The Stability Analysis of a Multi-Port Single-Phase Solid-State Transformer in the Electromagnetic Timescale

Rui Wang 1 ID , Qiuye Sun 1,2,*, Qifu Cheng 1,3 and Dazhong Ma 1 1 College of Information Science and Engineering, Northeastern University, Shenyang 110004, China; 1610232@stu.neu.edu.cn (R.W.); lowkeycheng@163.com (Q.C.); madazhong@ise.neu.edu.cn (D.M.) 2 State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China 3 State Grid Shenyang Electric Power Supply Company, Shenyang 110004, China * Correspondence: sunqiuye@ise.neu.edu.cn; Tel.: +86-139-9881-1006


Introduction
With the high expansion of distributed renewable energy utilization, the Energy Internet (EI), the future renewable electric energy delivery and management (FREEDM) system, and microgrids have been widely researched [1][2][3][4].Meanwhile, since the allowable weight and volume of a traction transformer are restricted, the solid-state transformer (SST) (also known as the Power Electronic Transformer, Energy Router, Intelligent Universal Transformer, etc.) has been one of the main elements in the EI, FREEDM, and microgrids [5].Compared with traditional transformers, the SST has several advantages, such as low weight, low volume, unity power factor, fault isolation, bidirectional power flow capability, high controllability, etc. [6].
Obviously, stability is a basic necessity to ensure the performance of the system.If the system is unstable, it is of no value to research its performance [7][8][9].Thus, the stability assessment of the multi-port single-phase solid-state transformer (MS3T) is a significant topic.As shown in Figure 1, from the viewpoint of the multi-timescale, the stability assessment can be divided into two parts [10], i.e., small-signal stability assessment and large-signal stability assessment.The non-linear behavior signal stability assessment pays more attention to the dynamic and transient events of the system.However, the small-signal stability assessment pays more attention to the steady state, which is studied in the electromagnetic timescale.According to the existing literature, small-signal stability assessment in the electromagnetic timescale always adopts a linearized method around a steady operating point [11][12][13].Compared with traditional power systems, the small-signal stability assessment of the MS3T is a critical issue due to the low-inertia and negative-impedance nature of systems dominated by power electric converters [14].Thus, this paper focuses on the small-signal stability assessment, which adopts a linearized method around a steady operating point, for the MS3T in the electromagnetic timescale.Traditionally, there are three stages in the MS3T, i.e., the AC-DC front-end side stage, dual active bridge DAB stage, and back-end side stage [15].Although the three stages are able to achieve the key appealing features for MS3T together, the MS3T is highly subject to instabilities on account of the interactions among the three stages.The SST stability analysis strategy based on each subsystem model was first proposed by M. Khazraei [16].Therein, each subsystem control strategy was so specialized that it was not easy for the common SST to utilize.Furthermore, the stability design criteria for a distribution system with SSTs were first proposed in References [17,18], and these stability criteria did not analyze the stability of the SST itself.Moreover, the premise of previous papers was that each subsystem of the SST was stable in the distribution system, and a practical method of stability analysis for the general SST was not proposed.Thus, the stability assessment of the SST in the electromagnetic timescale still remains a challenge that has rarely been researched.
Three main stability assessment methods were utilized to analyze the stability of the power electronics converter-dominated system in the electromagnetic timescale, i.e., a state-space-based approach, a transfer function approach, and an impedance-based approach.Compared with the state-space-based approach and transfer function approach, the impedance-based approach had several potential advantages to explicitly explain the impact of individual subsystems on the system stability [19].Thus, the impedance-based approach should be applied to assess the stability of the MS3T.The impedance-based approach was first proposed by Middlebrook [19][20][21][22][23][24].Furthermore, it is significant to check the right half-plane (RHP) poles of the transfer function matrix when using the generalized Nyquist criterion (GNC) for stability analysis.Several papers have been proposed to eliminate the step of checking the RHP poles, such as inverse generalized Nyquist criterion [21] and sum type criterion [22].In order to reduce the computational complexity of the stability assessment, norm-based impedance methods were put forward one after another [23].To deal with the bidirectional power flow, the sum type criterion was first proposed to assess the stability of the power electric converter [22].Furthermore, according to the different applications of the impedance-based approach, Mohammad Amin et al. proposed a stability assessment for high-voltage DC (HVDC) systems [24], and an impedance-based local stability criterion was proposed to assess the stability of DC distributed power systems [25].To sum up, it is advisable to apply the impedance-based approach to assess the stability of the MS3T.
To date, the overall practical stability assessment of the MS3T in the electromagnetic timescale has not been proposed.Thus, the objective of this paper is to propose a practical impedance-based Traditionally, there are three stages in the MS3T, i.e., the AC-DC front-end side stage, dual active bridge DAB stage, and back-end side stage [15].Although the three stages are able to achieve the key appealing features for MS3T together, the MS3T is highly subject to instabilities on account of the interactions among the three stages.The SST stability analysis strategy based on each subsystem model was first proposed by M. Khazraei [16].Therein, each subsystem control strategy was so specialized that it was not easy for the common SST to utilize.Furthermore, the stability design criteria for a distribution system with SSTs were first proposed in References [17,18], and these stability criteria did not analyze the stability of the SST itself.Moreover, the premise of previous papers was that each subsystem of the SST was stable in the distribution system, and a practical method of stability analysis for the general SST was not proposed.Thus, the stability assessment of the SST in the electromagnetic timescale still remains a challenge that has rarely been researched.
Three main stability assessment methods were utilized to analyze the stability of the power electronics converter-dominated system in the electromagnetic timescale, i.e., a state-space-based approach, a transfer function approach, and an impedance-based approach.Compared with the state-space-based approach and transfer function approach, the impedance-based approach had several potential advantages to explicitly explain the impact of individual subsystems on the system stability [19].Thus, the impedance-based approach should be applied to assess the stability of the MS3T.The impedance-based approach was first proposed by Middlebrook [19][20][21][22][23][24].Furthermore, it is significant to check the right half-plane (RHP) poles of the transfer function matrix when using the generalized Nyquist criterion (GNC) for stability analysis.Several papers have been proposed to eliminate the step of checking the RHP poles, such as inverse generalized Nyquist criterion [21] and sum type criterion [22].In order to reduce the computational complexity of the stability assessment, norm-based impedance methods were put forward one after another [23].To deal with the bidirectional power flow, the sum type criterion was first proposed to assess the stability of the power electric converter [22].Furthermore, according to the different applications of the impedance-based approach, Mohammad Amin et al. proposed a stability assessment for high-voltage DC (HVDC) systems [24], and an impedance-based local stability criterion was proposed to assess the stability of DC distributed power systems [25].To sum up, it is advisable to apply the impedance-based approach to assess the stability of the MS3T.
To date, the overall practical stability assessment of the MS3T in the electromagnetic timescale has not been proposed.Thus, the objective of this paper is to propose a practical impedance-based stability assessment of the MS3T.Compared with the previous existing literature, the proposed stability assessment has the following features: (1) This paper researches the structure of the MS3T and its subsystem control strategy; (2) The stability analysis of each of the MS3T's subsystems is achieved, such as the AC-DC front-end side converter, the DAB with an HF or MF transformer, and the back-end side incorporating DC-AC and DC-DC converters; (3) The overall practical stability assessment in the electromagnetic timescale, based on the measured impedance stability analysis, for the MS3T is investigated.Furthermore, this stability assessment only requires two current sensors and one external high-bandwidth small-signal sinusoidal perturbation current source; (4) The relative simulation and experiment results validate the performance of the stability assessment.
The rest of this paper is organized as follows.This paper establishes an MS3T structure model.Subsequently, each subsystem stability analysis of the MS3T is proposed in Section 2. Section 3 provides the overall stability assessment of the MS3T.In Sections 4 and 5, the proposed impedance-based assessment is validated through extensive simulation and hardware results.Finally, the conclusions are given in Section 6.

MS3T Structure Model
As shown in Figure 2, the MS3T structure model can be divided into four subsystems including an AC-DC front-end side converter subsystem [26], a DAB with an HF or MF transformer subsystem [27], a back-end side DC-AC converter subsystem [28], and a back-end side DC-DC converter subsystem [21].Z out,sub1 , Z in,sub2 , Z out,sub2 , Z in,sub3 , and Z in,sub4 are the output impedances of subsystem 1, the input impedance of subsystem 2, the output impedance of subsystem 2, the input impedance of subsystem 3, and the input impedance of subsystem 4, respectively.The AC-DC front-end side converter transforms the 220 V AC bus to a 400 V DC distribution bus.Furthermore, the DAB with an HF or MF transformer is used to reduce this DC voltage to a regulated DC voltage.Finally, the back-end side DC-AC converter produces an AC voltage to integrate an AC load, and the back-end side DC-DC converter produces a DC voltage to integrate a DC load [29].As this paper focuses on the small-signal stability of the MS3T, slow control strategies, such as second-level coordinated control, optimal energy management, and so on, are ignored [30].Therefore, the impedance-based stability approach can be applied to assess the overall stability of the MS3T.In light of the impedance-based approach, the system will be stable in the electromagnetic timescale if and only if the system satisfies following conditions: (1) each subsystem of the MS3T is stable; (2) the number of the counterclockwise encirclements of the (−1, j0) point by the ratio between the output source impedance and the input load impedance locus is equal to the number of the RHP poles of the minor loop gain.In reality, the second condition is usually judged only by checking whether the locus encircles the (−1, j0) point [21].The small-signal models of each subsystem in the MS3T are researched as follows.
Energies 2018, 11, x FOR PEER REVIEW 3 of 21 stability assessment of the MS3T.Compared with the previous existing literature, the proposed stability assessment has the following features: (1) This paper researches the structure of the MS3T and its subsystem control strategy; (2) The stability analysis of each of the MS3T's subsystems is achieved, such as the AC-DC frontend side converter, the DAB with an HF or MF transformer, and the back-end side incorporating DC-AC and DC-DC converters; (3) The overall practical stability assessment in the electromagnetic timescale, based on the measured impedance stability analysis, for the MS3T is investigated.Furthermore, this stability assessment only requires two current sensors and one external high-bandwidth small-signal sinusoidal perturbation current source; (4) The relative simulation and experiment results validate the performance of the stability assessment.
The rest of this paper is organized as follows.This paper establishes an MS3T structure model.Subsequently, each subsystem stability analysis of the MS3T is proposed in Section 2. Section 3 provides the overall stability assessment of the MS3T.In Sections 4 and 5, the proposed impedancebased assessment is validated through extensive simulation and hardware results.Finally, the conclusions are given in Section 6.

MS3T Structure Model
As shown in Figure 2, the MS3T structure model can be divided into four subsystems including an AC-DC front-end side converter subsystem [26], a DAB with an HF or MF transformer subsystem [27], a back-end side DC-AC converter subsystem [28], and a back-end side DC-DC converter subsystem [21].are the output impedances of subsystem 1, the input impedance of subsystem 2, the output impedance of subsystem 2, the input impedance of subsystem 3, and the input impedance of subsystem 4, respectively.The AC-DC front-end side converter transforms the 220 V AC bus to a 400 V DC distribution bus.Furthermore, the DAB with an HF or MF transformer is used to reduce this DC voltage to a regulated DC voltage.Finally, the back-end side DC-AC converter produces an AC voltage to integrate an AC load, and the back-end side DC-DC converter produces a DC voltage to integrate a DC load [29].As this paper focuses on the small-signal stability of the MS3T, slow control strategies, such as second-level coordinated control, optimal energy management, and so on, are ignored [30].Therefore, the impedance-based stability approach can be applied to assess the overall stability of the MS3T.In light of the impedancebased approach, the system will be stable in the electromagnetic timescale if and only if the system satisfies following conditions: (1) each subsystem of the MS3T is stable; (2) the number of the counterclockwise encirclements of the (−1, j0) point by the ratio between the output source impedance and the input load impedance locus is equal to the number of the RHP poles of the minor loop gain.
In reality, the second condition is usually judged only by checking whether the locus encircles the (−1, j0) point [21].The small-signal models of each subsystem in the MS3T are researched as follows.

The AC-DC Front-End Side Converter
Both the control strategy and the topological structure are shown in Figure 3.There are four main controllers in the AC-DC front-end side converter, i.e., voltage controller with virtual inertia, DC voltage controller, DC current feedforward controller, and current controller.Thus, the mathematical expression of the converter is obtained as follows: The current controller is described as G sub1i (s) = k sub1ip + k sub1ii /s.Thus, Equation (1) can be rewritten as follows:

The AC-DC Front-End Side Converter
Both the control strategy and the topological structure are shown in Figure 3.There are four main controllers in the AC-DC front-end side converter, i.e., voltage controller with virtual inertia, DC voltage controller, DC current feedforward controller, and current controller.Thus, the mathematical expression of the converter is obtained as follows: The current controller is described as Moreover, the small-signal model of the current controller can be shown as follows:   Moreover, the small-signal model of the current controller can be shown as follows: Neglecting the energy loss, the following equation can be obtained due to the power balance between two sides of the converter.
Furthermore, both U gd and I gd are equivalent to zero in the steady state [26].Thus, the small-perturbation Equation ( 5) can be obtained as follows: In light of the superposition theorem, the relations between u H (s) and i gd (s), u H (s) and i H (s), respectively, are obtained as follows: Neglecting the influence of the main grid voltage and virtual inertia controller, the DC output current feedforward controller and the voltage controller are applied, so the small-signal closed-loop transfer function can be obtained as follows: where

DAB with an HF or MF Transformer
As shown in Figure 3, the DAB is a bidirectional DC-DC power electrical converter with a phase-shift controller based on active bridges interfaced with a high/middle frequency transformer.Therefore, the expression of the transmitted power can be provided as follows: Defining Thus, Equation ( 8) can be rewritten as follows: Defining k DAB = I L /D s , where k DAB = nU H T s /4L T .Using Faraday's law, the capacitor voltage can be calculated as: Energies 2018, 11, 2250 The current controller is described as G sub2 (s) = k sub2p + k sub2i /s.The time delay is represented as G d (s) = 1/(T s s + 1).While a feedback loop (the feedback coefficient A = 1/k DAB ) is adopted in the DAB, the small-signal closed-loop transfer function of the DAB is given as follows:

The DC-AC or DC-DC Back-End Side Converter
The DC-AC back-end side converter controlled by the traditional voltage/current double controller is shown in Figure 3. Thus, the current controller mathematical equations of the converter are represented by Equation (12).
The current controller can be described as G sub3i (s) = k sub3ip + k sub3ii /s, and Equation ( 12) can be rewritten as: Moreover, when both the symmetrical characteristic and time delay are considered, the small-signal open-loop transfer function can be expressed as: Therefore, the small-signal closed-loop transfer function can be provided as follows: The similar analysis process for the voltage controller can be applied to that described previously to gain the voltage controller closed-loop transfer function.The voltage controller mathematical equations of the inverter are provided as follows: The current controller is described as G sub3v (s) = k sub3vp + k sub3vi /s, and Equation ( 17) can be rewritten as follows: Finally, when both the time delay and the symmetrical characteristic are considered, the open-loop transfer function and closed-loop transfer function of the overall DC-AC converter can be expressed as follows: As seen in Figure 3, the DC-DC converter adopts the traditional PI closed control strategy, the voltage controller is described as G sub4 (s) = k sub4p + k sub4i /s.Thus, the closed-loop transfer function of the overall DC-DC converter is expressed as follows:

The Stability Assessment of the Overall MS3T
Based on the aforementioned analysis, a practical impedance-based stability assessment in the electromagnetic timescale needs to be proposed to assess the stability of the overall MS3T.As shown in Figure 4, the total input-to-output transfer function of the cascade of two individual stable subsystems, which are module 1 and module 2, is:

The Stability Assessment of the Overall MS3T
Based on the aforementioned analysis, a practical impedance-based stability assessment in the electromagnetic timescale needs to be proposed to assess the stability of the overall MS3T.As shown in Figure 4, the total input-to-output transfer function of the cascade of two individual stable subsystems, which are module 1 and module 2, is: Since 1 G and 2 G do not have the right half-plane pole, the system is stable if and only if ( ) does not have the right pole.In other words, drawing the Nyquist plot of / out in Z Z , the system is stable if and only if the number of times that the trajectory encircles the (−1, j0) point in the clockwise direction is zero.The "black-box" unterminated modeling of a DC two-port network has been an attractive topic widely utilized in engineering [31].There is a demand in practice to acquire the linearized dynamics in the region "not far" from an operating point so as to assess the subsystem interaction and small-signal stability in the electromagnetic timescale.Apparently, the impedance of a general two-port network can be obtained by removing from the original environment and connecting to a high-bandwidth voltage source or current source.As mentioned above, this ratio must not encircle the (−1 + j0) point on Nyquist contours.Moreover, according to the Gershgorin theorem, if the eigenvalues of at all frequencies are restrained inside the unit circle, the stability of the AC microgrids can be guaranteed, i.e.,: Then, the necessary and sufficient condition satisfying Equation ( 22) is gained by Equation ( 21), which can be further transformed to Equation (23), since Moreover, according to the Kirchhoff laws, when the multiple paralleled subsystems are connected as in module 2, _ 2 in Z can be rewritten as follows: Since G 1 and G 2 do not have the right half-plane pole, the system is stable if and only if 1/(1 + Z out_1 /Z in_2 ) does not have the right pole.In other words, drawing the Nyquist plot of Z out_1 /Z in_2 , the system is stable if and only if the number of times that the trajectory encircles the (−1, j0) point in the clockwise direction is zero.The "black-box" unterminated modeling of a DC two-port network has been an attractive topic widely utilized in engineering [31].There is a demand in practice to acquire the linearized dynamics in the region "not far" from an operating point so as to assess the subsystem interaction and small-signal stability in the electromagnetic timescale.Apparently, the impedance of a general two-port network can be obtained by removing from the original environment and connecting to a high-bandwidth voltage source or current source.
As mentioned above, this ratio must not encircle the (−1 + j0) point on Nyquist contours.Moreover, according to the Gershgorin theorem, if the eigenvalues of Z out_1 /Z in_2 at all frequencies are restrained inside the unit circle, the stability of the AC microgrids can be guaranteed, i.e.,: Then, the necessary and sufficient condition satisfying Equation ( 22) is gained by Equation ( 21), which can be further transformed to Equation (23), since Moreover, according to the Kirchhoff laws, when the multiple paralleled subsystems are connected as in module 2, Z in_2 can be rewritten as follows: Energies 2018, 11, 2250 Thus, Equation ( 23) can be rewritten as follows: As shown in Figure 5, the stability of the MS3T can be assessed by an external high-bandwidth small-signal sinusoidal perturbation current ∧ i p , which is injected into the DC bus while the MS3T is in the steady state of the operating point.Furthermore, the small-signal response current in the source side and load side are measured.According to the aforementioned analysis, while the three stages are stable, the MS3T must be stable if Equations ( 26) and ( 27) are satisfied as follows: Thus, Equation ( 23) can be rewritten as follows: As shown in Figure 5, the stability of the MS3T can be assessed by an external high-bandwidth small-signal sinusoidal perturbation current p i ∧ , which is injected into the DC bus while the MS3T is in the steady state of the operating point.Furthermore, the small-signal response current in the source side and load side are measured.According to the aforementioned analysis, while the three stages are stable, the MS3T must be stable if Equations ( 26) and ( 27) are satisfied as follows: Furthermore, it is also possible to only measure the source-side response current to assess the stability of the MS3T, since The sufficient condition at which the MS3T is stable is gained by Equations ( 28) and ( 29): In summary, the proposed stability assessment based on impedance in the electromagnetic timescale, which is seen in Figure 6, can be described as follows: Step1: Divide the overall MS3T into four subsystems containing an AC-DC front-end side converter, a DAB with an HF or MF transformer, a DC-AC back-end side converter, and a DC-DC back-end side converter, and derive the expressions of each subsystem closed-loop transfer function (Equations ( 7), ( 11), (18), and ( 19)).
Step2: Check the right half-plane pole of Equations ( 7), ( 11), (18), and (19).If the number of the right half-plane pole is zero, the assessment process continues to Step3.If not, the system is unstable.Furthermore, it is also possible to only measure the source-side response current to assess the stability of the MS3T, since ∧ i p = ∆i out,sub1 + ∆i in,sub2 and ∧ i p = ∆i out,sub2 + ∆i in,sub3 + ∆i in,sub4 are satisfied.The sufficient condition at which the MS3T is stable is gained by Equations ( 28) and ( 29): In summary, the proposed stability assessment based on impedance in the electromagnetic timescale, which is seen in Figure 6, can be described as follows: Step1: Divide the overall MS3T into four subsystems containing an AC-DC front-end side converter, a DAB with an HF or MF transformer, a DC-AC back-end side converter, and a DC-DC back-end side converter, and derive the expressions of each subsystem closed-loop transfer function (Equations ( 7), ( 11), ( 18) and ( 19)).
Step2: Check the right half-plane pole of Equations ( 7), ( 11), ( 18) and (19).If the number of the right half-plane pole is zero, the assessment process continues to Step3.If not, the system is unstable.
Step3: As shown in Figure 5, the small-signal response current of the input subsystem 2 side is measured.If the inequality (28) can be satisfied, the assessment process continues to Step4.If not, there is a possibility that the MS3T is unstable.
Step4: Similar to Step3, the small-signal response current in the input subsystem 3 side and input subsystem 4 side are measured.If the inequality (29) can be satisfied, the MS3T is stable.If not, there is a possibility that the MS3T is unstable.Step3: As shown in Figure 5, the small-signal response current of the input subsystem 2 side is measured.If the inequality (28) can be satisfied, the assessment process continues to Step4.If not, there is a possibility that the MS3T is unstable.
Step4: Similar to Step3, the small-signal response current in the input subsystem 3 side and input subsystem 4 side are measured.If the inequality (29) can be satisfied, the MS3T is stable.If not, there is a possibility that the MS3T is unstable.
Is Eq. ( 28 To sum up, the proposed stability assessment of the overall MS3T in the electromagnetic timescale is sufficient, but not necessary.Nevertheless, this stability assessment only requires two current sensors and one external high-bandwidth small-signal sinusoidal perturbation current source.

Simulation Verification
To verify the effectiveness of the proposed overall practical stability assessment for the MS3T in the electromagnetic timescale, simulations in MATLAB/Simulink were conducted for a 10kW / 50Hz MS3T system.The system is depicted in Figure 3, and the parameters consisting of the AC-DC frontend side converter, DAB, and back-end side converters are shown in Tables 1-3, respectively.According to the closed transfer functions of each part of the MS3T in Section 2, each stage of the MS3T is stable if and only if the number of the right half-plane poles of each closed transfer function is zero.As shown in Table 4, where the poles reported can be obtained in MATLAB, there is no right half-plane pole for each stage of the closed transfer function.As a result, the three stages of the MS3T system are stable, and the stability of the overall MS3T depends on the interactions among the three stages.Of course, the 41 pole is very close to the imaginary axis in Table 4. Thus, the stability margin of the DC-DC back-end side converter should be provided.In the s-plane, it is very convenient to represent the stability margin by using the distance from the ordinate to the left ( 1 σ ).According to the DC-DC back-end side converter subsystem closed-loop transfer function in Section 2 and the parameters of subsystem 4 in Table 3, the characteristic equation can be obtained as follows: Figure 6.The flow chart of the proposed stability assessment method.
To sum up, the proposed stability assessment of the overall MS3T in the electromagnetic timescale is sufficient, but not necessary.Nevertheless, this stability assessment only requires two current sensors and one external high-bandwidth small-signal sinusoidal perturbation current source.

Simulation Verification
To verify the effectiveness of the proposed overall practical stability assessment for the MS3T in the electromagnetic timescale, simulations in MATLAB/Simulink were conducted for a 10 kW/50 Hz MS3T system.The system is depicted in Figure 3, and the parameters consisting of the AC-DC front-end side converter, DAB, and back-end side converters are shown in Tables 1-3, respectively.According to the closed transfer functions of each part of the MS3T in Section 2, each stage of the MS3T is stable if and only if the number of the right half-plane poles of each closed transfer function is zero.As shown in Table 4, where the poles reported can be obtained in MATLAB, there is no right half-plane pole for each stage of the closed transfer function.As a result, the three stages of the MS3T system are stable, and the stability of the overall MS3T depends on the interactions among the three stages.Of course, the 41 pole is very close to the imaginary axis in Table 4. Thus, the stability margin of the DC-DC back-end side converter should be provided.In the s-plane, it is very convenient to represent the stability margin by using the distance from the ordinate to the left (σ 1 ).According to the DC-DC

Parameters Values
The parameters of the dual active bridge (DAB).

Parameters
Values Substitute s = z − σ 1 into the above equation.Thus, the stability margin (σ 1 ) can be obtained as follows: 3.2006(z − σ 1 ) + 0.01212698 = 0 ⇒ σ 1 = −3.7890* 10 −3 (31) From the viewpoint of the improvement of the stability margin using the proposed stability assessment, the stability margin (σ 1 ) of the DC-DC back-end side converter can be presented as follows: Since k pwm and D are always constant, the stability margin can be enhanced by increasing the ratio of k sub4i and k sub4p .
Of course, with increasing of the order of the closed-loop transfer function, it is difficult to obtain the stability margin (σ 1 ).Phase Margin (PM) and Gain Margin (GM) can become an advisable choice.The Bode diagram of the DC-AC back-end side converter can be seen in Figure 7 as follows: Energies 2018, 11, x FOR PEER REVIEW 11 of 21 From the viewpoint of the improvement of the stability margin using the proposed stability assessment, the stability margin ( 1 σ ) of the DC-DC back-end side converter can be presented as follows: (1 ) Since pwm k and D are always constant, the stability margin can be enhanced by increasing the ratio of Of course, with increasing of the order of the closed-loop transfer function, it is difficult to obtain the stability margin ( 1 σ ).Phase Margin (PM) and Gain Margin (GM) can become an advisable choice.
The Bode diagram of the DC-AC back-end side converter can be seen in Figure 7 as follows: Thus, the Phase Margin of the DC-AC back-end side converter is 4.4 0 , and stability margin can also be improved by adjusting the voltage/current controller parameters.

Stable Case
In this section, the effectiveness of the proposed overall stability assessment for the MS3T can be verified.The stage system parameters are shown in Tables 1-3.The control strategy and the topological structure of the simulation text system are shown in Figure 3. Therein, the switch 1 (S1) is closed, and the switch 2 (S2) is opened.According to the aforementioned analysis in Section 3 and Table 4, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interaction among three stages.As shown in Figures 8 and 9, Step3 and Step4 in Section 3 are satisfied, so the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 10-12, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.2 in sub i Δ in a stable case.Thus, the Phase Margin of the DC-AC back-end side converter is 4.4 • , and stability margin can also be improved by adjusting the voltage/current controller parameters.

Stable Case
In this section, the effectiveness of the proposed overall stability assessment for the MS3T can be verified.The stage system parameters are shown in Tables 1-3.The control strategy and the topological structure of the simulation text system are shown in Figure 3. Therein, the switch 1 (S1) is closed, and the switch 2 (S2) is opened.According to the aforementioned analysis in Section 3 and Table 4, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interaction among three stages.As shown in Figures 8 and 9, Step3 and Step4 in Section 3 are satisfied, so the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 10-12, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.so the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 10-12, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.Although the proposed stability assessment in this paper is not suitable for analyzing dynamic/transient processes, when the load changes, it can be applied to assess the stability of the MS3T in the next steady-state operation point.A case in point is that the output DC load of the MS3T changes from 100 R = Ω to 50 R = Ω, and the other parameters are same as those in Tables 1-3.Since each controller of each subsystem is the same as that in the above simulation test system, the eigenvalues are also shown in Table 4. Thus, the stability of the MS3T depends on the interaction among the three stages.As shown in Figures 13 and 14, Step3 and Step4 in Section 3 are satisfied, so the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 15-17, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Thus, the performance of the proposed stability assessment can be verified.Although the proposed stability assessment in this paper is not suitable for analyzing dynamic/transient processes, when the load changes, it can be applied to assess the stability of the MS3T in the next steady-state operation point.A case in point is that the output DC load of the MS3T changes from R = 100 Ω to R = 50 Ω, and the other parameters are same as those in Tables 1-3.Since each controller of each subsystem is the same as that in the above simulation test system, the eigenvalues are also shown in Table 4. Thus, the stability of the MS3T depends on the interaction among the three stages.As shown in Figures 13 and 14, Step3 and Step4 in Section 3 are satisfied, so the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 15-17, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Thus, the performance of the proposed stability assessment can be verified.the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 15-17, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Thus, the performance of the proposed stability assessment can be verified.waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 15-17, where the input voltage/current of the AC-DC front-end side converter, the output voltage/current of the DC-AC back-end side converter, and the output voltage/current of the DC-DC back-end side converter coincide with the stable phenomenon.Thus, the performance of the proposed stability assessment can be verified.

Unstable Case
In this section, the effectiveness of the proposed overall stability assessment for the MS3T in the electromagnetic timescale can be verified.The system parameters of each stage are also shown in Tables 1-3.The stability margin will be reduced by adding the impedance of the output source subsystem [23].Thus, the control strategy and the topological structure of the simulation text system are shown in Figure 3 side converter, the closed-loop transfer functions of each subsystem are not affected, and each closedloop transfer function in an unstable case is same with that in a stable case.Furthermore, the eigenvalues for the unstable case are same with those for stable case, which are shown in Table 4. Thus, each subsystem is stable.Moreover, Step3 in the Section 3 is not satisfied, as shown in Figure

Unstable Case
In this section, the effectiveness of the proposed overall stability assessment for the MS3T in the electromagnetic timescale can be verified.The system parameters of each stage are also shown in Tables 1-3.The stability margin will be reduced by adding the impedance of the output source subsystem [23].Thus, the control strategy and the topological structure of the simulation text system are shown in Figure 3. Therein, switch 1 (S1) is opened, switch 2 (S2) is closed, and R add = 1.7 Ω as well as C add = 24 µF.Since R add and C add are only added in output side of the AC-DC front-end side converter, the closed-loop transfer functions of each subsystem are not affected, and each closed-loop transfer function in an unstable case is same with that in a stable case.Furthermore, the eigenvalues for the unstable case are same with those for stable case, which are shown in Table 4. Thus, each subsystem is stable.Moreover, Step3 in the Section 3 is not satisfied, as shown in Figure 18.Thus, Step4 is not executed.There is a possible that the MS3T is unstable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is unstable, as shown in Figures 19 and 20, where the input voltage/current of the AC-DC front-end side converter and the output voltage/current of the DAB coincide with the unstable phenomenon.To sum up, the MS3T is highly subject to instabilities on account of the interaction among the four subsystems, although each subsystem is stable.In other words, the instability of MS3T is caused by the interaction between subsystems.Therefore, the effectiveness of the proposed stability assessment is verified.
Energies 2018, 11, x FOR PEER REVIEW 15 of 21 18.Thus, Step4 is not executed.There is a possible that the MS3T is unstable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is unstable, as shown in Figures 19 and 20, where the input voltage/current of the AC-DC front-end side converter and the output voltage/current of the DAB coincide with the unstable phenomenon.
To sum up, the MS3T is highly subject to instabilities on account of the interaction among the four subsystems, although each subsystem is stable.In other words, the instability of MS3T is caused by the interaction between subsystems.Therefore, the effectiveness of the proposed stability assessment is verified.

Experimental Results
A 220 V, 50 Hz MS3T was built to verify the effectiveness of the proposed approach.As shown in Figure 21

Experimental Results
A 220 V, 50 Hz MS3T was built to verify the effectiveness of the proposed approach.As shown in Figure 21

Experimental Results
A 220 V, 50 Hz MS3T was built to verify the effectiveness of the proposed approach.As shown in Figure 21, the whole platform of the MS3T is controlled by TMS320F28335DSP + XC6SLX9FPGA.Furthermore, the communication circuit is RS232&RS485, and the system switch frequency is 19.2 kHz.The model of the IGBTs in the AC-DC front-end side stage and back-end side stage is 2MBI75U4A-120, and the model of the IGBTs in the DAB stage is CM400DU-24NFH.The system controller parameters are the same as those in Section 4, and the other parameters are shown in Table 5.As shown in Table 6, there is no right half-plane pole for each stage of the closed transfer function.As a result, the three stages of the MS3T system are stable, and the stability of the overall MS3T depends on the interactions among the three stages.5.As shown in Table 6, there is no right half-plane pole for each stage of the closed transfer function.
As a result, the three stages of the MS3T system are stable, and the stability of the overall MS3T depends on the interactions among the three stages.In this section, the effectiveness of the proposed overall stability assessment for the MS3T is verified.According to the aforementioned analysis in Section 3 and Table 6, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interactions among the three stages.As shown in Figures 22 and 23, Step3 and Step4 in Section 3 are satisfied, so the MS3T should be stable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is stable, as shown in Figures 24-26, where the input voltage of the AC-DC front-end side converter, the output voltage of the DC-AC back-end side converter, and the output voltage of the DC-DC back-end side converter coincide with the stable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.

Unstable Case of the Experimental Test System
In order to verify the effectiveness of the proposed overall stability assessment for the MS3T, Furthermore, the eigenvalues for the unstable case are same as those for stable case, which are shown in Table 6.Moreover, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interactions among the three stages.As shown in Figure 27, Step3 in Section 3 is not satisfied, so there is a possibility that the MS3T is unstable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is unstable, as shown in Figures 28 and 29, where the input voltage of the AC-DC front-end side converter and the output voltage of the DAB coincide with the unstable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.

Unstable Case of the Experimental Test System
In order to verify the effectiveness of the proposed overall stability assessment for the MS3T, R add and C add were added in the output side of the AC-DC front-end side converter, which is same as that in the unstable case of Section 4. Therein, R add = 1.7 Ω and C add = 1000 µF.According to the aforementioned analysis in Section 3, Since R add and C add are only added in the output side of the AC-DC front-end side converter, the closed-loop transfer functions of each subsystem are not affected, and each closed-loop transfer function in the unstable case is same as that in the stable case.Furthermore, the eigenvalues for the unstable case are same as those for stable case, which are shown in Table 6.Moreover, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interactions among the three stages.As shown in Figure 27, Step3 in Section 3 is not satisfied, so there is a possibility that the MS3T is unstable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is unstable, as shown in Figures 28  and 29, where the input voltage of the AC-DC front-end side converter and the output voltage of the DAB coincide with the unstable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.
add add AC-DC front-end side converter, the closed-loop transfer functions of each subsystem are not affected, and each closed-loop transfer function in the unstable case is same as that in the stable case.Furthermore, the eigenvalues for the unstable case are same as those for stable case, which are shown in Table 6.Moreover, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interactions among the three stages.As shown in Figure 27, Step3 in Section 3 is not satisfied, so there is a possibility that the MS3T is unstable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is unstable, as shown in Figures 28 and 29, where the input voltage of the AC-DC front-end side converter and the output voltage of the DAB coincide with the unstable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.AC-DC front-end side converter, the closed-loop transfer functions of each subsystem are not affected, and each closed-loop transfer function in the unstable case is same as that in the stable case.Furthermore, the eigenvalues for the unstable case are same as those for stable case, which are shown in Table 6.Moreover, the four subsystems of the MS3T are stable.Thus, the stability of the MS3T depends on the interactions among the three stages.As shown in Figure 27, Step3 in Section 3 is not satisfied, so there is a possibility that the MS3T is unstable.The stability of the MS3T can be directly verified by checking the waveform in the time domain.Apparently, the MS3T is unstable, as shown in Figures 28 and 29, where the input voltage of the AC-DC front-end side converter and the output voltage of the DAB coincide with the unstable phenomenon.Therefore, the effectiveness of the proposed stability assessment is verified.

Conclusions
With the high expansion of distributed renewable energy utilization, the MS3T has been widely applied.Meanwhile, the stability of the MS3T in an electromagnetic timescale is a basic requirement to ensure the performance of the MS3T.Thus, this paper proposed a practical stability assessment based on the impedance method for the MS3T in an electromagnetic timescale.First of all, the control strategy of the MS3T was investigated, and the stability analysis of each of its subsystems was researched.According to the small-signal method, the transfer function of each subsystem was built, and the stability of each subsystem was distinguished by the eigenvalue.Furthermore, the modified impedance stability criterion in the electromagnetic timescale was proposed by the Gershgorin theorem and Kirchhoff laws.Eventually, the stability assessment of the overall MS3T was verified through the simulation and experiment.Compared with the existing literature, this paper researched the structure of the MS3T and its subsystem control strategy, and the stability analysis of each of the MS3T's subsystems was achieved.Furthermore, the overall practical stability assessment in the electromagnetic timescale, based on measured impedance stability analysis, for the MS3T was investigated.Moreover, this stability assessment only requires two current sensors and one external

Conclusions
With the high expansion of distributed renewable energy utilization, the MS3T has been widely applied.Meanwhile, the stability of the MS3T in an electromagnetic timescale is a basic requirement to ensure the performance of the MS3T.Thus, this paper proposed a practical stability assessment based on the impedance method for the MS3T in an electromagnetic timescale.First of all, the control strategy of the MS3T was investigated, and the stability analysis of each of its subsystems was researched.According to the small-signal method, the transfer function of each subsystem was built, and the stability of each subsystem was distinguished by the eigenvalue.Furthermore, the modified impedance stability criterion in the electromagnetic timescale was proposed by the Gershgorin theorem and Kirchhoff laws.Eventually, the stability assessment of the overall MS3T was verified through the simulation and experiment.Compared with the existing literature, this paper researched the structure of the MS3T and its subsystem control strategy, and the stability analysis of each of the MS3T's subsystems was achieved.Furthermore, the overall practical stability assessment in the electromagnetic timescale, based on measured impedance stability analysis, for the MS3T was investigated.Moreover, this stability assessment only requires two current sensors and one external high-bandwidth small-signal sinusoidal perturbation current source.

Figure 1 .
Figure 1.Extending the multi-timescale classification of a power electronic converter.

Figure 1 .
Figure 1.Extending the multi-timescale classification of a power electronic converter.

Figure 2 .
Figure 2. The multi-port single-phase solid-state transformer (MS3T) structure model with three stages.Figure 2. The multi-port single-phase solid-state transformer (MS3T) structure model with three stages.

Figure 2 .
Figure 2. The multi-port single-phase solid-state transformer (MS3T) structure model with three stages.Figure 2. The multi-port single-phase solid-state transformer (MS3T) structure model with three stages.

2 SFigure 3 .
Figure 3.The overall control strategy and topological structure of the MS3T.Figure 3. The overall control strategy and topological structure of the MS3T.

Figure 3 .
Figure 3.The overall control strategy and topological structure of the MS3T.Figure 3. The overall control strategy and topological structure of the MS3T.

Figure 4 .
Figure 4. Interconnection of two stable independent systems.

Figure 4 .
Figure 4. Interconnection of two stable independent systems.

Figure 5 .
Figure 5. Setup for the terminal characterization of the MS3T.

Figure 5 .
Figure 5. Setup for the terminal characterization of the MS3T.

Figure 6 .
Figure 6.The flow chart of the proposed stability assessment method.

Figure 7 .
Figure 7.The Bode diagram of the DC-AC back-end side converter.

Figure 8 .
Figure 8. Relationship between measured currents of p i ∧

Figure 7 .
Figure 7.The Bode diagram of the DC-AC back-end side converter.

Figure 8 .
Figure 8. Relationship between measured currents of p i ∧

Figure 10 .Figure 10 .Figure 11 .
Figure 10.The input voltage/current of the AC-DC front-end side converter in a stable case.

Figure 11 .Figure 12 .
Figure 11.The output voltage/current of the DC-AC back-end side converter in a stable case.Figure 11.The output voltage/current of the DC-AC back-end side converter in a stable case.Energies 2018, 11, x FOR PEER REVIEW 13 of 21

Figure 13 .Figure 12 .
Figure 13.Relationship between measured currents of p i ∧

Figure 13 .
Figure 13.Relationship between measured currents of p i ∧

Figure 14 .
Figure 14.Relationship between measured currents of p i ∧

Figure 13 .
Figure 13.Relationship between measured currents of

Figure 13 .
Figure 13.Relationship between measured currents of p i ∧

Figure 14 .
Figure 14.Relationship between measured currents of p i ∧

Figure 16 .Figure 15 .Figure 15 .
Figure 16.The output voltage/current of the DC-AC back-end side converter in the next steady-state operation point.

Figure 16 .Figure 16 .Figure 16 .Figure 17 .
Figure 16.The output voltage/current of the DC-AC back-end side converter in the next steady-state operation point.

Figure 17 .
Figure 17.The output voltage/current of the DC-DC back-end side converter in the next steady-state operation point.

Figure 19 .
Figure 19.The input voltage/current of the AC-DC front-end side converter in an unstable case.

Figure 20 .
Figure 20.The output voltage/current of the DAB in an unstable case.

Energies 2018 ,
11, x FOR PEER REVIEW 16 of 21 kHz.The model of the IGBTs in the AC-DC front-end side stage and back-end side stage is 2 75 4 -120 MBI U A , and the model of the IGBTs in the DAB stage is 400 -24 CM DU NFH .The system controller parameters are the same as those in Section 4, and the other parameters are shown in Table

Figure 24 .Figure 25 .
Figure 24.The input voltage of the AC-DC front-end side converter in a stable case of the experiment test system.

Figure 24 .
Figure 24.The input voltage of the AC-DC front-end side converter in a stable case of the experiment test system.

Figure 24 .Figure 25 .
Figure 24.The input voltage of the AC-DC front-end side converter in a stable case of the experiment test system.

Figure 25 .Figure 26 .
Figure 25.The output voltage of the DC-AC back-end side converter in a stable case of the experiment test system.Energies 2018, 11, x FOR PEER REVIEW 18 of 21 400V 300V 200V 100V 0V -100V -200V -300V -400V the output side of the AC-DC front-end side converter, which is same as that in the unstable case of Section 4. Therein, in the output side of the AC-DC front-end side converter, the closed-loop transfer functions of each subsystem are not affected, and each closed-loop transfer function in the unstable case is same as that in the stable case.

Figure 26 .
Figure 26.The output voltage of the DC-DC back-end side converter in a stable case of the experiment test system.

Figure 27 .Figure 28 .
Figure 27.Relationship between measured currents of p i ∧

Figure 27 .
Figure 27.Relationship between measured currents of

Figure 27 .Figure 28 .
Figure 27.Relationship between measured currents of p i ∧

Figure 28 .Figure 29 .
Figure 28.The input voltage of the AC-DC front-end side converter in an unstable case of the experiment test system.Energies 2018, 11, x FOR PEER REVIEW 19 of 21 800V 700V 600V 500V 400V 300V 200V 100V 0V

Figure 29 .
Figure 29.The output voltage of the DAB in an unstable case of the experiment test system.

Table 1 .
-end side converter subsystem closed-loop transfer function in Section 2 and the parameters of subsystem 4 in Table3, the characteristic equation can be obtained as follows: The parameters of the AC-DC front-end side converter. back

Table 3 .
The parameters of the AC-DC front-end side converter.

Table 4 .
The poles of the closed transfer functions.

Table 5 .
The experimental parameters of the MS3T.
Figure 21.Experimental hardware test setup.

Table 5 .
The experimental parameters of the MS3T.

Table 6 .
The poles of the experimental text system closed transfer functions.