State Estimation

The state vectors of many dynamic systems evolve according to some linear or nonlinear equality constraints. For example, in ground target tracking [65, 123, 125], if we treat roads as curves without width, the road networks can then be described by equality constraints. In the quaternion-based attitude estimation problem, the attitude vector must have a unit norm [15]. In a compartmental model with zero net inflow [27], mass is conserved. In an undamped mechanical systems, such as one with Hamiltonian dynamics, energy conservation law holds. Likewise, in circuit analysis, Kirchhoff’s current and voltage laws hold. For state estimation with noise-free measurements due to equality constraints, numerous results and methods are available [51, 61, 66, 105, 116, 119, 124, 125, 131]. For example, the reparameterization method simply reparameterizes the system model so that the equality constraints are not required any more. It has several disadvantages. First, the physical meaning of the reparameterized system state may vary and be different at different time instants. Second, the interpretation of the reparameterized system state is less natural and more difficult [105]. Another popular method for equality constrained estimation is the projection method [51, 61, 66, 105, 124, 125], in which the estimate is projected onto the constraint subspace. Unfortunately, it has problems and limitations. Its main idea is to apply classical constrained optimization techniques to the constrained estimation problem. Some other methods, e.g., maximum probability method and mean square method, were also discussed in [105]. They are not free of the limitations, either. Also, all existing work processes the noisy measurement first and then the equality constraint. Is it the only choice or a good choice? If there are more than one choice, how should the end user choose among them? Unfortunately, such questions have not been answered in theory.

The present paper considers the state estimation problem subject to structural uncertainty - unknown or changeable dimension of the system state space. The system state is estimated when the structure and the true parameters of the system model are unknown but they belong to an uncertainty domain. A solution to this problem by another multiple-model algorithm is given in [7]. The requirements for its applicability under struc- tural uncertainty are formulated in [7]. These requirements are here extended for the IMM filter - a powerful scheme [2,5] for estimation of hybrid (continuous-discrete) systems. The IMM filter belongs to the group of the multiple-model algorithms that recently are very popular [1, 2, 8, 10, 11]. In most cases the IMM estimator is applied under parametric model/noise uncertainty [1-5, 9]. In contrast to this, here the problem with structural uncertainty is studied. The overall state estimate is a weighted sum of q partial estimates, generated by a bank of Kalman filters for q models with different structure from the uncertainty domain. At the same time the IMM model probabilities can be used for model order determination.

5 There are various data sources providing traffic data measurements. The most common are the induction loop detectors, which are roadside sensors that provide flow data. However, there are not enough induction loop detectors installed, especially in urban networks, to rely upon and deliver a full traffic state estimation. The main reason is the high installation and maintenance cost for an adequately dense sensor network. The data they provide is also not entirely reliable, as they are also prone to measurement errors (e.g. Briedis & Samuels, 2010 and Martins, 2008). Errors may occur either due to malfunctioning (e.g. in Herrera et al. (2010) it is mentioned that in California 30% of the 25,000 installed induction loop detectors does not work properly) or due to the nature of the sensor. For example, in a dual induction loop detector setup, a vehicle approaches and changes lane, passing over only one of the two loop detectors of that lane, resulting in an erroneous measurement. Additionally, a single induction loop detector setup requires additional assumptions to be made, e.g. for the average vehicle length. Therefore, uncertainty for the speed measurement values provided by such a setup is higher.

State estimation is a mathematical tool in which a set of measurements is combined with an assumed mathematical model of a system so that the resulting set of equations relating measurements to the system are satisfied in the least squares sense. That is, the system states are estimated using an over determined set of equations whose right- and left-hand sides agree with a minimum squared difference. Mathematically, the measurements are arranged in a vector and related to the system states by the vector valued function (of vector valued argument). Attention turns to the development of a three-phase distribution state estimator. The basic design of the estimator is non-iterative, purely real, linear zed and in full phase detail. In the distribution system application, it is assumed that coincident demands (example: smart meter) and selected synchronous measurements will be used. In the formulation below, data shall be represented in complex, rectangular, phasor form. Three-phase unbalanced voltages and current measurements are envisioned along with some branch active and reactive power flow measurements. Measurement vector, the state vector, and the process matrix are complex quantities where the subscripts and refer to real and imaginary components. Note that differentiation with respect to a complex variable is generally non analytic. Therefore is rewritten. Note that residual vector may be separated into real and image.

than the conventional wired alternative; reasonably, some of these issues deduce from the wireless networks struc- tures which we should address them in our design pro- cedure. In order to make a practical sense, the custom- ized wireless network architecture is proposed for this Wireless Networked Control System (WNCS) and the problem of transmission delays and packet losses which induced by this scheme is studied. Meanwhile the time- varying delay of the TCP based shared network is esti- mated by fuzzy state estimation technique ([3-4]). There- after the kalman filtering is applied to address the prob- lem of optimal filtering for this continuous-time system where the observations are communicated to the estima- tor via an unreliable channel resulting in timevarying delays. The filtering problem for NCSs has received much attention during the past few years ([5-7]). In [8] the re-organized innovation analysis approach is pro- posed for linear estimation [8], which is based on projec- tion and a re-organized innovation sequence. This tech- nique is adopted in [9] to address the problem of Kalman filtering for continuous-time systems with timevarying delay. We apply the proposed approach in [9] to tackle the network induced time-varying delay for optimal estima- tion of the states in NCS framework. The simulation re- sults show the applicability of the proposed approach.

Systematic observer design methodologies for invariant systems on general Lie groups have been proposed that lead to strong stability and robustness properties. Partic- ularly, Bonnabel et al. [33–35] consider observers which consist of a copy of the system and a correction term, along with a constructive method to find suitable symmetry-preserving correction terms. The construction utilizes the invariance of the system and the moving frame method, leading to local convergence properties of the observers. Also, [28] extends those constructive methods in order to apply them to a wider class of systems on Lie groups. This leads to development of a so-called Invariant Extended Kalman Filtering approach with provable local stability properties. Methods proposed in [92–94] to achieve almost globally convergent ob- servers. A key aspect of the design approach proposed in [92–94] is the use of the invariance properties of the system to ensure that the error dynamics are globally defined and are autonomous. This leads to a straight forward stability analysis and excellent performance in practice. More recent extensions to early work in this area was the consideration of output measurements where a partial state measurement is generated by an action of the Lie group on a homogeneous output space [33– 35, 92, 93, 102]. Also, [123, 143] develop a rigorous theory for designing minimum energy observers on Lie groups with near optimal performance. Recently, [69, 70] proposed state estimation methods on the specific Lie groups SO ( 3 ) and SE ( 3 ) based on the Lagrange–d’Alembert principle.

Having defined and illustrated the problem of covariant state estimation, we now recall Holevo’s theorem [15, Theorem 3.1], which states that for every loss function that is invariant under the group transformation g, the minimax risk as well as the average risk attain their minima at a covariant measurement. But, the analysis in reference [15] is done for loss functions expressed as functions of the true parameter and the estimator of the parameter. It can be recast in terms of the general framework involving estimators that are functions of the parameter, ρ ˆ : Θ 7→ D(H) by simply choosing the domain of the loss function to be the set of density matrices S (H) as opposed to the parameter space Θ. We thus state it as the following lemma which is the main ingredient of the proof of Result 2.3.

Monitoring of system operation conditions is essential for secure operation of power systems. In monitoring process, system data are acquired from measurement devices, which are distributed in the entire system, and they are transmitted to the control system through communication systems. After that, received data are processed by some computer aided tools called Energy Management System (EMS). State Estimation (SE) is one of the EMS functions which have been known as basis of EMS since it provides creditable data from raw data supplied by measurement devices. Indeed, due to the fact that other EMS functions utilize obtained creditable data, SE should be considered as kernel of EMS.

Power system state estimation is a central component in power system Energy Management Systems. A state estimator receives field measurement data from remote terminal units through data transmission systems, such as a Supervisory Control and Data Acquisition (SCADA) system. Based on a set of non-linear equations relating the measurements and power system states (i.e. bus voltage, and phase angle), a state estimator fine-tunes power system state variables by minimizing the sum of the residual squares. This is the well-known WLS method.

An efficient decoupled dynamic power system model has been described and investigated while based on introducing a transformation of ordinary DAE model using decoupled algorithm. We also used the classical method of E.K.F to real time dynamic state estimation of power systems while including some numerical approximation for the calculation of the Jacobian and which was preceded by a convergence analysis. The results show well the appropriate choice of the dynamic DM in terms of robustness, speed and computing time and, in a very clear way, the high quality of estimation offered by the Modified EKF.

In the discrete event literature, observability has been defined by Ramadge [38], for example, who derives a test for current state observability. Oishi et al. [37] derive a test for immediate observability in which the state of the system can be unambiguously recon- structed from the output associated with the current state and last and next events. ¨Ozveren et al. [22] and Caines [13, 14] propose discrete event observers based on the construction of the current-location observation tree that is impracticable when the number of locations is large, which is our case. Observability is also considered in the context of distributed monitoring and control in industrial automation, where agents are cooperating to perform system-level tasks such as failure detection and identification on the basis of local informa- tion [39]. Diaz et al. consider observers for formal on-line validation of distributed systems, in which the on-line behavior is checked against a formal model [29]. In the context of sen- sor networks, state estimation covers a fundamental role when solving surveillance and monitoring tasks in which the state usually has several components, such as the position of an agent, its identity, and its intent (see for example [17] or [11]).

Currently, micro-electro mechanical systems (MEMS) have shown remarkable popularity in the engineering industry because of their several advantages such as order of magnitude, smaller size, better performance, possibilities for batch fabrication, cost effective integration with electronic systems, and low power consumption [1]. Electrostatically actuated MEMs devices such as, micro actuators [2, 3], mems capacitive microphone [4, 5], sensors [6, 7], capacitive micro-plate [8, 9], micro tunable capacitor [10-12], and micro- mirrors [13, 14] are broadly designed, fabricated, used and analyzed. With the fast growth of micro scale technology, necessity for state estimation of these devices will be taken into consideration for controlling, fault detecting and identifying structures.

consideration at the distribution level [61]. Primarily, this is because measurement data in distribution networks tend to be very scarce and often nonexistent beyond the substa- tions. Moreover, transmission and distribution networks generally have different features. The former ones transfer large amounts of power and are characterized by mesh topolo- gies, high line reactance-resistance (X/R) ratios and a quite limited number of lines and buses. On the contrary, distribution systems typically transfer limited amounts of power, exhibit a radial topology consisting of many nodes with low X/R values and unbalanced loads. These circumstances have changed in recent years, due to the increasing pene- tration of distributed energy resources, which may introduce variability, uncertainty and even instabilities. Consequently, there is a growing interest in Distribution System State Estimation methods based on the joint use of pseudo-measurements and measurements from PMUs possibly using cheaper instrument than those available nowadays [49, 73]. Currently, one of the main state estimation methods also for DSSE is the same WLS- based algorithm, described in Chapter 5. However, to correctly capture the dynamic phenomena that characterize the distribution level some SE solutions rely explicitly on three-phase branch currents [74, 63], linearized models for pseudo-measurements [75], or unsynchronised phasor measurements [76]. The introduction of PMUs has increased the accuracy of SE algorithms, by measuring not only magnitudes, but also phase angle dif- ferences between voltages phasors at different nodes [69]. As explained in Chapter 5, a system is observable if at least so many independent measurements as the number of state variables are used. This suggests that a PMU should be used in each node of the network. Unfortunately, for economic and technical constraints, this is not feasible. Thus, the best solution is to combine traditional and PMU measurements. The inclusion of PMUs in state estimation creates two critical problems [7]:

This paper has presented a distribution state estimation method based on combination of real-time and pseudo measurement devices for conventional distribution system.. It applied state e[r]

Most of the SE algorithms used in industry are based on weighted least square (WLS) approach [1-6]. Even a single bad measurement will distort the estimate, as these algorithms minimize the weighted sum of error squares. In addition, the assignment of relatively larger weightages and round off errors cause numerical problems and might make the system ill-conditioned. Various bad data detection and identification methods have therefore been developed [7-8]. Recently an alternate SE algorithm, which is based on weighted least absolute value (WLAV) minimization technique, has been applied to power system problems [9-10]. Unlike WLS method, there is no explicit formula for the solution of WLAV algorithm but it can be reformulated as a linear programming (LP) problem. The estimate is then obtained by solving a sequence of LP problems. It is well known that this estimator has been capable of automatically rejecting bad measurements, as long as the bad measurements are not leverage points, and hence is found to be more robust than WLS estimator [11]. But this estimator requires large computation time and is not suitable for real-time applications. The need for an efficient algorithm that occupies minimum memory and requires lower computation time has led to the development of fast decoupled state estimation (FDSE) [12-17] based on P   and Q  V decoupling used in fast decoupled power flow. The rate of convergence is strongly influenced by the initial voltage, which some times has a large  and a poor V , and the coupling between P   and Q  V mathematical models. This coupling increases with system loading levels and branch r / x ratios, and consequently the convergence rate has been found to decrease [18]. The decoupled method either fails to provide a solution or results in oscillatory convergence on ill-conditioned power systems [19]. Various formulations based on WLS and WLAV algorithms have been used to obtain SE solutions [20-29].

This dissertation addresses the state estimation problem of spatio-temporal phe- nomena which can be modeled by partial differential equations (PDEs), such as pollutant dispersion in the atmosphere. After discretizing the PDE, the dynami- cal system has a large number of degrees of freedom (DOF). State estimation using Kalman Filter (KF) is computationally intractable, and hence, a reduced order model (ROM) needs to be constructed first. Moreover, the nonlinear terms, external distur- bances or unknown boundary conditions can be modeled as unknown inputs, which leads to an unknown input filtering problem. Furthermore, the performance of KF could be improved by placing sensors at feasible locations. Therefore, the sensor scheduling problem to place multiple mobile sensors is of interest.

How the accuracy of state estimation impacts on the cou- pled model parameter estimation is an interesting and chal- lenging research topic. The spatial and temporal dependence of atmospheric and oceanic circulations could further com- plicate the issue. For example, the Kuroshio meander in the south of Japan is very different to the Kuroshio mean- der across the Luzon Strait. The Kuroshio across the Luzon Strait is easily interrupted by the monsoon, but the mean- der in the south of Japan is a self-sustained dynamic system having multiple equilibria with non-periodic state changes (Taft, 1972; Yu et al., 2013); the uncertainty of the latter comes from the accumulation of the negative vorticities in the ocean. Further, we have already known that the method on a particular frequency can increase the opportunity of success. When such a real problem is addressed through the PE with a CGCM, we may need to make efforts on both adaptive mea- surements and spectral separation. The PE method shall be improved to perform separately at different timescales. How to speed up the convergent rate in the coupled model PE pro- cess is also an important issue. All of these require further research work in order to be clarified.

The parametric categorization represents models in which the principles behind the Lighthill– Whitham–Richards (Lighthill & Whitham, 1955) model, also known as a first order traffic flow and kinematic wave theory model are used. The two principles of LWR are that traffic is modelled and simulated conform: a fundamental diagram and the traffic flow conservation law. These models are therefore based on plausible theoretical assumptions on traffic behaviour in time (Van Lint, 2011). The inputted data consists of e.g. flows, OD-pairs and/or turn rates, for which the models determine macroscopic parameters (e.g. link capacities or parameters related to a fundamental diagram) or microscopic parameters (e.g. car-following behaviour or lane change behaviour). As these models try to incorporate real world car traffic theory such as e.g. queueing theory, car following theory and/or shockwave theory (Van Lint, 2011), it becomes inevitable that vast calibration of parameters is required as to assure that the model results comply with the real-world. Examples of these model types are; Newell’s simplified kinematic wave model (Newell, 1993), cell transmission models (Daganzo, 1994), the variational kinematic wave theory (Daganzo, 2005), link transmission models (Yperman, 2007) and more recently a Lagrangian based approach (Laval & Leclercq, 2013). These parametric based traffic estimation models do come with some limitations. Firstly traffic flow theory represents abstracted versions of real-world phenomena, as for example a fundamental diagram can only be approximated (Seo, 2015). This is due to individual driving behaviour e.g. differences in desired acceleration and deceleration, vehicle types, vehicle lengths, platooning, lane changing and changing traffic states. Determination of a realistic fundamental diagram for each segment of the network is therefore difficult, especially in an urban environment. The second reason is related to this urban environment, as already mentioned, traffic flow conservation law does not necessarily hold, making these models relatively unsuitable for urban traffic state estimation.

Control design, diagnosis and monitoring of power systems become a major concern and an important economic issue during the last decades, in particular, in industrialized countries. Indeed, the main reasons are not only to optimize energy consumption but also to reduce the "Black Out" in vulnerable regions. It is important to observe, preferably in real-time, relevant variables of the power network. To achieve this goal it is necessary to use state estimation techniques or state observers from a database using physical sensors. Indeed, it is very difficult or impossible (for accessibility, technical and/or cost reasons) to measure through hardware sensors the excessive number of state variables in large scale systems. It is therefore important to develop software sensors that can produce a reliable estimate of the necessary variables. Unfortunately, these techniques are often based on a static model of the network around an operating point, and are off line due to large computational requirements. Therefore, the Energy Management Systems (EMS) based on these techniques are, in general, inefficient and lead to poor management of the networks [1].

out as a possible solution a priori. When the pendulum ini- tial conditions are imperfectly known, the range of possible pendulum trajectories expands greatly with time, even if the forcing evolution is perfectly known (background shading, Fig. 1). Normally distributed initial perturbations to the truth with standard deviation of 1 rad s −1 in angular velocity and 0.5 rad in the initial angle lead to a divergence of roughly 200 rad between extreme trajectories (background shading, Fig. 1). The angle is not renormalized when the angle is greater or less than π , and thus the angle records a history of how many times the pendulum has rotated. If no informa- tion about the initial angular velocity is available, a reason- able assumption is that ω = 0 with some large error, but the pendulum trajectory with this initial velocity and the correct initial angle diverges from truth in less than 5 s (first dashed line, Fig. 1). In the case where the initial velocity is known perfectly but the initial angle is observed with an initial error of 0.5 rad (second dashed line, Fig. 1), the trajectory follows truth for 30 s before eventually diverging. As the time inter- val of interest increases, any uncertainty in the initial condi- tions will ultimately lead to a divergence between truth and this first-guess model trajectory. While these sample model trajectories may seem overly naive, the first-guess trajectory used for ocean state estimation usually has similar charac- teristics: usage of an observation at the initial time, some prior knowledge of the forcing, and a freely running forward model.