Weighted Bandwidth Method for Stability Assessment of Complex DC Power Systems on Ships

Abstract

In shipboard DC grids, tightly controlled load converters can impair the system stability, thus provoking the ship blackout. Conversely, load converters regulated by low control bandwidths are capable of inducing a stabilizing action. This compensation is verifiable if the loads are few. On the contrary, the balancing of control dynamics is hardly evaluated if the bus feeds multiple (i.e., hundreds or more) DC controlled loads. In this paper, the weighted bandwidth method (WBM) is presented to assess the small-signal stability of a complex shipboard power system by aggregating the multiple converters into two sets of controlled loads. Once the validity of the aggregation is proven, a stability study is performed on the two-loads system. As the last system is more inclined to instability than the initial multiple-loads system, the verification of the two-loads stability criterion guarantees that the shipboard DC grid also remains stable. Finally, emulations on HIL verify the proposed stability assessment thus providing the first unique verification of WBM.

1. Introduction

Developments in power electronics promote the exploitation of DC technology both in on-land distribution systems [1,2] and in shipboard power grids [3,4,5]. By focusing on the marine context, the DC paradigm offers several advantages, both in low (LV) and medium voltage (MV) applications [6,7,8]. Regarding the MVDC case, the IEEE Std. 1709 has described points of strength but also the open challenges [9]. Among the pros, a pervasive power conversion to DC provides a reduction in power system volume and an enhanced modularity in ship design [10,11]. Conversely, DC applications can exhibit an unstable behavior, investigated since the 90s [12,13] until the present day [14]. When the penetration of power electronics is high, as in DC grids, the dynamics interactions between controlled converters and filters can deteriorate the system’s stable operation. Indeed, if an LC input filter feeds a tightly controlled converter modeled as constant power load (CPL), small voltage sags on the DC bus can trigger voltage oscillations, and therefore instability. In this regard, refs [15,16] are important contributions for identifying the CPL instability issue and a possible control strategy to compensate for the destabilizing effect. From these milestones, several approaches have been proposed as effective in solving the CPL destabilizing effect. First, a discussion about the stability issue on shipboard DC grid is in [17]. Then, ref [18] discussed the CPL effect on terrestrial DC microgrids with an emphasis on stabilizing techniques. Other important contributions to DC stability are in [19,20], whereas [21] proposes the concept of a smart resistor to maintain the proper stability margin with minimized filter capacitors. Important work the one proposed in [22], where the loop-cancellation technique is tested with experimental results. Also the authors of the present paper have had a good experience with the linearization via state feedback. Indeed [23,24] are examples where this complex strategy is capable of ensuring system stability even in risky conditions (e.g., generating system disconnection, wrong parameters estimation). Albeit the ideal CPL model is not the worst case from a control standpoint [25], the DC stability assessment is usually performed by adopting the nonlinear CPL on the load side, as it prefigures a well-recognized destabilizing case. Another case in which the CPL hypothesis is conveniently assumed is in [26], where the researchers put the focus on the method for designing a fault-tolerant stabilizing system. CPL is also adopted for marine applications [27,28], while [29] considers a limitation in CPL bandwidth. Other ideas to overcome the ideal CPL are in [30,31], where reduced-order models are defined. Although these models can take into account how the control bandwidths influence the stability of two DC-DC cascaded converters, the methodology will be different when modeling large DC systems. In fact, if the embarked controlled loads are hundreds, as in naval electric ships [32], the DC shipboard microgrid becomes indeed very complex [33,34], consequently compromising the analytical stability assessment. This last aspect is also faced in [35] for a DC electric vehicles recharging infrastructure, where small-signal stability is negatively influenced by multiple supply stations with the same power and control bandwidth. Based on these examples, it is important to observe that the complexity in analytically evaluating the stability of a multiconverter DC power system is not related to the adopted stability criteria, but to issues of modeling activities. Indeed, in the last twenty years an important experience about DC stability has been developed. First, a valuable contribution on stability metrics is provided in [36]. Then, a complete treatise about criteria is given in [37], whereas other important works have specifically discussed about the possible methods for the stability assessment. For giving an idea, it is possible to enumerate the passivity-based stability criterion [38], the impedance-based system method [39,40,41] and the eigenvalue-based method [42,43]. Conversely, in authors knowledge, there is a sort of lack in defining a practical methodology to aggregate the stabilizing/destabilizing effect of controlled loads. As this desirable aggregation can simplify the stability analysis of complex DC grids, therefore the paper is interested in proposing this new Weighted Bandwidth Method (WBM). In authors opinion, this methodology can give a first analytical view about stability issues in isolated-radial DC grids with a pervasive presence of DC-DC controlled converters.

In this paper, the WBM is proposed as an original methodology to weigh how a controlled converter (i.e., its bandwidth) impacts on overall system stability. The WBM is therefore a novel technique for analytically assessing the small-signal stability of multiconverter DC power systems in the presence of very dispersed power and control bandwidth ratings. Such a method models a radial shipboard DC microgrid by means of an approximated DC system. The WBM makes it possible to aggregate multiple-controlled loads into two resulting loads, whose control bandwidths are linear combinations of the original ones. The approximated WBM-based model maintains the total power of the initial complex DC power system and results in smaller stability margins. Therefore, the stability assessment of the WBM-based model is expected to be less conservative. In other words, when the WBM-based model is working at the stability margin, the multiple-loads model is certainly stable. If the stability criteria [36,37,38,39,40,41,42,43] are satisfied for the WBM-based model, a stable evolution results when the initial multiple-loads model is perturbed. The stability assessment based on WBM-based model is then validated by means of emulations on high-fidelity Typhoon HIL platform. These tests prove the WBM capability in aggregating the dynamics interactions of a complex DC grid into two resulting controlled loads. If a perturbation does not compromise the stability of the two-loads aggregated system, also the complex DC shipboard power system results stable in the same perturbed condition.

The paper is organized as follows. Section 2 presents the DC power system, while the WBM-based model is described in Section 3 after the aggregation of multiple controlled loads has been defined. The methodology to study the DC stability of a two-load system is presented in Section 4. In Section 5, the HIL emulations on the initial multiconverter DC system verify the validity of WBM stability assessment. Finally, Section 6 reports the conclusions.

2. DC Microgrid Modeling

The DC power system topology is described in Section 2.1, while Section 2.2 shows the assumptions to simplify the system modeling. In Section 2.3, the analytical model is discussed with particular attention on the output impedance and the input admittance.

2.1. Power System Topology

The DC controlled grid is shown in Figure 1. This radial topology has N generating systems (left part) and M load systems on the right of the DC central bus. Each h-generating system (h = 1…N) is composed by the cascade of an AC synchronous machine, diode rectifier and DC-DC interface converter. The AC generator voltage is controlled by an Automatic Voltage Regulator operating on the excitation system, while the duty cycle D_gh is the DC-DC converter’s control signal. Both AC-DC and DC-DC buck converters outputs are filtered, thus a first order arrangement for the diode converter while LC stages are adopted for the interface to the bus. The k-system (k = 1…M) is a filtered DC-DC buck converter feeding the R_Lk load. The latter is an embarked load (e.g., propulsion, instrumentation, hotel load, low-voltage load), whose voltage is regulated by the DC-DC converter k (i.e., control signal D_k). As specified in [9], a multiconverter DC microgrid like the one depicted in Figure 1 can well represent a future onboard DC system. The authors are therefore interested in studying such a radial example with several interface converters.

2.2. Modeling Assumptions and Range of Validity

Numerous control loops are required to manage the DC system in Figure 1. The 2N loops regulate the DC bus voltage (N on AC machines and N on DC-DC converters), while M feedbacks control the voltage on DC loads. The converters’ current controls add extra M loops for the loads and additional N in the generating section. Overall, the DC grid works on the interacting action of B = 3N + 2M control loops. In a complex shipboard DC grid, the B number is therefore in the hundreds, so it follows that the stability assessment is almost unachievable as closed-form expressions. By considering this, three hypotheses simplify the DC model for a particular range of control bandwidths. The consequent stability analysis is also limited to the same range of validity.

The first hypothesis is on the current loop of each DC-DC converter. As the current control bandwidth is supposed to be ten times larger than the voltage control bandwidth, the inner current loop is negligible as well as its dynamics effect. The second assumption is for each LC filter on DC load. When the U_k voltage is regulated by a first-order dynamics with ω_k bandwidth, the k-filter does not affect the bus stability if ω_k is sufficiently smaller than the filter’s resonance frequency ω_fk. The load filter can be thus neglected as in [30,31]. A final condition is set by comparing voltage control bandwidths of generating/load DC-DC converters. If the ω_h bandwidths are ten time smaller than the ω_k ones, then the h-converters are assumed to operate in steady state condition. Consequently each h-filter is supplied by a constant voltage E_gh (i.e., average value at h-converter output). Apart from the first hypothesis which is quite common in power electronics, the other two are synthesized as ω_h << ω_k << ω_fk, then defining the range of validity. The complex DC microgrid can be simplified as in Figure 2 supposing that the three assumptions are verified.

2.3. Analytical Model

Since the R_gh filter resistors are practically ignored (i.e., small values), the N filters can be aggregated in a Thevenin equivalent circuit as in Figure 3. Now, the E equivalent voltage source feeds the LC parallel-resulting components, while the M voltage-controlled converters are clamped on the capacitor then supplied from the DC bus voltage. Each k-converter is feedback controlled to regulate the U_k voltage on R_Lk load, thus imposing the U_k0 reference in steady state condition. To simplify the analysis, the fast nonlinear switching dynamics of each converter is disregarded, thus resulting in average value models [9]. As the switching frequency is high, the converter dynamics is indeed negligible because the related pole is placed far beyond the considered field of study in the Gauss plane. Then, the controlled U_k output is given by the product between V average bus voltage and D_k duty signal (yellow box), while the equations with 0 subscript are in steady state. The duty cycle is the output of an integral regulator (dashed box), whose K_Ik gain imposes a first-order behavior as control target for the U_k output voltage. Once the converter dynamics are neglected, the open-loop transfer function H_k(s) is the cascade of the integral regulator and V₀ gain. The K_Ik value is set for the 0 dB crossing in ω_k bandwidth (blue box). Finally, the I_L total current is the sum of M input currents I_k. Thanks to above mentioned assumptions, now 2 + M state Equation (1) model the filtered DC microgrid supplying M voltage controlled converters. The first two equations are apparent, the third is defined in Figure 3. Here, D_k first derivative is determined from the red scheme, while U_k, U_k0 and K_ik are in yellow/blue box. In the last equation, the M load currents are moved at the converters inputs (i.e., D_k action) to get the total I_L. For a small-signal stability study, Equation (1) is linearized in a stable equilibrium point given by 2 + M rated values (V₀, I₀, D_k0). The model in (2) is then processed by the Laplace transform to find functions V(s) and I_L(s), as in (3)–(4). Output impedance Z_o(s) and input admittance Y_i(s) are in (5), where ω₀ = (LC)^−0.5 is the equivalent filter’s resonant angular frequency. As in [36,37,38,39,40,41,42], the system stability is evaluated on T(jω) = Z_o(jω)·Yi(jω). Thus, Equation (5) is combined in (6) to define the real-imaginary part of T(s), where G_k = D_k0²/R_Lk is the k-conductance and ω the angular frequency.

$$\left\{\begin{array}{l}\stackrel{•}{I}=\frac{1}{L}\left(E-V\right)\\ \stackrel{•}{V}=\frac{1}{C}\left(I-I_L\right)\\ \stackrel{•}{D_k}={\omega }_k{D}_{k0}-\frac{{\omega }_k}{V_0}{D}_{k}V\\ I_L={\displaystyle \sum _{k=1}^M\left[\frac{1}{R_{Lk}}{D}_k^{2}V\right]}\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}\Delta \stackrel{•}{I}=\frac{1}{L}\left(\Delta E-\Delta V\right)\\ \Delta \stackrel{•}{V}=\frac{1}{C}\left(\Delta I-\Delta I_L\right)\\ \Delta \stackrel{•}{D_k}=-{\omega }_k\Delta D_k-\frac{{\omega }_k{D}_{k0}}{V_0}\Delta V\\ \Delta I_L={\displaystyle \sum _{k=1}^M\left[\frac{2D_{k0}{V}_0}{R_{Lk}}\Delta D_k+\frac{D_{k0}^2}{R_{Lk}}\Delta V\right]}\end{array}\right.$$

(2)

$$D_k(s)=-\frac{{\omega }_k{D}_k(s)}{s}-\frac{{\omega }_k{D}_{k0}}{s{V}_0}V(s)\Rightarrow D_k(s)=-\frac{{\omega }_k}{s+{\omega }_k}\frac{D_{k0}}{V_0}V(s)$$

(3)

$$I_L(s)={\displaystyle \sum _{k=1}^M\left[\frac{2D_{k0}{V}_0}{R_{Lk}}{D}_k(s)+\frac{D_{k0}^2}{R_{Lk}}V(s)\right]}={\displaystyle \sum _{k=1}^M\left[\frac{D_{k0}^2}{R_{Lk}}\cdot \frac{s-{\omega }_k}{s+{\omega }_k}\right]V(s)}$$

(4)

$$Z_o(s)=\frac{V(s)}{-I_L(s)}=\frac{sL}{1+s^{2}LC}=\frac{1}{C}\frac{s}{{\omega }_0^2+s^2}$$

(5a)

$$Y_i(s)=\frac{I_L(s)}{V(s)}={\displaystyle \sum _{k=1}^M\left[\frac{D_{k0}^2}{R_{Lk}}\cdot \frac{s-{\omega }_k}{s+{\omega }_k}\right]}$$

(5b)

$$\Im \left(\overline{Z_o}\cdot \overline{Y_i}\right)=-\frac{\omega }{C\left({\omega }_0^2-{\omega }^2\right)}\cdot {\displaystyle \sum _{k=1}^M\left[G_k\frac{{\omega }_k^2-{\omega }^2}{{\omega }_k^2+{\omega }^2}\right]}$$

(6a)

$$\Re \left(\overline{Z_o}\cdot \overline{Y_i}\right)=-\frac{2{\omega }^2}{C\left({\omega }_0^2-{\omega }^2\right)}\cdot {\displaystyle \sum _{k=1}^M\left[G_k\frac{{\omega }_k}{{\omega }_k^2+{\omega }^2}\right]}$$

(6b)

3. Weighted Bandwidth Method

Conventionally, the study on product T(jω) is adopted to verify the DC system stability [36,37,38,39,40,41,42]. The same methodology is used for the Figure 1 radial distribution, where the Y_i(jω) input admittance (5b) models M controlled converters feeding M loads. As M is hundreds in a ship, the Y_i(jω) definition becomes complicated, making the stability evaluation analytically impracticable. The loads aggregation approach adopted here can limit the terms in Y_i(jω), enabling a consequent linear modeling. By exploiting this aggregation, the WBM can analytically investigate the stability in complex DC grids, where the several converters can negatively interact while reducing the stability margin. As expressed in the following, the paper novelty is not on the application of well-known stability criteria. Conversely, the novelty is put on the methodology to obtain a simplified model from an initial multiconverter power grid. As the simplified model is representative of the complex one, a stability study on the simplified grid can provide important information about the stability of complex DC microgrid.

3.1. Set Membership

The M loads having ω_k bandwidth are gathered in two sets (i.e., S and D), once elected the ω_B bandwidth as splitter. The splitter is the bandwidth to put a uniform DC system (i.e., same parameters as study case, except M equal bandwidths,) at the stability boundary (i.e., ℜ[Z_o∙Y_i] = −1). When the identical ω_k are larger than ω_B, the uniform controlled system is unstable (i.e., ℜ[Z_o∙Y_i] < −1). The ω_B limit is thus identified by two steps. Firstly, ℑ[Z_o∙Y_i] is nullified in (7a) to find the critical angular frequency, ω = ω_cr = ω_B. By putting ω_cr in the real part of (7b) while adding up the M conductances as G_L, Equation (8) is consequent and ω_B the positive solution (9). In the not uniform system, a load belongs to S stabilizing-loads set when its control bandwidth is smaller than ω_B. Conversely, the D destabilizing-loads set groups all the controlled loads with ω_k > ω_B.

$$\Im \left(\overline{Z_o}\cdot \overline{Y_i}\right)=-\frac{\omega }{C\left({\omega }_0^2-{\omega }^2\right)}\cdot {\displaystyle \sum _{k=1}^M\left[G_k\frac{{\omega }_B^2-{\omega }^2}{{\omega }_B^2+{\omega }^2}\right]=0}$$

(7a)

$$\Re \left(\overline{Z_o}\cdot \overline{Y_i}\right)=-\frac{2{\omega }^2}{C\left({\omega }_0^2-{\omega }^2\right)}\cdot {\displaystyle \sum _{k=1}^M\left[G_k\frac{{\omega }_B}{{\omega }_B^2+{\omega }^2}\right]=-1}$$

(7b)

$$C{\omega }_B^2+G_L{\omega }_B-C{\omega }_0^2=0$$

(8)

$${\omega }_B=\frac{-G_L+\sqrt{G_{{}^L}^2+4C^2{\omega }_0^2}}{2C}$$

(9)

3.2. Loads Order and Power Equivalence

To form the S-D sets, the initial M loads are firstly rearranged in increasing order of bandwidth by renaming their subscript (i.e., now ω₁ < … < ω_B < … < ω_M). Similarly, also the subscripts of other parameters are redefined. Then, ω_B limit (9) subdivides the loads in the S-D sets, respectively counting M_S and M_D elements (i.e., M = M_S + M_D). The S loads have ω_k < ω_B where k = 1…M_S, while the bandwidths of D loads exceed the ω_B value (i.e., ω_k > ω_B, k = M_S + 1…M). The WBM is conceived to ensure the power equivalence between the initial M loads and the aggregated S-D loads. Thus, total load power P_L and total conductance G_L are split as in (10), where G_S and G_D are the equivalent conductances, while P_S and P_D the relative powers.

$$P_L={\displaystyle \sum _{k=1}^M{P}_k=}{\displaystyle \sum _{k=1}^{MS}{P}_k+}{{\displaystyle \sum _{k=MS+1}^M{P}_k=}}_{}{P}_S+P_D$$

(10a)

$$G_L={\displaystyle \sum _{k=1}^M{G}_k=}{\displaystyle \sum _{k=1}^{MS}{G}_k+}{\displaystyle \sum _{k=MS+1}^M{G}_k{=}_{}}{G}_S+G_D$$

(10b)

3.3. Linear Combinations of Control Bandwidths

In the definition of S-D sets, the WBM must assure power equivalence and bandwidths aggregation. If the first target is achieved by (10a), the second deserves attention. The bandwidths aggregation law must produce a WBM-based model whose stability margins are lower than the ones of the initial complex DC model. When comparing the multiple-loads model and the WBM-based model, the second one must be more inclined to instability: in other words it reaches the stability boundary when feeding a smaller load. On the other hand, the aggregation law must be basic, thus based on a linear mathematical relationship, in order to simplify the stability assessment during the power system design. By starting from the S set, the Y_i(jω) admittance and the Y_S(jω) aggregated admittance are in (11).

$$Y_i(j\omega )={\displaystyle \sum _{k=1}^{Ms}\left[G_k\frac{j\omega -{\omega }_k}{j\omega +{\omega }_k}\right]}={\displaystyle \sum _{k=1}^{Ms}\left[G_k\frac{-2{\omega }_k}{j\omega +{\omega }_k}\right]+G_S}$$

(11a)

$$Y_S(j\omega )=G_S\frac{j\omega -{\omega }_S}{j\omega +{\omega }_S}=G_S\frac{-2{\omega }_S}{j\omega +{\omega }_S}+G_S$$

(11b)

$${\omega }_S={\displaystyle \sum _{k=1}^{Ms}{m}_k}{\omega }_k$$

(12a)

$${\omega }_D={\displaystyle \sum _{k=Ms+1}^M{m}_k}{\omega }_k$$

(12b)

The Y_i(jω) in (11a) models the M_S loads (i.e., ω_k bandwidths, k = 1…M_S, ω_k < ω_B), while the Y_S(jω) represents a single stabilizing controlled load having a hypothetical ω_S control bandwidth. The complexity in (11a) equation is evident when the sum of M_S fractions is considered. On the contrary, the aggregated admittance in (11b) has a structure similar to (11a) but a single ω_S bandwidth term. The latter is defined in (12a) as a linear combination of the S control bandwidths, weighted by the conductance ratio m_k = G_k/G_S = P_k/P_S. Thus, ω_S is a sort of center of gravity, whose definition derives from an analogy to the approximated method (i.e., current-line length products) to calculate the aggregated voltage drop of distributed loads in power systems [44,45]. The same procedure on D loads provides the ω_D aggregated control bandwidth in (12b). The linear combinations in (12) and the equivalent conductances in (10b) are the data used to specify the WBM-based model. Once the laws are defined in (12), the validity of the aggregated control bandwidths can be proven as in Section 3.4.

3.4. WBM Less Conservativeness

This Section verifies that the WBM is less conservative [46,47] and its application provides a resulting model with stability margins smaller than the ones of the initial DC complex model. To this aim, the 12 test-model and the related WBM-based model are compared to provide proof. In the 12 model, an LC filtered voltage E feeds two parallel connected DC-DC converters, named 1 and 2. A single controlled converter is shown in Figure 3. The two control bandwidths ω₁ and ω₂ are stabilizing (i.e., ω_k < ω_B), while G₁ and G₂ are the load conductances from the converters inputs. Consequently, each conductance takes into account both the supplied resistive load and the steady state duty cycle on feeding converter (i.e., G_k = D_k0²/R_Lk). By applying the WBM and the small-signal hypothesis (i.e., now the voltage source is ΔE), the 2-loads grid is aggregated into a S single load system, which has the same voltage input ΔE, the equivalent conductance G_S (i.e., G_S = G₁ + G₂ = m₁G_S + m₂G_S) and the aggregated bandwidth ω_S = m₁ω₁ + m₂ω₂ (12a). To compare 12 and S models, Equation (2) is rearranged to define the transfer functions from ΔE to ΔV, as in (13), (14). By using (11), the last equations are modified in (15), (16). The W₁₂(s) transfer function (15) is particularized when m₁ = 1 and m₂ = 0, thus ω_S = ω₁ (12a). In this case, W₁₂(s) equates (16). Equality appears also in the second limit case, when m₁ = 0 and m₂ = 1. In order to demonstrate that the WBM was less conservative in the middle condition (m₁ = m₂ = 0.5), a possible approach identifies the maximum G_S after which the two systems are unstable. For the S model, the G_S^S limit conductance is found in (17) by studying Z_o(jω)∙Y_S(jω) if q = 1. The imaginary part of the last product is nullified when ω = ω_cr = ω_S. Such a value is thus substituted in the real part to finally find the G_S^S. An identical procedure is followed in (18) to define the G_S¹² conductance in the middle condition, where ω_cr = (ω₁∙ω₂)^−0.5 is the critical angular frequency for the 12 model. If m₁ = m₂ = 0.5 and q = 1, the difference ΔG = G_S¹² − G_S^S in (19) is always positive, whatever the bandwidths ω₁ ≠ ω₂. Thus, the WBM’s lesser conservativeness is verified under three conditions (i.e., m₁ = 1 and m₂ = 0, m₁ = 0 and m₂ = 0, m₁ = m₂ = 0.5) for the S stabilizing loads (i.e., q = 1). Similar remarks for the D loads, where initial as well as aggregated bandwidths are above the ω_B limit. Indeed, by replaying the same approach when q > 1, the WBM can be again tested also on the D loads set, being G_D¹² > G_D^D. As ΔG results positive in the S-D cases, both aggregated controlled loads are definitely more inclined to instability in the three exemplifying positions. To complete the proof, the two studies on ΔG can be extended to the entire m₂ range, since the Routh-Hurwitz criterion [30,31] analytically identifies the limit conductances for each m₂ value. Once it is verified that the WBM is less conservative with few controlled loads, the same methodology is then directly applicable on numerous loads by updating Equations (10) and (12).

$$W_{12}(s)=\frac{1}{s^2\left(LC\right)+sL\cdot Y_{12}(s)+1}$$

(13)

$$W_S(s)=\frac{1}{s^2\left(LC\right)+sL\cdot Y_S(s)+1}$$

(14)

$$W_{12}(s)=\frac{{\omega }_0^{2}C\left(s+{\omega }_1\right)\left(s+{\omega }_2\right)}{C\left(s^2+{\omega }_0^2\right)\left(s+{\omega }_1\right)\left(s+{\omega }_2\right)+s{m}_1{G}_S\left(s-{\omega }_1\right)\left(s+{\omega }_2\right)+s{m}_2{G}_S\left(s+{\omega }_1\right)\left(s-{\omega }_2\right)}$$

(15)

$$W_S(s)=\frac{{\omega }_0^{2}C\left(s+{\omega }_S\right)}{C\left(s^2+{\omega }_0^2\right)\left(s+{\omega }_S\right)+s{G}_S\left(s-{\omega }_S\right)}$$

(16)

$$\left\{\begin{array}{l}\Im \left(\overline{Z_o}\cdot \overline{Y_S}\right)=-\frac{{\omega }_{cr}}{C\left({\omega }_0^2-{\omega }_{{}^{cr}}^2\right)}\cdot G_S\frac{{\omega }_S^2-{\omega }_{{}^{cr}}^2}{{\omega }_S^2+{\omega }_{{}^{cr}}^2}=0\\ \Re \left(\overline{Z_o}\cdot \overline{Y_S}\right)=-\frac{2{\omega }_{{}^{cr}}^2}{C\left({\omega }_0^2-{\omega }_{{}^{cr}}^2\right)}\cdot G_S^S\frac{{\omega }_S}{{\omega }_S^2+{\omega }_{cr}^2}=-q\end{array}\right.\Rightarrow G_S^S=qC\frac{{\omega }_0^2-{\omega }_S^2}{{\omega }_S}$$

(17)

$$\left\{\begin{array}{l}\Im \left(\overline{Z_o}\cdot \overline{Y_{12}}\right)=0\\ \Re \left(\overline{Z_o}\cdot \overline{Y_{12}}\right)=-q\end{array}\right.\Rightarrow G_S^{12}=qC\frac{{\omega }_0^2-{\omega }_1{\omega }_2}{2{\omega }_1{\omega }_2}{\left[\frac{0.5{\omega }_1^{}}{{\omega }_1^2+{\omega }_1{\omega }_2}+\frac{0.5{\omega }_2^{}}{{\omega }_2^2+{\omega }_1{\omega }_2}\right]}^{-1}$$

(18)

$$\Delta G=G_S^{12}-G_S^S=qC{\omega }_0^2\frac{{\left({\omega }_2-{\omega }_1\right)}^2}{2\left({\omega }_1+{\omega }_2\right){\omega }_1{\omega }_2}>0_{} \forall {\omega }_1\ne {\omega }_2$$

(19)

4. DC Stability Analysis

Section 3 has conceived the WBM to gather the controlled loads in two sets. Since aggregated loads are equivalent in power to the initial ones, but more inclined to instability, this Section investigates the DC grid stability by means of the WBM-based model.

4.1. Stability Criterion

The Nyquist criterion is able to assess the stability of the initial DC grid feeding M controlled loads. As in [36], this criterion proves the system stability when the curve Z_o(jω)∙Y_i(jω) does not clockwise-encircle the point (−1,0) on the Gauss plane. A consequent approach determines the stability behavior by checking if ℜ[Z_o∙Y_i] > −1 when ℑ[Z_o∙Y_i] = 0. When the real part is compliant with this condition, the Nyquist criterion is verified then the stability is guaranteed. On the base of this statement, Equation (20) result consequent if adopting (6)-(10)-(12) to define the WBM-based model. The first Equation (20a) is rearranged as in (21) to find the square ω_cr, whereas the system stability is ensured when the actual ψ term is less than 1 in (20b).

$$\Im \left(\overline{Z_o}\cdot \overline{Y_i}\right)=-\frac{{\omega }_{cr}}{C\left({\omega }_0^2-\omega {}_{cr}{}^2\right)}\cdot \left[G_S\frac{{\omega }_S^2-{\omega }_{cr}^2}{{\omega }_S^2+{\omega }_{cr}^2}+G_D\frac{{\omega }_D^2-{\omega }_{cr}^2}{{\omega }_D^2+{\omega }_{cr}^2}\right]=0$$

(20a)

$$\Re \left(\overline{Z_o}\cdot \overline{Y_i}\right)=-\frac{2\omega {}_{cr}{}^2}{C\left({\omega }_0^2-\omega {}_{cr}{}^2\right)}\cdot \left[G_s\frac{{\omega }_S^{}}{{\omega }_S^2+{\omega }_{cr}^2}+G_D\frac{{\omega }_D^{}}{{\omega }_D^2+{\omega }_{cr}^2}\right]=-\Psi$$

(20b)

$$\left(G_S+G_D\right){\omega }_{cr}^4+\left(G_S-G_D\right)\left({\omega }_D^2-{\omega }_S^2\right){\omega }_{{}^{cr}}^2-\left(G_S+G_D\right){\omega }_D^2{\omega }_S^2=0$$

(21)

4.2. Iterative Process for WBM Stability Assessment

By considering Equation (20), the simple iterative process of Figure 4 is developed for two targets. First, the stability behavior of the WBM-based model is evaluated when ω_S and ω_D are initially defined. Subsequently, the process could be used to properly redefine the two control bandwidths if the initial assessment proved the instability. The first step of the iterative process finds the square ω_cr (21), once assigned power system data (green box) and control bandwidths (red/blue boxes). Then, the square ω_cr is substituted in (20b) to determine the real coordinate ψ at step 2. This value is compared to the stability target Λ at step 3. When ψ ≤ Λ, the process is stopped as the stability is certainly ensured being Λ ≤ 1 by definition. Conversely, when ψ > Λ the iterative process (dashed lines) can also redesign ω_S and ω_D to lower the ψ till the Λ value. As the DC stability is impaired by large bandwidth controlled loads [30,31], the iterative process applies a bandwidths reduction to recover the stability requirement. To follow this strategy, the error e is at first determined as ψ−Λ (step 4). Then, both blue and red paths (step 5) can minimize e, then forcing a position near the point (−Λ, 0). By focusing on the blue process, the error is the information to reduce step-by-step the ω_S. As this bandwidth gradually decreases at each iterative cycle, the error e similarly moves towards zero. When e is smaller than a threshold (e.g., 10⁻³), the related ω_S is thus able to assure the target Λ. Similar considerations are used for the red path, while lowering the ω_D. The basic idea of the iterative process is considered as an advantage, as its application results also simple as a consequence. A complex process to establish the stability performance would be less useful when pursuing the effective implementation in the marine operative context. Moreover, the tuning (i.e., up/down) on ω_S and ω_D is definitely an effective, although elementary, approach to reaching the stability goal, while preserving the dynamics requirements for as long as possible. Future research activities will be based on this redesign, thus introducing the possibility of smart tuning the control bandwidths.

5. Validation of WBM Stability Assessment

The WBM aggregates the controlled loads in two sets, whose stabilizing-destabilizing effect is valued by the iterative process. Therefore, this process can provide a first check on stability performance. Then, some studies on poles placement and real-time HIL emulations are able to confirm the process results. When the iterative process is recognized as effective in studying the stability, its application is consequent and well-received.

5.1. Power System Data

A DC system feeding four controlled loads (M = 4) with a radial topology as in Figure 1 and a total power P_L = 24 MW is analyzed here. This DC grid is assumed to be supplied by two generating systems (N = 2), already designed in [23,24]. These groups have a total power P_g1 + P_g2 = 15.75 + 10.5 = 26.25 MW. Once we disregard the R_gh as in Section 2.3, the Thevenin equivalent parameters of input filters are L = 1.048 mH and C = 577.26 μF (i.e., ω₀ = 1286 rad/s). For the studied case, four R_Lk loads are fed by DC-DC controlled buck converters. Each load converter is filtered by an LC stage to ensure the power quality [9]. As in [23,24], there are several inputs to size the filters: converter rated power (P_nk), input-output rated voltage (V_n, U_nk), rated duty cycle (D_nk), rated output current (I_nk) and switching frequency (f_sk). Secondly, ΔV_% and ΔI_% (peak-peak voltage/current ripple) are the filter goals, while ΔP_% models the converter losses (percentage). The design as in [23,24] provides R_fk, L_fk and C_fk for each filtered converter, whilst ω_fk = (L_fkC_fk)⁻¹ is the k-filter resonance frequency and R_Lk is the k load resistance as square U_nk subdivided by power P_nk. The power system data are summarized in

Table 1. Parameters for modeling the load section.

	k = 1	k = 2	k = 3	k = 4
Pnk [MW]	5	4	5	10
Vn [V]	6000	6000	6000	6000
Unk [V]	3000	4500	5000	5000
Dnk	0.5	0.75	0.83	0.83
Ink [A]	1600	862	960	1900
fsk [Hz]	2500	2500	2500	2500
ΔP%k	4	3	4	5
ΔV%k	7	7	7	7
ΔI%k	30	30	30	30
Rfk [mΩ]	78.1	161.4	217.0	138.5
Lfk [mH]	1.2	1.7	1.2	0.6
Cfk [µF]	114.3	41.0	41.1	81.4
ωfk [rad/s]	2646	3742	4583	4583
RLk [Ω]	1.80	5.06	5.00	2.50

.

5.2. Test Setup

The DC grid has four controlled loads. Depending on the bandwidth, each load can foster stability, or it can provoke unstable behaviors. To evaluate the bandwidths’ effects, three different cases are configured in

Table 2. Control bandwidths configurations.

	Case 1	Case 2	Case 3
ω1 [rad/s]	200	200	200
ω2 [rad/s]	500	200	200
ω3 [rad/s]	1000	1000	914
ω4 [rad/s]	1200	1200	1097

. From the bandwidths of Case 1, the ω₂ is lowered in Case 2. Then, the bandwidths on load 3–4 are additionally decreased in Case 3. By observing the trend on bandwidths, the system stability improves from Case 1 to Case 3. The WBM stability assessment is able to evaluate this enhancement. A loads modification behaves as a perturbation to check system stability. From the initial condition with four stable loads (i.e., P_L = 24 MW), the disconnection of the lowest bandwidth (i.e., stabilizing) converter reduces the total power (i.e., P_L = 19 MW) and deteriorates the stability margins. In the following, the last point is discussed.

5.3. Stability Assessment

The ω_B splitter groups the loads in the S-D sets for what concerns the control bandwidths. The ω_B values in

Table 3. Control bandwidths configurations.

	Before	After
ωB [rad/s]	832	907
PS [MW]	9	4
PD [MW]	15	15
m1 [·]	0.56	0.00
m2 [·]	0.44	1.00
m3 [·]	0.33	0.33
m4 [·]	0.67	0.67

confirm that ω₂ always behaves as stabilizer since it is smaller than ω_B both before and after the perturbation. Similarly, also load 1 helps the stability balance until its disconnection. Conversely, converters 3–4 are always destabilizers as the related bandwidth overtakes the ω_B splitter. Once classified the loads before/after the perturbation, the P_S and P_D total powers in Table 3 are the sum of stabilizing/destabilizing converters powers (Table 1). Such values are used to relativize the single converter power as m_k ratio (e.g., m₃ = P₃/P_D = 5/15 = 0.33). From Equations (10)–(12), the ω_S and ω_D bandwidths are calculated as in

Table 4. Control bandwidths configurations.

	Case 1	Case 2	Case 3
ωS [rad/s] before/after	333/500	200/200	200/200
ωD [rad/s]	1133	1133	1036

. As the perturbation changes the power ratios of stabilizing loads (i.e., m₁ and m₂), consequently also the ω_S is characterized by two values, before and after the perturbation. Conversely, the ω_D is not modified by the load disconnection, as the related power ratios of destabilizing loads (i.e., m₃ and m₄) are not influenced by the perturbation.

Once the ω_S-ω_D aggregated bandwidths are defined as in Table 4, the iterative process of Figure 4 is run to establish the stability performance of each case. Particularly, the Nyquist diagram of Figure 5 highlights three intersections. The instability is made evident for the Case 1, where the perturbation is capable of imposing an intersection Ψ = 1.18 > 1. Conversely, the bandwidth reduction on ω₂ (i.e., 500–200) in Case 2 is enough to ensure the system stability after perturbation (i.e., Ψ = 0.8 < 1). Finally, Case 3 where the additional reduction on destabilizing bandwidths (i.e., ω₃ and ω₄) is able to guarantee the best stability result (i.e., Ψ = 0.65 < 1). Being P_D more than three times the P_S, a tiny reduction (−8.6%) in ω_D is sufficient to get the stability target in the last case. Finally, Figure 6 and Figure 7 show the poles of transfer functions W(s) from ΔE to ΔV (15) and (16), when load 1 is OFF in Case 1 and 3. In both cases, the aggregated complex poles SD are always slightly on the right of initial complex poles 234. By comparing their real parts, the relative errors (i.e., 10% in Case 1, 3.4% in Case 3) prove the less conservativeness of WBM stability assessment.

5.4. Hardware in the Loop Validation

To prove the validity of the WBM stability assessment, a DC shipboard system with four voltage-controlled DC-DC converters (Table 1) is emulated by Typhoon HIL 604 real-time platform (Figure 8). The controlled DC grid is modeled by means of the Software-In-the-Loop approach, thus the system-control code is compiled to run real-time simulation where the detailed switching behavior is made visible. The four controlled DC-DC converters are synthesized in HIL schematic editor (Figure 8), where core coupling elements subdivide the numerical task in three cores. In such a way, the platform can offer real-time transients notwithstanding the small simulation time step (0.5 μs). In the HIL tests, a perturbation (i.e., disconnection of load 1) is applied at t = 0 s to establish the analyzed scenarios. First, the instability of Case 1 (Ψ = 1.18) results evident in Figure 9, where the bus voltage presents a limit cycle (range of ±17%) thus forcing the consequent intervention of protections. Conversely, the best stability performance (Ψ = 0.65) is highlighted in the Case 3 results (Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14), confirming the stability assessment. Although the stability analysis disregards the presence of load filters, it results coherent with emulations, which on the contrary also considered the filtering stages. The control validity is confirmed on the bus voltage transient (Figure 10), where the rated value (6000 V) is restored in about 150 ms by an underdamped evolution. In a steady state condition, the peak-peak voltage ripple ΔV_% results equal to 4% thus in accordance to the requirements in [9]. The controlled load voltages in Figure 11 also display a well-filtered behavior as the LC components on loads are sized to ensure ΔV_% = 7%. Regarding the actuators, Figure 12 shows the duty cycle signals to impose a stable steady state condition on the bus voltage, while avoiding the saturation (i.e., Y-axis = 1). When the dynamics transient is concluded in t > 150 ms, the final duty values are greater than the rated D_nk (Table 1) to compensate for the voltage drop on filter resistances R_fk. The converter’s currents are in Figure 13 while Figure 14 shows the loads currents. In both transients, a settling time of 150 ms is visible. The currents in the filter inductors (Figure 13) exhibit ripples (ΔI_% < 30%) in compliance with the desired power quality (Table 1), conversely smaller ripples (ΔI_% ≈ 4%) are visible on loads currents (Figure 14).

5.5. Considerations on Stability Assessment and HIL Results

The results and figures of this paper are able to demonstrate the potentiality of the proposed approach. First, the Nyquist diagram in Figure 5 can provide an interesting overview about the stability performance of the three cases (i.e., three bandwidths combination) under study. When the perturbation affects the power grid in Case 1, the intersection in −1.18 means instability. Different scenarios are visible in the other two cases, where the reductions in bandwidths are able to restore the system stability even after the perturbation. Then, Figure 6 and Figure 7 are crucial because they confirm the WBM model, named SD, as less conservative. Indeed, the related complex poles (red) are always on the right of the initial complex poles (blue). This means that the aggregated model is more inclined to instability. If the multiconverter control is designed to maintain the aggregated poles on the left plane, certainly also the poles of initial complex system are on the left, thus representative of stable evolutions. Finally, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 show the real-time behavior by implementing the Typhoon HIL emulation. These transients are important because they validate the previous consideration: when the control bandwidths are not harmonized as in Case 1, the instability is consequent as in Figure 9. Differently, when the iterative process is applied on the WBM aggregated model, a convenient reduction in bandwidths is the solution for achieving the stability, even after the perturbation. This stable performance is made evident in the bus voltage transient of Figure 10, as well as in all the other transients, from Figure 11, Figure 12, Figure 13 and Figure 14.

6. Conclusions

The system stability is an important requirement in shipboard MVDC power systems, where undesired voltage oscillations can lead to blackouts. In this context, the paper has studied a methodology to assess system stability in a radial multiple-loads DC grid. As DC shipboard systems are complex (i.e., tens/hundreds controlled loads), some assumptions (e.g., load filters disregarding) are initially established to simplify the modeling while identifying the range of validity. Since a low-bandwidth controlled converter can balance the destabilizing action of a high-performance converter, WBM is proposed to aggregate the multiple controlled loads in the two stabilizing/destabilizing sets. Once WBM is proved to be less conservative, load aggregation is the base on which the iterative process for the stability assessment is developed. By taking into account the compensation provided by low-performance converters, attention is spent on the stability criterion’s verification. In particular, the reduction on bandwidths results effective in reestablishing the DC stability, if the destabilizing load quota is hypothetically known from the electrical balance or Power Management System. In order to test the weighted bandwidth method, a DC shipboard power system consisting of four controlled converters is used as the study case on which to evaluate the effects of control bandwidths on system stability. Although the methodology is conceived on a DC system with a limited number of controlled loads, the results are directly transferable to a realistic DC shipboard grid, thus the study contributes with high engineering value. Finally, HIL emulations test the preemptive stability assessments by verifying the restoration of stable evolutions towards equilibrium points.