Enlarged operational area of an Interline DC power Flow controller via adaptive droop control for Multi-Terminal HVDC systems

Enlarged operational area of an Interline DC power Flow controller via adaptive droop control for Multi-Terminal HVDC systems Mirhamed Pourmirasghariyan a,* , G.B Gharehpetian b , Oriol Gomis-Bellmunt c , David Campos-Gaona a , Panagiotis N. Papadopoulos d a Department of Electronic and Electrical Engineering (EEE), University of Strathclyde, Glasgow, G1 1XQ, UK b Department of Electrical Engineering, Amirkabir University of Technology, Tehran 15875-4413, Iran c CITCEA-UPC, Universitat Polit` ecnica de Catalunya, Barcelona 08028, Spain d School of Electrical and Electronic Engineering, The University of Manchester, Manchester, M60 1QD, UK ARTICLE INFO Keywords: Adaptive Droop Control HVDC IDC-PFCs MMCs Optimal Power Flow ABSTRACT The effective performance of Interline DC Power Flow Controllers (IDC-PFCs) in Modular Multilevel Converters (MMC)-based High Voltage Direct Current (HVDC) grids is restricted by 1) the current limitation of the HVDC cables/lines, 2) the HVDC buses’DC voltages and 3) the IDC-PFC capacitor’s DC voltage limit. The pivotal remedy for this issue is to utilize an adaptive droop control for the MMCs that varies its droop gain to maximize the IDC-PFC operation range. In this paper, 3D curves of the IDC-PFC’s important characteristics are used to assess the flexibility of the IDC-PFC control. By using this approach, a new degree of freedom for IDC-PFC controllability is achieved. The performance of the optimal-adaptive-droop-controlled strategy presented in this paper is validated using power flow studies. The results demonstrate that a wider operational area is conceivable for the IDC-PFC when this technique is applied as a combination of MMC converters’droop control and IDC-PFC duty cycle. 1. Introduction 1.1. Motivations Future meshed Modular Multilevel Converter (MMC)-based High Voltage Direct Current (HVDC) grids, targeting the integration of Renewable Energy Sources (RESs) into main power systems promise enhanced integration stability, without concerns regarding reactive power requirements [1–3]. Moreover, to enhance the flexibility of the Power Flow (PF) of the MMC-based HVDC grids, the Interline DC Power Flow Controllers (IDC-PFCs) have been proposed in the research literature [4–6]. Although IDC-PFCs are a promising technology, the region of the IDC-PFC’s operation is somehow limited. In fact, the operation of IDC-PFCs is highly dependent on the current limitation of HVDC cables/ lines, the DC voltage limitations of interconnected buses, and the IDCPFC capacitor’s DC voltage boundaries [5]. Therefore, the main goal of the present paper focuses on introducing a new degree of freedom to overcome the mentioned issues of the IDCPFCs’operation in MMC-based HVDC grids and enhancing IDC-PFC’s contribution to DC-PF. 1.2. Literature Review There is a growing interest in interconnecting Voltage Source Converters (VSCs) or MMCs for the purpose of achieving Multi-Terminal HVDC structure (MT-HVDC) grids, or meshed HVDC grids. The meshed HVDC grids have many HVDC cables/lines with complex control systems and operation modes [7] and [8]. One major concern of the MTHVDC grids is Power Flow studies. In case of poor control, congestion, and bottlenecks overloading would occur in HVDC cables/lines. Accordingly, DC-PFCs, which are equivalent to the Flexible AC Transmission System (FACTS) devices and have the same tasks, are introduced. DC-PFCs can manipulate the DC Power Flow (DC-PF) equations and bring about flexibility and controllability in power/current flow. Generally, DC-PFCs are known in three categories series, cascaded, and interline. Among these three categories, since the IDC-PFCs have a simple control system, structure, and economic advantages, they are under heavy attention [8]. The IDC-PFCs inject DC voltage in series into * Corresponding author. E-mail address: [email protected] (M. Pourmirasghariyan). Contents lists available at ScienceDirect International Journal of Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes https://doi.org/10.1016/j.ijepes.2024.110430 Received 7 September 2024; Received in revised form 10 November 2024; Accepted 15 December 2024 Electrical Power and Energy Systems 164 (2025) 110430 Available online 20 December 2024 0142-0615/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). their interconnected HVDC cables/lines by setting appropriate control variables (duty cycle) and consequently vary the current/power of their interconnected HVDC cables/lines. There are several methods for power sharing and current control in MT-HVDC grids apart from the presence of DC-PFCs. One effective way that enables the operators to appropriately utilize the capacity of converters, is adaptive droop control. In [9], a generalized droop-control strategy has been proposed that has the ability to operate under three possible modes (constant power, constant voltage, and droop-controlled voltage-power) according to the demand of operators with rational power sharing. The generic method of [9] paves the way for easy maneuverability over three different operation modes. The authors of [10], have proposed a scheme for the droop control of VSCs that not only acts to reach a reasonable power-sharing but also avoids converters’ power and DC voltage limit violations. The studied method of [10] is adaptive in which droop values of the converters are set automatically to reach a stable and safe operation for the system. Moreover, in [11], a droop control strategy has been introduced that considers frequency deviation and power sharing according to the characteristics of the voltage-current-frequency relationship. Also, in [12], the authors have developed a new coordinated-predictive droop control that avoids High Voltage Ride-Through (HVRT) by following an optimal droop coefficient. Further, the performance of droop techniques in terms of power sharing and transient stability, have been evaluated in [13]. The combination of droop control of MMCs/VSCs with the IDC-PFCs has some important advantages, which have never been studied in the previous droop-controlled strategies. Manipulating droop values of those converters in which an IDC-PFC is connected, can remove or at least mitigate the limitations of the IDC-PFC. More importantly, removing these limitations would pave the way to reach a more flexible, widened, and controllable operation of the IDC-PFCs and the grid as well. The aforementioned restrictions are dealt with partially in [5] by using only one degree of freedom (the duty cycle), however, a complete method for overcoming IDC-PFC limitations has not yet been proposed in the open literature. Therefore, this paper aims to overcome those restrictions with the help of adaptive droop control of IDC-PFC’s interconnected converters. Consequently, an adaptive-optimal droop control strategy is introduced, which considers the mentioned limitations and adapts the converters’droop gains accordingly to widen the IDC-PFCs’ operation capabilities. 1.3. Research Objectives and contributions The contributions of this paper are as follows: •In section II, an adaptive-droop-controlled strategy is proposed in which not only the DC voltage and power limits of the MMCs/VSCs are considered but also takes the limitations of the IDC-PFC into account. Hence, the droop gains of the MMCs/VSCs are set in the presence of IDC-PFCs within the DC network. By addressing these limitations, the duty cycle of the IDC-PFC can be swung extensively leading to a flexible operation. •In section III, the influence of variable droop control on the behavior of an IDC-PFC in a three-terminal CIGRE HVDC test grid is evaluated. Here, the 3D characteristics of the IDC-PFC with two autonomous degrees of freedom (duty cycle of the IDC-PFC and the variable droop) are analyzed. Afterwards, the results are compared to those of the [5], and the benefits of the proposed strategy are discussed. •In section V, an Optimal-Adaptive-Droop-Controlled Power Flow (OADC-PF) study is conducted in a steady-state situation. Optimal results for determining adaptive droop values mean the MMCs/VSCs are used to attain a successful IDC-PFC limitation removal. •Finally, in section VI, dynamic and OADC-PF case studies are provided. 2. The IDC-PFC under adaptive droop control 2.1. Operation Principles of IDC-PFC There are several recent publications regarding new topologies for IDC-PFC. However, in this paper, the basic topology presented in [14], is chosen to focus fundamentally on the effect that the variable droop gain has on the behavior of the IDC-PFC, as well as obtaining its 3D characteristics in the presence of duty cycle (D) and the new degree of freedom (KDroop) which provides an adaptable gain for the droop controller. Fig. 1 (a) illustrates the topology of the IDC-PFC, which is built up of a reduced dual H-bridge and a DC capacitor. The DC capacitor of the IDC-PFC can operate under positive or negative DC voltage, and its value (EC) is dependent on the voltages of its interconnected buses and the control variable setting D. The IDC-PFC is located through HVDC cable t (master cable) and HVDC cable u(slave cable) that injects predetermined-compensating voltages Us,t= (1−D)ECand Us,u= − DEC in series with the interconnected HVDC cables tand u, respectively, as shown in Fig. 1 (b). The performance of the IDC-PFC involves exchanging power between the two (master and slave) HVDC cables. Thus, the more compensating voltage injection increases through the main HVDC cable t, the more power/current flowing through the main HVDC cable tincreases, and vice versa. The IDC-PFCs are considered minor-sized converters in comparison with VSCs or MMCs. Therefore, the losses of the IDC-PFCs are slight (0.002 % losses of the MMCs’)[15]. Hence, in this research, the IDCPFCs are regarded to be lossless. The HVDC cables are modeled as π −Lump model, and the parameters Zt,Zu,Yt, and Yuare the series impedance of HVDC cable t, series impedance of HVDC cable u, shunt admittance of HVDC cable t, and shunt admittance of HVDC cable u, respectively. Moreover, for the steady-state analysis of the grid, the cable parameters are Rt,Gt,Ru, and Guwhich represent the HVDC cables t’s and u’s resistance and shunt conductance, respectively. Nevertheless, because the existence of the shunt admittance does not participate in PF results significantly, thus, they are crowded out of the formulations (hereafter) [16]. Based on Fig. 1 (a), considering that the IDC-PFC is placed between buses i,j, and k, the current flowing from bus-i(Ii) has two possible paths to flow between the HVDC cable tand HVDC cable u. There are two switching combinations, one for a positive current passage pair and the other one for a negative current passage pair. The switching combinations represented in Fig. 1 (a) are called positive current pairs in which the switches ({S2,S4,S6}) and the diodes ({D1,D3,D5}) are utilized. For the negative current directions, the switches ({S1,S3,S5}) and the diodes ({D2,D4,D6}) are used, for further information refer to [5]. For the current directions shown in Fig. 1 (a), the current of HVDC cable t(Iij) passes through the switches ({D1,EC,S4}and {S2,EC,D3}), while for the current of HVDC cable u(Iik) the switches ({S2,EC,D5}and {D1,EC,S6}) are involved. Considering, the duty cycle (D) for the closed state of the switches ({S2,S4}) for the current (Iij) passage, and the closed state of the switches ({S2,S6}) with the complementary duty cycle (1 −D) for the current (Iik) passage, the average current passing through the IDC-PFC’s capacitor (IC) must be zero. Under the given circumstances, the average current of the capacitor can be obtained as: IC=1 T∫T 0 ICdt =1 T(− DIij + (1−D)Iik)(1) The parameter Tand ICare the operation cycle of the IDC-PFC and average IDC-PFC capacitor’s current, respectively. The average current needs to be zero (IC=0) for the stable operation of the IDC-PFC. Then, one can achieve: IC=0→D=Iij Iij +Iik (2) M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 2 With simple KVL/KCL law and neglecting the shunt admittance (Y), the currents of HVDC cable t(Iij) and HVDC cable u(Iik) can be achieved as follows: It=Iij = − Iji =1 Zt ((UDC,i−UDC,j) + (1−D)EC)(3) Iu=Iik = − Iki =1 Zu ((UDC,i−UDC,k) − DEC)(4) The parameters UDC,i,UDC,j,UDC,k,EC,It, and Iuare the voltage of bus-i, voltage of bus-j, voltage of bus-k, IDC-PFC’s capacitor’s DC voltage, current of HVDC cable t, and current of the HVDC cable u, respectively. Now, by substituting equations (3) and equation (4) into equation (2) and further manipulation, the IDC-PFC’s capacitor’s DC voltage can be obtained: EC=TiUDC,i+TjUDC,j+TkUDC,k(5) Ti=DZt− (1−D)Zu ZtD2+Zu(1−D)2(6) Tj=(1−D)Zu ZtD2+Zu(1−D)2,Tk=−DZt ZtD2+Zu(1−D)2 The terms Ti,Tj, and Tkare the coefficients that relate the dependency of the IDC-PFC’s capacitor on the duty cycle (D) and its interconnected bus voltages (UDC,i,UDC,j, and UDC,k). Moreover, the powers of the IDC-PFC interconnected HVDC cables/ lines (tand u) are expressed, as follows: Ps,t=Us,tIt,Ps,u=Us,uIu;Us,t= (1−D)EC,Us,u= − DEC(8) PExchange IDC−PFC =Ps,t+Ps,u(9) The symbols Ps,tand Ps,uare the manipulated power of the HVDC cable t, and the manipulated power of the HVDC cable u, respectively. Additionally, the parameter PExchange IDC−PFC presents the exchanged power between the HVDC cables tand u. According to the equation (3) and Fig. 1 (b), increasing voltage injection into HVDC cable t(Us,t= (1−D)EC)), will increase the current flowing through the cable. Moreover, based on the active power balance condition, any increment in HVDC cable tcurrent will reduce the current of HVDC cable u, and vice versa. With the given description and based on equation (9), a positive value for PExchange IDC−PFC means the power is exchanged from the HVDC cable uto the HVDC cable t, and vice versa. 2.2. Necessity of Integrating IDC-PFC characteristics into the adaptive droop gains of the MMCs/VSCs The IDC-PFC operation has some restrictions. Regarding the fact that the IDC-PFC only operates by one degree of freedom (duty cycle), it might face some limitations including interconnected DC voltages of its interconnected buses, IDC-PFC capacitor’s DC voltage violation, and current limitations. Hence, the duty cycle of the IDC-PFC would not be allowed to swing thoroughly from 0 to 1 or at least more extensively. Therefore, to avoid and escape these restrictions as much as possible, adaptive droop control of the IDC-PFC’s interconnected converters could be a proper solution. The adaptive droop could mitigate this issue at some level. Therefore, in the present paper, a new strategy of adaptive droop control with the consideration of the IDC-PFCs is proposed and the important characteristics of the IDC-PFC with variable droop are analyzed. 2.3. Droop Concept Considering the imbalance of the inflow power and outflow power, the generalized droop control equation of the converter with the droop (KDroop) is presented below: Fig. 1. (a) IDC-PFC installed between HVDC cables tand uin a meshed HVDC grid, and (b) IDC-PFC modeled by dependent voltage sources through its interconnected HVDC cables tand u. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 3 (P*−P) + KDroop(U* DC −UDC) = 0 (10) In (10), the parameters UDC,U* DC,Pand P*are the DC voltage, DC voltage reference, power, and power reference, respectively. Moreover, the symbol KDroop represents the droop gain of an MMC. According to (10), an MMC-HVDC terminal might operate under one of the three modes: constant power, constant voltage, and voltage-power droop-controlled as shown in Fig. 2. The constant power mode maintains the power of the converter at a constant reference (KDroop =0) regardless of the DC voltage swing. For the constant DC voltage mode, the converter keeps the voltage at a fixed amount (KDroop =∞) regardless of power change. Finally, the droop-controlled mode is a combination of the two previous control modes in which the DC voltage of the converter changes according to a droop value (slope, m) for changing a specific power of the converter [11]. 2.4. Integrating IDC-PFC into the formulations of adaptive droop of the MMCs/VSCs Generally, there are some considerations for adaptive droop control of MMCs/VSCs. For example, even though the fixed droop gains of the converters are chosen in accordance with their ratings, this does not imply that they are operating at their full capacity. In other words, usually, there would be some available headroom for sharing the additional power imbalance. As such, one application of adaptive droop control is related to achieving reasonable power sharing among converters preventing converters’power limit violations. Moving forward, the second application is the voltage deviation of converters which is attained by proper droop swinging, which won’t let the voltage violation happen. Hence, based on the given concepts, the below coefficients are considered for MMC’s/VSC’s droop gain that is not in connection with any IDC-PFC: KAdaptive Droop,i=λKDroop Pmax − |Pi| ⏞⏟⏟⏞ A σ − |ΔUDC,i| ⏟⏞⏞⏟B ;ΔUDC,i=1− |UDC,i|(11) The term Aindicates the available headroom of the i-th converter, while the term Bdenotes the DC voltage deviation of the i-th converter from its reference value and it is only considered to have a σ =5 % deviation. The symbol λis a user-defined factor to have further control over the droop value. In this equation, when the converter operates near its full capacity, the amount of Agoes to zero and causes the converter to operate at constant power mode (KAdaptive Droop =0). Furthermore, when the converter operates close to its DC voltage limit, the term Bdecreases and this causes the converter to operate at constant voltage mode (KAdaptive Droop = ∞). On the other hand, whenever the IDC-PFC is connected to a droopcontrolled converter, the problem of limitations of the IDC-PFC’s operation could be avoided by assigning optimum adaptive droop control. Therefore, the following terms are also defined to bring the limitations of the IDC-PFC into the adaptive droop gain formulations where the IDCPFC is installed: KAdaptive Droop,i=λiKDroop,i× (Pmax − |Pi| ⏞⏟⏟⏞ Ai )×((Imax,t− |It|) + (Imax,u− |Iu|)) ⏞⏟⏟⏞ Ci ( σ − |ΔUDC,i| ⏟⏞⏞⏟ Bi )×(EC,n− |EC|) ⏟⏞⏞⏟ d (12) KAdaptive Droop,j=λjKDroop,j×(Pmax − |Pj| ⏞⏟⏟⏞ Aj )×(Imax,t− |It|) ⏞⏟⏟⏞ Cj ( σ − |ΔUDC,j| ⏟⏞⏞⏟Bj )×(EC,n− |EC|) ⏟⏞⏞⏟d (13) KAdaptive Droop,k=λkKDroop,k×(Pmax − |Pk| ⏞⏟⏟⏞ Ak )×(Imax,u− |Iu|) ⏞⏟⏟⏞ Ck ( σ − |ΔUDC,k| ⏟⏞⏞⏟Bk )×(EC,n− |EC|) ⏟⏞⏞⏟d (14) The parameters EC,n,It,max, and Iu,max are the nominal value of the IDCPFC capacitor’s DC voltage, the maximum current of HVDC cable t, and the maximum current of HVDC cable u, respectively. The above equations (12–14) represent the adaptive droop gains of the buses (i-th, j-th, and k-th) which are in connection with an IDC-PFC. The terms Ai, Aj, and Akare responsible for reasonable power-sharing of the converters i-th, j-th, and k-th, respectively. Moreover, the terms Bi,Bj, and Bkfor the converters i-th, j-th, and k-th, orderly, are considered for preventing DC voltage violations. Also, the symbols λi,λj, and λkare user-defined factors. While the terms Ai,Aj, and Akare associated with the powers of converters, the terms Ci,Cj, and Ckare defined to relate the droop gains of the converters to the currents of the IDC-PFC’s interconnected cables. For the adaptive droop gain of i-th converter (KAdaptive Droop,i), the term Ci embeds the current limitations of the HVDC cables tand u(It,max and Iu,max) into the adaptive droop gain of i-th converter which is in connection with HVDC cables tand u. The term Cireduces the droop gain (KAdaptive Droop,i(to prevent the i-th converter from having a major contribution to power-sharing when the currents of the IDC-PFC’s interconnected HVDC cables are bottlenecked. In other words, when the HVDC cables tand uare overloaded, the term Citends to fall forcing the i-th converter to operate as a constant power bus (KAdaptive Droop,i=0) not to absorb/inject power/current anymore. This procedure also happens for the adaptive droop gains (KAdaptive Droop,jand KAdaptive Droop,k) by the terms Cjand Ckfor the j-th, and k-th converters, respectively. In addition, since the IDC-PFC capacitor’s DC voltage (5) is dependent on the DC voltage of the buses i-th, j-th, and k-th, the term dis the same for the adaptive droop gains of all IDC-PFC’s interconnected converters. If the value of ECis close to its limit (EC,n), the droop gains rise (KAdaptive Droop,i=KAdaptive Droop,j=KAdaptive Droop,k=∞) refusing the DC voltage limit of IDC-PFC’s capacitor (EC) to be violated. At this time, considering the droop value of bus-iin (12), the effect of the proposed nominator (Ai×Ci) and the denominator (Bi×d) on the Fig. 2. Three possible operation modes of MMCs: (a) Constant voltage mode, (b) Constant power mode, and (c) Droop-controlled mode. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 4 overall droop value can be investigated. Fig. 3 shows that when the nominator (Ai×Ci) is low (close to zero), the converter is working close to its nominal power rating (therefore droop is low) and should not receive/inject more power, no matter what the denominator is. Based on Fig. 3, the less the term (Ai×Ci) is, the less the droop gain will be, which means the loading of the bus-iand its interconnected cables/lines will be less. This will help the IDC-PFC not to be trapped in HVDC lines/cables and converter limits. On the other hand, when the denominator (Bi×d) is low (close to zero), the droop value tends to rise rapidly to avoid voltage deviation of more than the specified amount, as seen in Fig. 3. This will prevent the IDC-PFC’s capacitor’s voltage from violating its limit which eventually will lead to further operational area for the IDCPFC. At last, these advantages will enlarge the operational area of the IDC-PFC. The rest of the droop values of IDC-PFC’s interconnected buses (KAdaptive Droop,jand KAdaptive Droop,k) expose the same behavior. Moreover, the IDC-PFC’s capacity to exchange power in this study is 8 MW (EC×IC). Since both the IDC-PFC capacitor’s DC voltage and its current are considered in the proposed droop gains, violating the IDCPFC capacity limitation of exchange power is prevented. In previous DC power flow studies with the presence of IDC-PFC [5,7,14,15,17], only the magnitude of the IDC-PFC capacitor’s DC voltage (EC) was controlled by swinging the duty cycle (D) within a limited operational area. Moreover, the droop-control power-sharing strategies [9–13] have not considered the presence of any kind of IDCPFC to study the benefits. However, in this paper, to avoid restrictions on the IDC-PFC’s operation, adaptive droop is exerted as a new degree of freedom. 3. IDC-PFC operation analysis under duty cycle and variable droop 3.1. Three-terminal HVDC test Grid: A comparison study to [5] The three-terminal HVDC test grid of CIGRE is studied in this section specifically to compare the results of the proposed droop strategy (as a new degree of freedom to contribute to the IDC-PFC’s operation) to the results of [5] (which only had considered duty cycle for the IDC-PFC operation), see Fig. 4. The grid information is derived from [5]. The HVDC bus-3is a constant power bus (connected to an offshore wind farm) with a power of 800 MW, while the bus-1is a slack bus. The HVDC bus-2is connected to an onshore-side converter. The aims of this study are listed below: •Firstly, since the IDC-PFC’s master (t) and slave (u) HVDC cables are connected to the bus-2(which affects both cables’currents), the droop coefficient of the bus-2is considered to be variable (which was constant power bus in [5] with 400 MW power) to study the effect of variable droop on the behavior of the IDC-PFC. For this purpose, all the possible operation of the system is swept up for various feasible droop gains. Therefore, the main characteristics of the IDC-PFC are extracted with two degrees of freedom (duty cycle and droop gain). •Secondly, the advantages of the proposed strategy are compared to the results of [5], and the broadened operational routes are analyzed. In other words, it is shown that the system can operate in an extensive route if the droop gain is chosen appropriately. •Finally, a figure that segregates the widened operational area caused by IDC-PFC and the combination of IDC-PFC and variable droop is provided. This figure illustrates the widened operational area in detail. 3.2. DC power Flow of Three-Terminal test HVDC grid in the presence of IDC-PFC with variable droop In this section, DC Power Flow (DC-PF) for the three-terminal HVDC test grid in the presence of variable droop gain is presented. Considering that the Bus-1is a slack-bus, the DC-PF can be stated as follows: UDC,1−U* DC,1=0,slack bus (15) I2−P2 UDC,2 =0,I3−P* 3 UDC,3 =0 (16) I1−I12 −I13 =0,I2+I12 +I23 =0,I3+I13 −I23 =0 (17) 1 R12 (UDC,1− (1−D)EC−UDC,2) − I12 =0 (18) 1 R13 (UDC,1−UDC,3) − I13 =0,1 R23 (UDC,3−UDC,2−DEC) − I23 =0 (19) P2= − KDroop,2× (UDC,2−U* DC,2)(20) P* 3=−UDC,2UDC,3−UDC,3DEC+U2 DC,3 R23 +−UDC,1UDC,3+U2 DC,3 R13 (21) For the given DC-PF equations, the parameters U* DC,1,U* DC,2,P* 3,R12,R13, and R23 are slack-bus reference DC voltage, the DC voltage reference value of adaptive-droop-controlled bus-2, constant power of bus-3 (offshore wind farm), and resistances of cable-1,cable-2, and cable-3, respectively. Moreover, considering the active power balance of the IDCPFC, the relation between duty cycle (D) and the droop of the bus-2 (KDroop,2) can be stated as follows: P2=UDC,2I2= − KDroop,2× (UDC,2−U* DC,2),I2D−I12 =0 (22) KDroop,2=UDC,2I12 D(U* DC,2−UDC,2)(23) The presented DC-PF is solved by the Newton-Raphson method [14] in steady-state conditions. 3.3. 3D analysis of the IDC-PFC Characteristics: Duty cycle and droop gain are autonomous degrees of freedom In this section, the contribution of variable droop as a new degree of freedom on the IDC-PFC’s behavior is analyzed and the results are compared to those of the [5], see Fig. 5 and Fig. 6. The following 3D curves are the outputs of DC-PF studies sweeping all the possible duty cycle and droop gain of bus-2(in MATLAB 2023b: m.file coding). By comparing the results of the IDC-PFC characteristics in Fig. 5 (with only duty cycle as a control variable) and Fig. 6 (with duty cycle and variable droop as control variables) in brief, the following advantages are achieved: •Comparing Fig. 5 (a) and Fig. 6 (a), the DC voltage of bus-2is swingable within the range 203871 V ≤UDC,2≤205991 V(as the variable droop intervenes as an additional degree of freedom), while Fig. 3. 3D response of the proposed adaptive droop (KAdaptive Droop,i) concerning power/current and voltage deviation from nominal values. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 5 it was 201854 V ≤UDC,2≤202674 V when only the duty cycle was in charge of the IDC-PFC operation. •Comparing Fig. 5 (b) and Fig. 6 (b), the IDC-PFC’s capacitor’s DC voltage is fully swingable within its rated value (as the variable droop intervenes as an additional degree of freedom), while it was 0 V ≤ |EC| ≤ 3155 V when only the duty cycle was in charge of the IDC-PFC operation. •Comparing Fig. 5 (c) and Fig. 6 (c), the IDC-PFC’s exchangeable power between HVDC cable tand HVDC cable uis swingable within the range −0.36699 MW ≤PExchange IDC−PFC ≤6.45541 MW (as the variable droop intervenes as an additional degree of freedom), while it was −1.23579 MW ≤PExchange IDC−PFC ≤1.89367 MW when only the duty cycle was in charge of the IDC-PFC operation. According to the result, the range of exchanged power has moved to the positive part, which means most power is transferred from the HVDC cable uto the HVDC cable t. •Comparing Fig. 5 (d) and Fig. 6 (d), it is concluded that the swingability of the HVDC cable t’s DC current, has changed from −2000 <It<−1000 A (descending) to −2000 <It<2000 A (as the variable droop intervenes as an additional degree of freedom). The swing-ability of the HVDC cable u’s DC current has shifted from −2000 <Iu<−1000 A (ascending) to −254 ≤Iu≤574 A (as the variable droop intervenes as an additional degree of freedom), respectively. Based on the results, the operation range for the HVDC cable uis slightly reduced with variable droop and duty cycle (828 A) which was (1000 A) with only duty cycle control. However, the HVDC cable tis operating at full capacity with the variable droop and duty cycle (4000 A) which was (1000 A) with only duty cycle control. •More importantly, the duty cycle can be swung within 0.45268 ≤ D≤0.88248 (as the variable droop intervenes as an additional degree of freedom), while it was within 0.36854 ≤D≤0.68871 where the only control variable was the duty cycle. Furthermore, the adaptive droop can be swung within 14.66527 ≤KDroop,2≤ 34.98027 considering system current and voltage limitations. Finally, to obtain an overall outlook over the operational area of the three-terminal test HVDC grid, the following figure which segregates the widened operational areas thanks to the presence of IDC-PFC and both IDC-PFC with droop control strategy is illustrated, see Fig. 7. The grey squares are representative of the initial operational area where there is no installed IDC-PFC. Moreover, the white squares are the current limitations of the HVDC cables. Since the IDC-PFC redistributes extra currents of overloaded HVDC cables/lines to the neighborhood HVDC cables/lines, the white squares become operable. In the next, the blue squares are indicating the IDC-PFC capacitor’s DC voltage limit (EC,n). Without an appropriate droop setting of the IDC-PFC’s interconnected converters, the IDC-PFC capacitor’s DC voltage might violate its limit. However, by assigning adaptive droop gains of the IDC-PFC’s interconnected converters, the blue squares become operable. As it was shown in Fig. 6 (b) the IDC-PFC capacitor’s DC voltage was completely operable thanks to the setting appropriate droop gain of the IDC-PFC’s interconnected converters, the operational area of the system expanded Fig. 4. Three-terminal test HVDC grid equipped with an IDC-PFC [5]. Fig. 5. IDC-PFC characteristics with duty cycle (one autonomous control variable) [5]: (a) DC voltage of bus-2, (b) IDC-PFC’s capacitor’s DC voltage, (c) Exchange power between master HVDC cable t and slave HVDC cable u, and (d) the currents of master and slave HVDC cables t and u. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 6 (the blue squares became operable), see Fig. 7. The darkest grey squares are non-operational areas where the power limitations of the converters will not allow further operation for the system. 4. State-Space HVDC system Modeling: Control structure & stability analysis 4.1. State-Space Modelling of Multi-Terminal HVDC grid with control structure To assess the accuracy, efficiency, and role of the proposed strategy in escaping IDC-PFC’s limitations, a control structure needs to be devised. For the dynamic simulations (in MATLAB-Simulink), the electromagnetic linearized transient model is implemented. Taking the three-terminal HVDC grid as an example in Fig. 4, the state-space equations are as follows: State-Space Equations: To generate state-space equations, variables are considered to be operating at their assumed linearized points: X≃X0+dX dt (24) Fig. 6. IDC-PFC characteristics with variable droop and duty cycle (two autonomous control variables): (a) DC voltage of bus-2, (b) IDC-PFC’s capacitor’s DC voltage, (c) Exchange power between master HVDC cable tand slave HVDC cable u, and (d) the currents of master and slave HVDC cables tand u. Fig. 7. Widened Operational area thanks to the combination of variable droop and duty cycle. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 7 The parameters X,X0, and dX/dt are variable, linearized point, and variation of the related variable, respectively. The HVDC transmission cables/lines are simulated in π −model, where Rij and Lij are the resistance and inductance of the cable/line, respectively. Nevertheless, the existence of the shunt resistors and capacitances (Y) does not contribute significantly to the results. Thus, they are eliminated from the spacestate equations [16]. Finally, the linearized model of the presented space-state equations is depicted as (25). dX dt =AX +BU (25) The differential equations describing the modeled three-terminal test VSC-HVDC grid are illustrated as follows: dUDC,1 dt =1 Cʹ 1 (P1 UDC,10 −UDC,1P10 U2 DC,10 −I12 −I13) ≈ 0,slack −bus (26) dUDC,2 dt =1 Cʹ 2 (−KAdaptive Droop,2 UDC,20 UDC,2+I12 +I23)(27) dUDC,3 dt =1 Cʹ 3 (P3 UDC,30 −UDC,3P30 U2 DC,30 +I13 −I23)(28) dI12 dt =1 L12 (UDC,1−UDC,2−R12I12 − (D0−1)EC−EC0D)(29) dI13 dt =1 L13 (UDC,1−UDC,3−R13I13)(30) dI23 dt =1 L23 (UDC,2−UDC,3−R23I23 −D0EC−EC0D)(31) dEC dt =1 C((D0−1)I12 +D0I23 + (I120 +I230)D)(32) where Aand Bare state and input vector coefficients, respectively: A= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −P10 Cʹ 1U2 DC,10 0 0 −1 Cʹ 1 −1 Cʹ 1 0 0 0−KAdaptive Droop,2 UDC,20Cʹ 2 01 Cʹ 2 01 Cʹ 2 0 0 0 −P30 Cʹ 3U2 DC,30 01 Cʹ 3 −1 Cʹ 3 0 1 L12 −1 L12 0−R12 L12 0 0 1−D0 L12 1 L13 0−1 L13 0−R13 L13 0 0 01 L23 −1 L23 0 0 −R23 L23 −D0 L23 0 0 0 D0−1 C D0 C0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (33) B=⎛ ⎜ ⎜ ⎜ ⎝ 1 Cʹ 1UDC,10 1 UDC,20Cʹ 2 0 0 0 0 0 0 0 0 −EC0 L12 0−EC0 L23 I120 +I230 C ⎞ ⎟ ⎟ ⎟ ⎠ T (34) The parameter Xis the state vector and the parameter U(U= {P1, KAdaptive Droop,2,D}) is the input vector of the state-space equations. The parameters Cʹ 1,Cʹ 2, and Cʹ 3are the converters’i,j, and kHVDC link capacitors, respectively. Moreover, the parameter Crepresents the IDCPFCs capacitor. The parameter (P1) decides how much power to be injected into the three-terminal HVDC grid. In addition, the duty cycle (D) and the adaptive droop gain of the bus-i(KAdaptive Droop,2) are the control variables. For the bus-i, the control structure representing equation (12) is given, see Fig. 8 (a). In the proposed droop control, the terms Ai,Bi,Ci, and drepresent the limits of the converter-ipower, currents of master and slave HVDC cables where the IDC-PFC is installed, DC voltage deviation of the converter-i, and IDC-PFC capacitor’s DC voltage, respectively. Fig. 8 (a) generates the equation (12) followed by a PI controller and a limiter. The droops for the bus-jand bus-kare set similarly following the equations (13) and (14), respectively. The PI controller is tuned/optimized by MATLAB 2023b PI-tuning toolbox. Also, the limiter considers the droop gain permissible operation route. Based on the given state-space model, the transfer function that relates the duty cycle (D) to the master HVDC cable tcurrent (I12) is achieved (G(s) = I12/D). Therefore, based on the transfer function G(s), the following control structure for master HVDC cable t(I12) current is shown in Fig. 8 (b). The following controller consists of four components including a PI controller (tuned by MATLAB Simulink Tuning Toolbox), a damper (for high-frequency oscillation damping), a limiter (maintains the duty cycle to operating within 0 ≤D≤1) and finally, the transfer function extracted by state-space equations. The control system of Fig. 8 (b) sets the duty cycle to follow the given reference for the master HVDC cable t(I12). Moreover, to ensure the proposed adaptive droop control strategy works well, the measurement signals must be faster than the action of the control system to separate the dynamics of different control actions. 4.2. Stability analysis In this section, the stability aspects of the proposed adaptive droop controller and master HVDC cable tcurrent controller are analyzed for the three-terminal HVDC grid. To make sure the control structure with variable droop and duty cycle will operate smoothly, the closed-loop transfer function of the control structure is assessed. Closed - loop transfer function =G(s)K(s) 1+G(s)K(s)(35) In the following, the pole map of the closed-loop transfer function is illustrated, see Fig. 9. The pole map of the closed-loop transfer function depicts two sets of poles, one neutral and the other sensitive to the change of bus-2droop gain. For the sensitive poles, it is seen that by swinging the droop value within the identified permissible operation Fig. 8. Control Structure: (a) droop control structure for the bus-iand (b) master HVDC cable tcurrent control structure. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 8 route (14.66 <KDroop,2<34.98), the poles move toward the stable part (more negative side), which shows that the proposed control structure is stable. One important aspect of the proposed strategy is the interaction of the droop control dynamic with the duty cycle dynamic. According to the control structure in Fig. 8 and equation (35), there is only one transfer function that relates the master HVDC cable tcurrent to the duty cycle. Therefore, the dynamics of the system are dictated by the existing transfer function. As the duty cycle imposes its dynamics on the output master HVDC cable tcurrent, these dynamics directly show themselves in the adaptive droop control strategy. Therefore, since their dynamics are the same, there are no interactions between the duty cycle and adaptive droop control strategy. 5. Steady-State Analysis: Optimal-Adaptive ¡Droop-Controlled power Flow (OADC-PF) 5.1. Importance of applying OADC-PF in the presence of an IDC-PFC The proposed adaptive droop control structure of Fig. 8 tries to adapt the droop value of the converters to avoid limitations of the IDC-PFC. However, this is not optimal. It is worthwhile knowing the optimal value of the converters’droops. In other words, the OADC-PF with the optimal results means that the converters’droops are set optimally, therefore, they contribute to the IDC-PFC limitation removal as much as possible. For solving the optimization problem, Sequential Quadratic Programming (SQP) of MATLAB software is utilized. Moreover, Genetic Algorithm (GA) is also used to solve the optimization problem ensuring the results are global optimal points. 5.2. Power injection model of the IDC-PFC It is a well-known method to generate the impact of the IDC-PFC in DC-PF formulations using power injection models (PIMs). The PIM formulations for the IDC-PFC are studied on the basis of the lumpedπ model of the interconnected HVDC lines. The effect of IDC-PFC in the DC-PF formulations is achieved by comparing power equations in the presence and absence of IDC-PFC [17], as follows: P(i) = − ((1−D)UDC,iEC Rt −DUDC,iEC Ru )(36) P(j) = (1−D)UDC,jEC Rt (37) P(k) = − DUDC,kEC Ru (38) In the equations (36)-(38), P(i),P(j), and P(k)present the impacts of an IDC-PFC that are artificially injected into the buses i,jand k, respectively. These equations represent the effects of an IDC-PFC on DC power flow studies. For further information regarding the modeling process of an IDC-PFC, refer to [17]. 5.3. Multi-Objective function &optimization problem In this paper, two goals will be followed: (i) minimizing the DC voltage deviations of the HVDC grid’s buses and (ii) minimizing the current index of all the HVDC cables and overhead lines. Thus, the DC voltage deviation term (FV) and the current mitigation term (FI) are described as below: FV=∑ Nbus i=1 |(UDC,i−1)i−1|;i=1,2,3, ..., Nbus (39) FI=∑ NLine t=1 |IDC−t| |IDC−t,max|;t=1,2,3, ..., NLine (40) For the above equations, the parameters Nbus and NLine indicate the number of HVDC buses and HVDC cable/overhead lines, respectively. The parameters IDC−tand IDC−t,max are the HVDC cable/ overhead line current and current limitation, respectively. For more clarification, the term FVdemonstrates the HVDC buses’ DC voltage deviation from their reference values (nominal values), while the term FIis for avoiding bottleneck occurrence in HVDC cables/ overhead lines. The more deviation of the systems’voltages is from their nominal value, the more the system has a broad operational area. In other words, the system must have had a broader operational area to be able to deviate systems’DC voltages from their nominal values. Therefore, because the DC voltages of the buses are allowed to swing (0.95 ≤UDC,i≤1.05), the DC voltage of each bus is subtracted to 1 and then the accumulated value is subtracted to 1 again, see equation (39). This will clarify whether the DC Power Flow (DC-PF) is able to find a proper solution in a broader operational area. Moreover, the equation (40) illustrates the loading of the multiterminal HVDC system. The less the term FIis, the less the loading of the system is. In other terms, having a small FImeans the injected powers from the Offshore Wind Farm (OWF)s into the multi-terminal HVDC transmission system have been distributed in a reasonable manner. Thus, all the HVDC cables/lines contribute to carrying the produced bulk power to the onshore side rather than a small number of HVDC cables/lines carrying a great deal of power. 5.4. Equality and inequality constraints The optimization problem is solved subject to several equality and inequality constraints. The inequality constraints include some operational terms including DC voltage (UDC,i), currents (IDC−t), onshore-side converter power (Pi), duty cycle (D), and IDC-PFC capacitor’s DC voltage. On the other hand, equality constraints are the power balance of the HVDC grid and the IDC-PFC capacitor’s DC voltage. The inequality and equality constraints are described as follows: Inequality constraints: Umin <UDC,i<Umax (41) IDC−t≤IDC−t,max (42) −Pmax ≤Pi≤Pmax (43) 0≤D≤1 (44) Emin ≤EC≤Emax (45) The DC voltage boundaries are set to be withinUmin =0.95 andUmax = 1.05. In addition, the IDC-PFC capacitor’s DC voltage is allowed to swing 0.05UDC,i(Emin=-0.05UDC,iandEmax =0.05UDC,i). Further, the onshoreFig. 9. Poles of closed-loop transfer function of the control structure. M. Pourmirasghariyan et al. International Journal of Electrical Power and Energy Systems 164 (2025) 110430 9