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Development approach of a programmable and open software package for power system frequency response calculation
Protection and Control of Modern Power Systems volume 2, Article number: 18 (2017)
Abstract
Dynamic behaviour of frequency is crucial for power system operation and control. Several frequency response models have been proposed to reveal frequency dynamics from different aspects. A comprehensive software package incorporating major frequency response models is needed for analysis and control of power system frequency dynamics. In this paper, an approach for developing a programmable and open software package for frequency response studies is proposed. The framework of the package is extendable with reduced frequency response models. Essential models for frequency response study are included, e.g., generator, load, and underfrequency load shedding (UFLS). The provided application program interfaces (APIs) enable simulation with highlevel languages by calling dynamic link library and makes the package programmable. An advanced application module is developed for quantitative assessment of transient frequency deviation. APIs can also be used for model extension and secondary development. To demonstrate the usage of the package, several examples are illustrated to explain how to perform simulations with the package, and to perform advanced applications using scripting with the provided APIs.
Introduction
As one of the most important electrical parameters of power systems, frequency and its dynamic characteristic is crucial for power system operation and control [1]. Frequency dynamics interacts with many devices in power systems in two ways. First, the performance of many devices are affected by frequency, e.g., speed governor of synchronous generators, induction motors, power system stabilizer (PSS) with frequency as input, and reactance and susceptance of transmission lines or shunt components. Second, the dynamic behaviour of system frequency is also affected by those frequencydependent devices. Besides, dynamic characteristic of frequency is a key factor influencing power system protection. Generators are protected against abnormal frequency deviation with overspeeding and underspeeding protective relays. Underfrequency load shedding (UFLS) is an important resort to prevent power system collapse in the event of large generation deficit [2, 3]. The cooperation between the generating unit protective relays and UFLS is important for power system frequency stability [4]. Moreover, investigation into some blackouts indicates that large frequency deviation is a main factor pushing power systems to the edge [5]. In the process of power systems restoration, frequency deviation should be carefully restricted by gradually starting generators and loads to avoid large frequency deviation and subsequent system failure [6]. Furthermore, with large scale of renewable energy integrated into power grids, power fluctuation from renewables will lead to continuous power system frequency fluctuation and deviation [7]. Primary and secondary frequency regulation have important effect on preventing frequency deviation. The participation of renewable energy generation in frequency regulation also plays an important role in frequency security, regulation, and control. Therefore, it is necessary to study the dynamic behaviour of power system frequency for improving the operation performance of modern power systems.
There are two ways to obtain frequency response of power systems. One is the measurement from devices such as phasor measurement units (PMUs) [8] and digital fault recorders from high voltage levels, and frequency disturbance recorder (FDR) [9, 10] and PMU Lights [11] from low voltage levels. The measured frequency reveals the actual dynamic behaviour of power systems. However, without the knowledge of event type and location, system frequency behaviour can hardly be examined with the measured frequency. In most situations, power systems are operated in ambient mode with little frequency deviation. Dynamic behaviour of frequency with large frequency deviation can be rarely observed. Therefore, the frequency dynamic behaviour can hardly be studied using measurement data.
The other way to get frequency response is to perform numerical simulations with mathematical models of devices. It can easily create scenarios with large frequency deviation by setting up appropriate events. It is the most used technique for studying frequency dynamic characteristics and designing proper control strategies. Numerical simulation methods can be classified into two categories: detailed methods and reduced methods. Full timedomain simulation is the widely used detailed method and provides detailed models of the network and dynamic equipment with appropriate simplification. With coupled active powerfrequency dynamics and reactive powervoltage dynamics, frequency, voltage, and angle dynamics can be studied at one time. With detailed network, full timedomain simulation can easily reveal the spacetime distribution characteristics of frequency dynamics [12]. With area interconnection and integration of large numbers of devices, computational burden of full timedomain simulation is significantly increased. Thus, full timedomain simulation is not suitable for such circumstance as online security evaluation and emergency control. To supplement the study of frequency dynamics, reduced models such as average system frequency model (ASF) [13], single machine model and system frequency response model (SFR) [14] are adopted. Unlike full timedomain simulation, the reduced methods consider only the active powerfrequency dynamics to reduce computation burden.
During the long history of power systems research and operation, full timedomain simulation is extensively used in commercial power system analysis software such as PSS/E and DigSILENT Power Factory. The availability of those commercial software package greatly promotes the research and operation of modern power systems. The reduced models, however, have been developed by researchers for specific studies, and there is no comprehensive software package incorporating these reduced models. The aim of this paper is to propose a framework for developing a programmable and open software package for frequency dynamics study with reduced models.
The rest of the paper is organized as follows. A direct current power flow based frequency response model (DFR) is proposed and reduced frequency response models are reviewed in Section 2. A software package is proposed in Section 3 to incorporate major reduced frequency response models. To demonstrate the usage of the package, the IEEE 9bus model, NPCC 140bus model, and a 1000bus model from China are tested in Section 4. The features of the proposed package are summarized and conclusions are drawn in Section 5.
Reduced frequency response models
DFR model
Full timedomain simulations can simulate frequency response of power systems in detail. However, due to the coupled active powerfrequency dynamics and reactive powervoltage dynamics, both frequency and voltage need to be considered when studying power system dynamic behaviour. The influence of frequency and voltage can hardly be distinguished. Consequently, the DFR model is proposed in this paper to decouple frequency and voltage dynamics and to consider the influence of the network. In the DFR model, system network is simulated by direct current power flow so the redistribution of imbalanced power between different generators and the spacetime distribution characteristics of frequency can be considered. To focus on active powerfrequency dynamics, some assumptions are made as follows. (1) Excitation and regulation system is strong enough to hold the generator terminal voltage and thus, the dynamics of the excitation and regulation systems and the PSS can be eliminated for its negligible influence on active powerfrequency dynamics. (2) Generators swing equations are reserved while the influence of transient process of the internal windings on system frequency change can be neglected due to the constant generator terminal voltage. Since turbinegovernors have significant effect on power system frequency dynamics, details of the turbinegovernor are modelled in the DFR model.
In the DFR model, the dynamic behaviour of frequency is only influenced by active power change. With constant bus voltages, direct current power flow is introduced to simulate the network when calculating active power flow under initial operating condition [15]:
where P is the active power injection, θ is the voltage angle of all buses except the slack bus, and B is the network susceptance matrix.
The assumption of constant voltage leads to the simplification of load models. With constant terminal voltage, reactive power of loads can be ignored and the polynomial load model [16] can be reduced as a static active power load with frequency dependency:
where P _{ L } is the actual load, P _{0} is active power of the load under initial condition, K _{ pf } is the load regulation coefficient, and Δf is frequency deviation.
Other models can also be simplified with appropriate assumptions. For example, high voltage direct current links (HVDC) can be represented as loads for sending and receiving ends with or without frequency dependency.
With the simplifications of the generating units, network, loads and other equipment, the DFR model can be shown in Fig. 1. Quantities in Fig. 1 are listed as follows. ω _{ i }, δ _{ i }, P _{ mi }, and P _{ ei } are rotor speed, rotor angle, mechanical power, and electrical power of generating unit i. Δf _{ j } and P _{ j } are the bus frequency and active power of load j.
Similar to full timedomain simulation, the DFR model can be expressed in terms of differentialalgebraic equations (DAEs), and can be solved by stepbystep integration such as implicit trapezoid integration. Comparing with full timedomain simulation, the computational burden of the DFR model is greatly reduced and it achieves a better computational efficiency with acceptable accuracy. The DFR model can be used to analyse events of load change, generator tripping, etc. It can be also applied to fast frequency response calculation for active power disturbances and event screening.
With the introduction of direct current power flow, the DFR model is applicable to systems in which the network reactance is significantly greater than the resistance, e.g., high voltage transmission systems. The DFR model is primarily useful for cases where frequency stability is the main concern and angle stability and voltage stability can be maintained.
ASF model
In real systems, the frequency difference among buses is trivial if generators remain in synchronism during transient process [13]. Thus frequency at different buses can be treated as uniform and spacetime distribution of frequency can be neglected. By neglecting the network, the DFR model reduces to ASF model from which uniform frequency can be achieved. The general diagram of the ASF model is shown in Fig. 2(a) where turbinegovernors and loads are modelled explicitly. P _{ mΣ} and P _{ eΣ} are total mechanical power and total active power load of the system. Δω is the uniform frequency of the system which is generated from the equivalent swing equation. In addition, all loads can be aggregated into an equivalent load model to simplify the ASF model. It can be applied in applications such as spinning reserve allocation, load frequency control, etc [17, 18].
The ASF model can be modelled with DAEs and solved with stepbystep integration. With network neglected, the computational burden of the ASF model is much less than that of the DFR model.
Single machine model
Single machine model can be treated as a special case of the ASF model, as shown in Fig. 2(b). It is obtained by further aggregating all turbinegovernors and loads in the ASF model. The nonlinearity of the turbinegovernors, such as the valve limits and dead bands, is reserved. The structure of aggregated turbinegovernors is usually the same as normal turbinegovernors. For example, for standalone system with most of electricity generated by thermal generating units, steam turbinegovernor is preferred for the aggregated model. Stepbystep integration is also used to solve the nonlinear single machine model.
SFR model
Nonlinearity of the turbinegovernors is considered in the DFR model, ASF model and the single machine model. No analytical expression can be directly obtained and stepbystep integration is the most popular method to get discrete response. By neglecting the nonlinear blocks and small time constants, SFR model was proposed in [14] to derive an analytical expression of frequency dynamics for standalone systems, in which the generators are dominated by reheat steam turbines. The block diagram of the SFR model is shown in Fig. 3 where P _{ d }, P _{ m }, H, D, R, F _{ H }, T _{ R }, and K _{ m } are disturbance, mechanical power, inertia, damping, droop, fraction of total power generated by highpressure turbine, time constant of reheater, and mechanical power gain factor of the aggregated system. Using the analytical expression given in [14], the largest frequency deviation, its corresponding time, and steady frequency under a given active power disturbance can be calculated. Several research adopts SFR model for adjusting UFLS [19, 20].
Discussion
The frequency dynamic characteristics can be categorized in different ways. For applications depending on the overall dynamic characteristics of frequency, e.g., frequency regulation, uniform frequency is usually assumed and the frequency at different locations is treated as the same. In this case, network can be neglected, and ASF model, single machine model, and SFR model are appropriate. The spacetime distribution feature of frequency during event is of most interest for applications such as event location and oscillation detection where the difference between the generators at different locations should be taken into account. In this case, the influence of network should be retained to get the spacetime characteristics, and the DFR model is suitable for such applications.
For detailed study of power system dynamic characteristics, the coupling between active powerfrequency dynamics and reactive powervoltage dynamics should be included, resulting in the complex full timedomain simulation. However, for cases where frequency dynamic characteristic is of most concern and voltage dynamic is of little interest or voltage can be held at desired levels, the active powerfrequency dynamics can be decoupled from the reactive powervoltage dynamics for simplification. It makes activepower the only factor affecting frequency, and the reduced models introduced above are suitable to examine the key impact of active power on frequency.
Architecture of software package incorporating reduced frequency response models
Framework of the software package
The framework of the proposed software package is shown in Fig. 4 with the following modules:

(1)
Model library: Models with great impact on the frequency dynamics should be modelled in the package. The models implemented in the package are discussed in the next section in detail.

(2)
Data assembler: The data file in supported formats such as the IEEE and PSS/E data formats can be recognized and imported into the memory.

(3)
Dynamic equivalence: The function of this module is to supply equivalence calculation for model parameters.

(4)
Event library: From the information contained in event library, power system malfunctions or failures, such as generator tripping, load shedding/increasing and continuous loads variation can be set up.

(5)
Frequency response calculation: The function of this module is to implement frequency response calculation. The reduced models, DFR, ASF, single machine and SFR model, are implemented in the package using the stepbystep integration method or analytical solution to calculate frequency response.

(6)
Frequency security assessment: Security assessment module is implemented for advanced applications. Details of the frequency security assessment module can be found in Section 3.4.

(7)
Application program interfaces (API): The APIs are used to provide interface functions for advanced applications and secondary development.
Model library
The following models are implemented in the package with appropriate simplifications.

(a)
Conventional generator: With decoupling of active powerfrequency dynamics and reactive powervoltage dynamics, detailed generator models with damping windings are not required. In DFR model, generators are usually modelled as classical model with swing equation and constant internal voltage behind transient or subtransient reactance. For ASF and single machine model, no transient or subtransient reactance is modelled. Only aggregated swing equations are kept in the ASF and single machine model.

(b)
Turbinegovernor: Turbinegovernors provide the mechanical power for generators and are modelled in detail in the package. Typical turbinegovernor models are only concerned with active powerfrequency dynamics, with reactive powervoltage dynamics ignored. Thus, turbinegovernor models can be reserved without simplification.

(c)
Load: Considering only active power, two types of load models are implemented in the package: static load model considering frequency dependency and dynamic load model considering induction motors with active powerfrequency dynamic response [21].

(d)
HVDC: With more and more HVDC projects deployed, the control of HVDC should be modelled for frequency studies, which contains active power modulation, dead band of frequency deviation and active power order submodules.

(e)
UFLS: The control strategies are important for power system frequency stability. Underfrequency load shedding is an important frequency control measure and is modelled in the package in detail.

(f)
Wind generator: For wind farm participating into frequency regulation, electrical control is ignored whereas pitch control of the wind turbines is implemented for frequency regulation [22]. Meanwhile, in order to make the wind turbine dynamic process similar to conventional generating units, virtual inertia control can be considered.

(g)
Photovoltaic (PV) generation and battery energy storage: With largescale photovoltaic generation connected to power systems, its fluctuation determines the demands for battery allocation. The PV is usually modelled as negative loads and energy storage is modelled as loads with response to frequency changes.

(h)
Protective relay: Generators are equipped with protective relaying devices and their malfunctions can cause serious active power disturbance and large frequency deviation. So the under/over frequency protective relays are included in this package.

(i)
Boiler and automatic generation control (AGC): For the purpose of medium and long term simulation, boiler dynamics and AGC should be considered [23, 24].

(j)
Userdefined model: If other power system models are required, userdefined models can be added in the model library to extend the function of the package.
Dynamic equivalence module
There are many types of generators in power systems. For the single machine and SFR models, turbinegovernors and loads should be aggregated as single turbinegovernor and load. Swing equation equivalence [25] and turbinegovernor equivalence [26] are implemented in the dynamic equivalence module. For turbinegovernor equivalence, two submodules are developed for different purposes:

(a)
Equivalence of the same type of turbinegovernors. When aggregating several turbinegovernors of the same type, model structure is retained and the parameters can be generally summed up with weights to give appropriate responses.

(b)
Equivalence of different types of turbinegovernors. When aggregating several turbinegovernors of different types, appropriate model structure should be first selected and then parameters are optimized to match the overall dynamic characteristics.
There are many algorithms to deal with model parameter equivalence, such as particle swarm optimization method (PSO) [27], dynamic aggregation [28, 29], weighted summation method and least square method [30, 31]. Appropriate algorithms can be implemented for desired applications.
Frequency security assessment
The output results from the frequency response calculation module can be further analysed with the frequency security assessment module. This module provides transient frequency deviation security (TFDS) assessment and frequency security margin index based on twoelement table [32]. According to the requirements of power system operation and control, frequency deviation constraints in extent and duration are given as twoelement table [f _{ cr }, t _{ cr }] where f _{ cr } is the deviation extent and t _{ cr } is the corresponding maximum duration. For a given frequency trajectory and twoelement table [f _{ cr }, t _{ cr }], the frequency security index can be calculated by considering the cumulative effect of frequency deviation.
Frequency evaluation can be used for further frequency control decisionmaking. For example, for the setting of UFLS scheme, it can be used to check the feasibility of the scheme and provide guidance for how to optimize.
API module
The API module is divided into several submodules to realize different functional requirements. For example, data assembler APIs are used to read data, and equivalence APIs are used for dynamic parameter equivalence calculation. For security assessment and UFLS setting evaluation, APIs are also implemented for model extension and secondary development.
To improve the programmability, all APIs can be called via dynamic link library (DLL). The advantage of DLL over graphic user interface (GUI) is the freedom to prepare scripts for specific applications. Since almost all highlevel languages support loading DLL, the package can be further implemented in other software to extend their functionality of frequency dynamics study.
Case study
A programmable and open software package with the framework proposed in Section 3 is implemented in this paper with C++. Dynamic models of PSS/E are supported. The compiled DLL is called in Python modules named pydfr for DFR model, pyasf for ASF model, pysm for single machine model, pysfr for SFR model, pyeqv for model equivalence, and pyevl for transient frequency deviation security evaluation.
Demonstration of APIs
The following example shows how a simulation with Python codes calling APIs is performed.
The API read() imports model data (including power flow and dynamic models) into the program. DC power flow is solved with API solve(), and power flow results are saved with API save(). Prior to running dynamic simulation, dynamic data should be imported with API dyre(), and the channels to be exported are set with API chan(). Events information can be imported by API dist(). If detailed action of the power system is expected, a dynamic action file can be set up with API progress(). Before running the simulation with API run(), API strt() is called to initialize the dynamic models. The simulation results are outputted automatically during simulation.
Model equivalence
To perform simulations with single machine and SFR models, model equivalence should be performed for multimachine systems. In this paper, dynamic aggregation and weighted summation method are implemented for model equivalence calculation. The following codes show how an equivalence for the single machine model with typical steam turbinegovernor model IEEEG1 is obtained.
The data imported into the package by the pydfr module can be accessed by the pyeqv module. To get the proper equivalent model, the same type of turbinegovernors are first aggregated by API aggregate_st(). The aggregated models can be used for ASF model. If single machine model is to be called, the API aggregate_dt() needs to be called to reduce turbinegovernors of different types to a single model.
Comparison of different models
Frequency responses of the reduced models are examined in this section with the IEEE 9bus model and a 1000bus model from China. Dynamic frequency of the two cases are shown in Figs. 5 and 6, respectively.
For the IEEE 9bus model, PSS/E is adopted to perform the full timedomain simulation with complete models of generators, exciters, and turbine governors. Loads are modelled as 40% constant impedance load plus 60% constant power load with frequency dependency. In Fig. 5, the response captured by DFR deviates greater than that by PSS/E since the actual load in full timedomain simulation decrease slightly with the drop of voltage which is neglected in the DFR model. The DFR model generally reflects the overall frequency dynamics when voltage dynamics is neglected. For simplified models, the uniform frequency of the ASF model is almost the same as that of the DFR model, and both models can reflect frequency response process with good accuracy. The single machine model has similar tendency as the DFR and ASF models. However, the aggregation of the two steam turbines and one hydraulic turbine in the 9bus model produces some errors, especially for the overshoot part around 8s. The linear SFR model gives different response from the nonlinear models where spinning reserve is limited. The values of the TFDS index η are calculated by the security assessment module and are shown in Fig. 5 based on a twoelement table of [59.75Hz, 2.0s].
For the 1000bus model from China, similar conclusions can be drawn. Due to the lack of spinning reserve, the frequency response of the equivalent SFR model deviates greatly from the results of other models and is not illustrated in Fig. 6. The twoelement table for the 1000bus model is [49.9Hz, 0.55s].
To compare the computation efficiency between the reduced models, time consumption for the generator tipping event in Fig. 6 is compared with simulation time span of 50s. On a PC with CPU of 2.83GHz, the simulation time of the DFR model, ASF model, and single machine model are 3.259s, 0.531s, and <1ms, respectively. With the improvement of computational efficiency, the package is suitable for online frequency response analysis.
UFLS control
To demonstrate the applications of the package in UFLS control, a UFLS scheme is set up for the 1000bus model. After tripping 5% of total generation, the first step of UFLS is activated when frequency drops beyond 49.25Hz with time delay of 0.2s. Four percent of total load is shed to recover frequency. The dynamics of frequency and total load are shown in Fig. 7.
Load variation
With large scale integration of renewable generation, the fluctuation of renewables will lead to frequency variation. A load variation model is provided in this package to check the impact of variable loads and renewables (negative load) on frequency dynamics. On the NPCC 140bus model, perturbation of renewable power is applied with the load variation shown in Fig. 8 (a). The frequency response is shown in Fig. 8 (b).
Advanced applications
With the APIs provided, advanced applications can be conducted by calling the APIs with highlevel languages. The following codes show the application of searching critical load shedding amount during a sudden generation trip to prevent further activation of UFLS. The general process of simulation and evaluation is redefined in the new function sim_evl().
Figure 9 shows the iteration process to calculate the critical amount of load shedding. The twoelement table of the 1st step of UFLS ([49.25Hz, 0.2s]) is used to check the frequency security index. When the index is zero, the 1st step of UFLS is critically activated. With the initial guess (0% of iteration 1, and 5% of iteration 0), the searching progress converges after 6 iterations, and 1.71% loads should be shed 0.2 s after the event to prevent the activation of the 1st step of UFLS. The percentage shown in Fig. 9 are the load shedding amount of each iteration.
Conclusion
Development approach of a programmable and open software package for power system frequency response calculation is proposed in this paper where reduced frequency response models are incorporated. With the modularized framework, the package can be easily extended by adding new functions to the modules. APIs are provided in DLL to be called by other highlevel languages. The programmability makes the package suitable for advanced applications and secondary development. An implementation of the proposed framework is introduced with support of PSS/E data formats. Simulations show that the software package is easy to use and APIs can be reorganized to perform simulations for specific purpose.
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Acknowledgement
This work was supported by National Natural Science Foundation of China (No: 51477092).
Authors’ contributions
HZ proposed the idea and helped to prepare and revise the draft. CL coded the simulation modules of DFR and ASF in C++, and prepared the draft. YX coded the aggregation module, evaluation module in C++, and the Python modules. Most simulations were conducted by YX. HS helped to revise the draft and reorder the sequence of the draft. The model library part was revised based on HS’s suggestion. All authors read and approved the final manuscript.
About the authors
Yuzheng Xie (1991), M.E. candidate. Major in power system security, stability assessment and control.
Hengxu Zhang (1975), Ph.D. and professor. Major in power system security and stability assessment, power system monitoring and numerical simulation.
Changgang Li (1984), Ph.D. and associate research fellow. Major in power system dynamic and control, and widearea measurement and control.
Huadong Sun (1975), Ph.D and professor level senior engineer. Major in power system security assessment, stability control and numerical simulation.
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Xie, Y., Zhang, H., Li, C. et al. Development approach of a programmable and open software package for power system frequency response calculation. Prot Control Mod Power Syst 2, 18 (2017). https://doi.org/10.1186/s4160101700451
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Keywords
 Frequency response
 Frequency control
 Software engineering
 Underfrequency load shedding
 Transient frequency deviation security
 Power systems