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Optimal Controller Design for Stabilizing a Power System with Multiple Converter-interfaced Generators
2019 IFAC Workshop on 2019 IFAC Workshop Control of Smart Gridon and Renewable Energy Systems 2019 IFAC Workshop on Control of Smart and RenewableAvailable Energy Systems online at www.sciencedirect.com 2019 Workshop on Jeju, IFAC Korea, JuneGrid 10-12, 2019 Control of Smart Grid and Renewable 2019 IFAC Workshop on Jeju, Korea, JuneGrid 10-12, 2019 Control of Smart and Renewable Energy Energy Systems Systems Jeju, June 10-12, 2019 Control of Smart and Renewable Energy Systems Jeju, Korea, Korea, JuneGrid 10-12, 2019 Jeju, Korea, June 10-12, 2019
ScienceDirect
IFAC PapersOnLine 52-4 (2019) 75–80
Optimal Controller Design for Stabilizing a Power System with Multiple Optimal Controller Design for Stabilizing a Power System with Multiple Optimal Controller Converter-interfaced Design for Stabilizing aa Power System with Multiple Generators Optimal Design for Stabilizing Power System with Multiple Generators Optimal Controller Controller Converter-interfaced Design for Stabilizing a Power System with Multiple Converter-interfaced Generators Converter-interfaced Generators Youngho Converter-interfaced Cho*. Ryangkyu Kim**. Mingyu Song**. Kwang Y. Lee***. Generators
Youngho Cho*. Ryangkyu Kim**. Kwang Y. Lee***. Kyeon Mingyu Hur** Song**. Youngho Youngho Cho*. Cho*. Ryangkyu Ryangkyu Kim**. Kim**. Mingyu Song**. Kwang Kwang Y. Y. Lee***. Lee***. Kyeon Mingyu Hur** Song**. Youngho Cho*. Ryangkyu Kim**. Kyeon Mingyu Hur** Song**. Kwang Y. Lee***. Kyeon Hur** Kyeon Hur** *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) **Electrical Engineering Department, [email protected]) University, Seoul, Korea, *Hyundai Electric, Seoul, Korea, (e-mail: *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) **Electrical Engineering Department, Yonsei University, [email protected]) Seoul, Korea, (e-mail: [email protected], [email protected], *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) **Electrical Engineering Department, Yonsei University, Seoul, **Electrical Engineering Department, Yonsei University, Seoul, Korea, Korea, (e-mail: [email protected], [email protected], [email protected]) ***Balyor University, Waco, TX 76798 USAYonsei (e-mail: [email protected]) **Electrical Engineering Department, University, Seoul, Korea, (e-mail: [email protected], [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) ***Balyor University, Waco, TX 76798 USA (e-mail: [email protected]) (e-mail: [email protected], [email protected], [email protected]) ***Balyor ***Balyor University, University, Waco, Waco, TX TX 76798 76798 USA USA (e-mail: (e-mail: [email protected]) [email protected]) ***Balyor University, Waco, TX 76798 USA (e-mail: [email protected]) Abstract: This paper performs optimization-based controller design to avoid control interaction among Abstract: This paper performsgenerators optimization-based controller design to avoid control between interaction among multiple converter-interfaced (CIGs). To assess the control interaction CIGs, an Abstract: This paper performs optimization-based controller design to avoid control interaction among Abstract: This paper performs optimization-based controller design to avoid control interaction among multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection is Abstract: This paper performs optimization-based controller design to avoid control interaction among multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection is investigated in a test case where three CIGs are connected to a microgrid. Parameter sensitivity is multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection is impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection investigated in a test case where three CIGs are connected to a microgrid. Parameter sensitivity is performed to in determine which control parameter has more influence on from the stability, particle swarm impedance-based stability analysis. Unexpected instability resulting theParameter CIGanddisconnection is investigated a test case where three CIGs are connected to a microgrid. sensitivity investigated autilized test case where three CIGs parameters are to a microgrid. Parameter sensitivity is performed to in determine which control parameter hasconnected more influence onthe the stability stability, and particle swarm optimization is to adjust the control to enhance with computational investigated in a test case where three CIGs are connected to a microgrid. Parameter sensitivity is performed to to is determine which control parameter has more more influence influence onthe the stability stability, with and particle particle swarm performed determine which control has on the stability, and swarm optimization utilizedresults to adjust the parameter control to tuning enhance computational efficiency. Simulation demonstrated thatparameters the procedure helpswith ensure the stability performed to determine which control parameter hasproposed more influence onthe the stability stability, and particle swarm optimization is utilized to adjust the control parameters to enhance computational optimization is utilized to adjust the control parameters to enhance the stability with computational efficiency. Simulation results demonstrated that the proposed tuning procedure helps ensure the stability of the test case if aresults CIG isdemonstrated disconnected from theproposed grid. to tuning optimization iseven utilized to adjust the control parameters enhance the stability with computational efficiency. Simulation that the procedure helps ensure the stability efficiency. Simulation thetheproposed of the test case even if aresults CIG isdemonstrated disconnectedthat from grid. tuning procedure helps ensure the stability efficiency. Simulation results demonstrated that the Control) proposed tuningby procedure helps the stability of case even is from grid. © the 2019,test IFAC (International of Automatic Hosting Elsevier Ltd. Allensure rights reserved. of the test case even if if aa CIG CIGFederation is disconnected disconnected from the the grid. stability, Keywords: converter-interfaced generation (CIG), system impedance-based stability, particle of the test case even if a CIG is disconnected from the grid. stability, impedance-based stability, particle Keywords: converter-interfaced generation (CIG), system swarm optimization (PSO). Keywords: converter-interfaced generation Keywords: converter-interfaced generation (CIG), (CIG), system system stability, stability, impedance-based impedance-based stability, stability, particle particle swarm optimization (PSO). Keywords: converter-interfaced generation (CIG), system stability, impedance-based stability, particle swarm optimization (PSO). swarm optimization (PSO). swarm optimization (PSO). Because the CIG impedance is defined by control parameters, 1. INTRODUCTION Because the passive CIG impedance is defined by delays, control parameters, 1. INTRODUCTION along with elements, and time the system Because the CIG impedance is defined by control parameters, Because the CIG impedance is defined by control parameters, 1. INTRODUCTION along with passive elements, and time delays, the system 1. INTRODUCTION stability can be secured by tuning the control parameters. In Owing to the increase of distributed energy resources (DERs) Because the CIG impedance is defined by control parameters, along with passive elements, and time delays, the system 1. INTRODUCTION along with passive elements, andor time delays, the system Owing stability canthe beCIG secured by tuning the control parameters. In to the increase of distributed energy generation resources (DERs) particular, malfunctions maintenance should be consisting mainly of converter-interfaced (CIG), along with passive elements, and time delays, the system stability can canthebe beCIG secured by tuning tuning or themaintenance control parameters. parameters. In Owing to the increase of energy resources (DERs) stability secured the control In Owing toand the increase of distributed distributed energy generation resources to (DERs) consisting of converter-interfaced malfunctions be mainly (CIG), carefullycan considered inby tuning processparameters. asshould a CIG utilities DER developers are paying attention their particular, stability beCIG secured bythe tuning themaintenance control In Owing to the increase of distributed energy generation resources (DERs) particular, the malfunctions or should be consisting mainly of converter-interfaced (CIG), particular, the CIG malfunctions or maintenance should be carefully considered in the tuning process as a CIG consisting mainly of converter-interfaced generation (CIG), utilities DER developers areways paying attention their disconnection may result in impedance changes in the system, impact onand the grid and the feasible to improve theto hosting particular, the CIG malfunctions or maintenance should be consisting mainly of converter-interfaced generation (CIG), carefully considered in the tuning process as a CIG utilities and DER developers are paying attention to their carefully distortions, considered inininstability. the tuning changes processin as a CIG utilities developers areways paying attention their disconnection impact onand the DER grid and the feasible improve theofto hosting may result impedance the system, and capacity. Hosting capacity is defined asto the amount DERs carefully considered inin the tuning processin as a CIG utilities and DER developers areways paying attention to their waveform disconnection may impedance impact on the grid and the feasible to improve the hosting disconnection may result result ininstability. impedance changes changes in the the system, system, impact on the grid and the feasible ways to improve the hosting waveform distortions, and capacity. Hosting capacity is defined as the amount of DERs that canon beHosting accommodated without adversely impacting power disconnection may result ininstability. impedance changes in the system, impact the grid and the feasible ways tothe improve theofhosting waveform distortions, and capacity. capacity is defined as amount DERs waveform and instability. capacity. capacity is defined as theconfigurations amount of power DERs To tune thedistortions, control parameters, particle swarm optimization that canor beHosting accommodated adversely impacting quality reliability under without existing control and To waveform and instability. capacity. Hosting capacity is defined as the amount of power DERs that can be accommodated without adversely impacting tune isthedistortions, control parameters, particle swarm optimization that can be accommodated without adversely impacting power (PSO) one of the attractive methods due to its quality or reliability under existing control configurations and To tune the control parameters, particle swarm without requiring infrastructure upgrades inconfigurations EPRI (2016). It is (PSO) that canor bereliability accommodated without adversely impacting power To tune the control parameters, particle swarminoptimization optimization quality under existing control and is one of the attractive methods due to its quality or reliability under existing control configurations and implementation and computational advantages Lee (2008), without requiring infrastructure upgrades in EPRI (2016). It is To tune the control parameters, particle swarm optimization (PSO) is one of the attractive methods due to its not uncommon however, observe the measures are takenIt to quality or reliability undertoexisting control configurations and (PSO) is one of the attractive methods due to the its without requiring infrastructure upgrades in EPRI (2016). is implementation and computational advantages in Lee (2008), without requiring infrastructure upgrades in EPRI are (2016). is (PSO) Azzouz Previous studies have shown not uncommon however, to observe the measures takenIt to is(2016). one and of computational the research attractiveadvantages methods due to its implementation in Lee (2008), disconnect the DERs for system security reasons. without requiring infrastructure upgrades in EPRI (2016). It is implementation and computational advantages in Lee (2008), not uncommon however, to observe the measures are taken to Azzouz (2016). Previous research studies have shown the not uncommon however, to observe the measures broad usage of PSO tocomputational improve control performance in Azzouz disconnect the DERs for system security reasons. are taken to implementation and advantages in Lee (2008), Azzouz (2016). Previous research studies have shown the not uncommon however, to observe the measures taken to Azzouz Previous research have shown the disconnect the DERs DERs for system system security reasons. are usage ofhelps PSO to improve control performance in Azzouz disconnect the for security reasons. (2016). It(2016). also minimize the totalstudies harmonic distortion and The extensive adoption of converter-interfaced generation broad Azzouz (2016). Previous research studies have shown the broad usage of PSO to improve control performance in Azzouz disconnect the DERs for system security reasons. generation broad usage of PSO to improve control performance in Azzouz (2016). It also helps minimize the total harmonic distortion and The extensive adoption of converter-interfaced the stability. (CIG) results in aadoption differentof type of stability phenomenon in the enhances broad usage ofsystem PSO to improvethe control performance in Azzouz (2016). It also helps minimize total harmonic distortion and The extensive converter-interfaced generation (2016). It the alsosystem helps minimize The extensive of converter-interfaced generation (CIG) results aadoption different type of harmonic stability phenomenon in the enhances stability. the total harmonic distortion and power system.in Unlike traditional problems focused (2016). It the alsosystem helps minimize the total harmonic distortion and The extensive of converter-interfaced generation enhances stability. (CIG) results in aaadoption different type of stability phenomenon in the enhances the systemthe stability. (CIG) results in different type of stability phenomenon in the This study utilizes PSO for tuning the control parameters power system. Unlike traditional harmonic problems focused on resonance of atype combination passive elements enhances the systemthe stability. (CIG) results inbecause a different of harmonic stabilityofphenomenon in the This power system. Unlike traditional problems focused study utilizes PSO for tuning the control parameters power Unlike traditional harmonic focused by setting a stability margin as an objective function and on resonance because of cable a combination of problems passive elements This study utilizes the PSO for tuning the control parameters such assystem. line inductance, capacitance, shunt capacitors power system. Unlike traditional harmonic problems focused by This study utilizes the PSO for tuning the control parameters on resonance because of a combination of passive elements setting a stability margin as an objective function and on resonance because of a combination of passive elements further proposes a control parameter tuning procedure to such as line inductance, cable capacitance, shunt capacitors This study utilizes the PSO for as tuningobjective the control parameters by setting aa stability margin function and and harmonic filters, of thecable broad further on resonance because a high-frequency combination of switching, passive elements by setting stability margin as an an objective function and such as line inductance, capacitance, shunt capacitors proposes a control parameter tuning procedure to such as line inductance, cable capacitance, shunt capacitors high-frequency secure system stability. Impedance-based stability analysis is and harmonic filters, the switching, broad by setting a stability margin as an objective function and further proposes aa control parameter tuning procedure control and control interaction of switching, power electronic such as bandwidth line inductance, cable capacitance, shunt capacitors further proposes control parameter tuning procedure to and harmonic filters, the high-frequency broad secure system stability. Impedance-based stability analysis to is and harmonic filters, the high-frequency switching, broad introduced to assess the stability margin of the CIG-connected control bandwidth and control interaction of power electronic further proposes a control parameter tuning procedure to secure system stability. Impedance-based stability analysis is devices distorts filters, theand waveforms and causesofstability problems. and harmonic the high-frequency switching, broad introduced secure system stability. Impedance-based analysis is control bandwidth control interaction power electronic to assess the stability margin of the stability CIG-connected control bandwidth and control interaction ofstability power electronic grid in Section 2. In Section 3, variations in stability the margin devices distorts the waveforms and causes problems. secure system stability. Impedance-based stability analysis is introduced to assess the stability margin of the CIG-connected This interaction is also known as the harmonic instability in control bandwidth and control interaction of power electronic introduced to assess the stability margin of the CIG-connected devices distorts the waveforms and causes stability problems. grid in Section 2. In Section 3, variations in stability margin devices distorts the waveforms and causes stability problems. are analyzed particularly when CIGs are disconnected from the This interaction is also known as the harmonic instability in introduced to assess the stability margin of CIG-connected grid in Section 2. In Section 3, variations in the stability margin Wang (2014). Without a careful investigation of the devices distorts the waveforms and causes stability problems. inParameter Section 2.sensitivity In Section variations in the stability margin are particularly when CIGs are disconnected from This interaction is also known as instability in This interaction isWithout also as the the harmonic harmonic instability in grid Wang (2014). a careful investigation of the grid.analyzed is3, investigated in Section 4the to in Section 2. In Section 3,also variations in the stability margin are analyzed particularly when CIGs are disconnected from distribution system to known understand harmonic stability This interaction isWithout also known as the the harmonic instability in grid are analyzed particularly when CIGs are disconnected from the grid. Parameter sensitivity is also investigated in Section 4the to Wang (2014). a careful investigation of the Wang (2014). Without a careful the investigation of the select an appropriate control parameter that is more influential distribution system to understand harmonic stability are analyzed particularly when CIGs are disconnected from the grid. Parameter sensitivity is also investigated in Section 4 to phenomenon, harmonic instability reduces the hosting Wang (2014). Without a careful the investigation of the select grid. Parameter sensitivity is also investigated in Section 4 to distribution system to understand harmonic stability an appropriate control parameter that is more influential distribution system to power understand the harmonic theParameter dominant pole.control Then, the introduction and application phenomenon, harmonic instability reduces the stability hosting on grid. sensitivity is also investigated in Section 4 to select an appropriate parameter that is more influential capacity to regulate the quality of the system. distribution system to understand the harmonic stability select an appropriate control parameter that is more influential phenomenon, harmonic instability reduces the hosting on the dominant pole. Then, the introduction and application phenomenon, harmonic instability the hosting of PSO toappropriate adjust control parameters are that explained inapplication Section 5. capacity to regulate the power quality ofreduces the system. select an control parameter is more influential on the dominant pole. Then, the introduction and phenomenon, harmonic instability reduces the hosting on PSO the dominant pole. Then, the introduction andinapplication capacity to the quality of the to6 adjust parameters are Section in 5. capacity to regulate regulate the power power quality of applicable the system. system.method to of Section provescontrol the effectiveness of explained the tuning process Impedance-based stability analysis is an on the dominant pole. Then, the introduction andinapplication of PSO to adjust control parameters are explained Section 5. capacity to regulate the power quality of applicable the system.method to Section of PSO to adjust control parameters are explained in Section 5. 6 proves the effectiveness of the tuning process in Impedance-based stability analysis is an securing simulation resultsare ofexplained thetuning PSOinapplication assess the stabilitystability of a CIG-connected grid in Sun (2009). The of PSO to6stability adjust control parameters Section in 5. Section proves the effectiveness of the process Impedance-based analysis is an applicable method to Section 6 proves the effectiveness of the tuning process in Impedance-based stability analysis is an applicable method to securing stability simulation results of the PSO application assess the stability of a CIG-connected grid in Sun (2009). The followed by concluding remarks in Section 7. relationship between the impedance of each CIG and the grid Section 6 proves the effectiveness of the tuning process in Impedance-based stability analysis is an applicable method to securing by stability simulation results of the the7.PSO PSO application application assess the of aathe CIG-connected grid Sun (2009). The securing stability simulation results of assess the stability stability ofdetermine CIG-connected grid inCIG Sun (2009). The followed concluding remarks in Section relationship impedance ofstability. eachin and the grid impedance isbetween used to system The important stability simulation results of the7.PSO application assess the stability of athe CIG-connected grid inCIG Sun (2009). The securing followed by concluding remarks in Section relationship between impedance of each and the grid followed by concluding remarks in Section 7. relationship between the impedance of each CIG and the grid impedance used to determine system stability. The important part in thisis analysis is obtain theofCIG that is followed 2. IMPEDANCE-BASED STABILITY ANALYSIS by concluding remarks in Section 7. relationship between thetoimpedance eachimpedance CIG and the grid impedance is used to determine system stability. The impedance isanalysis used to determine system stability. The important important part in this is (Point to obtain the CIG impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS observed from the PCC of Common Coupling). impedance isanalysis used to determine system stability. The important part in this is to obtain the CIG impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS part in this analysis is (Point to obtain the CIG Coupling). impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS observed from the PCC of Common part in this analysis is to obtain the CIG impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS observed observed from from the the PCC PCC (Point (Point of of Common Common Coupling). Coupling). observed ©from PCC (Point of Common Copyright 2019the IFAC 75 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 2019, IFAC (International FederationCoupling). of Automatic Control) Copyright 2019 responsibility IFAC 75 Control. Peer review©under of International Federation of Automatic Copyright © 75 Copyright © 2019 2019 IFAC IFAC 75 10.1016/j.ifacol.2019.08.158 Copyright © 2019 IFAC 75
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Fig. 3. System configuration of test case. Three CIGs are connected to the system through the line. 𝐺𝐺𝑐𝑐𝑐𝑐 = 𝑌𝑌𝑐𝑐 =
Fig. 1. System details of CIG: (a) entire circuit and control system and (b) control block diagram.
𝐺𝐺𝑃𝑃𝑃𝑃 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃
𝐿𝐿𝑐𝑐 𝐿𝐿𝑔𝑔 𝐶𝐶𝑓𝑓 𝑠𝑠 3 −𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 𝐿𝐿𝑔𝑔 𝑠𝑠+(𝐿𝐿𝑔𝑔 +𝐿𝐿𝑐𝑐 )𝑠𝑠+𝐺𝐺𝑃𝑃𝑃𝑃 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃
𝐿𝐿𝑐𝑐 𝐶𝐶𝑓𝑓 𝑠𝑠 2 −𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 +1 3 𝐿𝐿𝑐𝑐 𝐿𝐿𝑔𝑔 𝐶𝐶𝑓𝑓 𝑠𝑠 −𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 𝐿𝐿𝑔𝑔 𝑠𝑠+(𝐿𝐿𝑔𝑔 +𝐿𝐿𝑐𝑐 )𝑠𝑠+𝐺𝐺𝑃𝑃𝑃𝑃 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃
(3) (4)
Here, Gcl is the closed-loop transfer function, and Yc is the CIG admittance, which is derived from the relationship between VPCC and Ig,d. Equations (2)–(4) imply that a CIG can be modeled as a parallel combination of the current source and CIG admittance. 2.2 Equivalent Modeling and Analysis Fig. 2 illustrates the CIG equivalent model for a number of CIGs connected in cascade, where a series impedance separates each CIG. Each CIG can be modeled as a parallel circuit of the admittance and the current source. When the total admittance of the CIG side at PCC is addressed as Yeq, the current flowing into the grid can be obtained as follows:
Fig. 2. Equivalent model of CIGs. This model consists of CIG admittance and current sources.
𝐼𝐼𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 =
2.1 CIG Impedance Modeling The entire system model and its control block diagram are shown in Fig. 1. An LCL filter is used for current damping and harmonic filtering. The voltage across the filter capacitor and current flowing into the PCC are measured for grid synchronization and current control. Each CIG operates in a constant power and unity power factor mode for a basic approach. A DC source is used to assume constant DC voltage.
𝑠𝑠
1−1.5𝑇𝑇𝑑𝑑 2 1+1.5𝑇𝑇𝑑𝑑
𝑠𝑠 2
𝑌𝑌𝑒𝑒𝑒𝑒 𝐼𝐼 𝑌𝑌𝑔𝑔 +𝑌𝑌𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑐𝑐
(5)
2.3 Consideration of nonlinear power system If the power system is a nonlinear system, the investigation of the grid stability can be more complex due to the higher-order Yg. One way to investigate the stability of this system is to approximate the frequency response characteristics with a transfer function by vector fitting in Gustavsen (1999). In this case, the validity of the stability determination depends on the accuracy of the approximation that usually makes the higherorder transfer function. In this paper, the equivalent admittance of the power system is regarded as a first order transfer function to focus on the methodology of the controller design.
(1)
Because of the unity power factor operation, only d-axis current control is considered (i.e., zero q-axis current). The daxis grid side current with the converter controller GPI = kp +ki/s can be obtained as follows: 𝐼𝐼𝑔𝑔,𝑑𝑑 = 𝐺𝐺𝑐𝑐𝑐𝑐 𝐼𝐼𝑔𝑔,𝑑𝑑,𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑌𝑌𝑐𝑐 𝑉𝑉𝑃𝑃𝑃𝑃𝑃𝑃,𝑑𝑑
−
where Yg is the admittance of the source impedance (1/ Zg), Icon is the total current generated by the CIGs, and Isrc is the current generated by the grid. Based on the assumption that both currents Icon and Isrc are stable in Sun (2009), the stability of the current is determined by the common denominator on the right side of (5), which is 1/(Yg + Yeq).
Converter switching can be expressed as a time-delay function. According to Wang (2014), there is a 1.5 sampling period delay, which can be approximated by a Padė approximation as follows: 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑒𝑒 −1.5𝑇𝑇𝑑𝑑𝑠𝑠 ≈
𝑌𝑌𝑔𝑔 𝐼𝐼 𝑌𝑌𝑔𝑔 +𝑌𝑌𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐
(2) 76
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3.2 Stability Assessment for CIG Disconnection To observe the system stability problems due to malfunctions or maintenance of the CIGs, all cases of one or more CIG disconnections will be considered. Fig. 4 shows the stability assessment when only one CIG remains connected to the grid. As the electrical distance between the grid and the CIG increases, the stability margin decreases, but the system still operates in a stable condition. Fig. 5 shows the system stability assessment when two or more CIGs are in operation. Only when CIG1 disconnected is the system becomes unstable. By contrast, the other two cases are more stable than the base case despite the CIG disconnections. The two assessment results imply that a greater electrical distance between the grid and the nearest CIG results in a lower stability margin. In addition, having fewer CIGs in the system results in more stability. A test system model simulation carried out in PSCAD/EMTDC verifies the instability phenomenon in the time domain. Fig. 6 shows waveforms of the current flowing into the grid when CB1 is opened at 0.5 s and CB2 is opened at 0.65 s sequentially. CB refers to the circuit breaker. Before the CIG disconnection, the current waveforms are clean and stable. However, as a consequence of disconnecting CIG1, the current waveforms become distorted and the system becomes unstable. After the additional trip of CIG2 at 0.65 s, the system becomes stable again.
Fig. 4. Stability assessment when only one CIG maintains the connection.
4. CONTROL PARAMETER SENSITIVITY ANALYSIS A parameter sensitivity analysis to identify adjustable parameters to help reduce the computational complexity of the optimization by removing low-impact parameters. The quantification of the impact of a control parameter p on the stability margin is achieved by a sensitivity analysis, as follows:
Fig. 5. Stability assessment when two or more CIGs maintain the connection.
1
∆𝑋𝑋
𝑆𝑆𝑃𝑃 = 𝑛𝑛 ∑𝑛𝑛𝑡𝑡=1 |∆𝑃𝑃𝑡𝑡| 𝑡𝑡
(6)
where Δ Xt is the moved pole displacement, and Δ Pt is the variation in a parameter. As the variation in the pole displacement is nonlinear, the absolute value is averaged for the sensitivity analysis. According to (5) and (6), the stability margin of the system can be changed by the control parameters.
Fig. 6. Waveforms of current flowing into grid when CIG1 and CIG2 are disconnected from grid at 0.5 s and 0.65 s, respectively.
The proportional gain kp,r and integral gain ki,r (for the r-th CIG, r = 1, 2, 3) of the current controller are target parameters for the sensitivity analysis in this study. In addition, the following equation defines the relative sensitivity to quantify the impact of kp,r relative to ki,r:
3. SYSTEM STABILITY CHANGES ACCOMPANIED BY CIG DISCONNECTION
𝑅𝑅𝑅𝑅𝑟𝑟 =
3.1 System Configuration of Test System Fig. 3 shows a base case where three CIGs are connected to the grid and are in operation, where CIGs are numbered from the nearest to the PCC to the remotest. In the base case, all CIGs have the same parameters. The line impedance is expressed as a series connection of resistance and inductance. The distance between CIGs is 1.6 km. The q-axis voltage and current are set to 0. The parameters are shown in Table 5.
𝑆𝑆𝑘𝑘𝑝𝑝,𝑟𝑟 𝑆𝑆𝑘𝑘
𝑖𝑖,𝑟𝑟
(7)
A target pole for the first sensitivity analysis is pole 1 in the “CB1 open” case, as indicated in Fig. 5. Except for the parameters of CIG1, the control parameters of CIG2 and CIG3 are considered to calculate the sensitivity, where kp,r varies from 0.75 to 0.85, and ki,r varies from 100 to 1000. The amounts of change in parameters are 0.01 and 100, 77
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Table 1. Sensitivity analysis of pole 1 in “CB1 open” case Parameters kp,2 kp,3 ki,2 ki,3
Magnitude 180.9648 84.2525 0.0631 0.0293
Real part 180.4010 84.1847 0.0058 0.0013
Imag. part 14.2508 3.3781 0.0629 0.0293
Table 2. Sensitivity analysis for poles 2 and 3 in base case Parameters Pole 2 kp,1 Pole 3 Pole 2 ki,1 Pole 3
Magnitude 237.7854 32.4185 0.0714 0.0117
Real part 236.6427 25.3783 0.0071 0.0067
Imag. part 23.2647 20.1372 0.0710 0.0096
Table 3. Relative sensitivity analysis of both cases Parameters Pole 2 RS1 Pole 3 RS2 RS3
Fig. 7. Location changes of pole 1 by variations in parameters: (a) kp,2 variation, (b) kp,3 variation, (c) ki,2 variation, and (d) ki,3 variation.
Magnitude 3330.33 2770.81 2867.90 2875.51
Real part 33329.96 3787.81 31103.62 64757.46
Imag. part 322.67 2097.63 226.56 115.29
A second sensitivity analysis focuses on the control parameters of CIG1 and their impact on the two nearest poles (poles 2 and 3) on the imaginary axis in the base case under the same parameter variations as in the previous analysis. The details of the sensitivity analysis results for poles 2 and 3 are listed in Table 2, and the relative sensitivity analysis results of both cases are listed in Table 3. Similarly, the impact of kp,1 on the sensitivity of the real part is still larger than that of ki,1. The difference is that the relative sensitivity RS1 of the real part of pole 3 is approximately 10 times lower than RS2 of the real part, as observed in Table 3. This implies that the impact of ki,1 relative to kp,1 on pole 3 is larger than in the other cases. Fig. 8 shows the tendencies of the movement of the two nearest poles to the imaginary axis with respect to the control parameters. For pole 3, the displacement by ki,1 is larger than that by kp,1. However, the displacement of pole 2 by ki,1 is smaller than that by kp,1. In addition, the increase in ki,1 causes a change in the dominant pole and even worsens the stability because poles 2 and 3 move toward each other. The change in the dominant pole from pole 3 to pole 2 limits the capability for ki,1 to enhance the stability. Therefore, kp,1 is a more suitable and capable parameter for optimization to secure the system stability.
Fig. 8. Location changes of poles 2 and 3 by variations in parameters: (a) kp,1 variation and (b) ki,1 variation. respectively. Table I presents the sensitivity analysis results of pole 1 with respect to the control parameters, kp,r and ki,r (r = 2, 3). It is obvious that the sensitivity of pole 1 by kp,r is much larger than by ki,r. If the sensitivity is divided into real and imaginary parts for detailed information, the sensitivity of the real part of the pole 1 by kp,r is higher than the sensitivity of the imaginary part. By contrast, ki,r has more impact on the sensitivity of the imaginary part than that of the real part. Fig. 7 shows the tendencies of pole 1’s movements with respect to the control parameter variations. Red arrows indicate the directions of the pole movements as the parameters increase. The two figures in the upper part prove that kp,r can be utilized to secure the stability. On the other hand, the two figures in the lower part show that ki,r is not capable of enhancing the stability. ki,r causes relatively large variations in the pole location in the imaginary part rather than the real part. Only the proportional gains are adjusted to improve the stability in the “CB1 open” case.
5. PARTICLE SWARM OPTIMIZATION FOR CIG PARAMETER TUNING 5.1 Introduction of Particle Swarm Optimization The PSO theory is based on the social behavior of fish schools or bird flocks, which tend to move to their optimal positions by using their own or others’ experience in Lee (2008). Each particle position is evaluated by an objective function to find 78
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2) Select the appropriate parameters from the sensitivity analysis results for the PSO.
the best position. The next velocity is determined by the current velocity, current position, local best position, and global best position. The next position of the particle is the summation of the current position and next velocity. The relationships between these conditions can be expressed as the following equations:
3) For the case having the lowest stability margin among all cases investigated in 1), perform the PSO with the parameters selected in 2). The parameters of disconnected CIGs are not optimized in this step.
𝑣𝑣𝑖𝑖 [𝑘𝑘 + 1] = 𝑤𝑤𝑣𝑣𝑖𝑖 [𝑘𝑘] + 𝑐𝑐1 𝑟𝑟1 (𝑥𝑥𝑙𝑙𝑙𝑙,𝑖𝑖 [𝑘𝑘] − 𝑥𝑥𝑖𝑖 [𝑘𝑘]) + 𝑐𝑐2 𝑟𝑟2 (𝑥𝑥𝑔𝑔𝑔𝑔,𝑖𝑖 [𝑘𝑘] − 𝑥𝑥𝑖𝑖 [𝑘𝑘])
w = 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 − (𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 )
4) Replace the parameter values with the optimized values. (8)
𝑘𝑘
𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚
𝑥𝑥𝑖𝑖 [𝑘𝑘 + 1] = 𝑥𝑥𝑖𝑖 [𝑘𝑘] + 𝑣𝑣𝑖𝑖 [𝑘𝑘 + 1]
5) Perform the PSO again for the case with no CIG disconnection. The PSO finds the optimal value for the rest of the parameters: The first PSO ensures stability under the most severe condition, whereas the second maximizes the stability margin under the normal operating condition.
(9) (10)
Here, i is the particle number, vi is the velocity, w is the weighting factor, c1 and c2 are weighting coefficients, r1 and r2 are random numbers between 0 and 1, xi is the current position, xlb,i is the local best position among the current and previous positions of the i-th particle, xgb,i is the global best position among the current and previous xlb,i values of the i-th particle, wmax and wmin are the maximum and minimum values of w, k is the current iteration, and kmax is the maximum iteration number. The weighting factor determines the impact of the current velocity, and the weighting coefficients determine the impact of the difference between the current position and the local or global best position. According to Lee (2008), the typical weighting coefficients and maximum and minimum values of w are as follows: 𝑐𝑐1 = 𝑐𝑐2 = 2, 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 = 0.9, 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 = 0.4
6. SIMULATION RESULTS As shown in Fig. 5, the case of “CIG2, CIG3 Connected” has the lowest stability margin and is even unstable. Also, the sensitivity analysis proves that the proportional gains are more influential parameters than the integral gains in the test system. Therefore, kp,2 and kp,3 are first optimized without considering kp,1. Then, kp,1 is subsequently optimized by the second-round PSO with fixed values of kp,2 and kp,3 derived in the first-round PSO. Table 4 lists the parameter tuning results of the PSO. Fig. 8 shows the stability assessment results for the cases where only one CIG is connected to the grid after parameter adjustment by PSO. The adjustment of the proportional gain causes the real parts of the poles near the fundamental frequency to increase, whereas the real part of the dominant poles in Fig. 4 decreases. As a result, the dominant poles are replaced, and the stability margins of all cases are changed.
(11)
5.2 Objective Function for Parameter Tuning For the application of PSO in a stability assessment, the objective function is defined as follows: 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑀𝑀𝑀𝑀𝑀𝑀 (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 (𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 (
1
𝑌𝑌𝑔𝑔 +𝑌𝑌𝑒𝑒𝑒𝑒
)))
79
Although the stability margin decreases when only CIG2 maintains a connection, all cases in Fig. 8 still have enough of a margin to be stable. Fig. 9 shows the stability assessment results of other cases that kept two or more CIGs connected after PSO. Compared to the results before PSO in Fig. 5, the stability margins in these cases are increased. In particular, the case where CIG1 is disconnected becomes stable.
(12)
It indicates the real part of the dominant pole of the common denominator in (5). When the position of the pole is changed by the control parameters, the PSO aims to maximize the stability margin of the common denominator by minimizing the objective function because of its negative sign. When the dominant pole reaches the rarely-unmovable pole by adjusting the parameters, it has the maximum stability margin. Because the stability has priority over the converter performance, which is closely related to various grid requirements, the trade-off between the stability and the converter performance is not investigated further in this study.
To examine the difference in system stability before and after PSO, Fig. 10 shows current waveforms under the same condition as in Fig. 6 after parameter tuning. In contrast to Fig. 6, the current waveform maintains its stability despite of the CIG1 disconnection. Therefore, the simulation results demonstrate that the PSO based control parameter tuning process successfully helps ensure the system stability. 7. CONCLUSION This paper proposes a parameter tuning procedure based on an impedance-based stability analysis to ensure system stability and potentially improve the hosting capacity by utilizing PSO. The converter admittance derived from the converter current control consists of the control parameters and passive elements of the converter filter. The stability can be assessed by a common denominator that is the inverse of the summation of the grid admittance and the equivalent admittance. The tuning
5.3 PSO Application to Parameter Tuning To secure stability in all cases even with malfunctions of CIGs, the parameters are tuned by the following procedure: 1) Assess the system stability of the multi-CIG system. All cases of CIG disconnection should be analyzed.
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Table 4. Parameter tuning results of each PSO process Parameters 1st PSO 2nd PSO
kp,1 0.63476
kp,2 0.42443 (0.42443)
kp,3 0.5423 (0.5423)
Table 5. System parameters Parameters Grid voltage Grid frequency Short circuit capacity Grid impedance angle Distance between converters Switching frequency Switching delay (Td) Line Rline impedance Lline Filter Lg impedance Cf Lc PI control kp,n ki,n
Fig. 8. Stability assessment when only one CIG maintains connection after PSO.
Values 400 V 60 Hz 0.3 MVA 15º 1.6 km 1980 Hz 20 0.4375 /km 0.875 mH/km 0.6 mH 100 F 2.0 mH 0.85 500
changes the system status to be stable. This study should provide a systematic framework for identifying the problematic yet controllable design parameters to ensure the desired stability margin and secure the desired DER hosting capacity. This study should provide a systematic framework for identifying the problematic yet controllable design parameters to ensure the desired stability margin and secure the desired DER hosting capacity.
Fig. 9. Stability assessment when two or more CIGs maintain connection after PSO.
REFERENCES EPRI. (2016). Defining a roadmap for successful implementation of a hosting capacity method for New York state, in EPRI, Palo Alto, CA, USA, Tech. Rep. Wang, X., Blaabjerg, F. and Wu, W. (2014). Modeling and analysis of harmonic stability in an ac power-electronicsbased power system, vol. 29, no. 12, pp. 6412–6432, IEEE Trans. Power Electron.. Sun, J. (2009). Small-signal methods for ac distributed power systems-a review, vol. 24, no. 11, pp. 2545–2554, IEEE Trans. Power Electron.. Lee, K. Y. and El-Sharkwai, M. A. (2008). Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems, Wiley. Azzouz, M. A., Saadany, E. F. E. (2016). New impedance reshaping algorithm for grid-connected dgs, pp. 1–5, in 2016 IEEE Power and Energy Society General Meeting. Gustavsen, B. and Semlyen, A. (1999). Rational approximation of frequency domain responses by vector fitting, vol. 14, no. 3, pp. 1052–1062, IEEE Trans. Power Delivery.
Fig. 10. Waveforms of current flowing into grid in “CB1 open” case after parameter tuning. of control parameters affects the converter admittance, which then secure the stability margin. For the test system, the CIG disconnection can result in instability even if the system is stable under its normal condition. To solve this problem, a control parameter sensitivity analysis first identifies which control parameter has most influence on the stability. After that, a parameter optimization process begins with the most severe case having the lowest stability margin. Then, PSO is carried out again to optimize the rest of the parameters. The simulation results show that the proposed parameter tuning process enhances the stability of the test system and even 80
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Optimal Controller Design for Stabilizing a Power System with Multiple Optimal Controller Design for Stabilizing a Power System with Multiple Optimal Controller Converter-interfaced Design for Stabilizing aa Power System with Multiple Generators Optimal Design for Stabilizing Power System with Multiple Generators Optimal Controller Controller Converter-interfaced Design for Stabilizing a Power System with Multiple Converter-interfaced Generators Converter-interfaced Generators Youngho Converter-interfaced Cho*. Ryangkyu Kim**. Mingyu Song**. Kwang Y. Lee***. Generators
Youngho Cho*. Ryangkyu Kim**. Kwang Y. Lee***. Kyeon Mingyu Hur** Song**. Youngho Youngho Cho*. Cho*. Ryangkyu Ryangkyu Kim**. Kim**. Mingyu Song**. Kwang Kwang Y. Y. Lee***. Lee***. Kyeon Mingyu Hur** Song**. Youngho Cho*. Ryangkyu Kim**. Kyeon Mingyu Hur** Song**. Kwang Y. Lee***. Kyeon Hur** Kyeon Hur** *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) **Electrical Engineering Department, [email protected]) University, Seoul, Korea, *Hyundai Electric, Seoul, Korea, (e-mail: *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) **Electrical Engineering Department, Yonsei University, [email protected]) Seoul, Korea, (e-mail: [email protected], [email protected], *Hyundai Electric, Seoul, Korea, (e-mail: [email protected]) **Electrical Engineering Department, Yonsei University, Seoul, **Electrical Engineering Department, Yonsei University, Seoul, Korea, Korea, (e-mail: [email protected], [email protected], [email protected]) ***Balyor University, Waco, TX 76798 USAYonsei (e-mail: [email protected]) **Electrical Engineering Department, University, Seoul, Korea, (e-mail: [email protected], [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) ***Balyor University, Waco, TX 76798 USA (e-mail: [email protected]) (e-mail: [email protected], [email protected], [email protected]) ***Balyor ***Balyor University, University, Waco, Waco, TX TX 76798 76798 USA USA (e-mail: (e-mail: [email protected]) [email protected]) ***Balyor University, Waco, TX 76798 USA (e-mail: [email protected]) Abstract: This paper performs optimization-based controller design to avoid control interaction among Abstract: This paper performsgenerators optimization-based controller design to avoid control between interaction among multiple converter-interfaced (CIGs). To assess the control interaction CIGs, an Abstract: This paper performs optimization-based controller design to avoid control interaction among Abstract: This paper performs optimization-based controller design to avoid control interaction among multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection is Abstract: This paper performs optimization-based controller design to avoid control interaction among multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection is investigated in a test case where three CIGs are connected to a microgrid. Parameter sensitivity is multiple converter-interfaced generators (CIGs). To assess the control interaction between CIGs, an impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection is impedance-based stability analysis. Unexpected instability resulting from the CIG disconnection investigated in a test case where three CIGs are connected to a microgrid. Parameter sensitivity is performed to in determine which control parameter has more influence on from the stability, particle swarm impedance-based stability analysis. Unexpected instability resulting theParameter CIGanddisconnection is investigated a test case where three CIGs are connected to a microgrid. sensitivity investigated autilized test case where three CIGs parameters are to a microgrid. Parameter sensitivity is performed to in determine which control parameter hasconnected more influence onthe the stability stability, and particle swarm optimization is to adjust the control to enhance with computational investigated in a test case where three CIGs are connected to a microgrid. Parameter sensitivity is performed to to is determine which control parameter has more more influence influence onthe the stability stability, with and particle particle swarm performed determine which control has on the stability, and swarm optimization utilizedresults to adjust the parameter control to tuning enhance computational efficiency. Simulation demonstrated thatparameters the procedure helpswith ensure the stability performed to determine which control parameter hasproposed more influence onthe the stability stability, and particle swarm optimization is utilized to adjust the control parameters to enhance computational optimization is utilized to adjust the control parameters to enhance the stability with computational efficiency. Simulation results demonstrated that the proposed tuning procedure helps ensure the stability of the test case if aresults CIG isdemonstrated disconnected from theproposed grid. to tuning optimization iseven utilized to adjust the control parameters enhance the stability with computational efficiency. Simulation that the procedure helps ensure the stability efficiency. Simulation thetheproposed of the test case even if aresults CIG isdemonstrated disconnectedthat from grid. tuning procedure helps ensure the stability efficiency. Simulation results demonstrated that the Control) proposed tuningby procedure helps the stability of case even is from grid. © the 2019,test IFAC (International of Automatic Hosting Elsevier Ltd. Allensure rights reserved. of the test case even if if aa CIG CIGFederation is disconnected disconnected from the the grid. stability, Keywords: converter-interfaced generation (CIG), system impedance-based stability, particle of the test case even if a CIG is disconnected from the grid. stability, impedance-based stability, particle Keywords: converter-interfaced generation (CIG), system swarm optimization (PSO). Keywords: converter-interfaced generation Keywords: converter-interfaced generation (CIG), (CIG), system system stability, stability, impedance-based impedance-based stability, stability, particle particle swarm optimization (PSO). Keywords: converter-interfaced generation (CIG), system stability, impedance-based stability, particle swarm optimization (PSO). swarm optimization (PSO). swarm optimization (PSO). Because the CIG impedance is defined by control parameters, 1. INTRODUCTION Because the passive CIG impedance is defined by delays, control parameters, 1. INTRODUCTION along with elements, and time the system Because the CIG impedance is defined by control parameters, Because the CIG impedance is defined by control parameters, 1. INTRODUCTION along with passive elements, and time delays, the system 1. INTRODUCTION stability can be secured by tuning the control parameters. In Owing to the increase of distributed energy resources (DERs) Because the CIG impedance is defined by control parameters, along with passive elements, and time delays, the system 1. INTRODUCTION along with passive elements, andor time delays, the system Owing stability canthe beCIG secured by tuning the control parameters. In to the increase of distributed energy generation resources (DERs) particular, malfunctions maintenance should be consisting mainly of converter-interfaced (CIG), along with passive elements, and time delays, the system stability can canthebe beCIG secured by tuning tuning or themaintenance control parameters. parameters. In Owing to the increase of energy resources (DERs) stability secured the control In Owing toand the increase of distributed distributed energy generation resources to (DERs) consisting of converter-interfaced malfunctions be mainly (CIG), carefullycan considered inby tuning processparameters. asshould a CIG utilities DER developers are paying attention their particular, stability beCIG secured bythe tuning themaintenance control In Owing to the increase of distributed energy generation resources (DERs) particular, the malfunctions or should be consisting mainly of converter-interfaced (CIG), particular, the CIG malfunctions or maintenance should be carefully considered in the tuning process as a CIG consisting mainly of converter-interfaced generation (CIG), utilities DER developers areways paying attention their disconnection may result in impedance changes in the system, impact onand the grid and the feasible to improve theto hosting particular, the CIG malfunctions or maintenance should be consisting mainly of converter-interfaced generation (CIG), carefully considered in the tuning process as a CIG utilities and DER developers are paying attention to their carefully distortions, considered inininstability. the tuning changes processin as a CIG utilities developers areways paying attention their disconnection impact onand the DER grid and the feasible improve theofto hosting may result impedance the system, and capacity. Hosting capacity is defined asto the amount DERs carefully considered inin the tuning processin as a CIG utilities and DER developers areways paying attention to their waveform disconnection may impedance impact on the grid and the feasible to improve the hosting disconnection may result result ininstability. impedance changes changes in the the system, system, impact on the grid and the feasible ways to improve the hosting waveform distortions, and capacity. Hosting capacity is defined as the amount of DERs that canon beHosting accommodated without adversely impacting power disconnection may result ininstability. impedance changes in the system, impact the grid and the feasible ways tothe improve theofhosting waveform distortions, and capacity. capacity is defined as amount DERs waveform and instability. capacity. capacity is defined as theconfigurations amount of power DERs To tune thedistortions, control parameters, particle swarm optimization that canor beHosting accommodated adversely impacting quality reliability under without existing control and To waveform and instability. capacity. Hosting capacity is defined as the amount of power DERs that can be accommodated without adversely impacting tune isthedistortions, control parameters, particle swarm optimization that can be accommodated without adversely impacting power (PSO) one of the attractive methods due to its quality or reliability under existing control configurations and To tune the control parameters, particle swarm without requiring infrastructure upgrades inconfigurations EPRI (2016). It is (PSO) that canor bereliability accommodated without adversely impacting power To tune the control parameters, particle swarminoptimization optimization quality under existing control and is one of the attractive methods due to its quality or reliability under existing control configurations and implementation and computational advantages Lee (2008), without requiring infrastructure upgrades in EPRI (2016). It is To tune the control parameters, particle swarm optimization (PSO) is one of the attractive methods due to its not uncommon however, observe the measures are takenIt to quality or reliability undertoexisting control configurations and (PSO) is one of the attractive methods due to the its without requiring infrastructure upgrades in EPRI (2016). is implementation and computational advantages in Lee (2008), without requiring infrastructure upgrades in EPRI are (2016). is (PSO) Azzouz Previous studies have shown not uncommon however, to observe the measures takenIt to is(2016). one and of computational the research attractiveadvantages methods due to its implementation in Lee (2008), disconnect the DERs for system security reasons. without requiring infrastructure upgrades in EPRI (2016). It is implementation and computational advantages in Lee (2008), not uncommon however, to observe the measures are taken to Azzouz (2016). Previous research studies have shown the not uncommon however, to observe the measures broad usage of PSO tocomputational improve control performance in Azzouz disconnect the DERs for system security reasons. are taken to implementation and advantages in Lee (2008), Azzouz (2016). Previous research studies have shown the not uncommon however, to observe the measures taken to Azzouz Previous research have shown the disconnect the DERs DERs for system system security reasons. are usage ofhelps PSO to improve control performance in Azzouz disconnect the for security reasons. (2016). It(2016). also minimize the totalstudies harmonic distortion and The extensive adoption of converter-interfaced generation broad Azzouz (2016). Previous research studies have shown the broad usage of PSO to improve control performance in Azzouz disconnect the DERs for system security reasons. generation broad usage of PSO to improve control performance in Azzouz (2016). It also helps minimize the total harmonic distortion and The extensive adoption of converter-interfaced the stability. (CIG) results in aadoption differentof type of stability phenomenon in the enhances broad usage ofsystem PSO to improvethe control performance in Azzouz (2016). It also helps minimize total harmonic distortion and The extensive converter-interfaced generation (2016). It the alsosystem helps minimize The extensive of converter-interfaced generation (CIG) results aadoption different type of harmonic stability phenomenon in the enhances stability. the total harmonic distortion and power system.in Unlike traditional problems focused (2016). It the alsosystem helps minimize the total harmonic distortion and The extensive of converter-interfaced generation enhances stability. (CIG) results in aaadoption different type of stability phenomenon in the enhances the systemthe stability. (CIG) results in different type of stability phenomenon in the This study utilizes PSO for tuning the control parameters power system. Unlike traditional harmonic problems focused on resonance of atype combination passive elements enhances the systemthe stability. (CIG) results inbecause a different of harmonic stabilityofphenomenon in the This power system. Unlike traditional problems focused study utilizes PSO for tuning the control parameters power Unlike traditional harmonic focused by setting a stability margin as an objective function and on resonance because of cable a combination of problems passive elements This study utilizes the PSO for tuning the control parameters such assystem. line inductance, capacitance, shunt capacitors power system. Unlike traditional harmonic problems focused by This study utilizes the PSO for tuning the control parameters on resonance because of a combination of passive elements setting a stability margin as an objective function and on resonance because of a combination of passive elements further proposes a control parameter tuning procedure to such as line inductance, cable capacitance, shunt capacitors This study utilizes the PSO for as tuningobjective the control parameters by setting aa stability margin function and and harmonic filters, of thecable broad further on resonance because a high-frequency combination of switching, passive elements by setting stability margin as an an objective function and such as line inductance, capacitance, shunt capacitors proposes a control parameter tuning procedure to such as line inductance, cable capacitance, shunt capacitors high-frequency secure system stability. Impedance-based stability analysis is and harmonic filters, the switching, broad by setting a stability margin as an objective function and further proposes aa control parameter tuning procedure control and control interaction of switching, power electronic such as bandwidth line inductance, cable capacitance, shunt capacitors further proposes control parameter tuning procedure to and harmonic filters, the high-frequency broad secure system stability. Impedance-based stability analysis to is and harmonic filters, the high-frequency switching, broad introduced to assess the stability margin of the CIG-connected control bandwidth and control interaction of power electronic further proposes a control parameter tuning procedure to secure system stability. Impedance-based stability analysis is devices distorts filters, theand waveforms and causesofstability problems. and harmonic the high-frequency switching, broad introduced secure system stability. Impedance-based analysis is control bandwidth control interaction power electronic to assess the stability margin of the stability CIG-connected control bandwidth and control interaction ofstability power electronic grid in Section 2. In Section 3, variations in stability the margin devices distorts the waveforms and causes problems. secure system stability. Impedance-based stability analysis is introduced to assess the stability margin of the CIG-connected This interaction is also known as the harmonic instability in control bandwidth and control interaction of power electronic introduced to assess the stability margin of the CIG-connected devices distorts the waveforms and causes stability problems. grid in Section 2. In Section 3, variations in stability margin devices distorts the waveforms and causes stability problems. are analyzed particularly when CIGs are disconnected from the This interaction is also known as the harmonic instability in introduced to assess the stability margin of CIG-connected grid in Section 2. In Section 3, variations in the stability margin Wang (2014). Without a careful investigation of the devices distorts the waveforms and causes stability problems. inParameter Section 2.sensitivity In Section variations in the stability margin are particularly when CIGs are disconnected from This interaction is also known as instability in This interaction isWithout also as the the harmonic harmonic instability in grid Wang (2014). a careful investigation of the grid.analyzed is3, investigated in Section 4the to in Section 2. In Section 3,also variations in the stability margin are analyzed particularly when CIGs are disconnected from distribution system to known understand harmonic stability This interaction isWithout also known as the the harmonic instability in grid are analyzed particularly when CIGs are disconnected from the grid. Parameter sensitivity is also investigated in Section 4the to Wang (2014). a careful investigation of the Wang (2014). Without a careful the investigation of the select an appropriate control parameter that is more influential distribution system to understand harmonic stability are analyzed particularly when CIGs are disconnected from the grid. Parameter sensitivity is also investigated in Section 4 to phenomenon, harmonic instability reduces the hosting Wang (2014). Without a careful the investigation of the select grid. Parameter sensitivity is also investigated in Section 4 to distribution system to understand harmonic stability an appropriate control parameter that is more influential distribution system to power understand the harmonic theParameter dominant pole.control Then, the introduction and application phenomenon, harmonic instability reduces the stability hosting on grid. sensitivity is also investigated in Section 4 to select an appropriate parameter that is more influential capacity to regulate the quality of the system. distribution system to understand the harmonic stability select an appropriate control parameter that is more influential phenomenon, harmonic instability reduces the hosting on the dominant pole. Then, the introduction and application phenomenon, harmonic instability the hosting of PSO toappropriate adjust control parameters are that explained inapplication Section 5. capacity to regulate the power quality ofreduces the system. select an control parameter is more influential on the dominant pole. Then, the introduction and phenomenon, harmonic instability reduces the hosting on PSO the dominant pole. Then, the introduction andinapplication capacity to the quality of the to6 adjust parameters are Section in 5. capacity to regulate regulate the power power quality of applicable the system. system.method to of Section provescontrol the effectiveness of explained the tuning process Impedance-based stability analysis is an on the dominant pole. Then, the introduction andinapplication of PSO to adjust control parameters are explained Section 5. capacity to regulate the power quality of applicable the system.method to Section of PSO to adjust control parameters are explained in Section 5. 6 proves the effectiveness of the tuning process in Impedance-based stability analysis is an securing simulation resultsare ofexplained thetuning PSOinapplication assess the stabilitystability of a CIG-connected grid in Sun (2009). The of PSO to6stability adjust control parameters Section in 5. Section proves the effectiveness of the process Impedance-based analysis is an applicable method to Section 6 proves the effectiveness of the tuning process in Impedance-based stability analysis is an applicable method to securing stability simulation results of the PSO application assess the stability of a CIG-connected grid in Sun (2009). The followed by concluding remarks in Section 7. relationship between the impedance of each CIG and the grid Section 6 proves the effectiveness of the tuning process in Impedance-based stability analysis is an applicable method to securing by stability simulation results of the the7.PSO PSO application application assess the of aathe CIG-connected grid Sun (2009). The securing stability simulation results of assess the stability stability ofdetermine CIG-connected grid inCIG Sun (2009). The followed concluding remarks in Section relationship impedance ofstability. eachin and the grid impedance isbetween used to system The important stability simulation results of the7.PSO application assess the stability of athe CIG-connected grid inCIG Sun (2009). The securing followed by concluding remarks in Section relationship between impedance of each and the grid followed by concluding remarks in Section 7. relationship between the impedance of each CIG and the grid impedance used to determine system stability. The important part in thisis analysis is obtain theofCIG that is followed 2. IMPEDANCE-BASED STABILITY ANALYSIS by concluding remarks in Section 7. relationship between thetoimpedance eachimpedance CIG and the grid impedance is used to determine system stability. The impedance isanalysis used to determine system stability. The important important part in this is (Point to obtain the CIG impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS observed from the PCC of Common Coupling). impedance isanalysis used to determine system stability. The important part in this is to obtain the CIG impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS part in this analysis is (Point to obtain the CIG Coupling). impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS observed from the PCC of Common part in this analysis is to obtain the CIG impedance that is 2. IMPEDANCE-BASED STABILITY ANALYSIS observed observed from from the the PCC PCC (Point (Point of of Common Common Coupling). Coupling). observed ©from PCC (Point of Common Copyright 2019the IFAC 75 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 2019, IFAC (International FederationCoupling). of Automatic Control) Copyright 2019 responsibility IFAC 75 Control. Peer review©under of International Federation of Automatic Copyright © 75 Copyright © 2019 2019 IFAC IFAC 75 10.1016/j.ifacol.2019.08.158 Copyright © 2019 IFAC 75
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Fig. 3. System configuration of test case. Three CIGs are connected to the system through the line. 𝐺𝐺𝑐𝑐𝑐𝑐 = 𝑌𝑌𝑐𝑐 =
Fig. 1. System details of CIG: (a) entire circuit and control system and (b) control block diagram.
𝐺𝐺𝑃𝑃𝑃𝑃 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃
𝐿𝐿𝑐𝑐 𝐿𝐿𝑔𝑔 𝐶𝐶𝑓𝑓 𝑠𝑠 3 −𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 𝐿𝐿𝑔𝑔 𝑠𝑠+(𝐿𝐿𝑔𝑔 +𝐿𝐿𝑐𝑐 )𝑠𝑠+𝐺𝐺𝑃𝑃𝑃𝑃 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃
𝐿𝐿𝑐𝑐 𝐶𝐶𝑓𝑓 𝑠𝑠 2 −𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 +1 3 𝐿𝐿𝑐𝑐 𝐿𝐿𝑔𝑔 𝐶𝐶𝑓𝑓 𝑠𝑠 −𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 𝐿𝐿𝑔𝑔 𝑠𝑠+(𝐿𝐿𝑔𝑔 +𝐿𝐿𝑐𝑐 )𝑠𝑠+𝐺𝐺𝑃𝑃𝑃𝑃 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃
(3) (4)
Here, Gcl is the closed-loop transfer function, and Yc is the CIG admittance, which is derived from the relationship between VPCC and Ig,d. Equations (2)–(4) imply that a CIG can be modeled as a parallel combination of the current source and CIG admittance. 2.2 Equivalent Modeling and Analysis Fig. 2 illustrates the CIG equivalent model for a number of CIGs connected in cascade, where a series impedance separates each CIG. Each CIG can be modeled as a parallel circuit of the admittance and the current source. When the total admittance of the CIG side at PCC is addressed as Yeq, the current flowing into the grid can be obtained as follows:
Fig. 2. Equivalent model of CIGs. This model consists of CIG admittance and current sources.
𝐼𝐼𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 =
2.1 CIG Impedance Modeling The entire system model and its control block diagram are shown in Fig. 1. An LCL filter is used for current damping and harmonic filtering. The voltage across the filter capacitor and current flowing into the PCC are measured for grid synchronization and current control. Each CIG operates in a constant power and unity power factor mode for a basic approach. A DC source is used to assume constant DC voltage.
𝑠𝑠
1−1.5𝑇𝑇𝑑𝑑 2 1+1.5𝑇𝑇𝑑𝑑
𝑠𝑠 2
𝑌𝑌𝑒𝑒𝑒𝑒 𝐼𝐼 𝑌𝑌𝑔𝑔 +𝑌𝑌𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑐𝑐
(5)
2.3 Consideration of nonlinear power system If the power system is a nonlinear system, the investigation of the grid stability can be more complex due to the higher-order Yg. One way to investigate the stability of this system is to approximate the frequency response characteristics with a transfer function by vector fitting in Gustavsen (1999). In this case, the validity of the stability determination depends on the accuracy of the approximation that usually makes the higherorder transfer function. In this paper, the equivalent admittance of the power system is regarded as a first order transfer function to focus on the methodology of the controller design.
(1)
Because of the unity power factor operation, only d-axis current control is considered (i.e., zero q-axis current). The daxis grid side current with the converter controller GPI = kp +ki/s can be obtained as follows: 𝐼𝐼𝑔𝑔,𝑑𝑑 = 𝐺𝐺𝑐𝑐𝑐𝑐 𝐼𝐼𝑔𝑔,𝑑𝑑,𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑌𝑌𝑐𝑐 𝑉𝑉𝑃𝑃𝑃𝑃𝑃𝑃,𝑑𝑑
−
where Yg is the admittance of the source impedance (1/ Zg), Icon is the total current generated by the CIGs, and Isrc is the current generated by the grid. Based on the assumption that both currents Icon and Isrc are stable in Sun (2009), the stability of the current is determined by the common denominator on the right side of (5), which is 1/(Yg + Yeq).
Converter switching can be expressed as a time-delay function. According to Wang (2014), there is a 1.5 sampling period delay, which can be approximated by a Padė approximation as follows: 𝐺𝐺𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑒𝑒 −1.5𝑇𝑇𝑑𝑑𝑠𝑠 ≈
𝑌𝑌𝑔𝑔 𝐼𝐼 𝑌𝑌𝑔𝑔 +𝑌𝑌𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐
(2) 76
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3.2 Stability Assessment for CIG Disconnection To observe the system stability problems due to malfunctions or maintenance of the CIGs, all cases of one or more CIG disconnections will be considered. Fig. 4 shows the stability assessment when only one CIG remains connected to the grid. As the electrical distance between the grid and the CIG increases, the stability margin decreases, but the system still operates in a stable condition. Fig. 5 shows the system stability assessment when two or more CIGs are in operation. Only when CIG1 disconnected is the system becomes unstable. By contrast, the other two cases are more stable than the base case despite the CIG disconnections. The two assessment results imply that a greater electrical distance between the grid and the nearest CIG results in a lower stability margin. In addition, having fewer CIGs in the system results in more stability. A test system model simulation carried out in PSCAD/EMTDC verifies the instability phenomenon in the time domain. Fig. 6 shows waveforms of the current flowing into the grid when CB1 is opened at 0.5 s and CB2 is opened at 0.65 s sequentially. CB refers to the circuit breaker. Before the CIG disconnection, the current waveforms are clean and stable. However, as a consequence of disconnecting CIG1, the current waveforms become distorted and the system becomes unstable. After the additional trip of CIG2 at 0.65 s, the system becomes stable again.
Fig. 4. Stability assessment when only one CIG maintains the connection.
4. CONTROL PARAMETER SENSITIVITY ANALYSIS A parameter sensitivity analysis to identify adjustable parameters to help reduce the computational complexity of the optimization by removing low-impact parameters. The quantification of the impact of a control parameter p on the stability margin is achieved by a sensitivity analysis, as follows:
Fig. 5. Stability assessment when two or more CIGs maintain the connection.
1
∆𝑋𝑋
𝑆𝑆𝑃𝑃 = 𝑛𝑛 ∑𝑛𝑛𝑡𝑡=1 |∆𝑃𝑃𝑡𝑡| 𝑡𝑡
(6)
where Δ Xt is the moved pole displacement, and Δ Pt is the variation in a parameter. As the variation in the pole displacement is nonlinear, the absolute value is averaged for the sensitivity analysis. According to (5) and (6), the stability margin of the system can be changed by the control parameters.
Fig. 6. Waveforms of current flowing into grid when CIG1 and CIG2 are disconnected from grid at 0.5 s and 0.65 s, respectively.
The proportional gain kp,r and integral gain ki,r (for the r-th CIG, r = 1, 2, 3) of the current controller are target parameters for the sensitivity analysis in this study. In addition, the following equation defines the relative sensitivity to quantify the impact of kp,r relative to ki,r:
3. SYSTEM STABILITY CHANGES ACCOMPANIED BY CIG DISCONNECTION
𝑅𝑅𝑅𝑅𝑟𝑟 =
3.1 System Configuration of Test System Fig. 3 shows a base case where three CIGs are connected to the grid and are in operation, where CIGs are numbered from the nearest to the PCC to the remotest. In the base case, all CIGs have the same parameters. The line impedance is expressed as a series connection of resistance and inductance. The distance between CIGs is 1.6 km. The q-axis voltage and current are set to 0. The parameters are shown in Table 5.
𝑆𝑆𝑘𝑘𝑝𝑝,𝑟𝑟 𝑆𝑆𝑘𝑘
𝑖𝑖,𝑟𝑟
(7)
A target pole for the first sensitivity analysis is pole 1 in the “CB1 open” case, as indicated in Fig. 5. Except for the parameters of CIG1, the control parameters of CIG2 and CIG3 are considered to calculate the sensitivity, where kp,r varies from 0.75 to 0.85, and ki,r varies from 100 to 1000. The amounts of change in parameters are 0.01 and 100, 77
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Table 1. Sensitivity analysis of pole 1 in “CB1 open” case Parameters kp,2 kp,3 ki,2 ki,3
Magnitude 180.9648 84.2525 0.0631 0.0293
Real part 180.4010 84.1847 0.0058 0.0013
Imag. part 14.2508 3.3781 0.0629 0.0293
Table 2. Sensitivity analysis for poles 2 and 3 in base case Parameters Pole 2 kp,1 Pole 3 Pole 2 ki,1 Pole 3
Magnitude 237.7854 32.4185 0.0714 0.0117
Real part 236.6427 25.3783 0.0071 0.0067
Imag. part 23.2647 20.1372 0.0710 0.0096
Table 3. Relative sensitivity analysis of both cases Parameters Pole 2 RS1 Pole 3 RS2 RS3
Fig. 7. Location changes of pole 1 by variations in parameters: (a) kp,2 variation, (b) kp,3 variation, (c) ki,2 variation, and (d) ki,3 variation.
Magnitude 3330.33 2770.81 2867.90 2875.51
Real part 33329.96 3787.81 31103.62 64757.46
Imag. part 322.67 2097.63 226.56 115.29
A second sensitivity analysis focuses on the control parameters of CIG1 and their impact on the two nearest poles (poles 2 and 3) on the imaginary axis in the base case under the same parameter variations as in the previous analysis. The details of the sensitivity analysis results for poles 2 and 3 are listed in Table 2, and the relative sensitivity analysis results of both cases are listed in Table 3. Similarly, the impact of kp,1 on the sensitivity of the real part is still larger than that of ki,1. The difference is that the relative sensitivity RS1 of the real part of pole 3 is approximately 10 times lower than RS2 of the real part, as observed in Table 3. This implies that the impact of ki,1 relative to kp,1 on pole 3 is larger than in the other cases. Fig. 8 shows the tendencies of the movement of the two nearest poles to the imaginary axis with respect to the control parameters. For pole 3, the displacement by ki,1 is larger than that by kp,1. However, the displacement of pole 2 by ki,1 is smaller than that by kp,1. In addition, the increase in ki,1 causes a change in the dominant pole and even worsens the stability because poles 2 and 3 move toward each other. The change in the dominant pole from pole 3 to pole 2 limits the capability for ki,1 to enhance the stability. Therefore, kp,1 is a more suitable and capable parameter for optimization to secure the system stability.
Fig. 8. Location changes of poles 2 and 3 by variations in parameters: (a) kp,1 variation and (b) ki,1 variation. respectively. Table I presents the sensitivity analysis results of pole 1 with respect to the control parameters, kp,r and ki,r (r = 2, 3). It is obvious that the sensitivity of pole 1 by kp,r is much larger than by ki,r. If the sensitivity is divided into real and imaginary parts for detailed information, the sensitivity of the real part of the pole 1 by kp,r is higher than the sensitivity of the imaginary part. By contrast, ki,r has more impact on the sensitivity of the imaginary part than that of the real part. Fig. 7 shows the tendencies of pole 1’s movements with respect to the control parameter variations. Red arrows indicate the directions of the pole movements as the parameters increase. The two figures in the upper part prove that kp,r can be utilized to secure the stability. On the other hand, the two figures in the lower part show that ki,r is not capable of enhancing the stability. ki,r causes relatively large variations in the pole location in the imaginary part rather than the real part. Only the proportional gains are adjusted to improve the stability in the “CB1 open” case.
5. PARTICLE SWARM OPTIMIZATION FOR CIG PARAMETER TUNING 5.1 Introduction of Particle Swarm Optimization The PSO theory is based on the social behavior of fish schools or bird flocks, which tend to move to their optimal positions by using their own or others’ experience in Lee (2008). Each particle position is evaluated by an objective function to find 78
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2) Select the appropriate parameters from the sensitivity analysis results for the PSO.
the best position. The next velocity is determined by the current velocity, current position, local best position, and global best position. The next position of the particle is the summation of the current position and next velocity. The relationships between these conditions can be expressed as the following equations:
3) For the case having the lowest stability margin among all cases investigated in 1), perform the PSO with the parameters selected in 2). The parameters of disconnected CIGs are not optimized in this step.
𝑣𝑣𝑖𝑖 [𝑘𝑘 + 1] = 𝑤𝑤𝑣𝑣𝑖𝑖 [𝑘𝑘] + 𝑐𝑐1 𝑟𝑟1 (𝑥𝑥𝑙𝑙𝑙𝑙,𝑖𝑖 [𝑘𝑘] − 𝑥𝑥𝑖𝑖 [𝑘𝑘]) + 𝑐𝑐2 𝑟𝑟2 (𝑥𝑥𝑔𝑔𝑔𝑔,𝑖𝑖 [𝑘𝑘] − 𝑥𝑥𝑖𝑖 [𝑘𝑘])
w = 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 − (𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 )
4) Replace the parameter values with the optimized values. (8)
𝑘𝑘
𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚
𝑥𝑥𝑖𝑖 [𝑘𝑘 + 1] = 𝑥𝑥𝑖𝑖 [𝑘𝑘] + 𝑣𝑣𝑖𝑖 [𝑘𝑘 + 1]
5) Perform the PSO again for the case with no CIG disconnection. The PSO finds the optimal value for the rest of the parameters: The first PSO ensures stability under the most severe condition, whereas the second maximizes the stability margin under the normal operating condition.
(9) (10)
Here, i is the particle number, vi is the velocity, w is the weighting factor, c1 and c2 are weighting coefficients, r1 and r2 are random numbers between 0 and 1, xi is the current position, xlb,i is the local best position among the current and previous positions of the i-th particle, xgb,i is the global best position among the current and previous xlb,i values of the i-th particle, wmax and wmin are the maximum and minimum values of w, k is the current iteration, and kmax is the maximum iteration number. The weighting factor determines the impact of the current velocity, and the weighting coefficients determine the impact of the difference between the current position and the local or global best position. According to Lee (2008), the typical weighting coefficients and maximum and minimum values of w are as follows: 𝑐𝑐1 = 𝑐𝑐2 = 2, 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 = 0.9, 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 = 0.4
6. SIMULATION RESULTS As shown in Fig. 5, the case of “CIG2, CIG3 Connected” has the lowest stability margin and is even unstable. Also, the sensitivity analysis proves that the proportional gains are more influential parameters than the integral gains in the test system. Therefore, kp,2 and kp,3 are first optimized without considering kp,1. Then, kp,1 is subsequently optimized by the second-round PSO with fixed values of kp,2 and kp,3 derived in the first-round PSO. Table 4 lists the parameter tuning results of the PSO. Fig. 8 shows the stability assessment results for the cases where only one CIG is connected to the grid after parameter adjustment by PSO. The adjustment of the proportional gain causes the real parts of the poles near the fundamental frequency to increase, whereas the real part of the dominant poles in Fig. 4 decreases. As a result, the dominant poles are replaced, and the stability margins of all cases are changed.
(11)
5.2 Objective Function for Parameter Tuning For the application of PSO in a stability assessment, the objective function is defined as follows: 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑀𝑀𝑀𝑀𝑀𝑀 (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 (𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 (
1
𝑌𝑌𝑔𝑔 +𝑌𝑌𝑒𝑒𝑒𝑒
)))
79
Although the stability margin decreases when only CIG2 maintains a connection, all cases in Fig. 8 still have enough of a margin to be stable. Fig. 9 shows the stability assessment results of other cases that kept two or more CIGs connected after PSO. Compared to the results before PSO in Fig. 5, the stability margins in these cases are increased. In particular, the case where CIG1 is disconnected becomes stable.
(12)
It indicates the real part of the dominant pole of the common denominator in (5). When the position of the pole is changed by the control parameters, the PSO aims to maximize the stability margin of the common denominator by minimizing the objective function because of its negative sign. When the dominant pole reaches the rarely-unmovable pole by adjusting the parameters, it has the maximum stability margin. Because the stability has priority over the converter performance, which is closely related to various grid requirements, the trade-off between the stability and the converter performance is not investigated further in this study.
To examine the difference in system stability before and after PSO, Fig. 10 shows current waveforms under the same condition as in Fig. 6 after parameter tuning. In contrast to Fig. 6, the current waveform maintains its stability despite of the CIG1 disconnection. Therefore, the simulation results demonstrate that the PSO based control parameter tuning process successfully helps ensure the system stability. 7. CONCLUSION This paper proposes a parameter tuning procedure based on an impedance-based stability analysis to ensure system stability and potentially improve the hosting capacity by utilizing PSO. The converter admittance derived from the converter current control consists of the control parameters and passive elements of the converter filter. The stability can be assessed by a common denominator that is the inverse of the summation of the grid admittance and the equivalent admittance. The tuning
5.3 PSO Application to Parameter Tuning To secure stability in all cases even with malfunctions of CIGs, the parameters are tuned by the following procedure: 1) Assess the system stability of the multi-CIG system. All cases of CIG disconnection should be analyzed.
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Table 4. Parameter tuning results of each PSO process Parameters 1st PSO 2nd PSO
kp,1 0.63476
kp,2 0.42443 (0.42443)
kp,3 0.5423 (0.5423)
Table 5. System parameters Parameters Grid voltage Grid frequency Short circuit capacity Grid impedance angle Distance between converters Switching frequency Switching delay (Td) Line Rline impedance Lline Filter Lg impedance Cf Lc PI control kp,n ki,n
Fig. 8. Stability assessment when only one CIG maintains connection after PSO.
Values 400 V 60 Hz 0.3 MVA 15º 1.6 km 1980 Hz 20 0.4375 /km 0.875 mH/km 0.6 mH 100 F 2.0 mH 0.85 500
changes the system status to be stable. This study should provide a systematic framework for identifying the problematic yet controllable design parameters to ensure the desired stability margin and secure the desired DER hosting capacity. This study should provide a systematic framework for identifying the problematic yet controllable design parameters to ensure the desired stability margin and secure the desired DER hosting capacity.
Fig. 9. Stability assessment when two or more CIGs maintain connection after PSO.
REFERENCES EPRI. (2016). Defining a roadmap for successful implementation of a hosting capacity method for New York state, in EPRI, Palo Alto, CA, USA, Tech. Rep. Wang, X., Blaabjerg, F. and Wu, W. (2014). Modeling and analysis of harmonic stability in an ac power-electronicsbased power system, vol. 29, no. 12, pp. 6412–6432, IEEE Trans. Power Electron.. Sun, J. (2009). Small-signal methods for ac distributed power systems-a review, vol. 24, no. 11, pp. 2545–2554, IEEE Trans. Power Electron.. Lee, K. Y. and El-Sharkwai, M. A. (2008). Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems, Wiley. Azzouz, M. A., Saadany, E. F. E. (2016). New impedance reshaping algorithm for grid-connected dgs, pp. 1–5, in 2016 IEEE Power and Energy Society General Meeting. Gustavsen, B. and Semlyen, A. (1999). Rational approximation of frequency domain responses by vector fitting, vol. 14, no. 3, pp. 1052–1062, IEEE Trans. Power Delivery.
Fig. 10. Waveforms of current flowing into grid in “CB1 open” case after parameter tuning. of control parameters affects the converter admittance, which then secure the stability margin. For the test system, the CIG disconnection can result in instability even if the system is stable under its normal condition. To solve this problem, a control parameter sensitivity analysis first identifies which control parameter has most influence on the stability. After that, a parameter optimization process begins with the most severe case having the lowest stability margin. Then, PSO is carried out again to optimize the rest of the parameters. The simulation results show that the proposed parameter tuning process enhances the stability of the test system and even 80