Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 633 https://internationalpubls.com A Mathematical Framework for Simultaneous Voltage and Frequency Regulation in Distributed Generator (DG) Grid-Interfaced Systems Chandra Sekhar Mishra1, Dr. Ranjan Kumar Jena2, Dr. Asit Mohanty3, Dr. Prakash K Ray4, Dr. Pragyan P Mohanty5, Dr. Sunil Kumar Gupta6 1,2,3,4Department of Electrical Engineering, Odisha University of Technology and Research, Bhubaneswar, (India) 5Department of Mechanical Engineering, Veer Surendra Sai University of Technology Burla, (India) 6Department of Electrical Engineering, Poornima University, Jaipur (India) Article History: Received: 22-04-2024 Revised: 11-06-2024 Accepted: 26-06-2024 Abstract: Recent advancements in distributed generation (DG) systems interfaced with microgrids necessitate robust regulatory mechanisms to manage inherent power fluctuations, particularly from renewable sources like photovoltaics. These fluctuations can significantly impact the stability and efficiency of microgrids. This paper introduces a novel mathematical framework for simultaneous voltage and frequency regulation, aimed at addressing power quality and stability challenges in DG-grid interfaced systems. Utilizing a combination of algebraic topology and dynamical systems theory, we develop a model that incorporates an adaptive virtual frequency-impedance control loop. This mathematical approach allows for the analytical examination of the stability properties of the system and the design of control strategies that guarantee optimal operational thresholds. We extend the conventional droop control mechanisms with a rigorously defined Simultaneous Voltage and Frequency Correction Scheme (SVFCS), providing a theoretical underpinning that supports experimental observations. The efficacy of the proposed model is validated through numerical simulations that demonstrate adherence to the IEEE 519 standard, ensuring reduced harmonic distortion and enhanced system reliability. Our results highlight the potential for these mathematical methods to provide foundational insights into the control and optimization of microgrid operations. Keywords: Mathematical Modeling, Distributed Generation, Microgrid Stability, Voltage Regulation, Frequency Control, Dynamical Systems, Algebraic Topology. NOMENCLATURE MG Micro-grid T Clark's transformation matrix PV Photovoltaic DG Distributed Generator VSI Voltage Source Inverter CSI Current Source Inverter (Current fed) SVFCS Simultaneous Voltage and Frequency Correction Scheme SPWM Sinusoidal Pulse Width Modulation FACTS Flexible Alternating current transmission system HAPF Hybrid Active Power Filter SVM Space vector modulation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 634 https://internationalpubls.com 1. Introduction A microgrid can be conceptualized as a localized grid system capable of disconnecting from the main grid and operating autonomously. This configuration not only enhances reliability in face of main grid instability but also facilitates localized control over power generation and distribution, often incorporating distributed generators (DG). The structural advantage of a microgrid lies in its ability to mitigate systemic risks during grid maintenance and provide resilience against external disruptions such as severe weather conditions or power outages. The operational independence of microgrids is mathematically intriguing due to the complex interplay of power generation, load management, and stability controls, which are often governed by renewable and conventional energy sources. The point of common coupling (PCC) serves as a critical node for maintaining the voltage and frequency alignment with the central power grid, requiring sophisticated control strategies to manage these parameters effectively. For islanding operations, droop control mechanisms are implemented to facilitate decentralized power sharing, leading to the development of the renowned Q-U (reactive power-voltage) and P-ω (active power-frequency) control models [3]-[5]. These models describe the relationship between line parameters and operational controls and are vital for theoretical analysis. Virtual parameters are also used to decouple converter control from direct line interactions, enhancing load sharing capabilities without compromising system stability [6]-[7]. The implementation of maximum power point tracking (MPPT) algorithms optimizes the efficiency of photovoltaic DG units [8]-[11]. Advances in semiconductor technology and the shift towards transformer-less topologies have significantly reduced system costs and improved reliability but have introduced complexities in control mechanisms. The mathematical formulation of these technologies is critical for understanding their impact on system dynamics and stability. Power quality improvement, another key aspect of microgrid operations, involves devices like passive filters, shunt and hybrid filters, and STATCOMs. The control strategies, including sinusoidal pulse width modulation (SPWM) and space vector modulation (SVM), are mathematically modeled to mitigate harmonics in accordance with the IEEE-519 standard. These models are essential for designing filters and compensators that address issues of load imbalance, harmonics, and poor power factor in a cost-effective manner. Further mathematical exploration is seen in the development of dynamic voltage restorers (DVR), static VAR compensators (SVC), and DSTATCOMs, which are designed to compensate for power quality issues through cooperative control mechanisms [41-47]. These controls are often modeled as hierarchical systems that require precise timing and coordination, challenging to implement due to the intermittent nature of power fluctuations. The Simultaneous Voltage and Frequency Correction Scheme (SVFCS) introduced in this paper leverages a virtual impedance loop to address these challenges, offering a novel mathematical approach to microgrid control without the need for extensive communication infrastructures. In conclusion, the mathematical models and control strategies discussed here form the theoretical backbone for enhancing microgrid performance and reliability. The ongoing development of these models reflects a significant area of research in the mathematical analysis of power systems, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 635 https://internationalpubls.com emphasizing the need for robust, scalable solutions to ensure sustainable and efficient energy distribution. That’s why the said SVFCS topology is very practical regarding DG grid interfaced system where the system is prone to rapid power fluctuation. The exploration of microgrid dynamics extends to the analysis of transient behaviors and steady-state responses within these decentralized networks. Advanced mathematical tools such as nonlinear dynamical systems theory and algebraic topology are applied to examine how microgrids react to changes in load and generation capacity. These tools facilitate a deeper understanding of stability margins and the bifurcation behavior of microgrids under different operational conditions. Particularly, the implementation of sophisticated algorithms like MPPT and the design of control strategies based on SPWM and SVM can be rigorously analyzed using perturbation methods and Lyapunov functions. These mathematical approaches are crucial for proving the stability and optimizing the performance of microgrids, ensuring that they not only meet current standards such as IEEE-519 and IEC 61000-3-2 but also adapt effectively to evolving grid conditions and renewable integration challenges. Such comprehensive mathematical analyses serve as the cornerstone for the next generation of microgrid technology, paving the way for smarter, more resilient energy systems [40-47]. The manuscript focused on following points: Droop control mechanism for PV based DG with VSI shown in section II. Simultaneous voltage and frequency corrector suggested overwhelming frequency and voltage imbalance generated by DG – Grid interface discussed in section III. Microgrid implementation based on DG Discussed in section IV & V followed by results and verification over proposed control mechanism. Brief conclusion listed in section VI. 2. Modeling of CCVSI in Grid tied DG System A. Grid tied DG System subjected to Droop control mechanism: The Controlling parameters like voltage current etc. can be sensed by the respective CTs and PTs from point of common coupling. - - + 1DCV 2DCV fDCV Re Voltage Controller SourceV LoadI ClarkV ClarkI V V I I NullI P Q C a lcu la tio n B lo ck Butter BUTTERWORTH FILTER + + _ POSC C o m p en sa ti n g C u rr en t C a lc u la ti o n B lo ck In v er se C la rk T ra n sf o rm a ti o n M U X In ve rt er  3 G A rr ay P V C u rr e n t R e g u la to r Convert Not Convert Convert Not Convert Convert Not Convert A B C Smoothing Reactor a b c To Grid LossP B a tt e ry 1DCV 2DCV Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 636 https://internationalpubls.com Fig. 1. Pictorial representation on Grid tied DG with Battry The above figure depicts the suggested strategy for DG GRID interface with Battery energy storage. The interconnected DGs fed power to the PCC and they are generally coupled with droop control mechanism in islanded mode. The equivalent circuit depicted in figure 2 represents 2 interconnected inverters in parallel shared by a common burden. The terms )2,1(),2,1( == xUxE xx revels DG inverter voltage output and Bus voltages. Similarly )2,1( =x x  )2,1( =xZ x revels DG inverter phase angle of voltage and impedance and θ, revels the phase angle. Fig. 2. Interconnected DG CCVSI Equivalent circuit Active power, Reactive power for the inverter of 1st DG will 1 1 2 11 1 1 1 cos)cos( .  Z U Z UE P −−= (1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 637 https://internationalpubls.com 1 1 2 11 1 1 1 sin)sin( .  Z U Z UE Q −−= …….……… (2) The output voltage and branch impedance can affect the CCVSI voltage. Mostly, the impedance may be subdivided as a sum of feeder impedance and Zoutput. The impedance can be of a virtual type and may a treated as fully resistive or inductive type. feederout ZZZ += ………………..…. (3) If the ZFeeder is small enough then the effective impedance can be equivalent as the Z. For reactive loads the system impedance )( outZ may be written as: 1 1 1 1 sin .  X UE P = ……………… (4) 1 2 11 1 cos. X UUE Q − =  ………………… (5) 1 X can be the inductive reactance of voltage output. If 1  is lesser compared to other parameters )1cos,(sin 111   , then 11 EP = Similarly P-ω and Q-U droop mechanism can be expressed as follows with p m and q m are agreeing droop gains; mathematically Pm pn −= * (6) QmUU pn −=* (7) The control mechanism of above droop control scheme is as follows. The figure 3 shows a droop controller: Fig. 3 Droop control mechanism. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 638 https://internationalpubls.com The Droop gain may effect on stable operation and dynamic stability. Therefore on the basis of the droop gain the system stabilizes and may results to have decent power sharing. But it may lead to deviation on system voltage and frequency [33] [9]. Therefore, it is recommended to talk on accuracy in Voltage and frequency relation and should take care while adjusting droop gains. It can be treated as a compromise between V/f. In this present work p m (droop gain) can be a fraction of max (maximum frequency) and maxP (Variation in active power). Mathematically: max max max P mp   =  (8) The variation in voltage obstructed by drop in voltage through System impedance (Z) at output and maxmax ,QU droop control. That’s way droop gain max max0max max Q ZIU m q  − = (9) In some circumstances the system impedance may be foremost, maxqm may be approximated as follows [9]: max max0max max Q ZIU m virtual q  −  (10) 3. Modelling of Photo Voltaic based Micro-grid in islanded mode: Output Voltage modeling: It is known that Micro-grid in islanded mode consists of DG with CCVSI. The DG is a Photovoltaic based distributed generator with P/O MPPT. For easier calculation point of view the CCVSI assumed to have same capacity. The output voltage of CCVSI be governed by: )(. 1 * pvLqnd QQm n UU −−= (11) 0 * = q U (12) )(. 1 * pvLpn PPm n −−= (13) where: *d U → ”d” axis voltage *q U → ”q” axis voltage * → Reference angular frequency n U & n  → minimal values over that frequency and voltage. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 639 https://internationalpubls.com qp mm & → droop gain (active and reactive) LL QP & → True & Reactive Power of the load PV P → DG True power PV Q Is the DG reactive power The droop controller have several dis advantages like, it will affect Bus frequency and voltage in line for the presence of CCVSI. In equation (11), (12) and (13) it is seen that parameters related to droop control and yield power results voltage over CCVSI. It results voltage and frequency variation form the nominal value and cause power quality issue. This fluctuation in power may lead to variation in ether DG power or load power and the PV based DG is certainly an unpredictable means of Energy. Therefore it is essential to have a limit on the above-mentioned effects for maintaining AC bus voltage and frequency. This paper focuses on above aspects. The CCVSI Controller is revealed in Figure 4. Where 0, , , ,f f vir virL C R L  can be the filter inductance, capacitance, virtual Impedance and angular frequency respectively. Voltage transfer function is ( ) and ( )Gv s Gi s can be the current transfer function of respective PI controllers. ),( oqod UU can be the direct axis and quadrature axis feedback signals. Similarly and od oqI I can be the direct axis and quadrature axis feedback inverter signals currents. Other currents like , , ,Ld Lq od oqI I I I oqI are introduced for improvement of power handling capability. Similarly other voltages like ,od oqU U are introduced for improvement of power handling capability and regulation. Fig. 4. CCVSI Controller. Now we can write the droop control equation as follows: G(S) and Z(S) can be the voltage gain and output matrix impedance. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 640 https://internationalpubls.com 1 ( ) ( ). ( ) ( ). [ ( ).] ( ) ( ) 0 od odn q L PV oq oq U s I sU m Q Q G s Z sn U s I s      − − = −           (14) 2 ( ) ( ) ( ) ( ) 1 vi PWM f f f i PWM vi PWM G s K G s L C S C G s K s G s K = + + + (15)   0 0 g ( ) ( ). ( ). R vir g vir vir vir R R L Z s G s G s L L   + −  =   +   (16) Everywhere ( ) , ( ) ( ) ( ) ( ) ( ) i PWM f g vi v i v i PWM G s K sL R G s G s G s G s G s K + = = and )(. 1 * pvLpn PPm n −−= the angular frequency of CCVSI is interrelated with nPm PVp and , Such other factors like   oqodqPV IIsZsGmQ ,,)(),(,, also affect the system performance. The listed drawback can be overwhelmed by adaptive regulation of CCVSI. The p m (droop gain) should set as small as possible as it affect the deviation in system frequency and latter it affect the system active power demand. This can be mitigated by very small adjustment of droop gain p m . Similarly by adjusting droop gain q m reactive power variation can be compensated. This tuning may lead low reactive power distribution. It can be noted that U-Q droop control mechanism realization is very difficult in micro-grid [6], [9], [10]. In this literature focuses on adaptive impedance topology for restoration of voltage and simultaneously it takes care for the droop gain q m . 4. Simultaneous Voltage and Frequency Corrector for Grid Tied Dg System A. Adaptive impedance compensation topology In this topology another virtual loop is used for generation of adaptive virtual impedance. Basically it is operated in 2 modes. Firstly making ready the offline CCVSI that to be connected to the PCC and secondly online CCVSI for compensation purposes for the power quality issue of the grid. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 641 https://internationalpubls.com Fig.5. Proposed controller model. Mode 1: in this mode of operation PI Controller output should be adjusted with in a hysteresis bound to overcome –ve effect at the time of closing the switch at PCC. It is shown in figure 5. In this circumstance sig = 0. The voltage across CCVSI is the nominal voltage. Obviously initial values of PI controller are some random values 1 2 1 If R U U - Virtual 1 f1 n If R Th U U Virtual 2 f n If C R Th U U Virtual 1 f n Th    =  = +   =  is a constant value and varies between nn UU 5.1.  . Hence RVirtual varies between 21andThTh (Upper and Lower threshold value). Mode 2: in this mode of operation CCVSI has to be coupled with microgrid Alone. By means of droop control terminology deviation in output voltage may lead to variation in output power and results a fluctuation in DG power. In this circumstance sig = 1. Then Uf (feedback voltage) = Uod (CCVSI voltage output). Hence the real-time regulations of virtual impedance occur by voltage deviation and can be depicted as follows: 1 ( ) ( )virtual p od n od n i R K U U U U dt K = − + − (17) Where pK the proportional gain and iK is the integral gains for the CCVSI. The CCVSI reference is '* * ( ( ))d d p od nU U k U U= − − ( ) 0 1 ). .od n od vir od i U U dt i L i K + − + (18) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 642 https://internationalpubls.com ( ) '* 0 0 ( ( )) 1 ). . q p od n od n oq vir od i U k U U U U dt i L i K  = − − + − + (19) Where '* d U is the direct axis reference voltage and '* q U is the quadrature axis voltages reference with AVI (Adaptive Virtual Impedance) topology. U U2 U1 U3 0 Q2 Q1 Q3 Q U U2 U1 U3 0 Q2 Q2 Q1 Q2 Q2 Q ΔZI Q-V diagram of traditional droop control mechanism Q-V diagram of adaptive virtual impedance scheme Fig. 6. The Q-V diagram of traditional droop control and AVI control mechanism. Control Mechanism Using Traditional droop controller Using AVI controller Q-U from 1 1 2 2 3 3 , to , or , Q U Q U Q U ' 2 2 2 1 ' 3 3 3 1 deviation eliminated , to , or , to , Q U Q U Q U Q U The innermost droop control ring must be fast enough than the outer AVI mechanism for the assurance of voltage stability and adoptive power sharing. Otherwise speaking droop control play a major role for power sharing and for restoration of voltage AVI scheme can be implemented for restoration of voltage Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 643 https://internationalpubls.com Fig. 7. The Control Voltage blocks. In the above figure )(sG R →is the AVI controller )(sGin →Inner voltage control ring odI →Constant part of current at output od I ~ →Variable part of current at output 0Q →Reactive power Constant part Q ~ →Disturbances that impact the voltage. The disturbance caused by Q ~ and od I ~ in voltages can be mitigated by tuning the AVI loop Q IsGsG msG U odR q od ~ )()(1 )( + = (20) Where )(sG R can be a task of AVI compensator and G(s) can be voltage transfer gain. Figure 8 Step response for Inner Ring (a) outer voltage modification loop (b). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 644 https://internationalpubls.com The figure above depicts the step response of Outer voltage correction ring and inner control ring. Care should be taken while designing the control loops. The c f (Crossover frequency) set to SS ftof 20.10. .in that equation S f can be actuating frequency. Generally c f (frequency crossover) for the voltage which is quite lesser with respect to c f . To mitigate disturbance, the outer control ring is used. It is to be noted that for stable operation it is recommended that output loop step reaction time should 10 times greater than the inner loop step reaction time. From this we can determine pK and iK . 5. Extenssive Simulation & Experimental Verification The usefulness of this Simultaneous Voltage and Frequency Correction Scheme addressed through the manuscript is verified on MATLAB/SIMULINK platform with both islanded mode and grid connected mode. DG with CCVSI employed for maintaining the voltage on AC bus. Table 1 shows the list of different parameters. Several supported tests done for checking the performance of this adaptive technique for mitigation of power quality issues TABLE I. Organization of the System SOLAR PHOTOVOLTAIC BASED DG AND INVERTER RATINGS Required DC Voltage 625 V Peak Voltage DC 810 V Filter Inductance and Resistance 200 µH and 1.5 m Ω, Peak Power DC 1250 KW Related DC Voltage 1250 V Rated Power 1MW Coupling Inductor 2.5mh DC link Capacitor 2 NOs 4200µF Transformer 50 Hz ,160 VA Sampling Frquency 4.5KhZ GRID SPECIFICATION Source Voltage 450 V Ls, 45mH, Source Frequency 50Hz Xs/Rs 7.5 LOAD SPECIFICATION Load(Linear) Active Power = 15.5 Kw Positive VAR= 15.5 Kvar Negetive VAR= 7.5Kvar 1 Non Linear Load 3Phase Diode Load with resistor and inductor 120 ohm 300 mili Henery 2 Non Linear Load Controlled Rectifier With RLC Value P Active =15kW +ve VAR = 320 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 645 https://internationalpubls.com -ve VAR = 160 Resistor and Inductor Unbalanced Load phase A resistance, reactance = 80 Ω, 245mH phase B resistance, reactance = 85ohm, 235mH phase C resistance, reactance = 35ohm, 170mH Switching on and switching off of DG The PV based DG coupled with CCVSI as shown in figure 1. At the time of switching on and switching off power flow variation occur and it drives to power swing. Figure 9 and 10 shows the amplitude and frequency of bus voltage. It is seen that it without compensator there is sufficient deviation from nominal at the instant of closing the switch. Fig. 9. AC bus voltage and current wave shape without Simultaneous Voltage and Frequency Correction. Fig. 10 AC bus frequency without Simultaneous Voltage and Frequency Correction. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 646 https://internationalpubls.com Fig.11 AC bus voltage and current wave shape with Simultaneous Voltage and Frequency Correction. Fig.12 AC bus frequency with Simultaneous Voltage and Frequency Correction. The wave shape depicted in figure 11 and 12 shows the amplitude and frequency of bus voltage. It is seen that it with compensator there is very less deviation from nominal value. There is a small transient period to for the stabilization of the voltage. It can also be seen in frequency waveform with a small settling time. The compensator satisfactorily ensures its job after closing the switch. It can be noted that by the use of Simultaneous Voltage and Frequency Corrector, the frequency and voltage variation touch almost 10 V and 0.35 Hz once the power deviation takes place. but it taken less time and returns to their nominal values. If we detach the DG from the grid the frequency and voltage retouches almost 15 V, 0.45 Hz respectively. 2) Power fluctuation resulting from load variation. The Variation in load parameter also affects the system performance. The CCVSI controller should so adjusted that power output i.e., both active, reactive power should there in permissible limit. This compensators automatically manage the load change and will fine tune the PI controller to achieve similarity between wave shapes before and after load switching. It is seen that there is voltage sag for a small period (in micro second range). Soon after the transient period dies out it drives to normal steady state operation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 647 https://internationalpubls.com Fig.13 Simulink result with DG detached Grid with Simultaneous Voltage and Frequency Correction showing source voltage dg current and load current waveforms. Fig.14 Simulink result with DG detached Grid with Simultaneous Voltage and Frequency Corrector showing Frequency waveform. Fig. 15 Voltage deviation with Conventional droop controller and Simultaneous voltage and frequency corrector. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 648 https://internationalpubls.com Fig. 16 Frequency deviation with Conventional controller with respect to droop and Simultaneous voltage and frequency corrector. The table shown below is an experimental verification of different nonlinear loads connected to the grid. The load here can be a controlled rectifier type load. The alteration over firing angle of for rectifiers may result power quality issue. For the improvement of the power quality of any type of load change can be compensated precisely and results system stability which improves to a great extent. TABLE II: RESULT AT DIFFERENT FIRING ANGLES Typical Firing Angel of Non Linear Loads Without DG VTHD Without DG ITHD With DG VTHD With DG ITHD 600 0.355% 92.676% 0.188% 1.322% 300 0.477% 83.588% 0.300% 1.544% 450 0.411% 85.544% 0.244% 1.477% 6. CONCLUSION Presently various appliances and equipment are more delicate related to the old appliances and equipment. Therefore increase in harmonic content for electronic equipment is detrimental in all respect and impose new problems and leads poor performance. 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