MPS estimation model includes a set of robustly quick observable states, one per machine, which allows capturing the interaction with other generators, transmission line losses, unknown disturbances, and model errors. The result is a design superior to other related estimation techniques such as the extended Kalman filter (EKF) or Sliding Modes Perturbation Observer (SMPO) in terms of (i) a conventional-like simple pole placement-based tuning, (ii) low online computational load and disturbance rejection capability, and (iii) small gain-based convergence assessment. The performance of the proposed state estimation scheme is illustrated in a 3-machine power system under different operational conditions.


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## Body

## 1. Introduction

Multimachine power systems (MPSs) are one of the most representative examples of large-scale interconnected systems. An MPS comprises a set of generators (which have pretty different inertia constants) and loads interconnected through transmission lines.The main objective is to generate, transform, distribute, and consume electrical energy [1, 2].The MPS functioning may be affected by different disturbances, e.g., three-phase faults, load variations, and parameter uncertainty. Thus, different control approaches have been proposed to ensure optimal operating conditions [3–6]. However, these control schemes regularly assume the full knowledge of system state variables through sensors, an unnatural assumption due to physical and economic conditions. Therefore, despite the improved quality of the measurement devices, it is always necessary to count on a suitable tool to deliver information about the system when required. Therefore, a state estimator (SE) is a helpful tool for monitoring and controlling electrical networks.Dynamic system state estimators developed decades ago, and different local and centralized schemes are found in scientific literature, many of which are based on the Extended Kalman (EKF) approach [7–14] and some of its variants [15–22]. Despite its robust performance (in the presence of parametric errors and measurement noise) and well-known constructive design, these dynamic estimators have a heuristic tuning, not to mention the lack of formal robust convergence proofs. Also, for those designed under a centralized approach (Wide-Area) [7, 14, 21], there is a heavy computational load associated with integrating a set of nonlinear (NL) ordinary differential equations (ODEs) (in a number that grows quadratically with the number of states). In contraposition, the NL sliding mode perturbation observer (SMPO) [23] is built based on a decentralized model variant of the NL MPS model, with additional (called fictitious) states that account for intermachine state interaction. The NL SMPO has an efficient computational load (as the number of ODEs to be integrated grows linearly with the number of states) and formal robust convergence proof. However, the SMPO presents a complex structure, unclear tuning, and fragility to noisy measurements.Along with the estimation schemes mentioned above, Luenberger-like observers can also be found in scientific literature [24–26]. In [24, 25], the estimator design is based on the fully linearized representation of the MPS, ignoring the model errors. On the contrary, by employing a linear representation of MPS, the state vector is reconstructed through a centralized approach [26]. However, the observer gains are precalculated offline around a particular equilibrium point.Our work is the natural extension of [27]; for the lossless MPS system, we proposed a decentralized dynamic estimator based on load angle measurement that retains the advantages of the EKF and SMPO observers and overcomes their obstacles. In the present work, to be more practical, we have considered an MPS with lossy transmission lines through a comprehensive modeling approach, demonstrating that this assumption does not modify the conceptual idea behind the decentralized estimation methodology. The present study also includes designing a decentralized dynamic state estimator based on the relative speed measurement and a method focused on estimating the terminal voltage magnitudes from estimated state variables. In summary, the decentralized reconstruction of the state vector associated with each generator (load angle, relative speed, electrical power, and terminal voltage magnitudes) is carried out through two linear (L) decentralized (D) geometric estimators (GEs) (Luenberger-like observer) [28–30], one through employing the load angle as measured output and the other one based on the relative speed measurement. The proposed state estimation methodology (which also includes the construction of a tailored estimation model) is a point of departure to address the observer-based output-feedback control design problem with an application-oriented robust decentralized scheme. The developed methodology might be a novel alternative to those control approaches based on Lyapunov-like methods. Specifically, because these control schemes require the construction of Lyapunov functions, which for transmission systems with medium-length transmission line models (transfer conductances between buses) is not a trivial task [31–33].Firstly, the conventional NL-centralized (NL-C) model of the MPS (classical third-order dynamic model) is rewritten so that the electrical power injection of generators is a new state variable to include the transmission lines’ losses effect on the generator dynamics. In addition, the new NL-C MPS model is realized as a set of linear (L) decentralized (D) robustly observable models with augmented states that capture nonlinearity, parameter error, transmission line losses, and intermachine state interaction. Furthermore, either based on load angle or relative speed measurements (which is an advantage in the case that load angles of generators are not available as measurements [34, 35]), two linear geometric (Luenberger-like) observers with linear-decentralized measurement injection are built on the basis of the robust observability property of the linear model tailored for the estimation purpose at hand [28–30]. Then, the robust functioning of both estimators is assured with conditions coupled with a conventional-like tuning scheme. Finally, the proposed estimation methodology is illustrated and tested with a representative benchmark example employed in previous MPS studies [4, 5, 36, 37], finding that the proposed estimator yields a robust performance against modeling and measurement errors.


## 2. Problem Formulation

In this section, the classical third-order MPS dynamic model is rewritten by considering the electrical power injected by generators into the grid as a state variable instead of using generators' traditional transient internal voltages as one of them. Then, the estimation objective is presented.


### 2.1. A Variant of the Multimachine Power System Model

The dynamic behavior of a large-scale MPS, which consists ofN synchronous generators interconnected through lossy power lines, is represented by(1a)δ˙i=ωi,δi0=δi0,(1b)ω˙i=−Di2Hiωi+12HiPmi−Pi,ωi0=ωi0,(1c)E˙qi′=1Td0i′Efi−Eqi,Eqi′0=Eqi0′i=1,…,N,where(1d)Eqi=Eqi′+xdi−xdi′Idi,Pi=Eqi′Iqi,Qei=Eqi′Idi,(1e)Vi=xqiIqi2+Eqi′−xdiIdi2,and the direct Idi and quadratic Iqi axis currents considering lossy transmission lines are given by(1f)Iqi=Eqi′Gii+∑j≠iNEqj′Gijcosδi−δj+Bijsinδi−δj,(1g)Idi=−Eqi′Bii+∑j≠iNEqj′Gijsinδi−δj+Bijcosδi−δj,where δi is the load angle, ωi is the relative speed, Pi is the electrical power, Pmi is the mechanical input power which is considered known, Efi is the voltage field, xqi is the quadrature axis reactance, xdi and xdi′ are the direct and transient axis reactances, respectively, Di is the per unit damping factor, Hi is the inertia constant, Td0i′ is the direct axis transient short circuit time constant, respectively, Qei is the reactive power, Eqi′ is the transient electromotive force in the quadrature axis, Bij and Gij are, respectively, the susceptance and conductance of the generator “i”, Vi is the terminal voltage magnitude, and N is the number of generators connected to the grid.In order to adopt the electrical power injected by generators into the grid as a state variable, the time derivative ofPi with respect to time is obtained as follows:(2a)P˙i=1Td0i′EfiIqi−1Td0i′Eqi′Iqi−xdi−xdi′Td0i′IqiIdi+Eqi′I˙qi,where(2b)I˙qi=E˙qi′Gii+∑j≠iNE˙qj′Gijcosδi−δj+Bijsinδi−δj−∑j≠iNEqj′Gijsinδi−δj−Bijcosδi−δjωi,−∑j≠iNEqj′Gijsinδi−δj+Bijcosδi−δjωj.Thus, from (1a), (1b), and (2a), the large-scale MPS dynamic model can be represented by a variation of the standard three-dimensional flux decay model as follows:(3a)δ˙i=ωi,δi0=δi0,yi=δiorωi,i=1,…,N,(3b)ω˙i=−aiωi−biPi+dωi,ωi0=ωi0,(3c)P˙i=−ciPi+γiδ,ω,dγ,pi,pIi+ui,Pi0=Pi0,where(3d)γiδ,ω,dγ,pi,pIi=−cixdi−xdi′IqiIdi+Eqi′I˙qi,δ=δ1,…,δN⊤,ω=ω1,…,ωN⊤,dωi=12HiPmi,δi∈A=0,2π,ωi∈Ωi=0,ωi+,ui∈Ui=ui−,ui+,ai=Di2Hi,bi=12Hi,ci=1Td0i′,pi=ai,bi,ci,xdi,xdi′⊤,pIi=Gij+Bij,w=Idi,Eqi′,Eqj′,E˙qi′,E˙qj′,ωj⊤,ηi=dwi,Iqi⊤,dγ=w,ηi⊤,ui=ciIqiEfi.ηi and w are the measured vector and unmeasured vector input, respectively, yi is the measured output, ui is the control input associated with Efi, γi is a (Lipschitz-bounded) nonlinear function, pi and pIi are the local and interaction parameters, respectively.In per-machine vector notation, the MPS NL model (3a)–(3d) is written as(4a)x˙i=Aixi+bddωi+buγix,dγ+ui,xi0=xi0,(4b)yi=cyxi,i=1,…,N,where(5)xi=δiωiPi,Ai=0100−ai−bi00−ci,bu=001,bd=010,cy=1,0,0or0,1,0,γix,dγi is a Lipschitz-bounded function that englobes information about parameters, known and unknown inputs, nonlinearities, transmission losses, and intermachine interaction.In compact form, the MPS NL model (4a) and (4b) is rewritten as(5a)x.i=fixi,dxi,xi0=x0i,yi=cyxi,x0i∈X0i⊆Xi,xi∈Xi,dxi∈Dxi,where(5b)fix¯i,d¯xi=Aixi+bddωi+buui+γix,dγi=0,(5c)γixi,dγi−γix¯i,d¯γi≤lxγxi−x¯i+ldγγdγi−d¯γi,where x¯i is a uniquely robust unstable nominal steady-state (SS) associated with the nominal input d¯xi, and γi is a Lipschitz-bounded function (5c) about the nominal operation.Remark 1.
In comparison with our previous work [27] as a practical modification (to be more realistic) for estimation purposes, the medium-length transmission line model is adopted whose single-phase equivalent model can be represented by π configuration. However, this assumption does not modify the conceptual idea behind the proposed estimation methodology. Also, a general study of transmission line losses can be done through the power nodal balances.


### 2.2. Estimation Objective

Based on the measurement of load angle and relative speed of generators, the problem consists in proposing a state estimation comprehensive design methodology for MPS that allows the independent estimation of dynamic state variables of generatorsδi,ωi,Pi along with terminal voltage magnitudes Vi with the following features: (i) a robustly convergent, computationally efficient, and linearly-decentralized dynamic state estimator, (ii) systematic construction based on a linearly-decentralized realization of the nonlinear representation of MPS (3), (iii) identification of the underlying observability condition based on two candidates’ measured outputs, (iv) a priori (before simulation or testing) assurance of robust estimator functioning, and (iv) a simple conventional-type two-parameter per-machine gain tuning scheme with a transparent connection between gain choices and state estimate convergence features.


## 2.1. A Variant of the Multimachine Power System Model

The dynamic behavior of a large-scale MPS, which consists ofN synchronous generators interconnected through lossy power lines, is represented by(1a)δ˙i=ωi,δi0=δi0,(1b)ω˙i=−Di2Hiωi+12HiPmi−Pi,ωi0=ωi0,(1c)E˙qi′=1Td0i′Efi−Eqi,Eqi′0=Eqi0′i=1,…,N,where(1d)Eqi=Eqi′+xdi−xdi′Idi,Pi=Eqi′Iqi,Qei=Eqi′Idi,(1e)Vi=xqiIqi2+Eqi′−xdiIdi2,and the direct Idi and quadratic Iqi axis currents considering lossy transmission lines are given by(1f)Iqi=Eqi′Gii+∑j≠iNEqj′Gijcosδi−δj+Bijsinδi−δj,(1g)Idi=−Eqi′Bii+∑j≠iNEqj′Gijsinδi−δj+Bijcosδi−δj,where δi is the load angle, ωi is the relative speed, Pi is the electrical power, Pmi is the mechanical input power which is considered known, Efi is the voltage field, xqi is the quadrature axis reactance, xdi and xdi′ are the direct and transient axis reactances, respectively, Di is the per unit damping factor, Hi is the inertia constant, Td0i′ is the direct axis transient short circuit time constant, respectively, Qei is the reactive power, Eqi′ is the transient electromotive force in the quadrature axis, Bij and Gij are, respectively, the susceptance and conductance of the generator “i”, Vi is the terminal voltage magnitude, and N is the number of generators connected to the grid.In order to adopt the electrical power injected by generators into the grid as a state variable, the time derivative ofPi with respect to time is obtained as follows:(2a)P˙i=1Td0i′EfiIqi−1Td0i′Eqi′Iqi−xdi−xdi′Td0i′IqiIdi+Eqi′I˙qi,where(2b)I˙qi=E˙qi′Gii+∑j≠iNE˙qj′Gijcosδi−δj+Bijsinδi−δj−∑j≠iNEqj′Gijsinδi−δj−Bijcosδi−δjωi,−∑j≠iNEqj′Gijsinδi−δj+Bijcosδi−δjωj.Thus, from (1a), (1b), and (2a), the large-scale MPS dynamic model can be represented by a variation of the standard three-dimensional flux decay model as follows:(3a)δ˙i=ωi,δi0=δi0,yi=δiorωi,i=1,…,N,(3b)ω˙i=−aiωi−biPi+dωi,ωi0=ωi0,(3c)P˙i=−ciPi+γiδ,ω,dγ,pi,pIi+ui,Pi0=Pi0,where(3d)γiδ,ω,dγ,pi,pIi=−cixdi−xdi′IqiIdi+Eqi′I˙qi,δ=δ1,…,δN⊤,ω=ω1,…,ωN⊤,dωi=12HiPmi,δi∈A=0,2π,ωi∈Ωi=0,ωi+,ui∈Ui=ui−,ui+,ai=Di2Hi,bi=12Hi,ci=1Td0i′,pi=ai,bi,ci,xdi,xdi′⊤,pIi=Gij+Bij,w=Idi,Eqi′,Eqj′,E˙qi′,E˙qj′,ωj⊤,ηi=dwi,Iqi⊤,dγ=w,ηi⊤,ui=ciIqiEfi.ηi and w are the measured vector and unmeasured vector input, respectively, yi is the measured output, ui is the control input associated with Efi, γi is a (Lipschitz-bounded) nonlinear function, pi and pIi are the local and interaction parameters, respectively.In per-machine vector notation, the MPS NL model (3a)–(3d) is written as(4a)x˙i=Aixi+bddωi+buγix,dγ+ui,xi0=xi0,(4b)yi=cyxi,i=1,…,N,where(5)xi=δiωiPi,Ai=0100−ai−bi00−ci,bu=001,bd=010,cy=1,0,0or0,1,0,γix,dγi is a Lipschitz-bounded function that englobes information about parameters, known and unknown inputs, nonlinearities, transmission losses, and intermachine interaction.In compact form, the MPS NL model (4a) and (4b) is rewritten as(5a)x.i=fixi,dxi,xi0=x0i,yi=cyxi,x0i∈X0i⊆Xi,xi∈Xi,dxi∈Dxi,where(5b)fix¯i,d¯xi=Aixi+bddωi+buui+γix,dγi=0,(5c)γixi,dγi−γix¯i,d¯γi≤lxγxi−x¯i+ldγγdγi−d¯γi,where x¯i is a uniquely robust unstable nominal steady-state (SS) associated with the nominal input d¯xi, and γi is a Lipschitz-bounded function (5c) about the nominal operation.Remark 1.
In comparison with our previous work [27] as a practical modification (to be more realistic) for estimation purposes, the medium-length transmission line model is adopted whose single-phase equivalent model can be represented by π configuration. However, this assumption does not modify the conceptual idea behind the proposed estimation methodology. Also, a general study of transmission line losses can be done through the power nodal balances.


## 2.2. Estimation Objective

Based on the measurement of load angle and relative speed of generators, the problem consists in proposing a state estimation comprehensive design methodology for MPS that allows the independent estimation of dynamic state variables of generatorsδi,ωi,Pi along with terminal voltage magnitudes Vi with the following features: (i) a robustly convergent, computationally efficient, and linearly-decentralized dynamic state estimator, (ii) systematic construction based on a linearly-decentralized realization of the nonlinear representation of MPS (3), (iii) identification of the underlying observability condition based on two candidates’ measured outputs, (iv) a priori (before simulation or testing) assurance of robust estimator functioning, and (iv) a simple conventional-type two-parameter per-machine gain tuning scheme with a transparent connection between gain choices and state estimate convergence features.


## 3. State Estimation Methodology

This section follows the methods of [27]; briefly, the proposed dynamic state estimation methodology for MPSs includes the following: (i) derivation of the augmented L-D model for estimation, (ii) the conceptual estimation problem, (iii) robust observability assessment based on two candidates output functions, i.e., the load angle and relative speed of each generator, (iv) construction of the convergent estimator based on the load angle measurement, (v) development of the convergent estimator based on the relative speed measurement, (vi) tuning scheme, and (vii) comparison with other observer-based approaches.


### 3.1. Estimation Model

The MPS (4a)-(4b) in per-machine ιi-parametric form is expressed as:(6a)x.i=Aixi+bddωi+buui+ιi,xi0=xi0,(6b)yi=cyxi,ιi=γix,dγ,i=1,…,N,where(7)Ai=0100−ai−bi00−ci,eAit≤aie−λit,λi=ζimwim,λi=minζimϖim,ϖie≈λ1,2i=−ζimϖim±ϖim1−ζim2j,λ3i=−ϖie,and λ1,2i and λ3i are the three eigenvalues of Ai (6a), which can be uniquely solved in terms of i-th machine parameters as follows:(8)ζim,ϖim,ϖie=ℱai,bi,ci.Remark 2.
L-D estimation model. To derive a decentralized linear model with unknown-reconstructible input, the expressionγix,dγ (3d) is defined as a new state variable ιi which represents all the uncertainties, external disturbances, interaction among generator units, and known and unknown nonlinearities. This new variable ιi differs concerning our previous approach [27] in the following: (i) the new one considers the effect of the losses in the transmission lines on generator dynamics, (ii) the old one was represented by the high order nonlinear terms obtained from the expansion in Taylor series (TS) about the nominal state-input pair x¯,d¯γ of γix,dγ (corresponding to a purely inductive network), and (iii) the linear terms resulted from the TS expansion were included in the formulation of the matrix Ai for tuning purposes.
By assuming that the signalιi is in a slow-varying regime (SVR) concerning the exponential convergence speed ϖio of the observer (to be designed), the linear decentralized (L-D) estimation model has the following structure:(8a)x.i=Aixi+bddωi+buui+ιi,xi0=xi0,yi=cyxi(8b)ι˙i≈0,ιi0=ιi0,i=1,...,N,where ιi is an unknown exogenous input signal.
In compact notation, the augmented linear decentralized (L-D) system (8a) and (8b) is written as(9)χ.i=Aaχi+βddωi+βuui,yi=κδ,ωχi,i=1,…,N,where(10)χi=xi,ιi⊤,Aa=01000−ai−bi000−ci10000,βd=bd0,βu=bu0,dimχi=4.
In our particular case, for estimator design purposes, the output function vector can be given by the following two candidates functions:(11)κδ=1,0,0,0,κω=0,1,0,0.
The estimation model (the multimachine third-order model augmented with robustly quick observable states, one per machine) is a decentralized one (9). On the other hand, the current multimachine power system model is interconnected (3a)–(3d). The novelty of the proposed estimation approach resides in this fundamental difference between the actual plant and estimator models.


### 3.2. Estimation Problem

Driven by the measured input-output pair signalsyi,dit of the actual MPS (4a) and (4b), the dynamic data processor (dynamic state estimator) must produce a robustly convergent state estimates χ^it, i.e.,(12)χ^i0≈χi0⇒χ^it⟶rχit=τit,χi0,di⋅,yit∈X,yi=κδ,ωχit,with(13)χi=xi,ιi⊤,κδ=1,0,0,0,κω=0,1,0,0,di=dωi,ui⊤,dimχ^i=4,where χi is the augmented state variable vector considering the dynamic variables of each generator xi (load angle, relative speed, and electrical power) along with the new state variable ιi (one per machine), which is a robustly quick observable state that retains the information of interaction among generators, all the uncertainties, external disturbances, and known and unknown nonlinearities of the MPS system, di⋅ is the measured input vector, andyi is the output function which can either be the load angle or relative speed of each generator.Now, the main problem relies on robustly and instantaneously estimating for every time instant t the augmented stateχit (9) on the basis of the corresponding output vector (load angle or relative speed of generators), the known inputs di⋅ signals, and their time derivatives.


### 3.3. Robust Observability Assessment Based on Two Candidates Output Functions

The following subsections derive the observability conditions based on load angle and relative speed as output functions.


#### 3.3.1. Load Angle Measurement

For this, firstly, based on the load angle measurementyi=δi, the observability matrix of the state-augmented estimation model (9) is obtained as follows:(14)Oδi=κδκδAaκδAa2κδAa3=100001000−ai−bi00ai2biai+ci−bi,(15)detOδi≠r0⇔bi=DiMi≠r0,i=1,…N.The matrix pair (Aa, κδ) of (9) is robustly observable because the coefficient bi, associated with the inertia constant Mi of the i-th machine, is robustly strictly positive bi>r0.Remark 3.
Observability Property. The observability condition (15) is the same as the one obtained in [27]; (14) shows almost no variations with respect to the preceding work [27] due to the differences involved in the construction of the L-D estimation model (9) as mentioned in Remark 2.


#### 3.3.2. Relative Speed Measurement

A similar observability analysis is carried out under relative speed configuration as measured output functionyi=ωi. For this, as in the previous subsection, the L-D estimation model (9) is employed to obtain the observability matrix,(16)Oωi=κωκωAaκωAa2=100−ai−bi0ai2biai+ci−bi,(17)rankOωi=3,detOωi≠r0⇔bi=DiMi≠r0,i=1,…N.The successive time derivations of the output map of the estimation model (9) draw three algebraic equation sets [38].(17a)ψωit=Oωiχωi+ϱωit,χωi=ωi,Pi,ιi⊤,i=1,…N,where(17b)ψωit=yiy˙iy¨i,ϱωit=Tωiυi,Tωi=0000κωβd0κωAaβuκωAaβdκωβd,υi=ui,dωi,d˙ωi⊤.The observability condition (17) establishes that the state of (9) is in the one-dimensional time-varying surface, as shown as follows:(18)Xδit=χi∈X|ψωit=Oωiχωi+ϱωit,dimXδi=1.Accordingly, the restricted one-dimensional system is(18a)δi∗t=τxi∗t,δi0,ψωit,ϱωit∈Xδit,(18b)δ˙i∗t=ωit,δi∗0=δi0∗,i=1,…N,where (18b) and (18a) are defined as the unobservable dynamics and motion, respectively.Proposition 1.
If the unobservable motionδi∗t (18a) is exponentially (E)-stable,δi∗,ψωi,ϱωi produce an online reconstruction of χit based on relative speed measurement yi=ωit, the known inputs di=dωi,ui⊤, and their time derivatives (up to adequate order) [28, 29].The plant and its unobservable motions coincide [i.e.,δi∗t=χit], but the same is not necessarily true for their stability properties. If χit is stable, δi∗t is stable. If χit is unstable, δi∗t can be stable or unstable [28, 29].


### 3.4. Estimator Design Based on Load Angle Measurement

The construction of the proposed L-D GE estimator based on load angle measurement for MPS (3a)–(3d) includes the effect of the losses in the transmission lines on generator dynamics and closely follows the estimation methodology reported in [27]. In this present study, only the main construction steps are presented. Moreover, a method focused on estimating the terminal voltage magnitudes from estimated state variables is proposed.In summary, the robustly convergent observer inχi-coordinate is written as(19)χ^.i=fiχ^i,di+kδioζio,ϖioyi−κδχ^i,i=1,…,N,where(19a)kδioζio,ϖio=Oδi−14ζioϖio,4ζio2+2ϖio,4ζioϖio3,ϖio4⊤=kiδ,kiω,kiP,kiι⊤ζio,ϖio,(19b)kiδζio,ϖio=4ζioϖio,kiωζio,ϖio=4ζio2+2ϖio2,(19c)kiPζio,ϖio=−aibi4ζio2+2ϖio2−1bi4ζioϖio3,(19d)kiιζio,ϖio=−ciaibi4ζio2+2ϖio2−ci+aibi4ζioϖio3−1biϖio4.The vector gain inχi coordinates changed in comparison with the one derived in [27] on account that the observability matrix is different in both cases despite considering the same measured output (based on the load angle measurement). Mainly, the main difference relies on constructing the new L-D estimation model as mentioned in Remark 3.The robustly convergent estimation error of the MPS estimator (19) is EU-bounded as shown as follows:(20)χ˜it≤aiχe−λiotχ˜i0+aiχλioϵθi,i=1,…∞,N.The linear-decentralized observer inxi,ιi coordinates is given by(20a)x^.i=Aix^i+bddωi+buui+ι^i+kixζio,ϖioyi−cyx^i,x^i0=x^i0,(20b)ι^˙i=kiιζio,ϖioyi−cyx^i,ι^i0=ιi0,i=1,…,N,where(21)kixζio,ϖio=kiδ,kiω,kiP⊤ζio,ϖio,kiι=kiιζio,ϖio.In detailed form, the L-D GE estimator (20a) and (20b) in open-loop configuration for the MPS is written as follows:(21a)δ^˙i=ω^i+kiδζio,ϖioyi−δ^i,δ^i0=δ^i0,yi=δi,(21b)ω^˙i=−aiω^i−biP^i+dωi+kiωζio,ϖioyi−δ^i,ω^i0=ωi0,(21c)P^˙i=−ciPi+ι^i+ui+kiPζio,ϖioyi−δ^i,P^i0=P^i0,(21d)ι^˙i=kiιζio,ϖioyi−δ^i,ι^i0=ι^i0,i=1,…,N.Then, the successfully estimated values of the augmented stateχit from the proposed L-D GE estimator (20a) and (20b) are involved in the estimation of terminal voltage magnitudes.For this, given the estimation ofP^i and assuming that the known input Iqi≠0 (it is valid for normal operationally conditions [39]), an estimated value of Eqi′ can be computed as(22)E^qi′=P^iIqi,i=1,…,N,and(23)I^di=−E^qi′Bii+∑j≠iNE^qj′Gijsinδ^i−δ^j+Bijcosδ^i−δ^j,i=1,…,N.Then, we have(24)V^i=xqiIqi2+E^qi′−xdiI^di2i=1,…,N.The proposed L-D GE estimator (20a) and (20b) allows estimating of dynamic state variables (load angle, relative speed, and electrical power) associated with the N generators together with the terminal voltage magnitudes by employing the load angle as the unique measured output along with local information (composed by the known input pair di=dωi,ui⊤, as well as Iqi is assumed to be known).


### 3.5. Estimator Design Based on Relative Speed Measurement

If the unobservable motionδi∗t (18a) is robustly exponentially (RE)-stable (the stability of load angle of a third-order synchronous generator employed in MPS studies has been analyzed in [35, 40]; thus, the unobservable dynamics do not affect the stability of power systems), the online integration of the auxiliary estimator based on relative speed measurement is expressed as(25)χ^.i=fiχ^i,di+kωioζio,ϖioyi−κωχ^i,χ^i0=χ^i0≠χi0,χ^i∈Xδit,where(26)kωioζio,ϖio=0,Oωi−1⊤0,2ζio+1ϖio,2ζio+1ϖio2,ϖio3⊤,yields a motion χ^it that RE-converges to the plant motion χit according to the inequality(27)χ^i−χi≤aiχe−λiotχ˜i0+aiχλioϵθi,χ^i,χi∈Xδit,i=1,…,N.Inxi,ιi-coordinates (25), the L-D GE estimator is written as(28a)δ^˙i=ω^i,δ^i0=δ^i0,(28b)ω^˙i=−aiω^i−biP^i+dωi+gωiζio,ϖioωi−ω^i,ω^i0=ω^i0,(28c)P^˙i=−ciP^i+ι^i+ui+gpiζio,ϖioωi−ω^i,P^i0=P^i0,(28d)ι^˙i=gιiϖioωi−ω^i,ι^i0=ι^i0,where the output-driven corrector for the i-th subsystem is expressed as(29)gωiζio,ϖiogpiζio,ϖiogιiϖio=2ζio+1ϖio−aibi2ζioϖio−1bi2ζio+1ϖio2−aicibi2ζio+1ϖio−ai+cibi2ζio+1ϖio2−1biϖio3.Also, the estimation of terminal voltage magnitudes is obtained utilizing estimated state variables from (28a)–(28d) through (24).Remark 4.
Robust Convergence. A similar analysis of the estimation error dynamics of the proposed L-D GE estimators (19) and (25) based on the current MPS model (considering unmodeled parasitic dynamics) can be carried out as the one presented in [27]. Thus, the estimation error dynamics will be robustly exponentially (RE)-stable if the observer frequencies ϖio, one per machine, are chosen so that the stabilizing terms λsi dominate the potentially destabilizing ones λdi [27, 28, 30].


### 3.6. Tuning Scheme

The tuning guideline is derived from the pole placement approach regarding damping-frequency pairsζio,ϖio. For the L-D GE estimator based on load angle measurement, the reference gain kδio (19a) of (19) relies closely on the proposed methodology in [28–30]. This one is based on its counterpart in z-coordinates (one per machine), according to the characteristic polynomials of the prescribed linear, noninteractive, pole assignable (LNPA) output error dynamics(29a)Lipi,ϖioy˜i≔y˜i4+4ζioϖioy˜i3+4ζio2+2ϖio2y˜i2+4ζioϖio3y˜i1+ϖio4y˜i=0,where(29b)y˜i=y^i−yi,y^i=κδz^i,i=1,…,N,where the pole configuration (or frequency gain vector) can be set with a standard or tailored form (ITAE stands for integral of time-weight absolute error, Butterworth), [41], and Li is the associated linear operator.For the fourth-degree polynomial (29a) motivated by optimal linear-quadratic regulator (LQR) and nonlinear (NL) extended Kalman filter (EKF), as well as GE [28, 29], the prescribed pole pattern associated with (29a) is given by two complex-conjugate pole pairs with damping frequency pair as follows:(30)ν1,2i=−ζioϖio±ϖio1−ζio2j,ν3,4i=−ζioϖio±ϖio1−ζio2j,where(31)ζio=niζζim,niζ=nζ∈1,6≔Jζϖio=niϖϖim,niϖ=nϖ∈10,50≔Jϖ,Whereas for the L-D GE estimator based on relative speed measurement (28a)–(28d), the output-driven corrector κωio (26) is based on the prescribed vector gain associated with its counterpart in z-coordinates as well, which is defined according to the LNPA output error dynamics as(31a)Lipi,ϖioy˜i≔y˜i3+2ζio+1ϖioy˜i2+2ζio+1ϖio2y˜i1+ϖio3y˜i=0,where(31b)y˜i=y^i−yi,y^i=κωz^i,i=1,….N.The poles related to (31a) are assigned as a pair of complex-conjugate poles with reference characteristic frequency ϖio, and a sufficiently large ζio>0.71 damping factor and one pole is set to be real with characteristic frequency ϖio, as shown as follows:(32)ν1,2i=−ζioϖio±ϖio1−ζio2j,ν3i=−ϖio,where(33)ζio=niζζim,niζ=nζ∈1,6≔Jζϖio=niϖϖim,niϖ=nϖ∈10,50≔Jϖ.The proposed tuning scheme exploits the natural characteristics of MPS to obtain a better functioning in terms of an adequate compromise between speed reconstruction and tolerance to measurement noise with a conventional-like simple, transparent, and easy-to-apply tuning procedure based on two-parameter tuning.


### 3.7. Comparison with Other Observer-Based Approaches

In comparison with the conventional nonlinear-centralized (NL-C) EKF (based on the stochastic version of the MPS model by considering a global set of generators (3a)–(3d) in compact form) and nonlinear-decentralized (NL-D) SMPO [23] estimators for MPSs, the novelties are (i) from a theoretical perspective, the comprehensiveness of the methodology, and (ii) from an industrial applicability viewpoint; it is a more systematic and scalable design with a priory guarantee of reliable functioning and an admissible small online computational load (considerably smaller than the one of the NL-C EKF and similar to the one of the NL-D SMPO) and a substantially simpler easy-to-apply (preferably conventional-like) tuning scheme.Concerning the NL-D SMPO estimator [23], the aims are as follows: (i) to retain its low-dimensionality and ability to reject modeling errors, (ii) to remove the fragility due to measurement noise propagation, and (iii) to simplify its rather complex tuning scheme. Concerning the conventional NL-C EKF estimator, the aims are as follows: (i) to retain its tolerance to measurement noise, (ii) to significantly reduce its online computational load, and (iii) to overcome the drawback of not having formal convergence proof.


## 3.1. Estimation Model

The MPS (4a)-(4b) in per-machine ιi-parametric form is expressed as:(6a)x.i=Aixi+bddωi+buui+ιi,xi0=xi0,(6b)yi=cyxi,ιi=γix,dγ,i=1,…,N,where(7)Ai=0100−ai−bi00−ci,eAit≤aie−λit,λi=ζimwim,λi=minζimϖim,ϖie≈λ1,2i=−ζimϖim±ϖim1−ζim2j,λ3i=−ϖie,and λ1,2i and λ3i are the three eigenvalues of Ai (6a), which can be uniquely solved in terms of i-th machine parameters as follows:(8)ζim,ϖim,ϖie=ℱai,bi,ci.Remark 2.
L-D estimation model. To derive a decentralized linear model with unknown-reconstructible input, the expressionγix,dγ (3d) is defined as a new state variable ιi which represents all the uncertainties, external disturbances, interaction among generator units, and known and unknown nonlinearities. This new variable ιi differs concerning our previous approach [27] in the following: (i) the new one considers the effect of the losses in the transmission lines on generator dynamics, (ii) the old one was represented by the high order nonlinear terms obtained from the expansion in Taylor series (TS) about the nominal state-input pair x¯,d¯γ of γix,dγ (corresponding to a purely inductive network), and (iii) the linear terms resulted from the TS expansion were included in the formulation of the matrix Ai for tuning purposes.
By assuming that the signalιi is in a slow-varying regime (SVR) concerning the exponential convergence speed ϖio of the observer (to be designed), the linear decentralized (L-D) estimation model has the following structure:(8a)x.i=Aixi+bddωi+buui+ιi,xi0=xi0,yi=cyxi(8b)ι˙i≈0,ιi0=ιi0,i=1,...,N,where ιi is an unknown exogenous input signal.
In compact notation, the augmented linear decentralized (L-D) system (8a) and (8b) is written as(9)χ.i=Aaχi+βddωi+βuui,yi=κδ,ωχi,i=1,…,N,where(10)χi=xi,ιi⊤,Aa=01000−ai−bi000−ci10000,βd=bd0,βu=bu0,dimχi=4.
In our particular case, for estimator design purposes, the output function vector can be given by the following two candidates functions:(11)κδ=1,0,0,0,κω=0,1,0,0.
The estimation model (the multimachine third-order model augmented with robustly quick observable states, one per machine) is a decentralized one (9). On the other hand, the current multimachine power system model is interconnected (3a)–(3d). The novelty of the proposed estimation approach resides in this fundamental difference between the actual plant and estimator models.


## 3.2. Estimation Problem

Driven by the measured input-output pair signalsyi,dit of the actual MPS (4a) and (4b), the dynamic data processor (dynamic state estimator) must produce a robustly convergent state estimates χ^it, i.e.,(12)χ^i0≈χi0⇒χ^it⟶rχit=τit,χi0,di⋅,yit∈X,yi=κδ,ωχit,with(13)χi=xi,ιi⊤,κδ=1,0,0,0,κω=0,1,0,0,di=dωi,ui⊤,dimχ^i=4,where χi is the augmented state variable vector considering the dynamic variables of each generator xi (load angle, relative speed, and electrical power) along with the new state variable ιi (one per machine), which is a robustly quick observable state that retains the information of interaction among generators, all the uncertainties, external disturbances, and known and unknown nonlinearities of the MPS system, di⋅ is the measured input vector, andyi is the output function which can either be the load angle or relative speed of each generator.Now, the main problem relies on robustly and instantaneously estimating for every time instant t the augmented stateχit (9) on the basis of the corresponding output vector (load angle or relative speed of generators), the known inputs di⋅ signals, and their time derivatives.


## 3.3. Robust Observability Assessment Based on Two Candidates Output Functions

The following subsections derive the observability conditions based on load angle and relative speed as output functions.


### 3.3.1. Load Angle Measurement

For this, firstly, based on the load angle measurementyi=δi, the observability matrix of the state-augmented estimation model (9) is obtained as follows:(14)Oδi=κδκδAaκδAa2κδAa3=100001000−ai−bi00ai2biai+ci−bi,(15)detOδi≠r0⇔bi=DiMi≠r0,i=1,…N.The matrix pair (Aa, κδ) of (9) is robustly observable because the coefficient bi, associated with the inertia constant Mi of the i-th machine, is robustly strictly positive bi>r0.Remark 3.
Observability Property. The observability condition (15) is the same as the one obtained in [27]; (14) shows almost no variations with respect to the preceding work [27] due to the differences involved in the construction of the L-D estimation model (9) as mentioned in Remark 2.


### 3.3.2. Relative Speed Measurement

A similar observability analysis is carried out under relative speed configuration as measured output functionyi=ωi. For this, as in the previous subsection, the L-D estimation model (9) is employed to obtain the observability matrix,(16)Oωi=κωκωAaκωAa2=100−ai−bi0ai2biai+ci−bi,(17)rankOωi=3,detOωi≠r0⇔bi=DiMi≠r0,i=1,…N.The successive time derivations of the output map of the estimation model (9) draw three algebraic equation sets [38].(17a)ψωit=Oωiχωi+ϱωit,χωi=ωi,Pi,ιi⊤,i=1,…N,where(17b)ψωit=yiy˙iy¨i,ϱωit=Tωiυi,Tωi=0000κωβd0κωAaβuκωAaβdκωβd,υi=ui,dωi,d˙ωi⊤.The observability condition (17) establishes that the state of (9) is in the one-dimensional time-varying surface, as shown as follows:(18)Xδit=χi∈X|ψωit=Oωiχωi+ϱωit,dimXδi=1.Accordingly, the restricted one-dimensional system is(18a)δi∗t=τxi∗t,δi0,ψωit,ϱωit∈Xδit,(18b)δ˙i∗t=ωit,δi∗0=δi0∗,i=1,…N,where (18b) and (18a) are defined as the unobservable dynamics and motion, respectively.Proposition 1.
If the unobservable motionδi∗t (18a) is exponentially (E)-stable,δi∗,ψωi,ϱωi produce an online reconstruction of χit based on relative speed measurement yi=ωit, the known inputs di=dωi,ui⊤, and their time derivatives (up to adequate order) [28, 29].The plant and its unobservable motions coincide [i.e.,δi∗t=χit], but the same is not necessarily true for their stability properties. If χit is stable, δi∗t is stable. If χit is unstable, δi∗t can be stable or unstable [28, 29].


## 3.3.1. Load Angle Measurement

For this, firstly, based on the load angle measurementyi=δi, the observability matrix of the state-augmented estimation model (9) is obtained as follows:(14)Oδi=κδκδAaκδAa2κδAa3=100001000−ai−bi00ai2biai+ci−bi,(15)detOδi≠r0⇔bi=DiMi≠r0,i=1,…N.The matrix pair (Aa, κδ) of (9) is robustly observable because the coefficient bi, associated with the inertia constant Mi of the i-th machine, is robustly strictly positive bi>r0.Remark 3.
Observability Property. The observability condition (15) is the same as the one obtained in [27]; (14) shows almost no variations with respect to the preceding work [27] due to the differences involved in the construction of the L-D estimation model (9) as mentioned in Remark 2.


## 3.3.2. Relative Speed Measurement

A similar observability analysis is carried out under relative speed configuration as measured output functionyi=ωi. For this, as in the previous subsection, the L-D estimation model (9) is employed to obtain the observability matrix,(16)Oωi=κωκωAaκωAa2=100−ai−bi0ai2biai+ci−bi,(17)rankOωi=3,detOωi≠r0⇔bi=DiMi≠r0,i=1,…N.The successive time derivations of the output map of the estimation model (9) draw three algebraic equation sets [38].(17a)ψωit=Oωiχωi+ϱωit,χωi=ωi,Pi,ιi⊤,i=1,…N,where(17b)ψωit=yiy˙iy¨i,ϱωit=Tωiυi,Tωi=0000κωβd0κωAaβuκωAaβdκωβd,υi=ui,dωi,d˙ωi⊤.The observability condition (17) establishes that the state of (9) is in the one-dimensional time-varying surface, as shown as follows:(18)Xδit=χi∈X|ψωit=Oωiχωi+ϱωit,dimXδi=1.Accordingly, the restricted one-dimensional system is(18a)δi∗t=τxi∗t,δi0,ψωit,ϱωit∈Xδit,(18b)δ˙i∗t=ωit,δi∗0=δi0∗,i=1,…N,where (18b) and (18a) are defined as the unobservable dynamics and motion, respectively.Proposition 1.
If the unobservable motionδi∗t (18a) is exponentially (E)-stable,δi∗,ψωi,ϱωi produce an online reconstruction of χit based on relative speed measurement yi=ωit, the known inputs di=dωi,ui⊤, and their time derivatives (up to adequate order) [28, 29].The plant and its unobservable motions coincide [i.e.,δi∗t=χit], but the same is not necessarily true for their stability properties. If χit is stable, δi∗t is stable. If χit is unstable, δi∗t can be stable or unstable [28, 29].


## 3.4. Estimator Design Based on Load Angle Measurement

The construction of the proposed L-D GE estimator based on load angle measurement for MPS (3a)–(3d) includes the effect of the losses in the transmission lines on generator dynamics and closely follows the estimation methodology reported in [27]. In this present study, only the main construction steps are presented. Moreover, a method focused on estimating the terminal voltage magnitudes from estimated state variables is proposed.In summary, the robustly convergent observer inχi-coordinate is written as(19)χ^.i=fiχ^i,di+kδioζio,ϖioyi−κδχ^i,i=1,…,N,where(19a)kδioζio,ϖio=Oδi−14ζioϖio,4ζio2+2ϖio,4ζioϖio3,ϖio4⊤=kiδ,kiω,kiP,kiι⊤ζio,ϖio,(19b)kiδζio,ϖio=4ζioϖio,kiωζio,ϖio=4ζio2+2ϖio2,(19c)kiPζio,ϖio=−aibi4ζio2+2ϖio2−1bi4ζioϖio3,(19d)kiιζio,ϖio=−ciaibi4ζio2+2ϖio2−ci+aibi4ζioϖio3−1biϖio4.The vector gain inχi coordinates changed in comparison with the one derived in [27] on account that the observability matrix is different in both cases despite considering the same measured output (based on the load angle measurement). Mainly, the main difference relies on constructing the new L-D estimation model as mentioned in Remark 3.The robustly convergent estimation error of the MPS estimator (19) is EU-bounded as shown as follows:(20)χ˜it≤aiχe−λiotχ˜i0+aiχλioϵθi,i=1,…∞,N.The linear-decentralized observer inxi,ιi coordinates is given by(20a)x^.i=Aix^i+bddωi+buui+ι^i+kixζio,ϖioyi−cyx^i,x^i0=x^i0,(20b)ι^˙i=kiιζio,ϖioyi−cyx^i,ι^i0=ιi0,i=1,…,N,where(21)kixζio,ϖio=kiδ,kiω,kiP⊤ζio,ϖio,kiι=kiιζio,ϖio.In detailed form, the L-D GE estimator (20a) and (20b) in open-loop configuration for the MPS is written as follows:(21a)δ^˙i=ω^i+kiδζio,ϖioyi−δ^i,δ^i0=δ^i0,yi=δi,(21b)ω^˙i=−aiω^i−biP^i+dωi+kiωζio,ϖioyi−δ^i,ω^i0=ωi0,(21c)P^˙i=−ciPi+ι^i+ui+kiPζio,ϖioyi−δ^i,P^i0=P^i0,(21d)ι^˙i=kiιζio,ϖioyi−δ^i,ι^i0=ι^i0,i=1,…,N.Then, the successfully estimated values of the augmented stateχit from the proposed L-D GE estimator (20a) and (20b) are involved in the estimation of terminal voltage magnitudes.For this, given the estimation ofP^i and assuming that the known input Iqi≠0 (it is valid for normal operationally conditions [39]), an estimated value of Eqi′ can be computed as(22)E^qi′=P^iIqi,i=1,…,N,and(23)I^di=−E^qi′Bii+∑j≠iNE^qj′Gijsinδ^i−δ^j+Bijcosδ^i−δ^j,i=1,…,N.Then, we have(24)V^i=xqiIqi2+E^qi′−xdiI^di2i=1,…,N.The proposed L-D GE estimator (20a) and (20b) allows estimating of dynamic state variables (load angle, relative speed, and electrical power) associated with the N generators together with the terminal voltage magnitudes by employing the load angle as the unique measured output along with local information (composed by the known input pair di=dωi,ui⊤, as well as Iqi is assumed to be known).


## 3.5. Estimator Design Based on Relative Speed Measurement

If the unobservable motionδi∗t (18a) is robustly exponentially (RE)-stable (the stability of load angle of a third-order synchronous generator employed in MPS studies has been analyzed in [35, 40]; thus, the unobservable dynamics do not affect the stability of power systems), the online integration of the auxiliary estimator based on relative speed measurement is expressed as(25)χ^.i=fiχ^i,di+kωioζio,ϖioyi−κωχ^i,χ^i0=χ^i0≠χi0,χ^i∈Xδit,where(26)kωioζio,ϖio=0,Oωi−1⊤0,2ζio+1ϖio,2ζio+1ϖio2,ϖio3⊤,yields a motion χ^it that RE-converges to the plant motion χit according to the inequality(27)χ^i−χi≤aiχe−λiotχ˜i0+aiχλioϵθi,χ^i,χi∈Xδit,i=1,…,N.Inxi,ιi-coordinates (25), the L-D GE estimator is written as(28a)δ^˙i=ω^i,δ^i0=δ^i0,(28b)ω^˙i=−aiω^i−biP^i+dωi+gωiζio,ϖioωi−ω^i,ω^i0=ω^i0,(28c)P^˙i=−ciP^i+ι^i+ui+gpiζio,ϖioωi−ω^i,P^i0=P^i0,(28d)ι^˙i=gιiϖioωi−ω^i,ι^i0=ι^i0,where the output-driven corrector for the i-th subsystem is expressed as(29)gωiζio,ϖiogpiζio,ϖiogιiϖio=2ζio+1ϖio−aibi2ζioϖio−1bi2ζio+1ϖio2−aicibi2ζio+1ϖio−ai+cibi2ζio+1ϖio2−1biϖio3.Also, the estimation of terminal voltage magnitudes is obtained utilizing estimated state variables from (28a)–(28d) through (24).Remark 4.
Robust Convergence. A similar analysis of the estimation error dynamics of the proposed L-D GE estimators (19) and (25) based on the current MPS model (considering unmodeled parasitic dynamics) can be carried out as the one presented in [27]. Thus, the estimation error dynamics will be robustly exponentially (RE)-stable if the observer frequencies ϖio, one per machine, are chosen so that the stabilizing terms λsi dominate the potentially destabilizing ones λdi [27, 28, 30].


## 3.6. Tuning Scheme

The tuning guideline is derived from the pole placement approach regarding damping-frequency pairsζio,ϖio. For the L-D GE estimator based on load angle measurement, the reference gain kδio (19a) of (19) relies closely on the proposed methodology in [28–30]. This one is based on its counterpart in z-coordinates (one per machine), according to the characteristic polynomials of the prescribed linear, noninteractive, pole assignable (LNPA) output error dynamics(29a)Lipi,ϖioy˜i≔y˜i4+4ζioϖioy˜i3+4ζio2+2ϖio2y˜i2+4ζioϖio3y˜i1+ϖio4y˜i=0,where(29b)y˜i=y^i−yi,y^i=κδz^i,i=1,…,N,where the pole configuration (or frequency gain vector) can be set with a standard or tailored form (ITAE stands for integral of time-weight absolute error, Butterworth), [41], and Li is the associated linear operator.For the fourth-degree polynomial (29a) motivated by optimal linear-quadratic regulator (LQR) and nonlinear (NL) extended Kalman filter (EKF), as well as GE [28, 29], the prescribed pole pattern associated with (29a) is given by two complex-conjugate pole pairs with damping frequency pair as follows:(30)ν1,2i=−ζioϖio±ϖio1−ζio2j,ν3,4i=−ζioϖio±ϖio1−ζio2j,where(31)ζio=niζζim,niζ=nζ∈1,6≔Jζϖio=niϖϖim,niϖ=nϖ∈10,50≔Jϖ,Whereas for the L-D GE estimator based on relative speed measurement (28a)–(28d), the output-driven corrector κωio (26) is based on the prescribed vector gain associated with its counterpart in z-coordinates as well, which is defined according to the LNPA output error dynamics as(31a)Lipi,ϖioy˜i≔y˜i3+2ζio+1ϖioy˜i2+2ζio+1ϖio2y˜i1+ϖio3y˜i=0,where(31b)y˜i=y^i−yi,y^i=κωz^i,i=1,….N.The poles related to (31a) are assigned as a pair of complex-conjugate poles with reference characteristic frequency ϖio, and a sufficiently large ζio>0.71 damping factor and one pole is set to be real with characteristic frequency ϖio, as shown as follows:(32)ν1,2i=−ζioϖio±ϖio1−ζio2j,ν3i=−ϖio,where(33)ζio=niζζim,niζ=nζ∈1,6≔Jζϖio=niϖϖim,niϖ=nϖ∈10,50≔Jϖ.The proposed tuning scheme exploits the natural characteristics of MPS to obtain a better functioning in terms of an adequate compromise between speed reconstruction and tolerance to measurement noise with a conventional-like simple, transparent, and easy-to-apply tuning procedure based on two-parameter tuning.


## 3.7. Comparison with Other Observer-Based Approaches

In comparison with the conventional nonlinear-centralized (NL-C) EKF (based on the stochastic version of the MPS model by considering a global set of generators (3a)–(3d) in compact form) and nonlinear-decentralized (NL-D) SMPO [23] estimators for MPSs, the novelties are (i) from a theoretical perspective, the comprehensiveness of the methodology, and (ii) from an industrial applicability viewpoint; it is a more systematic and scalable design with a priory guarantee of reliable functioning and an admissible small online computational load (considerably smaller than the one of the NL-C EKF and similar to the one of the NL-D SMPO) and a substantially simpler easy-to-apply (preferably conventional-like) tuning scheme.Concerning the NL-D SMPO estimator [23], the aims are as follows: (i) to retain its low-dimensionality and ability to reject modeling errors, (ii) to remove the fragility due to measurement noise propagation, and (iii) to simplify its rather complex tuning scheme. Concerning the conventional NL-C EKF estimator, the aims are as follows: (i) to retain its tolerance to measurement noise, (ii) to significantly reduce its online computational load, and (iii) to overcome the drawback of not having formal convergence proof.


## 4. Results and Discussion

The Western System Coordinating Council (WSCC) 9-bus electrical power grid is used [2, 4, 5, 7, 8, 36] to demonstrate the estimator’s performance. The WSCC employed for evaluation purposes is shown in Figure 1, and the system’s nominal parameters and the operations used for the sample problem investigated are listed in Table 1.Figure 1
WSCC 3-machine, 9-bus system [2].Table 1
Machine data.
ParametersGenerator 1Generator 2Generator 3Di (p.u.), Td0i′ (s)6, 8.965, 62, 5.89Hi (s)23.16.13.01xdi′ (p.u.), xdi (p.u.), xqi (p.u.)0.0608, 0.146, 0.09690.1198, 0.8958, 0.86450.1813, 1.3125, 1.2578Efi (p.u.)1.0561.7891.403Pmi (p.u.)0.7161.630.85


### 4.1. Tuning

The adjustable vectorskδio (19a) and kωio (26) are set according to the tuning procedure mentioned previously. For the L-D GE estimators (17) and (25), the damping-frequency tuning pair is set with: ζ1o=0.71, ζ2,3o=1.5, ϖ1o=60, ϖ2o=35,andϖ3o=67.


### 4.2. Testing Scheme

The measurements are affected by fluctuations driven by low-amplitude and high-frequency (close to resonant one) sinusoidal inputs, as shown as follows:(34)y1=δ1+1e−4sin600t,y2=δ2+1e−4sin350t,y3=δ3+1e−4sin670t.Moreover, the robust testing scheme considers different operation conditions as given as follows: (i) when generator one is affected by an unknown electromagnetic disturbance involving the excitation winding, which suddenly appears att=1 [sec], the fault is cleared after 0.5 seconds (clearing time tcl); in this sense, the generator comes back to its prefault configuration, (ii) in parameter uncertainty, the value associated with the time constant Td1′ fluctuates around 15% and 25% from its original values through the time simulation, (iii) once the machines reach the steady-state regime, generator two is affected by a mechanical power variation at t=30 [sec], which is cleared after 0.1 seconds, and (iv) to evaluate the transient performance of the L-D GE estimators, initial condition uncertainty has been considered for both estimation schemes (with respect to operation points).The proposed robust testing scheme considers initial conditions errors, noisy measurements, and known/unknown disturbances. These testing conditions are more complex than those employed in previous estimation studies [7–10, 14, 23, 36].Figures2–7 depict the performance of estimators. Solid yellow lines represent the actual values of the load angle, relative speed, electrical power, and the nonlinear interconnection term of each machine; on the other hand, the dashed lines are used to show the estimated variables by both estimators of the L-D GE (19) in magenta and L-D GE (25) in cyan.Figure 2
Comparison between the L-D GE estimators: load angle and relative speed of generator one.Figure 3
Comparison between the L-D GE estimators: the electrical power and nonlinear interconnection term of generator one.Figure 4
Comparison between the L-D GE estimators: load angle and relative speed of generator two.Figure 5
Comparison between the L-D GE estimators: the electrical power and nonlinear interconnection term of generator two.Figure 6
Comparison between the L-D GE estimators: load angle and relative speed of generator three.Figure 7
Comparison between the L-D GE estimators: the electrical power and nonlinear interconnection term of generator three.As can be appreciated, the transient behavior related to the abovementioned conditions is correctly reconstructed by both L-D GE estimators, which have a fast convergence rate to the actual states. Also, as can be noticed, the performance of the L-D GE (28a)–(28d) is similar to the one obtained through the L-D GE (20a) and (20b) based on the load angle measurement. The latter is lightly affected by measurement noise, as shown in Figures 3 and 5, specifically in estimating the nonlinear interconnection terms. The estimated signals from the L-D GE estimators can be used for monitoring purposes, particularly for designing observer-based controllers to enhance the transient stability of each generator in a decentralized fashion. As mentioned earlier, generator one is affected by an unknown exogenous input at t = 1 [sec]; the proposed estimators correctly estimate the dynamic state variables due to the fact that the unknown disturbance is included in the nonlinear interconnection term ιi, and it is reconstructed along with the other state variables.Figure8 shows the estimation of terminal voltage magnitudes, which are correctly reconstructed from the estimated state variables of each generator. However, the noise effect is notorious in the estimated variables by the L-D GE estimator on the basis of the load angle measurement (19). Therefore, we want to highlight that the disturbance which affects generator two at t = 30 [sec] is reflected in an extreme condition of MPS (for comparative academic purposes). Despite this, the transient and steady-state regimes are suitably captured by both L-D GE estimators.Figure 8
Estimation of voltage magnitudes through the proposed L-D GE estimators.Remark 5.
Online Computational Load. The WSCC type 3-machine 9-bus system under the centralized extended Kalman filter approach (based on the stochastic version of the MPS model by considering a global set of generators (3a)–(3d) in compact form [27]) requires solving 54 ordinary differential equations (ODEs) due to the fact that the Riccati equations grow quadratically according to the number of state variables to be estimated. In the opposite case, the two proposed L-D GE estimators (19) and (25) only solve 12 ODEs (4 ODEs for each machine, considering the augmented state, which efficiently retains the interconnection information among machines). This difference can be made further significant in electrical power grids with many generators.


## 4.1. Tuning

The adjustable vectorskδio (19a) and kωio (26) are set according to the tuning procedure mentioned previously. For the L-D GE estimators (17) and (25), the damping-frequency tuning pair is set with: ζ1o=0.71, ζ2,3o=1.5, ϖ1o=60, ϖ2o=35,andϖ3o=67.


## 4.2. Testing Scheme

The measurements are affected by fluctuations driven by low-amplitude and high-frequency (close to resonant one) sinusoidal inputs, as shown as follows:(34)y1=δ1+1e−4sin600t,y2=δ2+1e−4sin350t,y3=δ3+1e−4sin670t.Moreover, the robust testing scheme considers different operation conditions as given as follows: (i) when generator one is affected by an unknown electromagnetic disturbance involving the excitation winding, which suddenly appears att=1 [sec], the fault is cleared after 0.5 seconds (clearing time tcl); in this sense, the generator comes back to its prefault configuration, (ii) in parameter uncertainty, the value associated with the time constant Td1′ fluctuates around 15% and 25% from its original values through the time simulation, (iii) once the machines reach the steady-state regime, generator two is affected by a mechanical power variation at t=30 [sec], which is cleared after 0.1 seconds, and (iv) to evaluate the transient performance of the L-D GE estimators, initial condition uncertainty has been considered for both estimation schemes (with respect to operation points).The proposed robust testing scheme considers initial conditions errors, noisy measurements, and known/unknown disturbances. These testing conditions are more complex than those employed in previous estimation studies [7–10, 14, 23, 36].Figures2–7 depict the performance of estimators. Solid yellow lines represent the actual values of the load angle, relative speed, electrical power, and the nonlinear interconnection term of each machine; on the other hand, the dashed lines are used to show the estimated variables by both estimators of the L-D GE (19) in magenta and L-D GE (25) in cyan.Figure 2
Comparison between the L-D GE estimators: load angle and relative speed of generator one.Figure 3
Comparison between the L-D GE estimators: the electrical power and nonlinear interconnection term of generator one.Figure 4
Comparison between the L-D GE estimators: load angle and relative speed of generator two.Figure 5
Comparison between the L-D GE estimators: the electrical power and nonlinear interconnection term of generator two.Figure 6
Comparison between the L-D GE estimators: load angle and relative speed of generator three.Figure 7
Comparison between the L-D GE estimators: the electrical power and nonlinear interconnection term of generator three.As can be appreciated, the transient behavior related to the abovementioned conditions is correctly reconstructed by both L-D GE estimators, which have a fast convergence rate to the actual states. Also, as can be noticed, the performance of the L-D GE (28a)–(28d) is similar to the one obtained through the L-D GE (20a) and (20b) based on the load angle measurement. The latter is lightly affected by measurement noise, as shown in Figures 3 and 5, specifically in estimating the nonlinear interconnection terms. The estimated signals from the L-D GE estimators can be used for monitoring purposes, particularly for designing observer-based controllers to enhance the transient stability of each generator in a decentralized fashion. As mentioned earlier, generator one is affected by an unknown exogenous input at t = 1 [sec]; the proposed estimators correctly estimate the dynamic state variables due to the fact that the unknown disturbance is included in the nonlinear interconnection term ιi, and it is reconstructed along with the other state variables.Figure8 shows the estimation of terminal voltage magnitudes, which are correctly reconstructed from the estimated state variables of each generator. However, the noise effect is notorious in the estimated variables by the L-D GE estimator on the basis of the load angle measurement (19). Therefore, we want to highlight that the disturbance which affects generator two at t = 30 [sec] is reflected in an extreme condition of MPS (for comparative academic purposes). Despite this, the transient and steady-state regimes are suitably captured by both L-D GE estimators.Figure 8
Estimation of voltage magnitudes through the proposed L-D GE estimators.Remark 5.
Online Computational Load. The WSCC type 3-machine 9-bus system under the centralized extended Kalman filter approach (based on the stochastic version of the MPS model by considering a global set of generators (3a)–(3d) in compact form [27]) requires solving 54 ordinary differential equations (ODEs) due to the fact that the Riccati equations grow quadratically according to the number of state variables to be estimated. In the opposite case, the two proposed L-D GE estimators (19) and (25) only solve 12 ODEs (4 ODEs for each machine, considering the augmented state, which efficiently retains the interconnection information among machines). This difference can be made further significant in electrical power grids with many generators.


## 5. Conclusions

In this work, two robust linear decentralized (L-D) geometric estimators (GE) are proposed, one considering the load angle as measurement (19) and another one based on relative speed as measured output (25). Methodologically speaking, (i) both estimators are constructed based on a new representation of the classical flux decay MPS dynamic model by adopting the electrical power as a state variable to include the effect of the transmission losses on generators’ dynamics, (ii) the new MPS dynamic model (3a)–(3d) is employed to construct an L-D estimation model (9), for which the state variables of generators are augmented by considering a set of new state variables (one per machine), these new variables stand for all the uncertainties, external disturbances, interaction among generator units, and known and unknown nonlinearities, (iii) the observability properties of the L-D estimation model (9) were identified for each measured output, for load angle (15) and the relative speed (17), respectively, (iv) both estimators have a precise and systematic industrial-like tuning based on two parameters associated with the generators’ dynamics, (v) the proposed L-D state estimation schemes are computationally efficient (the involved ODEs by the estimators grow linearly for states to be estimated, almost a quarter of the total ODEs of the conventional EKF [27]), and (vi) both estimators have a robust convergence based on input-to-state (IS) stability and small gain ideas (as one presented in [27]).The performance of the L-D GE estimators was evaluated via numerical simulations through a robust testing scheme considering known/unknown disturbances and noisy measurements. As a result, the developed dynamic estimators show a fast convergence rate to the current states of MPS and robust performance against measurement noises and initial condition uncertainties.The present study on MPS estimation is a point of departure to address the observer-based output-feedback control design problem with an application-oriented robust decentralized scheme.



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*Source: 1464297-2023-02-03.xml*