Dynamic Modeling, Stability, and Control of Power Systems with Distributed Energy Resources

Dynamic Modeling, Stability, and Control of Power Systems with Distributed Energy Resources

I Introduction

Significant infrastructural changes are currently being implemented on power system networks around the world by maximizing the penetration of renewable energy, by installing new transmission lines, by adding flexible loads, by promoting independence in power production by disintegrating the grid into micro-grids, and so on [1]. The shift of energy supply from large central generating stations to smaller producers such as wind farms, solar photovoltaic (PV) farms, roof-top PVs, and energy storage systems, collectively known as distributed energy resources (DERs) or inverter-interfaced resources (IBRs), is accelerating at a very rapid pace. Hundreds of power electronic devices are being added, creating hundreds of new control points in the grid. This addition is complimented by an equal progress in sensing technology, whereby high-precision, high sampling-rate, GPS-synchronized dynamic measurements of voltages and currents are now available from sensors such as Phasor Measurement Units (PMUs) [2]. With all these transformational changes happening in the grid, operators are inclining to explore new control methods that go far beyond how the grid is controlled today.

In the current state-of-art, power system controllers, especially the ones that are responsible for transient stability and power oscillation damping, are all operated in a decentralized and uncoordinated fashion using local output feedback only. A survey of these controllers is given in Appendix -A. With rapid modernization of the grid, these local controllers, however, will not be tenable over the long-term. Instead, system-wide coordinated controllers will become essential. Such controllers where signals measured at one part of the network are communicated to other remote parts for feedback are called wide-area controllers [3].

Wide-area control alone, however, will not be enough either. It may improve the damping performance of the legacy system, but will not be able to keep up with the unpredictable rate at which DERs are being added to the grid. Every time a new DER is added it will be almost impossible for an operator to retune all the wide-area control gains to accommodate the change in dynamics. DERs have high variability and intermittency, and need to be operated in a plug-and-play fashion. Accordingly their controllers need to be local, decentralized, and modular in both design and implementation. In other words, neither should the design of one DER controller depend on that of another nor should either of these two controllers need to be updated when a third DER is added in the future. The overall control architecture for the future grid needs to be a combination of these decentralized plug-and-play DER controllers and distributed wide-area controllers.

The objective of this article is twofold. The first objective is to present a suite of new control methods for developing this combined control architecture. For brevity, the design will be limited to only one particular application - namely, adding damping to the oscillations in power flows after both small and large disturbances, also called power oscillation damping (POD) in short [4]. POD is one of the most critical real-time control problems in today’s power grid, and its importance is only going to increase with DER integration. The applicability of the design, however, go far beyond just POD to many other grid control problems such as frequency control, voltage control, and congestion relief. Efficacy of the methods for enhancing transient stability as a bonus application will be illustrated via simulations. While many papers on decentralized DER control and distributed wide-area control exist in the literature (surveys on these two control methods are given in Appendix -B and Appendix -C, respectively), very few have studied the simultaneous use of both. Moreover, most DER controllers reported so far lack the modularity and plug-and-play characteristic explained above. The control methods presented in this article address all of these challenges. The second objective is to present a comprehensive list of mathematical models of the various components of a power grid ranging from synchronous generators, their internal controllers, loads, wind and solar farms, batteries to the power electronic device interfaces and associated control mechanisms for each of these components. While these models may be individually well-cited in the literature, very few references so far have collected all of them together to understand the holistic dynamic behavior of an entire grid.

The rest of the article is organized as follows. Section II describes the dynamic model of a power system, integrated with different types of DERs. A general framework for modeling is provided first, followed by details of each individual component model. Section III demonstrates the impacts of DERs on power system dynamics through numerical simulations. Motivated by these simulation results, Section IV develops new decentralized DER control laws using the idea of retrofit control [5], as well as distributed wide-area controllers for damping of low frequency oscillations using sparse optimal control. The effectiveness of this combined control strategy is demonstrated on the IEEE 68-bus power system with wind and solar farms. The article concludes with a list of open research problems.

Ii Power System Models

Symbol Numerical value Description
base angular speed for a 60Hz power system. Unit of is rad/sec.
number of buses
-th bus voltage
, active and reactive power injected from the -th component
state of the -th component
control input of the -th component
model parameter depending on operating point
admittance matrix
, , , , index set of the buses connecting to generators, loads, wind farms, solar farms, energy storages, and that of non-unit buses. These sets are disjoint, and
TABLE I: Nomenclature for power system models. The tie-line parameters for constructing of the IEEE 68-bus power model, which is a benchmark model used in simulation, are available in [6]. All power system variables are considered to be in per unit unless otherwise stated.
Fig. 1: (Top) Illustrative example of power system model. (Bottom) Schematic diagram of the power system.
Fig. 2: Signal-flow diagram of power system model (1) and (2)

First, the dynamic models of the four core components of a power system are developed - namely, generation, transmission, load, and energy storage. The generating units are classified into conventional power plants and DERs such as wind generators and PV generators. Each model follows from first-principles of physics. Note that in reality a generation facility, whether that be conventional generation or wind/solar generation, and energy storage facilities contain many generating units and storage devices inside them. In the following, the terms generator, wind farm, solar farm, and energy storage system are used to refer to an aggregate of those individual units representing the overall facilities. Similarly, the term load is used to refer to an aggregate of all consumers inside the associated demand area. Each aggregated unit comes with its own individual bus, such as a generator bus or a load bus. The buses are connected to each other through a network of power transmission lines. The power system may also contain buses where no generator, wind/solar farm, load, or energy storage system is connected. These buses are called non-unit buses. The term component is used to refer to either a unit with its bus or the non-unit bus. An example of these connected components is shown in Fig. 1.

As will be shown in the following, a general form for the dynamic model of the -th component of a power system, whether that component be a generator, load, storage, wind farm, or solar farm, can be written as


for , where the nomenclature of this model is summarized in Table I. Details of the two functions and for each component of the grid will be described shortly. Throughout the article, complex variables will be written in bold fonts, (for example ). All symbols with superscript will denote setpoints.

The components are interconnected by a transmission network. Let denote the admittance matrix of the network (for details of the construction of this matrix, please see “Construction of Admittance Matrices”). The power balance across the transmission lines follows from Kirchoff’s laws as


where is the element-wise multiplication, is the element-wise complex conjugate operator, , , and are the stacked representations of , and for . From (2), is determined for a given and . The overall dynamics of a power system can be described by the combination of (1) and (2). A signal-flow diagram of this model is shown in Fig. 2.

The power system model (1)-(2) is operated around its equilibrium. This is determined as follows. The steady-state value of , , , and , and parameter in (1) must satisfy


and (2). The steady-state value of is assumed to be zero without loss of generality. A standard procedure of finding the steady-state values consists of two steps: first, find for satisfying (2), and then, given the triple find the pair satisfying (3) for each . The first step is called power flow calculation, details of which are given in “Brief Tutorial on Power Flow Calculation”. The second step, often called an initialization step, will be described later in this section.

In the following, the state-space models of generators, loads, energy storage systems, wind farms, solar farms, and non-unit buses, conforming to the structure in (1), are derived. For easier understanding, each subsection starts with a qualitative description of the respective component model followed by its state-space representation. Some parts may refer to equations that appear later in the text. To simplify the notation, the subscript is omitted unless otherwise stated.

Ii-a Generators

Fig. 3: Signal-flow diagram of the model of a generator with its bus, where the constant signals , , and are omitted.

A generator consists of a synchronous machine, an energy supply system (or a prime-mover), and an excitation system [4]. The excitation system induces currents in the excitation winding, and thereby magnetizes the rotor. The prime-mover generates mechanical power to rotate the rotor in this magnetic field. The synchronous machine converts the mechanical power to electrical power, which is transmitted to the rest of the grid. The dynamics of the prime-mover is usually ignored because of its slow time-scale.

Synchronous Machine

While various types of synchronous machine models are available in the literature (for example see [4]), in this article a well-known model called the one-axis model or flux-decay model is used. This model consists of the electro-mechanical swing dynamics (4) and the electro-magnetic voltage dynamics (5). For simplicity, the mechanical power in (4) is assumed to be constant.

Excitation System

Typically, the excitation system consists of an exciter, an Automatic Voltage Regulator (AVR) that regulates the generator voltage magnitude to its setpoint value, and a Power System Stabilizer (PSS) that ensures small-signal stability. The exciter with AVR is modeled as (6), where is a control input representing an additional voltage reference signal to the AVR. The PSS is taken as a typical speed-feedback type controller which consists of two stage lead-lag compensators and one highpass washout filter [7].

The state-space representation of the overall generator model can be written as follows (definitions of , , , are given in Table I while those of the other symbols are provided in Table II):

Symbol Numerical value Description
rotor angle relative to the frame rotating at . Unit of is (rad)
frequency deviation, that is, rotor angular velocity relative to
q-axis voltage behind transient reactance
field voltage
PSS state
exciter field voltage
PSS output
additional voltage reference to AVR
30 inertia constant (sec)
0.1 damping coefficient
0.1 d-axis transient open-circuit time constant (sec)
, 1.8 d- and q-axis synchronous reactance
0.3 d-axis transient reactance
8.0 steady-state mechanical power
0.05 time constant of exciter (sec)
20 AVR gain
150 PSS gain
1.0 setpoints for the field voltage
1.0 setpoints for the bus voltage magnitude
10 washout filter time constant (sec)
, 0.02, 0.07 lead-lag time constants of the first stage of PSS (sec)
, 0.02, 0.07 lead-lag time constants of the second stage of PSS (sec)
TABLE II: Nomenclature for generator model. The values of the synchronous machine parameters below are typical, and rated at generator capacity. The parameters for the IEEE 68-bus test system, which is a benchmark power system model used later in this article, are available in [6].

Synchronous machine:
Electro-mechanical swing dynamics:


Electro-magnetic dynamics:


Excitation System:
Exciter with AVR:


where represents the setpoint of .





Therefore, for , the model of the generator at the -th bus can be written in the form of (1) with


and and in (1) follow from (4)-(8). The signal-flow diagram of this model is shown in Fig. 3. For a given triple , the pair satisfying (3) is uniquely determined as and in (9) with where

Remark 1

Relationships between the one-axis model (4)-(5) and some other standard models of synchronous generators are shown in Appendix -F.

Ii-B Non-unit Buses

Non-unit buses are simply modeled by the Kirchoff’s power balance law, namely for ,


In reference to (1) this means , , and are empty vectors.

Ii-C Loads

Loads are commonly modeled by algebraic power balance equations, although extensive literature also exists for dynamic loads (for example, see [8, 9]). The well-known static load models are:


where is the complex conjugate operator, , , and are constant. Therefore, for , in reference to (1) a load at the -th bus can be represented by and being empty vectors, being either , , or , and the output equation being either (11), (12) or (13). For the simulations later in this article, constant impedance loads will be used. For a given triple , the load impedance will be calculated such that .

Ii-D Wind Farms

Fig. 4: Physical structure of model of wind farm with its bus
Fig. 5: Signal-flow diagram of model of wind farm with its bus, where constant signals , , , , are omitted.

A wind farm model typically consists of a wind turbine, a doubly-fed induction generator (DFIG), and a back-to-back (B2B) converter with associated controllers. A battery with DC/DC converter can be added to the B2B converter if needed. Fig. 4 shows the physical architecture of a wind farm with its bus while Fig. 5 shows a signal-flow diagram of the model. When the battery is not connected, the current in Fig. 4 is regarded as zero. The symbols for the wind farm model are listed in Table III.

Wind Turbine

The wind turbine, as shown in Fig. 4, converts aerodynamic power coming from the wind to mechanical power that is transmitted to the DFIG. The turbine is typically modeled as a one-inertia or two-inertia model (the latter is followed in this article) consisting of a low-speed shaft, a high-speed shaft, and a gearbox [10]. For simplicity, the aerodynamic power in the two-inertia dynamics (15) is assumed to be constant as wind speeds usually change slowly.


The DFIG shown in Fig. 4 converts the mechanical power from the turbine into electrical power. The DFIG consists of a three-phase rotor and a three-phase stator. The stator is connected to the wind bus to transmit the electrical power into the grid while the rotor is connected to the B2B converter with associated controllers that control the rotor winding voltage. The stator and rotor are coupled electro-magnetically, which is reflected in the dynamics of the stator and rotor currents expressed in a rotating d-q reference frame [11].

B2B Converter with its Controllers

The B2B converter is used for regulating the DFIG rotor voltages , as well as the reactive power flowing from the stator to the converter. The B2B converter consists of two three-phase voltage source converters, namely, the rotor-side converter (RSC) and the grid-side converter (GSC), linked via a common DC line [12]. Each of the converters is equipped with a controller. The explanation of the models of GSC, RSC, and their controllers is as follows.

Following [12], the GSC dynamics is expressed as the variation of the AC-side current in d-q reference frame.

The GSC controller consists of an inner-loop controller and an outer-loop controller [12]. The objective of the outer-loop controller is to generate a reference signal of the GSC currents for regulating both of the DC link voltage and the reactive power flowing into the GSC to their respective setpoints. The outer-loop controller is designed as a PI controller as (19). The inner-loop controller aims at regulating to the generated reference signals , by the control of the duty cycles , . Following [12], in this article the controller is designed such that the transfer function from (or ) to (or ) is a desired first-order system when the duty cycles are not saturated. The controller is implemented as (20).

The RSC model is described as


where and are the DFIG rotor currents, and are the duty cycles of the RSC, and is the DC link voltage. In this article, the RSC resistance and inductance are considered to be negligible, that is, . This assumption is always satisfied by incorporating the two into the DFIG rotor circuit. Thus, the RSC model used in this article is described as (21).

The RSC is equipped with an inner-loop controller and an outer-loop controller. The outer-loop controller generates reference signals for the DFIG rotor currents and for regulating the stator voltage magnitude and the high-speed shaft speed to their setpoints while the inner-loop controller aims at regulating the RSC currents [13]. This control action is actuated through the control of the duty cycles of the B2B converter.

The RSC and GSC are connected by a DC link equipped with a capacitor whose dynamics is derived from the power balance through the B2B converter [12].

Battery and DC/DC converter

A battery is used for charging or discharging of electricity whenever needed. The battery comes with a DC/DC converter that steps up/down the battery terminal voltage. Both devices are also sometimes used for suppressing the fluctuations in the output power by controlling the DC/DC converter. The model for each is described as follows.

The DC/DC converter is modeled by buck (step-down) and boost (step-up) models. These models are widely available in the literature [14]. When the converter dynamics are sufficiently fast, simpler models where the output voltage and current are explicit functions of the duty ratio can be derived. In this article this simple model is used.

The battery circuit is shown as the dark yellow part in Fig. 4 [12]. Its dynamics can be represented as the variation of the battery voltage and the output current .

Interconnection to Grid

The net active and reactive power injected by the wind farm to the grid are determined as the sum of the power leaving from the stator and that consumed by the B2B converter.

Symbol Numerical value Description
Wind turbine
, angular velocity of low-speed shaft and high-speed shaft
torsion angle (rad)
aerodynamic power input depending on wind speed
, inertia coefficients of the low-speed and high-speed shafts (sec)
, friction coefficients of the low-speed and high-speed shafts
torsional stiffness (1/rad)
damping coefficient of turbine
90 gear ratio
mechanical synchronous frequency (rad/sec)
, d- and q-axis rotor currents
, d- and q-axis stator currents
, d- and q-axis rotor voltages
electromechanical torque converted by DFIG
power flowing from DFIG to bus
number of wind generators inside the farm
magnetizing reactance
, stator and rotor leakage reactance
, stator and rotor resistance
GSC and its controller
, d- and q-axis currents flowing from AC side to DC side
, d- and q-axis duty cycles
power flowing from bus to GSC
steady-state value of
steady-state DC link voltage
, inner-loop controller state
, outer controller state
, reference signal of and
, additional control input signals
, inductance and resistance of GSC
, P gains of outer controller
, I gains of outer controller
GSC current dynamics’ time constant to be designed
RSC and its controller
, d- and q-axis duty cycles
, inner-loop controller state
, reference signal of and
, additional control input signals
, P gains of outer controller
, P gains of the inner-loop controller
, I gains of the inner-loop controller
DC link
DC-side voltage
DC link capacitance
conductance representing switching loss of the B2B converter
Buck-and-Boost DC/DC converter
current injecting from DC/DC converter into DC link
voltage at battery side
step down/up gain
battery voltage
current injected from the battery
battery capacity
battery conductance
, resistance and inductance of battery circuit
TABLE III: Nomenclature for the wind farm model. The signs of the currents are positive when flowing in the direction of the corresponding arrows in Fig. 4. The values of the models parameters of a 2MW 690V wind turbine, DFIG are shown in [10, 11]. In the following list, values rated at 100MW, which is the system capacity used in the simulations, are shown.

Wind turbine:


where is defined in (16).



where is defined in (15), and in (21), and




where and are defined in (20), and in (24).

Outer-Loop controller of GSC:


where and are defined in (18) and (24).

Inner-Loop controller of GSC:


where and are defined in (18), and in (19), in (24), and is a saturation function whose output is restricted within the range of .



where and are defined in (23), and in (24).

Outer-Loop controller of RSC:


where is defined in (15).

Inner-Loop controller of RSC:


where and are defined in (16), and in (22), and in (24).

DC link:


where and are defined in (18), and in (21), and in (16), and in (25). When the battery and DC/DC are not connected, .

Buck-and-Boost DC/DC Converter:


where and are defined in (24) and (26).



where is defined in (25).

Interconnection to grid


where and are defined in (16), and and in (18).

In reference to (1), the wind farm model with the battery and DC/DC converter can be summarized as:


and and in (1) follow from (15)-(27) for . The steady-state is determined as follows. Note that, given the total generated power , there exists a degree of freedom for determining and satisfying (27) in steady state. Thus, not only the triple but also the pair needs to be known. In this setting, the pair satisfying (3) is uniquely determined.

Ii-E Solar Farms

Fig. 6: Equivalent circuit of the model of a solar farm and its bus
Fig. 7: Signal-flow diagram of the model of a solar farm and its bus, where the constant signal , , and are omitted.
Fig. 8: (a) PV array structure (b) I-V characteristics of PV cell KC200GT
Symbol Numerical value Description
DC/AC converter and its controller
, d- and q-axis currents flowing from AC-side to DC-side
, d- and q-axis duty cycles
power injecting from solar bus
, inner-loop controller state
, outer-loop controller state
, reference signal of and
, additional control input signals on duty cycles
number of PV generators inside farm
, inductance and resistance of DC/AC converter
steady-state power injecting from solar bus
, PI gains of d-axis outer-loop controller
, PI gains of q-axis outer-loop controller
0.7 design parameter representing time constant of converter current dynamics
DC link
DC link voltage
DC link capacitance
conductance representing switching loss of DC/AC converter
Buck-and-Boost DC/DC converter
current flowing from DC/DC converter to DC link
voltage at PV array side
step down/up gain so that solar farm is operated at MPP
PV array
current flowing from the PV array to the DC link
series resistance inside PV array model
voltage of constant voltage source inside PV array
TABLE IV: Nomenclature for the solar farm model. The signs of the currents are positive when flowing in the direction of the corresponding arrows in Fig. 6. The values of the PV array parameters are the case where and with the KC200GT PV cell [15]. The value of below is a typical one because that will change depending on power system operation conditions, as shown in (36). The values of all parameters in per unit are rated at 100MW.

A solar farm model consists of a PV array, a buck-and-boost DC/DC converter, a DC/AC converter with a controller, and a DC link [15], as shown in Fig. 6. The signal-flow diagram for the system is shown in Fig. 7. The dynamics of the DC/AC converter, its controller, and DC link are similar to those in the wind farm model, given in (31)-(34). The models of the PV array and DC/DC converter are described as follows.

PV array

The PV array is a parallel interconnection of circuits, each of which contains series-connected PV cells, as shown in Fig. 8 (a). Each PV cell is assumed to be identical. Typically, a PV cell has nonlinear I-V characteristics, as shown by the blue line in Fig. 8 (b) [15]. Assuming that the PV cell is operated around the so-called maximum power point (MPP) where the cell output power is maximized, the I-V curve around this point can be approximated by a linear function as shown by the red line. In that case, the PV array can be modeled as a series connection of a constant voltage source with value and a resistance whose value is . This PV array model is described as (29).

DC/DC converter

A buck-and-boost DC/DC converter is used to ensure PV cell operation around MPP. This can be done by determining the converter step down/up gain such that the steady-state PV array output voltage and current coincide with the maximum point on the I-V curve. While the gain can be dynamically regulated by controllers, for simplicity, the gain is supposed to be constant. The DC/DC converter model is described as (30).

PV array:


where is defined in (30).

Buck-and-Boost DC/DC converter:


where and are defined in (34) and (29).

DC/AC converter:


where and are defined in (33), and in (34).

Outer-Loop controller of DC/AC converter:


where and are defined in (31).

Inner-Loop controller of DC/AC converter:


where and are defined in (31), and in (32), and in (34).

DC link:


where and are defined in (31), and in (30).

In reference to (1) the solar cell model can be summarized as:


and and in (1) follow from (29)-(34) for . The steady-state value of and can be found as follows. Suppose that is at the MPP. Given , , , and , the pair satisfying (3) are then uniquely determined as and in (35) where


Ii-F Energy Storage Systems

The energy storage system consists of a battery, a buck-and-boost DC/DC converter, a DC/AC converter, and a controller, as shown in Fig. 9. The basic functions of these four components are to charge/discharge electricity, to step down/up the battery terminal voltage, to rectify the three-phase current to a DC current, and to regulate the DC voltage in between the converters, respectively. When the energy storage system is connected to DC line, the DC/AC converter is not needed. The dynamics of the DC/AC converter, its controller, DC/DC converter, and DC link are similar to those described in equations (31), (32)-(33), (30), and (34), respectively.

Fig. 9: Physical structure of the model of the energy storage system with its bus

Iii Impact of DERs on Power System Dynamics

Fig. 10: IEEE 68-bus, 16-machine power system model with one DER. The downward arrows represent load extractions.
Fig. 11: Variation of 13 dominant eigenvalues of PV-integrated power system.
Fig. 12: Trajectories of the frequency deviations of all synchronous generators.
Fig. 13: Variation of 14 dominant eigenvalues of wind-integrated power system.
Fig. 14: (a) Singular value plot of the frequency response of the linearized solar farm model from to . (b) The same plot of the linearized wind farm model.
Fig. 15: Variation of ten dominant eigenvalues of wind-integrated power system by changing I gains of the RSC controller of the B2B converter

Given a DER-integrated power system model (1)-(2), the question is - how does the penetration of DERs and their controllers dictate the stability and dynamic performance of the grid? This section demonstrates these impacts using numerical simulation of the IEEE 68-bus power system model [16]. The network diagram is shown in Fig. 10. The DER bus, denoted as Bus 69, connects to Bus 22. The reactance between Bus 22 and 69 represents the transformer for stepping down the grid voltage to the DER voltage. Its value is taken as .

First, consider the DER to be a solar farm as in (29)-(34) with . The other bus indices , , are shown in Fig. 10. The model of this PV-integrated power system is the combination of (1)-(2) where for , , and are defined as (4)-(8), (11), (10), and (29)-(34). Note that is the number of PV generators inside the farm. A question here is - how does affect small-signal stability of the grid model? Fig. 11 shows the 13 dominant eigenvalues of the linearized power system model at a desired equilibrium. The eigenvalues around for start moving to the right as the value of is increased, and finally cross the imaginary axis when , resulting in an unstable system. Each PV generator is rated at 2 MW; therefore means that the net steady-state power output of the solar farm is MW, which is 3.85% of the total generated power of the system. This may look like a small percentage, but in terms of the stability limit the amount of solar penetration is quite close to critical. This pole shift happens due to the fact that the equilibrium changes with . When a fault (modeled as an impulse function causing the initial conditions of and in (31) to move from their equilibrium values) is induced oscillations in the transient response of the states can easily be seen. Fig. 12 shows the frequency deviation of all 16 synchronous generators for the cases where . The results indicate that as increases the PV-integrated power system, without any DER control, becomes oscillatory with poor damping.

Next, the solar farm at Bus 69 is replaced by a wind farm without a battery or a DC/DC converter. Fig. 13 shows the first 14 dominant eigenvalues of the linearized wind-integrated power system at a desired equilibrium. The eigenvalues around