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Dissertation Qinhao ShorterVersion 1

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Introduction
With the reduction in supply of conventional energy resources on this planet, it is critical and necessary to launch an investigation into renewable energy for a sustainable future. The United States Department of Energy (DOE) projects that the U.S. will receive 20 percent of the its energy supply from renewable energy by 2030. This is equivalently to around a 300 GW capacity. Recently in Europe, wind generators have been constructed offshore of Denmark. With the rise of renewable energy, the configuration of power systems will have to change accordingly. Therefore, more advanced control techniques are needed for addressing the issues occurred in a renewable or hybrid power system. There may also need to improvements to energy conversion efficiency so that more energy can be extracted from the renewable energy resources. This dissertation is going to present the power module of MPPT and also examine the time delay issue that occurs in the microgrid or medium voltage DC system.
Currently, the grid is in the process of upgrading. The goal of is to ensure that the next generation grid is more resilient and smarter in dealing with voltage and power fluctuations. With increasing amounts of renewable energy entering the current grid network, there is going to be an increasing distributed, constant power load in the power grid as more distributed power generators enter the field. With such high performance required in power regulation, it is critical to eliminate any stability issues caused by a constant power load.

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This work is going to present a comprehensive method of addressing the constant power load in the system.
Topics of the dissertation
MPPT improvement and realization:
Solar and wind energy are promising energy resources especially considering the amount of the resource, the cost, and other environmentally friendly considerations. Many signs such as government incentive policies, the Middle East’s construction of large solar power plants, and the large manufacture amount of solar panels points to the conclusion that this source of renewable energy is going to be the dominant source of future energy. Currently, the efficiency of converting solar energy to electrical energy is around 25-30%. To improve the conversion ratio significantly requires the development of a new photon material in order to more efficiently convert solar to electrical energy. It would be highly beneficial to develop a maximum power point tracking algorithm which will facilitate energy extraction. Basically, these are the two dominant reasons are the factors contributing the low energy conversion ratio. The former is the more important factor though the latter can also contribute to a higher conversion ratio.
MPPT is an optimization-based algorithm designed to find the maximum power point operating point on the load side. This algorithm's primarily applied to unimodal systems and its goal is finding the extreme point within a system. Solar panel possess many unimodal characteristics but bring other potential challenges, for example, when there are multiple extreme points such as when a shadow is cast over the panel. In this work, the shadow problem will not be discussed. The less time it takes to find the extreme point, the better the efficiency of the solar panel. Since the overall amount of solar energy that can be exploited will be huge, even a 1 percent improvement of energy conversion would bring considerable profit and energy savings. One of the major applications of MPPT is solar and wind where there is only one extreme point in the system. Despite decades of research and analysis, there is still no ideal algorithm for tracking the maximum power point due to two factors: cost and efficiency. This dissertation will propose using the ripple correlation control MPPT algorithm for locating the maximum power point and the use of another control algorithm to better improve the transient dynamics of the solar panel once a new duty cycle is found.

Adaptive Control Theory
Adaptive control theory is a branch of modern control domain. Its first appeared in the 1950s. In adaptive control, model reference adaptive control is an important method used to tune the controller adaptively. As the name suggests, the controller’s parameters are adaptively changed according to the situation of the system and by changing the controller’s parameters online, to control the system effectively. The adaptive control regime uses the Lyapunov function to design the controller such that the systematic error follows the stability in terms of Lyapunov stability and therefore the error will asymptotically approaches to zero[6, 7]. The beauty of using model reference adaptive control in a lot of engineering applications is because adaptive control doesn’t necessary require complete information from the system to develop a controller algorithm. This advantage is very critical in power electronic device development because the system’s characteristic are numerous or unobtainable due to the nonlinear characteristics of the system. By integrating this algorithm into MPPT development, a significant improvement in performance can be realized. The following chapters will show how adaptive control law can be coupled with the MPPT algorithm in solar panels to achieve a more satisfactory transient performance in the solar conversion system.

Stability issue in the medium voltage DC microgrid transmission system
The rise of the renewable variable generator is also accompanying the rising development of DC microgrid construction. The development of the hybrid power grid has many renewable generators entering the AC system. When many power electronic inverters and motor drives are tightly regulated they act much like constant power loads. Constant Power Load (CPL) has a negative impedance increase while the relationship between the change of voltage versus the change of current is negative, namely negative incremental impedance instability.
Briefly put (as it will be discussed in greater detail in the following chapter) centralized control is utilized in the power transmission system by means of an automatic meter instrument (AMI). AMIs collect voltage data, current magnitudes, and send the data back to the DMS for deployment plan computation. Each constant power load, actually the inverter/converter based renewable generators, is controlled by the duty cycle. Along with this closed-loop feedback of control configuration, there is a time delay within the system and a solution to get eliminate or reduce the potential influence or instability caused by the time delay should be found. Therefore using the adaptive control and modern control could reduce the potential influence as much as possible.

LITERATURE REVIEW
The upgraded Power grid is upgrading to be smarter, more robust, and more resilient. The reason is because there will be more dynamics within the power system due the increasing number of variable generators entering the grid. Meanwhile there is the rising practice of installing power electronic devices with different rating settings into the system for different applications. Dynamics from different elements are going to destabilize the power system and also create more potential problems. Although these topics are not going to be covered in this dissertation work, several obvious changes are going to take place: harmonics, voltage regulation, variable distributed generator’s impact on the system.
One of the appealing features of the next generation of the grid is that it will be designed to be much more resilient and smarter in order to handle the issues when there are fluctuations in the power system. As a result, more power with a decent power factor or power quality will be delivered with a steady voltage profile. Another feature of the grid is that it will consist of a much higher percentage of renewable power resources supplied locally. Since the availability of power sources can be localized, according to the geographic resources distribution, a local power supply can be provided without remotely connecting to the main power grid. Such a method of planning will benefit many regions which are either lacking a convenient supply of sustainable energy or are cut off from the main grid as the result of a nature disaster. There are two major components of the renewable power supply so far: solar energy, which can be extracted and converted to electricity by using solar panel; and wind, which can be extracted by wind power generators. Wind power generators use wind blades to extract energy through wind and produce the AC energy.
There are opportunities and challenges from these energy sources: 1). These energy resources are free and unlimited; 2). The involve power electronic devices and the come with harmonic components within the system; 3). They require many regulators to control the voltage output and therefore typically can be treated as a constant power load; 4). Since wind and solar is randomly distributed it therefore requires forecasting to avoid voltage spikes in the system. For example, the weather often changes significantly during one day alone. The solar energy source profile is very likely to align to the load consumption profile whenever there is a sunny day, it is also less likely to have lots of wind. Somehow the wind resource profile is aligned opposite of the energy consumption profile in the system. One of the significant advantages to using renewable energy is that the energy source is plentiful and great enough to extract. Therefore in order to most efficiently extract and convert solar energy into electrical energy the conversion ratio needs to be high. By using of different control and regulation methods, these energy generators act like constant power load. As mentioned earlier, the negative incremental impedance leads to the stability of the system. Therefore we need a solution to address this issue.

MPPT ALGORITHM
Solar panel’s voltage/current versus power output characteristic is unimodal without considering the shadow situation. With changing parameters, such as the working temperature and solar radiation level, there is always one unique extreme point in the P-V curve. The maximum power point tracking algorithm (MPPT) is supposed to enable the solar panel to always operate at the nominal voltage magnitude. Therefore the power output will be at its maximum. The MPPT algorithm has been a pretty well known algorithm in the industry for at least two decades. Its major contribution to the power system, specifically, is that it finds the maximum power point when the system carrying the nonlinearity characteristic. For example, solar panels are distributed variable generators because they are more sparsely installed in the field and also because the output of the solar panels is directly related to solar radiation.
In practice, there are two topologies of the solar panel configurations: one is a stand-alone PV plant while the other one is grid tied PV plant. The difference between these two types of configurations lies in whether or not it is necessary for the PV generator to send and receive energy to and from the power grid. In the stand alone PV panel, it usually goes through the MPPT algorithm first. This algorithm is used to calculate the duty cycle when there is a change in the external environment. Then, there is a power converter which is going to elevate or decrease the voltage output and make the voltage magnitude match the load requirement.
The efficiency improvement of the power conversion system can significantly increase the power energy converting ratio by only one percentage point. The usage of solar energy will be unlimited.
Several MPPT algorithms have been reported in the literature. The most common algorithm is perturb and observe method[6-9]. This control strategy requires external circuitry to repeatedly perturb the voltage array and subsequently measure the resulting change in the output power. P&O is a proven technology and has been intensively used in an industry application. The advantage of P&O is that it is cost effective and relatively easy to implement. However, the algorithm is inefficient in the steady state because it needs a ripple component of the current and voltage. This in turn, requires extra energy for the purpose of searching for the optimal operating point. This method is also very inefficient as searching requires large consumption of energy. The P&O algorithm also fails under rapidly changing environmental conditions. This is because it cannot discern the difference between changes in power due to environmental effects versus changes in power due to the inherent perturbation of the algorithm.
The Incremental Conductance Method uses the fact that the derivative of the array power with respect to the array voltage is ideally zero at the MPP. Specifically, when the sign is positive, it means that the current operating point lies to the left of the MPP. On the other hand when the sign is negative, it means that the current operating point lies to the right of the MPP. This method has been shown to perform well under rapidly changing environmental conditions, such as in the shade of solar radiation, nevertheless it comes at the expense of increased response time due to complex hardware and software requirements. These then become the major drawback of this algorithm.
In [8, 10], the author mentioned an important issue concerning the existing PV panels in the field. The major concern is that the partial shading due to clouds, trees, buildings or overheads, such that the bypassed diodes across modules may make the system difficult to track the solar curve using the MPPT scheme. Therefore the curve of P-V or I-V in the solar panel exhibits a twisted curve when the external environment behaves as such. Since the characteristics of the solar panel’s operating point varies, the global extreme point in the solar panel is going to be hard to find because there are numerous local maximum and minimum power points appearing. The author presents a critical concern about the partial shading as a common cause of reduction in the power yield of a PV source. The author then presents the algorithm to solve the partial shading issue in the solar panel. This issue is very critical and has been ignored by many researchers because the solar panel is often assumed to be well exposed to the sunshine during it is working.
In [11], a T-S Fuzzy Logic Controller is proposed to solve the MPPT algorithm for stand-alone solar power generator systems. The advantage of using the Fuzzy Logic Control algorithm is that such method provides robustness to the dynamics of the system. The underlying principle of fuzzy logic is that the nonlinear system can be represented as a series of IF-THEN fuzzy rules. These rules have local linearity characteristics to equivalently represent the nonlinearity characteristic of the solar panel as an entire system. Then the variation of the current and the voltage values will be sent to the table of fuzzy rules to determine the controlling signal. The control signal will modify the magnitude of the voltage or current in the panel such that the panel is functioning at the maximum power level. The advantage of this fuzzy logic control is that this algorithm facilitates the solar panel to have high conversion. However, the drawback of this algorithm is that those IF-THEN fuzzy rules are predefined, in other words, the characteristic performance of the solar panel should be studied in detail so that the internal knowledge of the solar panel can be acquired. The information of the solar panel will be used afterwards to define these control rules. The characteristic of the solar panel may be quite numerous according to the manufacturer and it is very complicated to generalize the MPPT algorithm based on various solar panel brands and models. However, if it is assumed that all the PV panels are equivalently and follow the same principles, then the result of this algorithm is satisfied.
Here I listed one table to show the comparison of different given MPPT algorithms’ pros and cons and it will give a clearer view of these algorithms:
In general, there are many new algorithms proposed in the literature [7, 12-14] and the goal is to improve the efficiency of the power conversion from the solar panel to the load side or, in some other cases, when the PV panel is connected to the grid. The golden rule in the engineering research follows the philosophy simplicity. Being smart and being good is always key in designing an electronic device. All of these algorithms, either has became the proven technology in the solar panel or are still under the research and development and they all have some flaws to some extent. Therefore it is very important to use different algorithms and methodologies to cope with these drawbacks.
DEVELOPMENT OF MICROGRID INFRASTRUCTURE
As mentioned earlier, the development of the microgrid originates from each solar panel module. With a sufficiently renewable energy resource, such as wind or solar th future is going to have more and more distributed variable energy generators within the system. This is called a microgrid and it can either provide energy as an independent system or it also can be connected to the AC system. Since the microgrid is much closer to the load site, whenever there is energy short from the microgrid, the AC system can also direct the flow of energy to the load side. This is similar to the grid-tied PV system which is able to receive and send energy to the grid. The microgrid shares the similar functionality.
The figure below shows the basic configuration of the hybrid power grid system in future. Within this system, when studying one section it can be simplified down to the system in figure 3. In this figure, the left hand side represents the DC energy sources. The DC energy source is tightly regulated to produce constant power and there are constant power loads within the system. The plot of the constant power load V-I curve shows that there is a negative incremental impedance in the load and therefore it is going to produce within the system. There are two solutions to solve such an issue: one is to use the global control technique. For example this can be a slide mode control, a neural network etc. The other method is using much more conventional control technique and is called the linearization solution.
We linearize the nonlinear system at certain operating point and study the dynamic characteristic at that point. The controller is designed and implemented for that certain operating point as well. In global sense of the system, the first method must be more versatile as it actually can address the problem with more freedom. However, in the real implementation of the control loop for the system, it encounters the time delay issue as in the control configuration of the power system. Within the physical construction of the power grid, there is a control configuration which is used to control the power grid’s energy deployment plan. The control grid uses the data extracted and measured in the power grid nodes, such as the automatic meter instrument (AMI) or phase measurement unit (PMU), to develop the control plan by using the duty cycle in the converter connected to the DC energy generators.
In real time experiments, there is going to be a time delay within the control closed loop. The central server must calculate the duty cycle according to the measured data and send them back to the local SCADA communication tower. These towers will then send the duty cycle to the local converters. Within such a loop, the time delay is going to ruin the control effect as within the closed-loop. The feedback causes instability within the system and the results can be shown in a simulation. Therefore it is critical to point out that the time delay is a local situation and therefore it would be much more useful to use the linearization method rather than the global control technique to address the problem. It is suggested that users utilize the advanced control algorithm to change the poles and zeros of the system in order achieve a better system response overall. This dissertation is going to introduce several methods for addressing this problem in the following chapters. In each method, there are some advantages and disadvantages for the user. There is always a certain trade-off between the cost of the controller and the performance of the control effectiveness.

APPLICATION OF ADPAPTIVE CONTROL IN POWER SYSTEM
Our goal for next generation power grid is to make the grid more reliable, more robust and smarter. These needs are correlated with the development of new features the power grid will have in the future. There are going to be many more distributed small generators among users and people are going to connect to the power grid with different types of renewable power generators to supply power to the users. In the course of development, there are lots of power grid component modules which have this nonlinearity characteristic. One obvious example is that the solar panels of the system’s P-V curve is not linear. People are conventionally using PI control to regulate the voltage and current in the system and then generate the pulse signals for converting DC to AC energy.
The underlying challenge of using such a method is that all these conventional PI controller developments are under one background while all the details/parameters of the power system are given and the controller is developed based on that. Though we are still able to generate the controllers’ parameter, it will no longer be accurate as the system has a lot of nonlinearity. All those classical control technique will be inaccurate as we are unable to obtain the accurate system’s information or following the information of the system. However, many modern control techniques can address these challenges effectively.
As mentioned in the introduction chapter, there are several adaptive control methods used. They are: gain scheduling, model reference adaptive control, and self-turning regulation. Model reference adaptive control is used in this dissertation work. It is going to improve the conversion ratio of the power conversion system. The beauty of model reference adaptive control is the controller can modify the characteristic of the plant by only tuning the controller’s parameters based on the errors between the state variables and the system output. The properties of the plant, under the effect of the controller, approaches to the properties of the reference model. Usually, the reference model is predefined. The controller’s parameters are determined based on the output error signal and the tuning parameter’s error. The underlying control principle guarantees that the converged parameters making the entire system converges to zero, in other word, the system’s output error and controller parameters’ error are asymptotically converging to zero and therefore it ‘learned’ the characteristic from the reference model [15, 16].
This dissertation develops a two level MPPT control algorithm that utilizes the ripple correlation control in the first level and model reference adaptive control in the second level. Its ultimate goal is to eliminate or mitigate the transient oscillation and approach the maximum power point efficiently. In the figure 2, there are two layers of control loops. The first control layer consists of a feed forward path that contains the RCC, [12, 17-20] the plant, and the feedback loop. The second loop consists of feed forward path that contains the signal d(t), which is the duty cycle derived based on the RCC algorithm in order to get the maximum power point and the controller. It also contains the feed forward loop, which is used to obtain the error signal in order to tune the properties of the plant to the reference model. As a result, the entire photovoltaic power conversion systems are improved to eliminate any potential transient oscillations in the system’s output voltage. Transient oscillations in the system’s output voltage can result after the duty cycle has been updated to account for rapidly changing environmental conditions. To prevent the plant from displaying such oscillations, a critically damped system is implemented as the reference model in REF _Ref435641994 h Error: Reference source not found in the model block. In the adaptive control, the error from the output response, which is the error signal of the system’s response and the reference model’s output response, and the signals from the difference of the controller’s parameters change. Properly tuning the controller parameters enables the output of the plant to match the output of the reference model, at which point the error converges to zero asymptotically and the maximum power is obtained.
SYSTEM modeling
SOLAR PANEL Conversion system development
SYSTEM DESCRIPTION
Solar characteristic
Learning the characteristics of the Sun will better allow us to understand the time constant of the Sun’s variation in solar radiation. As a result, the time constants of the controller can be determined. The reason is that if the time constant of the Sun is on the order of second, it means that the Sun’s irradiation is changing every several seconds. For a simple example, if the solar radiation is changing every 5 seconds, that means in 5 seconds the corresponding power tracking problem should be solved and the maximum power point should be tracked, otherwise, the operating point will move on while the controller is still stuck in the original phase. Based on the literature review and public data on solar information, the Sun’s time constant is on the order of one minute [7]. That is why the sampling period of either observatory or research lab is always five minute for the smallest sampling unit. Based on the given finding of the Sun’s time constant, this will also add a filter on the ripple component of the voltage and current in the solar panel. As a result, it is equivalent to adding a band-pass filter to the voltage and current of the solar panel’s outputs such that the DC component in the voltage and the current values are eliminated and also the high frequency component due to the change of the Sun.
In conclusion, the given ripple component of the voltage and current in the RCC algorithm is within a stable frequency window. The information in these ripple components are steady enough to determine the corresponding duty cycle, which will be fed into the conversion system in the solar panel.

PV characteristics
Both REF _Ref435641994 h Error: Reference source not found and REF _Ref435642295 h Error: Reference source not found present the voltage-current characteristics of photovoltaic systems under various levels of solar insulation and the solar panel’s working temperatures. The maximum power point occurs at certain points which are highlighted in these figures. As shown in the figures, the curves may have different shapes when the external temperature or solar radiation are numerous. To find the maximum power point, the nominal voltage or current value needs to be found, which is denoted as vmm or im.
REF _Ref435642289 h Error: Reference source not found, shows the underlying structure of the photovoltaic system. It is equivalent to the current source parallel connected to two diodes. One photovoltaic module is constructed by connecting the solar cells in parallels or in series. In this analysis, it is recommended connecting them in series as it is easier to generate the voltage and current magnitudes closed to the load. Each solar panel is designed to bypass a connection in case one panel is out of use in practice. Whenever there is certain dysfunction or anything happen to certain panel, the bypass circuit can be activated to disconnect the panel from the entire system. REF _Ref435642289 h Error: Reference source not found shows the single solar cell’s model. It consists of the current source, labeled as Iph, and the parallel connected with two diodes. In some models, it uses one single diode model for representing the solar cell. We can regulate the voltage or current of the solar panel using a DC-to-DC converter interfaced with an MPPT controller to deliver the maximum allowable power [8]. The underlying mathematical equations of representing solar panel are shown in the REF EqLin1 h Error: Reference source not found
In the aforementioned equation, Iph is the solar induced current and Is is the saturation current of the first diode. Is2is the saturation current of the second. Vt is the thermal voltage in the equation. N is the quality factor of the first diode and N2 is the quality factor of the second diode. V is the voltage across the solar cell electrical ports. All these pieces of information can be found in the solar cell in Matlab Simulink module.
Figure 5 shows a symbolic representation of the solar panel’s configuration, integrated with the MPPT controller and the converter. As we see in this figure from left to the right, the energy flow starts from the very left and moves to the right. On the left, the solar panel is exposed to sunshine which varies all the time. The changing environment forces the characteristics of the solar panels’ P-V curves to change all the time. The MPPT algorithm is utilized when it extracts the current and voltage component through the high pass filter. Therefore the high frequency components from the solar panel’s output is used to determine whether we should add or subtract some magnitude of voltage/current to the system and change the operating point of the system. The MPPT’s output is the duty cycle and it will control the converter’s voltage output.
Depending on the application, other power converter topologies may be used in place of the boost converter. In the boost converter system, shown in the following figure, the MPPT controller senses the voltage and current of the solar panel and yields the duty cycle to the switching transistor S. Therefore the duty cycle of the transistor is related to the array voltage through:
Where VPV and IPV are the solar panel’s voltage and current, respectively. And ROis the load resistance. Both the array voltage and current consist of the average term (DC term) as well as the ripple term, which is caused by the switching frequency in the converter. The goal then is to design a controller that continually calculates the optimal value of the duty cycle so that VPV tracks VM and therefore obtains the maximum power point.

Converter Dynamics
The equation ( REF EqLin2 h Error: Reference source not found) provides the foundation for conventional MPPT algorithm used to compute the converter’s duty cycle in the steady states. However, to optimize transient response, the MPPT control must consider the dynamics between the duty cycle and array voltage. Since transient oscillations are undesirable and can lead to inefficient operation and waste energy, the MPPT control needs to eliminate such transient oscillations in the array. Voltage output after the duty cycle is updated to account for changing environmental conditions. A detailed dynamic model of the boost converter can be found in [6]. To simplify the analysis of the system’s transient response, we consider a small equivalent circuit. A resistor R is used to model the solar array with a small signal array voltage and the small signal array current across its terminals.
We can now derive the transfer function from the control signal to the array voltage in small signal operation around an operating point. This transfer function is characterized by the dynamics of the system. It should be noted that in the dynamic model there is load in the boost converter as battery storage. The energy storage usually stores the energy and its application can range from microgrid to an individual, residential use of solar panel on the roof. Such circuit topology will change the value of vPV in equation REF EqLin1 h Error: Reference source not found and move the operating point in the steady-state response. This will have little effect on system’s frequency response for the range of frequencies near the natural frequency. As a result, resonances or under-dampened oscillations will not be observed in the output response of the system.
On the right hand side of the circuit is the potential load, which can be either a battery, a resistive load, or a utility power grid. Typically there is a DC power link on the right hand side between the power converter and the load side. The purpose of this DC link is to maintain the constant value of the voltage output. The dynamic response from the battery storage will be ignored and our focus will be on the relationship between the change of duty cycle to the array voltage in small signal operation.
Here is the small signal circuit of the system shown in the REF _Ref435645804 h Error: Reference source not found. Here we insert certain disturbance signals to the equation and then we are able to obtain the small signal circuit of the system and the internal relationship/dynamics is achieved from REF _Ref435645804 h Error: Reference source not found:
Here is the relationship in the frequency domain:
Where S is the Laplace operator and drepresents the small signal variation around the converter’s duty cycle D at the operating point.
On the left hand side of the system, the resistor, RI represents the linearized solar panel at a certain operating point of the solar panel. The middle part in the circuit is the power conversion system where the voltage magnitude is tuned to the nominal maximum power point voltage. On the right hand side, the current source is representing the load. The load can be either represented as power grid or certain resistive load. Reorganize equation (3) above, the transfer function between the changes of duty cycle to the change of voltage output is shown in below:
As shown in the equation above, it is known that fDis equal to the expression that:
V is the steady DC output voltage of the boost converter. This relationship indicates that the DC steady-state relationship between fDand Vo is unaffected by the transient switching action. And therefore, the equation of ( REF EqLin4 h Error: Reference source not found) can be reconstructed into the following one:
The minus sign in equation ( REF EqLin6 h Error: Reference source not found REF EqLin5 h Error: Reference source not found REF EqLin5 h Error: Reference source not found) is an indication that the reverse characteristic relationship between the change of duty cycle and the change of the voltage output from the solar array. In other words, if the duty cycle increases, for example from 0.6 to 0.62, then the voltage would decrease down to certain extent. This transfer function is derived from a linearized circuit of the nonlinear system around a signal operating point. The characteristic of the transfer function of the small signal circuit has this critical term in the denominator:
where wn is the natural frequency and the natural frequency is determined by the electronics elements:wn=1LOCI and the damping ratio’s formula expression is ζ=12RILOCI.
As shown later, the resistor, which is used to represent the equivalent solar panel in the small signal circuit is the only variable in the system and also determines the damping ratio. In REF _Ref435648513 h Error: Reference source not found., it shows that the convergence ratio of the system according to the input duty cycle change has different transient dynamics. We believe that the overall system’s response can be optimized as the convergence curve of the voltage lies on the black line and therefore it will not consume any additional energy for searching the optimal operating point. Another thing to recall from the previous section, by using the Ripple Correlation Control, the duty cycle and the maximum power point that is found is the true maximum power point.
Dynamic characteristic of the linearized solar panel are numerous and unknown to users when it is functioning on the field. In the classical control, the transient dynamics of system are decided by the damping ratioζ. When ζ is less than 1, the system has under-damped characteristic and have oscillation during the transient phase. When damping ratio ζ is exactly or around one, then the system will converge to the steady state at the most decent rate without any oscillation. When the damping ratio ζ is larger than 1, then it is called overdamped and it converges to steady state slower than the system being critically damped. Therefore the goal is to design the controller’s parameter such that the overall transfer function has the critical damping characteristic. The contribution of introducing the model reference adaptive control is because the resistor RI cannot be found in the working condition with various solar radiation. We need to tune the controller in the solar panel which as a result has the critical damped characteristic anyways.
As seen in the figure, it is the characteristic of current and voltage of the solar panel. The tangent line represents the equivalent resistor in the small signal modeling. The slope of the tangent lines is unknown in most cases and we will use ripple correlation control to find the maximum power point in the system.
The analytical relationship to determine the value of this resistor can be approximately written as:
The ripple correlation control is used to determine the optimal duty cycle with the value of resistor changing. The details of ripple correlation control (RCC) will be introduced and discussed in the following section.

Ripple Correlation Control
As mentioned earlier, there is a ripple correlation control in this hierarchical control structure. This is mainly used to find the optimal duty cycle under various environmental conditions. Ripple correlation control is similar to the algorithm of perturb and observe, however, the major difference between these two methods is that the perturb and observe method is basically using external signal of the voltage and current output to determine the current operating point. Ripple correlation control uses the existing signal from the converter to determine the location of operating point. In other words, the P&O algorithm always needs extra energy to produce a perturbance even if the current voltage value is already around the nominal voltage value corresponding to the maximum power point.
The underlying principle of ripple correlation control is still quite similar to the P&O algorithm, however, by taking advantage of the existing ripple component in the voltage and current from the power converter, this algorithm saves a lot of energy used track the maximum power point. This is the major contribution of the RCC algorithm. Here the principle of RCC is formalized in the following. Recalling the output of the voltage and current due to the PWM, it always has lots of high frequency components which can be automatically utilized to do the calculation. After equation set (9), there is another flow chart which is commonly used for implementing the RCC algorithm.
In this control law, k is a negative constant number in ( REF EqLin10 h Error: Reference source not found). This control law can be qualitatively described as follows: when the optimal maximum power point lies to the right of the current operating point, the product of the ripple component of power and voltage is larger than zero. When the optimal maximum power point lies to the left to the current operating point, the product of ripple component of power and voltage is small than zero. When the product is equal to zero, it indicates that the current system operating point is the nominal maximum power point. So a small voltage incremental step along the positive or negative direction is added to the voltage and drives it to approach to the nominal voltage. Proceeding to the aforementioned statement, voltage value will increase or decrease based on the concerning operating point.

Development of microgrid stability
Shown in the figure below is the expansion in coupling offshore renewable energy. This system includes variable frequency drive based platforms and the microgrid theme. The proposed system architecture is provided in REF _Ref435649837 h Error: Reference source not found utilizing a DC backbone. The directions that many manufacturers of power system equipment are exploring with offshore technologies to harness and transmit electric power provides further encourages that the proposed research efforts and system architecture are viable [6].
The DC renewable generators with power electronic converters and motor drives can be treated as constant power loads (CPL). For the case of a motor drive, the inverter drives the motor and tightly regulates the speed as approaching the constant speed as required. Assuming the relationship between the torque and speed in the motor drives is linear. Then the motor torque will remain constant in the constant power consumption by the motor. In the CPL, it is easy to observe that the instantaneous value of impedance is positive: as the ratio between the voltage and current is positive. However, the incremental impedance is always negative, if you observe the V-I curve of the constant power load it is easy to find that dV/dI<0. In the literature, the latter is referred as negative incremental impedance instability.
CPL induced instability or oscillations can be resolved by modifying the DC system’s hardware structure, by adding resistors, filters, or energy storage elements. However, approaches based on feedback control can offer more practical and efficient solutions. In addition to linear controllers featured by simple architectures and designs, many research teams have chosen nonlinear based control approaches to stabilize CPL scenarios to avoid limitations of linearization (operating closely to the system equilibrium point) and large-signal stability is guaranteed [9]. The primary nonlinear approaches include the use of Lyapunov-based design, hysteresis control, nonlinear passivity-based techniques, and boundary control. However, some of the disadvantages of these techniques include the use of proportional-derivative (PD) controllers which can be sensitive to noise, current and voltage transient overshoots, and are difficult to implement.
In order to transmit the power from the DC source side to the CPL load side, it uses the duty cycle of the power converter for tuning the amount of power. The figure below depicts the flow diagram of the system where the network node collects the voltage, current data is directed to the centralized server where it calculates the duty cycle, and fed the calculated ones to the local converter through telecommunication system. Such configuration is realized by using the Simulink interfacing with C language in the simulation environment. However, the results obtained from the simulation are unstable and the underlying reason is because of the time delay effect in the system.
Fundamentals of Bidirectional DC/DC converters and system development.
The dual active bridge DC/DC converter shown in REF _Ref435653226 h Error: Reference source not found was first proposed in [11]. The converter topology has grown in popularity as demand in bidirectional power flow capability which has increased in research pursuits such as battery charger applications for electric vehicles. Research teams have devised new control techniques for improving system efficiencies [12] and using state of the art semiconductor devices for high frequency operation of the topology [13]. Research efforts have been primarily centered upon low power applications. In this work, dual active bridge is used as the interface between two medium voltage DC buses. As shown in the figure below:

Constant power load model development
The following figure is the key section of the overall power system and will be studied thoroughly in this work. Starting from left to right, the medium voltage DC bus has been replaced with an ideal voltage source (assuming a well regulated DC bus) and a bidirectional DC/DC converter interfaces with the medium voltage DC bus and with the induction motor inverter. A 7.2 kV rated, single core, XLPE insulated, PVC sheathed, unarmored cable bridges the power converters and is modeled as a coupled pi circuit as shown [15].
As described in [16], a common constant power load is a DC/AC inverter that drives an electric motor and tightly regulates the speed of the machine needs to be constant. Assuming a linear relationship between torque and speed, for every speed there is one and only one torque. For constant speed, torque will be constant as well as power. Therefore, the motor/inverter combination presents a constant power load characteristic to the DC/DC converter. With the understanding that the average current of the bidirectional DC/DC converter can be described by (2) and assuming that the DC/AC inverter and motor can be approximately represented as a constant power load (whose time constant is much smaller than that of the DC/DC converter), REF _Ref435649837 h Error: Reference source not found can be simplified to REF _Ref435653843 h Error: Reference source not found.
Based on the previous analysis, the task at hand is to design a controller compensator. Power system engineers are accustomed to using some form of PID controller. The integral (I) control is often used to eliminate steady-state error, while derivative (D) is used to control and improve stability and system damping, This section will first show that the proportional derivative (PD) controller cannot achieve satisfactory steady-state performance and dynamic (e.g. stable) response at the same time. Then the principles of model reference control (MRC) are used to stabilize the system.
To appreciate the damping control effect of a PD controller, we first consider a simplified second-order model of the plant. Because the distance between the DC/DC converter and motor inverter is short (application being on an offshore platform), the cable can be approximated with a line inductance in series (Fig. 8) [17]. Noting the output capacitor of the dual active bridge DC/DC converter, the system model can be approximated as a second-order system with output impedance transfer function described by:
Where Z2,out represents the simplified second-order output impedance, the meaning of KDC is given in (4), and the other circuit parameters are defined in Fig. 4.
The first choice for a controller compensator would be the PD controller because this type of compensation provides a phase lead, which creates a stabilizing effect. Generally for a second-order plant, the derivative (D) part of the PD control can act on the damping component of the closed-loop system and thus stabilize the plant. Fig. 9 shows a diagram with PD control in the closed loop.
And a standard PD compensator of the form KP + KDs is used to control the plant in (6), and KP and KD are the proportional and derivative gains, respectively. If we express the denominator of (39) as that of a standard second-order system with natural frequency ωn and damping ratio ς, we can calculate:
A couple of observations can be made on the performance of PD control in (7). First, under the premise of stability, as KP increases the damping ratio, ς, increases and natural frequency decreases; as KD increases, the natural frequency, ωn, and damping ratio decrease. This behavior of PD control is different from the conventional effect of PD control when applied to a stable and minimum-phase plant where, for example, bigger KD and KP usually achieve larger damping ratio and higher natural frequency, respectively.
Time Delay in the Power Transmission System

As derivations and simulations obtained above predict, the model reference control implemented in the DC power transmission system stabilizes the instability issue caused by the constant power load (CPL). There are several layers of control in the power system: 1). Primary control; 2). Secondary control 3). Tertiary control 4). Generation Rescheduling. The constant power load stability issue is related to the scope of primary control and when the power transmission deployment is implemented, it involves secondary control and the time to complete the deployment is around 0.2 second. Therefore it cannot be treated as negligible in the system modeling when the power deployment needs to be performed.
We assume that the power deployment mechanism is centralized power control and in the transmission system the amount of power transmitted through the transmission line is calculated online according to the relationship between the supply and need. On the supply side, people need to collect the information about the source of DC power, such as the weather, the solar radiation, the wind speed, and so on. On the demand side, the load consumption profiles need to be collected. Previously, there are three major factors to change/tune the power transmission capability that have been found which are: the blade control, the duty cycle in the power converter, and power reference of grid connected converter. In this work, the focus is on the control of the duty cycle according to the transmitted power amount. Here is the internal mathematical representation of the converter system regulated through the power converter. As shown in equation (24), the variation of duty cycle is going to change the amount of power transmitted:
In this equation, P is the power amount, n is the number of submodules, V subscripted as H and L are the voltage magnitude on the two side of transmission line. LT is the leakage inductor in the converter. The value of this leakage inductor is related to the transmission capability. In the analysis model, high end voltage is 5000V and low end voltage is 1000V, respectively. The period is 1/3000 second and n is 5. Since the duty cycle used in this power converter is essentially for the bidirectional converter. This means that if the duty cycle lies within the range of 0 to 0.5, the direction of the power transmitted in one direction. And when the duty cycle lies in 0.5 to 1, it means the other direction. Meanwhile the DC energy acts as a power generator or motor depends on the supply-need characteristic. Because the maximum power amount can be transmitted, we set the duty cycle to be 0.5. And we get the value of the inductor as putting every other component’s value into the equation (44):
We got the value of LT is 5.208 mH. Now with the various amount of power transmission tasks, we use this calculated inductor value to configure the system. Then with the different value of amount of power transmitted, it needs to calculate the corresponding duty cycle for each power flow situation. Since there are two roots for the duty cycle each of them indicates the direction of the transmitted power flow. For the study we analyze in this paper, the transmitted 100 kW power through the transmission system, then we will find the duty cycle’s magnitude:
We will find that one solution of the duty cycle is to represent the power flows from the DC energy source to the power grid. As you can see in the whole calculation process, the central power unit is calculating the duty cycle online and then send the signal to local converter and give demand of certain value of the duty cycle. There is the inevitable time delay in the power transmission deployment because of the mechanism of the telecommunication and it has been shown that there are several time delays in the signal communication. It has also been shown that the time delay has an influence on the power system when the time delay is within the closed-loop feedback. In this case, we’ve found that there is a significant impact on the system’s output characteristic by having the time constant and therefore we need to find a way to solve the problem.
In the following section, I’d like to introduce several methods to addressing the issue within the microgrid.

System Model Development by Using State Space Equation:

In a previous paper, the controller is used to address the instability caused by the negative incremental impedance in constant power load. The constant power load is the tightly regulated renewable generator with power converter. The controller’s output is the duty cycle and, by controlling the duty cycle, it is able to control the converter. The constant power load bears the nonlinearity characteristic and combined with the time delay, we need to linearize the system at a certain operating point range where we can treat the overall system as linear. Previous literature and research have found a global solution to solve the negative incremental impedance in the system, however, such a global control algorithm is not effective in addressing the time delay issue as well.
The system needs to be linearized at certain operating point and assume that the transmission system holds the dynamics at such an operating point. The solution of the controller is feasible when the system is running around the operating point. This method is feasible in reality as the power transmission system may have several different transmission tasks. Designers can calculate several controller’s parameters according to different transmission tasks. The duty cycle is an essential factor to control the power transmission system and unfortunately it has the time delay too because of the telecommunication device characteristic. It is impossible to somehow avoid the time delay because of the telecommunication system. The telecommunications system sends the duty cycle signal and causes the instability.
We’d like to introduce the state space equation for representing the entire system which includes the power transmission system, constant power load and the time delay. The time delay’s frequency domain representation is in exponential component term and it is very hard to analyze the exponent term in the transfer function and therefore the Pade approximation is introduced for building the entire system and developing a controller. The Pade approximation represents the system with different order of transfer function and the generic representation is shown in the following. The higher the order of the approximation, the more internal state variables that are introduced and associated calculation needs to be conducted when defining the controller. The positive side is that we are able to achieve a more accurate approximation and the controller will be more effective.
Therefore, the entire system can be studied by putting them in cascade form. The transfer function of the dynamics of the duty cycle to the voltage output on the transmission receiver side can be represented as:
As shown in the above derivation, the reason for the instability when the time delay is introduced is due to the positive zeros in the time delay approximation equation. As we assume that the given system in the denominator is stable, C(s)G(s), in other word, the poles of this system are negative poles. However, the additional terms related to the time delay may be potentially shifted in the poles from negative to positive. As long as the magnitude of the time delay exceeds one third of the time constant of the system, it will never be insignificant in the system.
With the high order of the system, it would be more straightforward to use the state space equation and state feedback control to manipulate the poles of the integrated system: constant power load with the time delay. We need to integrate two systems to cascade to one system and develop controller from there
The duty cycle which feeds into the power converter has the following relationship and we transfer the time delay representation in exponential term into transfer function term. Then eventually it switches to the state space equation representation for the controller development purpose.
This is the approximation of the time delay in the first order system and meanwhile we will introduce one additional state variable, x, in the state space equation. Recall the system configuration: the duty cycle goes into the DC-DC converter and generates an equivalent current source. This current source goes through the transmission line and there is another dynamics in the transmission system.
There is an internal gain associated with the dynamics between the duty cycle and the current of the converter. The current source is the input of the transmission system, which will be discussed later. Recall the dynamics between the duty cycle versus the change of the voltage output:
These manipulations provide the new dynamics between the changes of the duty cycle versus the change of the converter current. The input of the transmission system is the converter current. The following state space equations state the dynamics:
Y represents the resistor of the constant power load at certain operating point. The following figure shows the cascade configuration of the analyzed system with full state space feedback control. The controller’s inputs are a measurement of current, and voltage in the system which are measurable. The output of the controller is the duty cycle which to the converter in physical meaning.
There are four variables as state variables in the above equations, specifically: x= [Vc1, Vc2, iL, x]T. The integrated system’s state space equation is shown in the following:
The stability of the system is based on matrix A in the state space equation (51) and this matrix A is 4 by 4 in this case. The eigenvalues of matrix A is dependent on these parameters’ values which are the electronic device components, internal values within the power converter and the duration of the time delay in practice. These values are shown in the REF _Ref435691423 h Error: Reference source not found:
Based on REF _Ref435691423 h Error: Reference source not found, we are able to obtain zeros and poles of the system and plot them in the s-plane, as shown in the figure. From REF _Ref435691493 h Error: Reference source not found, it is shown that there are two features of these poles distribution which lead to instability:
One is that there is one positive pole among four eigenvalues. The other feature is the other poles having a small real part and significantly large imaginary part. The result is the system has huge oscillation and it converges to the steady state very slowly.
Full state feedback control is utilized to provide the gain control to change the poles in the overall system by introducing the vector K. Control vector, K, has the following form and it combines matrix A with vector B to generate a new matrix A-BK.
The new matrix of the state space equation is going to have poles designed by the user. The choice of the poles are suggested to be within the region shown in the REF _Ref435691712 h Error: Reference source not found. It has a relatively large real part in the pole and a small imaginary part. The underlying reason of such design philosophy is to build a system that has a faster convergence rate with small oscillation.
Namely, these four poles are the new matrix and the task here is to find the vector K such that dynamics of the system changes. The choice of location of these poles is not unique as there is freedom according to the users’ preference. If the user wants to achieve rigid transient performance with fast convergence and a small magnitude of the transient oscillation, the locations of these poles can be pure negative real values. If the user only wants to stabilize the system with a small budget, it is suggested to use the location of the poles in figure while the magnitude of these poles can be chosen larger.
The fourth value in the controller vector K corresponds to the internal time delay state variable x. This is different from the other three state variables which can be directly measured in the transmission system in that the forth state variable should be calculated. Recall the state space equation, when the Pade approximation was introduced and we were able to find the expression of this state variable:
There is only one state variable related to the time delay state. With the higher approximation transfer function’s order, there is going to be way more state variables introduced and more complicated mathematical expression should be calculated to find these state variables. We need to find the casual expression of such internal state so therefore they can actually be measured in practice.

System Model Development by Using Model Matching Control:

By finding the state feedback gain in the state space equation, we can see from the simulation result that the system goes to stable because of the time delay. It is worthwhile to mention that there is no way to fully eliminate the time delay in the system. Another common method, particularly used in addressing the time delay issue, is called the Smith Predictor Compensator. From the result figure, we can see that the system goes to stable although it takes a while to converge to the steady state. Also there is a voltage sag within first around 0.6 second mark. This is because the positive zeros or non-minimal phase zeros’ effect. Such issues can be further improved by introducing the model matching technique for designing the compensators. We do not want the system’s response to have such characteristic and the reason is because of the positive zeros in the system. Therefore we will introduce the model matching for improvement.
Non-minimal phase zeros are contributed to the transient dynamics of the system. It doesn’t cause any stability issues as only the location of poles are contributing to the system’s stability. Using full state space feedback is only effective in improving the locations of poles while the model matching technique can improve the location of poles and minimal phase zeros. In other words, there is no effective way to solve the non-minimal phase zeros so far with the limitation associated with techniques. Nevertheless, we are able to break down the system into the part which is controllable and the other part which is non-controllable.
Since the time delay of the duty cycle is in cascade to equation (54) and (55), it leads to the following overall system:
Due to the controllable matrix calculated based on the analysis system is full rank, the system is therefore entirely controllable. We need to check to be sure the overall system is controllable. Otherwise, either the full state feedback controller or model matching controller will not be effectively working as the control scheme is ineffective to those uncontrollable state variables. In equation (56) we were able to obtain the poles and zeros. The transfer function is decomposed into the zero-pole form. Then we’d like to find the reference model, go(s), which is based on equation (38) and user design. We want go(s) to be close to the original system, equation (38), as much as possible for the cost of controller. The transfer function derived from the system, g(s) and reference system go(s) should be implementable. The definition of implementable can be equivalent to the following points:
We need to find the compensators for the system with two freedom configurations. The following is the procedure to find the compensators’ parameters.
simulation results and discussion
REF _Ref435692081 h Error: Reference source not found shows the voltage output of bidirectional DC/DC converter. First, it is important to note that by using Full State Feedback Control (FSFC) and the Model Matching Control (MMC), the voltage output throughout the transmission system (including the time delay) becomes stable. Such results demonstrate that these two methods used in this paper are both effective. Second, in MMC by introducing new zeros we improve the transient characteristic of the voltage output to certain extent as it clearly shows that the voltage drop at the step response has been significantly decreased. Also the time taken to approach to the steady state has been increased. Such behavior is more recommended in a power system. The entire system’s response is quite slow and therefore it avoids unnecessary energy wasted in the controller and the control is smoother.
In summary, this dissertation introduces one method to addressing the instability issue caused by a constant power load’s negative incremental impedance and another two methods in addressing the instability issue caused by the constant power load and the telecommunication time delay. Each method has a certain connection with the other and indeed based upon the simulation result, the voltage output has been improved significantly. Also as shown with the comparison of voltage outputs by using different methods, it provides advantages and disadvantages of using these different controllers, according to user's choice.
One more thing needs to be noticed is that in each scenario, we consider the time delay to be 0.2 second because it is the maximum time delay in practice. The same as in the operating point of the constant power load, in other words, we had a strong assumption that the overall system is working in certain condition and will not change the operating point too often while in operation. The following methods are all in the domain of linear system analysis and locally effective. In the next paper, we are going to present the global controller development by introducing the adaptive control because we only give the overall system a desired performance and the controller will adaptively tune the system and control the system’s output. The advantage of using adaptive control will eliminate the numerous offline calculations if the system is operating at different operation points. Instead, it is going to do the online calculation and control the system.

CONCLUSIONS
The results in Chapter 5 demonstrates the effectiveness of using adaptive control in improving the energy conversion efficiency in the MPPT and therefore the voltage output shows a smoother convergence curve in the transient dynamics of the system. The MPPT algorithm’s realization is based on the foundation that the two control schemes can be coupled together. The full state feedback control and model matching control techniques are trying to mitigate the effect by redistributing the locations of the poles and minimal phase zeros in the system and therefore the stability of the system can be assured. Moreover, the better transient response can be improved.
As we are following the progress of the technology revolution we can expect more and more distributed variable renewable generators or energy sources in the field. Many realistic issues cannot be discovered through pure analysis and are more likely to be encountered through the experiment or real life. We may not find the time delay issue by just doing the simulation as there are always some assumptions that require us to simplify the problem or which are even beyond our consideration.
This work comprehensively illustrates an essential contribution by developing control techniques which use modern control design techniques. These methods are more versatile and effective to deal with the situations in practice.
As we have discussed in previous sections, more work is required to pick the truly valuable fruit from the tree. For example, to simplify the control implementation into adaptive control is a rather complicated task to reach computationally. Depending on the size of the system, it could be completed with a more advanced computing unit, such as super-computer. We need to develop a more simplified control realization in order to implement these advanced control schemes.

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