Outer Power Control Research Articles

Active Disturbance Rejection Control Combined with Improved Model Predictive Control for Large-Capacity Hybrid Energy Storage Systems in DC Microgrids

In DC microgrids, a large-capacity hybrid energy storage system (HESS) is introduced to eliminate variable fluctuations of distributed source powers and load powers. Aiming at improving disturbance immunity and decreasing adjustment time, this paper proposes active disturbance rejection control (ADRC) combined with improved MPC for n + 1 parallel converters of large-capacity hybrid energy storage systems. ADRC is utilized in outer voltage control loops, and improved MPC is employed in inner current control loops of n battery converters. Droop control is adopted to obtain power distribution between n battery converters, and a DC bus voltage compensator is used to compensate voltage deviations and maintain constant DC bus voltage. The low-pass filter (LPF) is adopted to obtain high-frequency power as the reference for the supercapacitor converter, ADRC is also utilized in the outer power control loop, and MPC is employed in the inner current control loop. Compared with traditional observers, the voltage expansion state observer of the proposed ADRC control is independent of the system model and parameters and consequently has strong disturbance immunity, and significantly reduces voltage overshoots during power fluctuations. The MPC-based inner current control loops of n + 1 converters accelerate current response speed and significantly decrease switching losses. Simulation and experimental results indicate that utilizing the proposed control strategies, large-capacity HESS has stronger anti-interference ability, shorter regulation time, smaller switching loss, and simultaneously maintains the stability of the DC bus voltage.

Small‐signal rotor angle stability of multi‐virtual synchronous machine (n‐VISMA) microgrid

AbstractAutonomous microgrid is known to lack appropriate inertia and damping for grid stabilization. Due to this, virtual synchronous machine (VISMA) has been introduced to provide necessary ancillary services through control of power converters. In a multi‐VISMA (n‐VISMA) microgrid, relative rotor angle stability of the power system is dependent on the active power balance after small perturbation. Thus, the use of relevant analytical models are essential issues for microgrid stability analysis. This paper presents a comprehensive small‐signal stability analysis to study inherent electromechanical oscillations in the virtual rotors. The subsystems of the microgrid consisting of VISMA, network, load and the outer power control were all modelled in Synchronous Reference Frame. The small‐signal model (SSM) was tested on IEEE‐9 bus system with VISMA replacing electromechanical synchronous machines on the network. To validate the developed numerical analytics, dynamic responses of the SSM are compared with those of the non‐linear (NL) system dynamics and the results reveal that the developed linearized SSM is sufficient to accurately characterize behaviour of the VISMA microgrid when operated in autonomous mode. Eigenvalues analysis and parameter sensitivities of the critical modes were investigated. Oscillatory participations of the VISMAs and steady state stability limit of the microgrid have also been investigated.

Large-Signal Stability Analysis for Islanded DC Microgrids with n+1 Parallel Energy-Storage Converters

In islanded DC microgrids, the negative impedance characteristics of constant power loads (CPLs) usually introduce instability influences; on the contrary, hybrid energy-storage systems (HESSs) constituted of batteries and supercapacitors (SCs) have stabilization advantages. To guarantee the large-signal stability of islanded DC microgrids with n+1 parallel energy-storage converters, an equivalent model is first constructed based on the control strategies of the converters. Then, according to the mixed potential function theory, a large-signal stability criterion, considering powers, inductors, capacitors, the DC bus voltage, the equivalent internal resistances of batteries, the proportional parameters of the inner current loop of n battery DC–DC converters, the proportional parameter of the outer power control loop of the SC DC–DC converter, and the proportional parameter of the inner current loop of the CPLs, is derived. Furthermore, the proposed large-signal stability criterion is optimized via the use of droop control for n battery converters, and coefficients related to the droop coefficients are also taken into account. These involved control parameters reveal the process of regulating the HESS and CPLs instead of ideal modeling and significantly reduce the conservatism of the criterion to some extent. In addition, on the basis of the large-signal stability criterion presented herein, the maximum CPL power that the islanded DC microgrids can stably support is obtained. Finally, simulation and experimental results verify the validity of the provided large-signal-stability criterion. The given procedure of analyzing large-signal stability is more consistent with planning and operating actual DC microgrids.

SOC Balancing Control Based on Predictive Power Model Amongst Supercapacitor Packs in MMC With Embedded Energy Storage System

Modular multilevel converter with supercapacitor (SC) packs-based energy storage system (MMC-SESS) can play a role in energy transition and renewable energy consumption. However, SESS's imbalance problem is caused by the inconsistency of its packs, which has a negative effect on the utilization of the MMC-SESS and has become a hot research topic. Confronting this problem, this paper proposes a state of charge (SOC) balancing control strategy amongst SC packs based on a discrete time-domain predictive power model in MMC-SESS based medium-voltage direct-current (MVDC). It is a double-loop control to accurately control the charging and discharging power. The outer power control establishes a discrete time-domain predictive power model for the SOC differences. The inner current loop designs a hybrid controller based on the continuous control set-model prediction control (CCS-MPC) and the feedback control strategy to accurately track the dynamic current reference. Simulations in PSCAD/EMTDC have shown that the proposed control can achieve SOC balancing amongst SC packs and improve the dynamic response as well as mitigate the SC pack current ripples.

Grid Interconnection Modeling of Inverter Based Resources (IBR) Plant for Transient Analysis

The increase in penetration levels of inverter-based resources (IBRs) is changing the dynamic performance of power grids of different parts of the world. IBRs are now being more and more integrated into the grid at a single connection point as an IBR plant. Due to the complex nature and dynamicity of each inverter model, it is not realistic to build and analyze full complex models of each inverter in the IBR plant. Moreover, simulating a large plant including detailed models of all the IBRs would require high computing resources as well as a long simulation time. This has been the main issue addressed in the new IEEE Std 2800-2022. This paper proposes a novel approach to model an IBR plant, which can capture the transient nature at the plant level, detailed IBR control at the inverter level, interactions of multiple IBR groups in a plant structure, and a collector system connecting the IBRs to the grid. The IBRs in the plant use a voltage source inverter topology combined with a grid-connected filter. The control structure of the IBR includes a cascaded loop control where an inner current control and outer power control are designed in the dq-reference frame, and a closed-loop phase-locked loop is used for the grid synchronization. The mathematical study is conducted first to develop aggregated plant models considering different operating scenarios of active IBRs in an IBR plant. Then, an electromagnetic transient simulation (EMT) model of the plant is developed to investigate the plant’s dynamic performance under different operating scenarios. The performance of the aggregated plant model is compared with that of a detailed plant model to prove the effectiveness of the proposed strategy. The results show that the aggregated EMT simulation model provides almost the same result as the detailed model from the plant perspective while the running time/computation burden is much lower.

Decoupled fractional-order harmonic filter for power quality enhancement of grid-connected DG units

The global expansion of power electronics interfaced distributed generator (DG) units has led to the incrementation of nonlinear loads in the distribution systems, which may impose negative effects on the power quality. To maintain the desired power quality while actively compensating the harmonics in the presence of frequency deviations, this work proposes a decoupled closed-loop fractional-order current controller that incorporates the harmonic alleviation with the DG power generation function. Accordingly, an optimal fractional-order proportional–integral–derivative (FOPID) is designed to control the DG unit fundamental current, while compensating the local nonlinear load harmonic current (LHC) effects. A noteworthy contribution of this paper is that the optimal control scheme delivers the desired performance without the need for LHC detection and effectively reduces the operational cost. Another contribution is the implementation of an optimal FOPI-based control scheme in the outer power control loop to extract the fundamental current reference with no phase-locked loop (PLL) components. This offers comprehensive solutions with several improvements, such as eliminating the power tracking steady-state error under fundamental current tracking errors and preventing harmonic disturbances from being imposed on the harmonic control by the power control part. The feasibility and effectiveness of the proposed power control scheme are validated in comparison with the conventional PI approach.

Control of single-phase photovoltaic H6 inverter in grid-connected and stand-alone modes of operation

This paper proposes a control strategy for single-phase transformerless photovoltaic H6 inverter with capability of operation in both grid connected and stand-alone modes. This control strategy enables seamless transition between two modes without any reconfiguration of control structure and need of any external communication. A multi-loop control structure including an inner voltage and current control with PR controller and outer power control with droop controller is utilised to make the inverter operate in both modes. In grid-connected mode, the power control loop regulates reference voltage and frequency for voltage control loop, to adjust power flow of inverter to the main grid while in islanded mode, reference voltage and frequency are determined according to droop characteristic of inverter to supply local load demand. The simulation and experimental result are used to verify the performance of the proposed control.

Comprehensive Modeling, Analysis, and Comparison of State-Space and Admittance Models of PLL-Based Grid-Following Inverters Considering Different Outer Control Modes

The phase-locked loop (PLL)-based grid-following inverters have been widely used as the interface between renewable energy resources (e.g., wind and solar power) and the utility grid. In addition to the basic PLL and inner current control loop that are responsible for tracking the phase angle of the point of common coupling (PCC) voltage and regulating the current injection according to the estimated phase angle, respectively, these grid-following inverters are commonly also equipped with outer power and voltage control loops. Some examples are outer active power/dc-link voltage control loops and outer reactive power/PCC voltage control loops, which provide the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dq$ </tex-math></inline-formula> -axis current references for the inner current control loop. However, the existing literature generally focuses on specific outer control modes within a specific small-signal modeling framework, e.g., state-space or admittance modeling framework. In addition, different research groups usually repeatedly derive these small-signal models without a systematic modeling and analysis procedure. Furthermore, the systematic and comprehensive comparative modal and admittance analysis of these outer control modes is still missing in the literature. To avoid reinventing the wheel, provide the small-signal stability analysis community with handy both state-space and admittance models, and investigate the effects of various outer control loops on the small-signal models, this article presents a comprehensive and systematic small-signal modeling, analysis, and comparison framework of the PLL-based grid-following inverters, where the state-space and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dq$ </tex-math></inline-formula> -domain admittance models of eleven control modes are derived step by step in a completely followable style. All of these derived small-signal models are verified by the time-domain step responses and frequency-domain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dq$ </tex-math></inline-formula> -domain admittance measurement. The insights into the effects of these outer control modes on the poles and residues of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dq$ </tex-math></inline-formula> -domain admittance model are also explored, which help to understand how these outer control modes contribute to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$dq$ </tex-math></inline-formula> -domain admittance shaping.

Research on Direct Power Control Strategy Based on Voltage Controlled Virtual Synchronous Generator

To support the “carbon peak and carbon neutrality” goal, new energy is poised to explode, and new energy power generation converter is simultaneously facing new challenges. The conventional current-controlled new energy converter can quickly transmit active power on the DC bus to the power grid. However, for the weak grid, the stability margin of the converter grid-connected system is reduced on the one hand, which can easily cause resonance oscillation; on the other hand, the current controlled converter cannot actively respond to system frequency and voltage fluctuation to offer support. The voltage controlled virtual synchronous generator (VVSG) is used to improve system small signal stability and frequency stability; however, its power response speed is too slow to meet the requirements of fast following power command. Although a voltage/current dual-mode switching control scheme is put forward to achieve characteristics complementary of current controlled converter and voltage-controlled converter, the control structure switching and intermediate variable following is required to realize mode switching, which is prone to large power shocks and switching failures. In view of the problem, a direct power control strategy based on VVSG is proposed. The control structure is raised based on a conventional VVSG outer active power control loop, and the output active power and frequency characteristics are analyzed. Compared with the voltage and current dual-mode control, VVSG with direct power control can perform large inertia characteristic in the weak grid and fast power following characteristics in the strong grid by adjusting λ, and without control structure switching and intermediate variable following. Moreover, the two characteristics can be smoothly transited. In addition, the active support ability of voltage source can be maintained under both characteristics. Finally, the effectiveness of the proposed control strategy is verified through simulation results.

Maximum power extraction from a switched reluctance generator based wind power generating system using optimization techniques

AbstractThis article advocates the use of two different optimization techniques, namely, gravitational search algorithm (GSA) and particle swarm optimization (PSO), to obtain suitable combinations of firing angles for the asymmetric half bridge converter (AHBC) to ensure optimum power extraction from a switched reluctance generator (SRG) based wind power generating system, at any given wind speed. As optimum power generation from an SRG turns out to be a multi‐dimensional search problem due to the nonlinear magnetic behavior of the machine, optimization techniques can be aptly employed for obtaining suitable values of control parameters. The techniques considered in this work, GSA and PSO, do not require any information about the geometrical features of the machine. Both techniques are implemented with an outer power control loop to propel the machine to perform optimally at variable wind speeds. Simulated results obtained by modeling these techniques in MATLAB environment are validated with the experimental results from a laboratory prototype consisting of a 1 hp, 2000 rpm, 8/6 SRM, and an AHBC. The results presented bring out the relative merits and demerits of the optimization techniques, thus highlighting their suitability for actual field implementation.

An improved control strategy for current source converter-based HVDC using fundamental frequency modulation

When pulse width modulation (PWM) is applied in a high-voltage current source converter (CSC) based on fully controlled devices, there are some concerns arising, such as large DC voltage fluctuations, high switching losses, and a narrow power operation range. To solve these problems, the CSC-based HVDC using fundamental frequency modulation (FFM-CSC) is proposed. First, the topology of the CSC is introduced. Then, an independent control strategy for FFM-CSC is presented, which can be used to realize univariate control of DC voltage or DC current. The mathematical model of CSC is also analyzed, and on this basis, a coordinated control strategy is proposed for FFM-CSC to decouple the active and reactive power. An outer loop power controller and an inner loop current controller are designed, and the power operation range is investigated. Finally, an FFM-CSC simulation model is built in PSCAD/EMTDC. The simulation results show that there are smaller DC voltage fluctuations, lower switching losses, and a wider power operation range under the proposed FFM-CSC control strategy.

Investigation of Damping Effect of PLL on Low-Frequency Harmonic Stability of Grid-Tied Inverter With αβ and dq Current Control Schemes

The choice of phase-locked loop (PLL) bandwidth in three-phase grid-tied inverters (GTIs) is always a tradeoff between robust performance under distorted grid conditions and fast detection of grid voltage phase angle during the low-voltage ride through. Besides, even though both the GTI with decoupled <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> -frame (CC1) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha \beta }$ </tex-math></inline-formula> -frame (CC2) current control schemes employ PLL for grid synchronization, and damping effect of PLL, which is a function of its control bandwidth, is not identical. As a consequence, the output impedance of GTI with CC1 and CC2 has diverse interaction scenarios with the weak grid impedance. Therefore, this work investigates the impact of PLL bandwidth on the output impedance of GTI in the two different current control schemes using impedance models in a common frame of reference ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> -frame here). Negative damping region exhibited by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula> -axis impedances are identified. It is found that CC2 has a less significant negative damping region, while that of CC1 widens with the increase in PLL bandwidth. Furthermore, the side effects of dissimilar damping regions on inverter-grid system stability are demonstrated through the impedance interaction analysis. The output impedance shaping caused by voltage feedforward and outer power control loops on CC1 and CC2 is also investigated. Finally, a compensator is designed to overcome the phase lag introduced by PLL and voltage feedforward loop. The above findings are validated through MATLAB/Simulink simulations and experiments.

Analysis of Harmonic Resonance Characteristics in Grid-Connected LCL Virtual Synchronous Generator

The virtual synchronous generator (VSG), which emulates the essential behavior of the conventional synchronous generator, has attracted great attention. This paper proposes to analyze the harmonic resonance characteristics in VSG using the state-space model. The analysis is based on a full-order state-space small-signal model that fully considers the dynamic of the inner loops and the VSG-based outer power control loop. Participation analysis is used to point out the contributions of different states to the eigenvalues. Moreover, eigenvalue locus and singular value decomposition (SVD) are applied together to evaluate the impact of the inner loop parameters on the harmonic resonance characteristics around the LCL filter resonance frequency. The analysis indicates that the harmonic resonance instability is mainly caused by decreasing the proportional gains of the current loop and the voltage loop. Finally, extensive numerical simulation and experimental results are given to verify the validity of the theoretical analysis. Both the simulation and experimental results indicate that the voltage of the common coupling point is unstable after decreasing the proportional gains of the current and voltage controllers. As Kpc decreases from 5 to 0.4 or Kpv decreases from 0.6 to 0.2, the harmonic distortion factor (HDF) around the LCL filter resonance frequency increases. Furthermore, the consistency of simulation results, experimental results, and the theoretical analysis results is validated.

Controller Parameters Optimization for Multi-Terminal DC Power System Using Ant Colony Optimization

Voltage source converter based multi – terminal DC grids (VSC – MTDC) are widely used for integration of renewable resources. Control of these grids requires vector control method, which includes inner and outer current controllers, tuning of these controllers is very important. Conventionally used tuning methods consider approximated linear model to tune the proportional – integral (PI) parameters of voltage source converter (VSC) which fails to produce optimum disturbance rejection results. In this research an Ant Colony Optimization (ACO) technique is used to get the optimum parameters of inner and outer power control layers for multi – terminal DC system. ACO is applied simultaneously on inner and outer control layers and results are compared with classical tuning method and well established meta - herustic technique, Particle Swarm Optimization (PSO) using MATLAB software. Dynamic models of VSC – MTDC are developed in PSCAD/EMTDC to exalt the qualities of the presented tuning method. In this paper, a four terminal VSC – MTDC has been used under different sequence of events of disturbances e.g. load change on AC side, active power reference change for inverter and rectifier mode and disconnection of one terminal to evaluate the robustness of ACO algorithm with respect to classical method. The proposed tuning method gives superior results under different disturbance compared to conventional method.

Transient Stability of Single-Loop Voltage-Magnitude Controlled Grid-Forming Converters

This letter presents a design-oriented analysis on the transient stability of grid-forming (GFM) converters using single-loop voltage-magnitude control scheme. It is revealed that the used voltage-magnitude controller has a critical impact on the transient stability. This differs from the GFM converters using the vector-voltage control, whose transient stability is dominated by the outer power control loops. The theoretical findings are verified by experimental tests.

Small-Signal Stability Analysis for Microgrids Under Uncertainty Using MALANN Control Technique

Microgrids are often considered as the solution for affordable and clean energy in the distribution sector. This article presents the small-signal stability analysis of a distributed generation unit in an autonomous microgrid operation. The purpose of the proposed strategy is to optimally improve the capacity of the power system to restore the reasonable operating condition following a small physical disturbance. The proposed strategy is the joined execution of both the modified antlion optimization algorithm (MALO) and artificial neural network (ANN), and hence it is abbreviated to MALANN. In this article, the proposed controller comprises two control loops, namely the inner current control loop and the outer power control loop. The MALO technique is incorporated to generate the dataset of possible proportional integral (PI) gain parameters. By using the accomplished dataset of MALO, the ANN is trained, and convincing estimate execution is brought out through the entire machine working condition. The proposed strategy is implemented in MATLAB/Simulink, and the results are examined with two test cases and compared with various solution techniques such as base method and ant-lion optimization. The results prove that the stability analysis is reasonably accurate, and the controller offers a reliable system's operation.

Cascaded Droop and Inverse Droop Regulation for Two-Layer Coordinated Power Flow Control in Series-Connected Power Cells

To interconnect string power cells into high voltage grid in a distributed manner, a cascaded power flow regulator is proposed for simultaneous control of the injected power to the grid and flexible power sharing among string power cells. First, the traditional frequency and voltage magnitude droop control at the central controller behaves as the outer power control loop for an accurate regulation of the system power flow to main grid. Then, by setting the frequency reference and voltage magnitude reference from outer power control loop as base values for each power cells local controller, inner voltage magnitude droop for apparent power S control and the frequency inverse droop for P/E control are implemented at each power cell local controller to achieve well-decoupled real and reactive power sharing among series-connected power cells. With this control architecture, the power cells and the entire grid-tied system can obtain the flexible power regulation in a fully voltage control manner. Accordingly, seamless grid-tied to islanding operation mode transfer and power sharing of the system with other converters in autonomous islanding operation can be easily obtained.

Effect of Reactive Power Characteristic of Offshore Wind Power Plant on Low-Frequency Stability

Oscillation phenomena of offshore wind power plant (OWPP) in a wide frequency range can be caused due to impedance interactions between grid-connected inverters (GCIs) and transmission cables. In this article, impedance model of GCI with outer power control loop, inner current control loop and phase-locked loop is first established in dq reference frame. The correctness is validated by frequency scanning method. Then, the effects of active and reactive power/current references on dq impedance characteristics of GCI with/without consideration of power control loop are investigated using complex space vectors and complex transfer functions. Furthermore, RLC circuit model of transmission cable considering frequency-dependent characteristics is also established for dq-domain IBSC. On the basis of them, it's found that low-frequency oscillation phenomena of OWPP under power control mode may occur if active power reference exceeds a certain threshold value, which can be mitigated by injecting a certain amount of negative reactive power. Impacts of PLL parameters, length of transmission cable and number of paralleled GCIs on required negative reactive power for low-frequency stabilization are further investigated. Both Matlab/Simulink-based simulation and OPAL-RT-based real-time verification are implemented in an OWPP with four permanent magnet synchronous generators to validate the correctness of the reactive power characteristic analysis results and the feasibility of mitigating low-frequency oscillation phenomena by negative reactive power injection.

Minimum Short Circuit Ratio Requirement for MMC-HVDC Systems Based on Small-Signal Stability Analysis

This paper determines the minimum short circuit ratio (SCR) requirement for a modular multilevel converter based high-voltage direct current (MMC-HVDC) transmission systems. Firstly, a simplified model of MMC is introduced; the MMC is represented by its AC and DC side equivalent circuit. Next, by linearizing the MMC subsystem and the DC network subsystem, the deduction of the small-signal models of MMC subsystem, the small-signal model of the DC network and MMC-HVDC are carried out successively. Thirdly, the procedure for determining the minimum SCR requirement of MMC-HVDC is described. Finally, case studies are performed on a two-terminal MMC-HVDC system under four typical control schemes. The results show that the restraint factors for the rectifier MMC is predominantly the voltage safety limit constraint, and the restraint factors for the inverter MMC are mainly the phase locked loop (PLL) or the outer reactive power controller. It is suggested that the minimum SCR requirement for the sending and the receiving systems should be 2.0 and 1.5 in the planning stage.

Analysis of Sub-Synchronous Band Oscillation in a DFIG System With Non-Smooth Bifurcation

With the interconnection of large-scale wind power, the sub-synchronous oscillation (SSO) which is mainly analyzed by the local small-signal method occurs in many power grids. This paper presents the phenomenon of the sub-synchronous band oscillation (SSBO) of the doubly-fed induction generator (DFIG) system due to limits, which belongs to global dynamics, and analyzes its relationship with certain limits from the viewpoint of non-smooth bifurcation, which can serve as one of the mathematical foundations of the SSBO. Specifically, the concept of non-smooth bifurcation, a power system interconnected with the DFIG and the related current limit (CL) of d-axis and q-axis in the outer power control loop (OPCL) of the DFIG’s rotor side converter (RSC) are introduced first. Secondly, the SSBO phenomenon induced by the CL of d-axis and q-axis is presented. And then, the influence of one parameter on the SSBO is analyzed, such as the short circuit capacity, the fault grounding resistance and the PI parameters of the OPCL, which reveals the non-smooth bifurcation with respect to relative one parameter. Finally, on the plane, the non-smooth bifurcation characteristics of PI controller parameters in the OPCL are given, showing that the non-smooth bifurcation/SSBO may be associated with saturation of different limits.