Modelica-Driven Power System Modeling, Parameter Identification and Physically-Based Model Aggregation

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1 -Driven Power System Modeling, Parameter Identification and Physically-Based Model Aggregation August 2, 203 JOAN RUSSIÑOL MUSSONS Master s Degree Project Stockholm, Sweden 203 XR-EE-ES 203:0

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3 January 203 to August 203 -Driven Power System Modeling, Parameter Identification and Physically-Based Model Aggregation Master Thesis Joan Russiñol Mussons Electrical Power Systems Division School of Electrical Engineering, KTH Royal Institute of Technology, Sweden Supervisor and Examiner Prof. Dr.-Ing. Luigi Vanfretti KTH Stockholm Supervisor Tetiana Bogodorova KTH Stockholm Stockholm, August 2, 203

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5 Abstract Historically, dynamic modeling and simulation in power systems community is performed using di erent and mostly incompatible softwares. Even though there have been great e orts in the development of the Common Information Model (CIM) for power system applications, there are still many challenges for power system dynamic modeling without any ambiguity. Therefore, exists a need to develop unambiguous models which can be reused in di erent simulation environments and become universally compatible. Moreover, there is a need for a formal mathematical language that can allow for dynamic model exchange without ambiguity. Such language could compliment the CIM standard and e orts. In recent research, the language has been proposed as the definitive solution to these challenges due to the flexibility and mathematical formalism that the language provides for dynamic model representation. is a relatively new object-oriented language specially born in order to meet the model development requirements. The language is an equation based language with a clear focus on model reutilization. In this project the modeling will play a major role, in addition the reader will find some practical advice on developing power system component models in. In addition, this thesis discusses power system model parameter identification and aggregation. Using synthetic measurement data derived from simulations, the parameters of a model are identified. This thesis illustrates the application of for power system modeling and simulation, as well as the ability for unambiguous model exchange. It also shows how and FMI technologies can be combined and utilize for power system simulation. Finally, this thesis shows examples of application of the RaPId Toolbox for power system parameter identification and physical-based model aggregation. iii

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7 Acknowledgments First of all, I would like to thank Prof.Dr.-Ing. Luigi Vanfretti for giving me this opportunity as his Master Thesis student and to Juan Antonio Martínez Velasco for providing me his contact. In addition, I would like to thank very specially Tetiana Bodogorova for her guidance. To continue, I would like to thank all the people who helped me and supported me during the duration of this project. In particular, I would like to thank specially all the members of the KTH SmarTS Lab. In addition, I would like to mention and thank again Prof.Dr.-Ing. Luigi Vanfretti and Farhan Mahmood for providing me the Simulink models which were used to obtain the synthetic measurement data. The original model of the feeder used for load model aggregation in Chapter 5 was provided by Dr. Alan Collinson of SP Energy Networks and implemented in SimPowerSystems/Simulink by Farhan Maahmood and Dr. Luigi Vanfretti of KTH SmarTS Lab. During this project I consider that I made friends who were of great support and I would like to mention them as well (in alphabetical order): Naveed Ahmad, Achour Amazouz, Viktor Appelgreen, Maxime Baudette, Rokibul Hasan, Farhan Mahmood, Dr. Rafael Segundo and many more people who made my stay at KTH a pleasure. Finally, I would like to thank my family for their support and for giving me their valuable advice, as well as all the friends I made in Kista (The Kista People) and my Spanish friends. This research project was supported by the itesla Collaborative R&D project funded by the European Commission. Joan Russiñol Mussons August 2, 203 v

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9 Contents Notation List of Figures ix xi. Introduction.. Background Problem Definition Objectives Contributions Overview of the Report , Equation-Based Modeling and Simulation Introduction to Re-usability Modeling and Simulation Software Environments Basics Simulation Parameters Implementing Power System Models in 3.. Introduction Connectors Power System Modeling Guidelines List of Symbols Generators Second Order Third Order Component to Network Interface Generator Controls Turbine Governor Automatic Voltage Regulator Loads Constant P and Q Voltage Dependent Load ZIP Load Frequency Dependent Load Exponential Recovery Load Jimma s Load... 9 vii

10 Contents Mixed Load Order I Induction Machine Photovoltaic Models Photovoltaic Generators Power Factory Photovoltaic Model Software-to-Software Validation Generator Validation Solar Photovoltaics, TG and AVR Load Validation System Identification and Parameter Estimation Introduction General Optimization Algorithms Gradient Descent Method Particle Swarm Optimization Algorithm, PSO Genetic Algorithms The itesla RaPId Toolbox Power System Model Identification Introduction Parameter Estimation Case Generator Methodology for Estimating Generator Parameters Results Load Aggregation Load Aggregate Model Identification Results Photovoltaic Panel Parameter Estimation Case Discussion Models Model Exchange System Identification RaPId Toolbox Thesis Results Conclusion and Future Work Conclusions Future Work Bibliography 63 A. Appendix 65 A.. Software-to-Software Model Validation Results viii

11 Notation AC AVR DC Alternating Current Automatic Voltage Regulator Direct Current EMTP Electro-Magnetic Transient Programs FMI FMU GUI KTH OSS P PF PMU Functional Mock-up Interface Functional Mock-up Unit Graphic User Interface Kungliga Tekniska Högskolan Open Source Software Active Power Power Factory Phasor Measurement Unit Power System Analysis Toolbox PV Q SPS TG ZIP Photovoltaic Reactive Power SimPowerSystems Turbine Governor Constant Impedance, Current, Power ix

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13 List of Figures 2.. Example power system model in A model displayed in text, icon and diagram view (shown using Wolfram System Modeler) Simulation options in System Modeler and Dymola Connection between two electrical power system models from [] Electrical connector from [] inopen Icon of the third order generator implemented Example on how to describe parameters Turbine governor block diagram [2] Automatic voltage regulator block diagram [2] Comparison between the model in Power Factory and Model of the PV cell ZIP-Jimma load software-to-software validation and models used to validate the generator models and models used to validate the Solar Photovolatics, TG and AVR models and models used to validate the loads models A system with inputs, outputs and perturbations System identification illustration model to be converted to FMU Generating an FMU from Dymola General diagram of the main internal computation loop used in RaPId Main Graphical User Interface (GUI) of the itesla rapid Toolbox Schematic of the testing model structure Comparison between machines with di erent inertias Comparison between machines with di erent damping coe cients Comparison between machines with di erent direct axis time constant Simulink model of the testing system Simulink model with the FMU of the model Methodology used for the identification process Comparison between the reference (Simulink) and the identified () model responses for the torque perturbation Comparison between the reference (Simulink) and the identified () model responses for the field voltage perturbation xi

14 List of Figures 5.0. Validation of the identified generator model parameters for the torque perturbation Validation of the identified generator model parameters for the field voltage perturbation Scottish Power distribution grid corresponding to the modeled load [3] model in the identification process RaPId Simulink model for the load aggregation case Exponential recovery aggregate load model results Voltage dependent aggregate load model results Frequency dependent aggregate load model Results ZIP Results model in the identification process RaPId Simulink model for the PV panel parameter identification Panel parameter identification statistical analysis Panel parameter identification simulation results against reference data A.. Second order synchronous machine software-to-software validation A.2. Third order synchronous machine software-to-software validation A.3. Automatic voltage regulator software-to-software validation A.4. Turbine governor software-to-software validation A.5. Constant P,Q load software-to-software validation A.6. ZIP load software-to-software validation A.7. ZIP-Jimma load software-to-software validation A.8. Voltage dependent load software-to-software validation A.9. Exponential recovery load software-to-software validation A.0.Frequency dependent load software-to-software validation A..Mixed load software-to-software validation A.2.Induction machine software-to-software validation A.3.Constant P Q generator software-to-software validation A.4.Constant P V generator software-to-software validation A.5.KTH PV model software-to-software validation (Irradiation change) A.6.KTH PV model software-to-software validation (Temperature change) xii

15 Introduction.. Background Power systems are very complex and they expose a high variety of dynamic phenomena. For these reasons, di erent modeling and simulation approaches have been proposed as the core of power systems simulation softwares, in order to meet di erent simulations requirements [4]. Simulation models are increasingly being used in problem solving and to aid in decisionmaking [5]. Many tools in order to create power systems simulation models exist, however they are incompatible. The reasons why they are not compatible can be categorized as follows [4]: Data format incompatibility. Di erent dynamics models. Di erent modeling approaches and predefined models. To overcome these di culties, recent research suggests the use of formal mathematical modeling languages, in particular, [4]. is a standardized language which is object-oriented and equation-based, specially born in order to create models of physical components. o ers several advantages against other tools, for example: Straightforward and open modification of models. A common standard modeling language. Unambiguous model exchange between simulation environments. In addition, modeling is only one part of the problem, the other part is to match the models behavior with the actual behavior recorded from the field measurements. This is known as model validation and calibration. To perform model validation, system identification techniques must be applied. These techniques allow the model response to match the system measured response by using a combination of tools to correct the model by, for example, optimally changing model parameters.

16 . Introduction.2. Problem Definition Modeling and simulation are becoming more important since engineers need to analyze complex systems, often using components from di erent domains [6]. This is becoming increasingly important in power systems, for example in the context of PMU-based wide-area control. o ers the possibility of multi-domain modeling, however, a suitable power systems library is needed to perform cyber-physical power system studies involving di erent domains, as in the wide-area control example. Several attempts for power systems modeling using the language have been made in recent years [7]. Despite these attempts, those libraries became proprietary and closed for modifications. Other Open Source Software (OSS) libraries have not been updated to support the latest language standard, which would require users to make substantial and often di cult changes to utilize them. In essence, there was no up to date OSS library in order to simulate power systems. Additionally, having good mathematical models is not enough. To be useful, the models must match the actual behavior as measured in the field, by for instance tunning their parameters. This model to reality fitting can be done using di erent techniques. The problems this thesis focuses on are: Contribute to the development of Open Source power systems models to be included in a library being developed within the FP7 itesla project. Propose methods for parameter estimation and model aggregation. Provide examples of the developed methods applied to power systems..3. Objectives In order to deal with the problems listed above the following objectives were set: Perform a literature review. Learn the language and model implementation specifics. Develop power system component models in. Develop some power system component modeling guidelines. Perform parameter estimation and aggregation on power system models. Provide illustrative examples of the power of and its ability for model exchange. These ambitious objectives were set to illustrate the benefits of using for power systems and will be the foundation for future work in the area. 2

17 .4. Contributions.4. Contributions This thesis is part of the European Project itesla, concretely work-packages 3.3 and 3.4. All the information that can be found in this report is a direct contribution towards it. In addition, the models implemented in this thesis contribute as well to the Power- Systems Open-Source library, which is being developed in collaboration with Grupo AIA (Spain), RTE (France) and KTH SmarTS Lab, with Prof.Dr.-Ing. Luigi Vanfretti as work-package leader. This thesis was also used as an example of the use of a new toolbox for system identification called RaPId. The toolbox was developed in KTH SmarTS Lab by Achour Amazouz and Prof.Dr.-Ing. Luigi Vanfretti, funded through the European Commission within the FP7 itesla project..5. Overview of the Report The report is divided in two main parts:. for power systems. 2. System identification for parameter estimation and physically-based power system model aggregation. The first part is the basis for all the thesis. It includes Chapter 2 and Chapter 3. In Chapter 2 an introduction to can be found, whereas, Chapter 3 contains the practical application of for power system modeling. The second half consists in system identification for parameter estimation and physical-based power system model aggregation. In Chapter 4 there is an introduction to the system identification techniques used and in Chapter 5 the practical applications of these techniques to power systems can be found. Finally, at the end of the report discussions and conclusions are provided. 3

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19 2, Equation-Based Modeling and Simulation 2.. Introduction to is a non-proprietary standardized language, thus is not protected by trademark, patent or copyright. is an object-oriented and declarative language. It has been developed in order to conveniently model dynamic behavior of complex physical systems, for example, mechanical, hydraulic, electric and other domains. The fact what distinguishes from other languages is that it is object-oriented equation based language, that means, that each model is described itself by equations. The use of equations allows a greater flexibility since they do not prescribe a certain data flow [8]. The four most important features of are [9]: is based on equations instead of assignment statements. This permits an acausal modeling that gives better reuse of classes since equations do not specify a certain data flow direction. has multi-domain modeling capability, meaning that model components corresponding to physical objects from several di erent domains can be described and connected. is an object-oriented language with a general class concept that unifies classes and general sub-typing into a single language construct. This facilitates reuse of components and evolution of models. has a strong software component model, with constructs for creating and connecting components. Thus, the language is ideally suited as an architectural description language for complex physical systems, and to some extent for software systems Re-usability A model is reusable in because it is modeled independently of the environment where it will be used. This is achieved by including in the definition of each component its 5

20 2., Equation-Based Modeling and Simulation equations using only local variables and connectors. Thus, there is no connection between a component and the rest of the system, except from the connector Modeling and Simulation Software Environments As it was stated before, is a non-proprietary language so it can be used for free by anyone. For this reason, a wide range of softwares based on is now available (see Table 2.). Table 2..: simulation enviroments Proprietary simulation environments Cy Dymola MOSILAB SimulationX LMS Imagine.Lab AMESim MapleSim OPTIMICA Studio Mworks Wolfram System Modeler Open source simulation environments Jmodelica.org c Open SimForge All the softwares are based on language, the biggest di erence among them is the interface they use. This variety allows the engineer to choose the tool more suitable to him/her. In order to pursue this thesis, System Modeler and Dymola were used Basics is a transparent programming language which allows the user to create very complex models. Like in other programming languages, in there exist the type parameter and the type variable. One parameter is a type which contains a value that is set at the start of the simulation and will remain constant during it. On the other hand, a variable is a type which contains a value that changes withing time. In, defining a parameter or a variable is straightforward, as shown in Fig.2.. The propierty that most characterizes is the way it allows the user to introduce equations that are always accurate. For that purpose, the programmer must state when the equation section starts. On the contrary, if the programmer wants to execute a routine in a specific order, an algorithm should be used. See Fig.2.. These two concepts are summarized with the following simple definitions: Equation: mathematical statement, example : F = ma. 6

21 2.2. Basics Algorithm: sequence of calculations that leads to a result. In Fig.2. a basic model can be found. Fig.2. shows how parameters, variables and equations can be used to design a power system component model. Figure 2..: Example power system model in Further details on the language can be found in [9]. When using based tools there exist three types of model views:. The text view. Is where the code is written. All the graphical modifications will have an impact on it. 2. The icon view. Is where the graphical representation of the model will be designed. In essence, allows to define the model s graphical appearance. 3. The diagram view. Is where the user can drag and drop previously made models, the models will appear as an instance with the appearance created in the icon view. To exemplify these three views, please observe Fig.2.2 which shows the same model displayed in di erent views. 7

22 2., Equation-Based Modeling and Simulation Figure 2.2.: A model displayed in text, icon and diagram view (shown using Wolfram System Modeler) 2.3. Simulation Parameters In programming environments the simulation solvers are decoupled from the model. This makes easier the exchange of models. Di erent softwares provide di erent solvers and options. The basic characteristics that will change the outputs of the simulations are the solver used, the integration time step and the tolerance. The engineer can choose between several kinds of solvers. In general, there are two distinct types of solvers: Solvers with automatic integration step: the step is adaptive, i.e. it changes according to the dynamics of the system, this allows for fast simulation time. Solvers with constant integration step: during the simulation the time step remains 8

23 2.3. Simulation Parameters constant. When simulating with a constant time step, choosing a large step will make the simulation take less time, but at the same time it will be less accurate, on the other hand, if the integration step is too small the simulation will be large. In Fig.2.3 the simulation options of System Modeler and Dymola are shown. Interval length (Step) Solver Tolerance Figure 2.3.: Simulation options in System Modeler and Dymola 9

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25 3 Implementing Power System Models in 3.. Introduction In this chapter a detailed explanation of the power system models implemented in is given. The models presented are based from another software used as reference, namely. is a power systems toolbox for Matlab developed by Federico Milano. Further information can be found in the manual [2]. Implementing the same models from in is a challenging task. The way works is completely di erent from. The steps to achieve a successful implementation are the followings: Understand the conceptual background of the model. Read the documentation of the model. Identify the main equations that define the dynamic behavior of the model. Locate the initialization equations. Write the model in. Perform a software-to-software validation of the model against the model. Validation is a key issue when implementing models. The procedure followed in order to validate the models is detailed at the end of this chapter. The software-to-software validation will consist in a comparison of the behavior when using the same test system in and. The results from the experiments performed can be found in Appendix A. When comparing both softwares two observations were made: simulates faster than and gives more options for solving the system of equations (additional solvers). Implementing a model in requires less time than in. In addition, the training required to implement a model in is lower than for. Finally, outstanding challenges of this part of the project were:

26 3. Implementing Power System Models in The aim of the study is to match the actual dynamic behavior of the system, the models developed are phasor-time domain or positive sequence models that disregard certain system dynamics. No power flow tool was developed to be integrated along, all the power flow solutions were taken from Connectors In order to interface di erent component models, has a predefined class called connector. This class ensures that the models can be reusable and independent from each other. It can host two types of variables: Standard connector (without any prefix): This type of connector ensures that the magnitude and the sign of the variable are the same. Flow connector (flow prefix): This type of connector is based in the principle of sum equal to zero. Notice that in the flow connection has a sign convention. It is defined as positive when the flow enters the connector and it is defined as negative when the flow exits the connector. An illustration of such type of connector is shown in Fig.3.. It is very important to understand that the models have to be defined according to the connectors. During this thesis two kinds of connectors have been used: Connectors for real signals. Connectors for electrical signals. To connect real signals, for example, the torque of a generator, the standard connector Real defined in has been used. In this case, this connector already exists in the Standard Library. This kind of connector is based in the principle of equality. For equality it is understood that when two components are connected their value is the same (magnitude and sign). On the other hand, for the electrical connections an specific connector has been used, this connector was first develop in [],[0]. The basic structure of the connector is shown in Fig.3.2. It is important to notice that the electrical connector is defined in terms of real part and imaginary part of the voltage and current phasors. This means that despite the electrical magnitudes are phasors, the library does not use complex values, instead it uses real values defined in Cartesian coordinates. 2

27 3.3. Power System Modeling Guidelines Figure 3..: Connection between two electrical power system models from [] Figure 3.2.: Electrical connector from [] in Open 3.3. Power System Modeling Guidelines As it was stated before, creating models in is not very complex. But in order to create models that can be reused, some criteria must be followed: The final model should be one simple block. In the case that the model was created by parts and then they were just connected, the programmer must aggregate them in one block and icon. The parameters that the user can modify must be stated on the top-layer block of the model. In models made up from di erent blocks, the user must not have to look inside them in order to change any parameter. In addition, the developer should define the model parameters with a default value in the top-layer of the model. The programmer must use labels to identify the blocks and their corresponding connectors, as shown in Fig.3.3. All the parameters must be described using strings, so the user knows what they are, as shown in Fig.3.4. Use the attribute public/protected. Public: visible and changeable by anything. Protected: cannot be changed by a modifier. Import constants from.constants. There are three ways to initialize the model; through auxiliary parameters or through initial equations or algorithms. Each of them have their own advantages and disadvantages. For example, when using an auxiliary parameter it is possible to check the 3

28 3. Implementing Power System Models in correctness of the value, but the code becomes more di cult to understand. The opposite happens when using initial equations or algorithms. Finally, when using auxiliary parameters to initialize the model it is recommended to label them with a string message. Figure 3.3.: Icon of the third order generator implemented Figure 3.4.: Example on how to describe parameters 4

29 3.4. List of Symbols 3.4. List of Symbols Table 3. contains the symbols and their meaning used in the equations below. When a parameter has the superscript 0 means that it refers to the power flow solution or initial value, for example v 0 is the power flow initial voltage Generators Two di erent models of synchronous generators have been implemented according to the mathematical models used in [2]. This models are the second order generator and the third order generator. The models are based in the Park-Concordia transformation. Detailed information can be found in [] Second Order This model is the classic electro mechanical model. The di erential equations are: = b (w ) (3.) where ẇ =(p m p e D(w ))/M (3.2) b =2 f (3.3) p e =(v q + r a i q )i q +(v d + r a i d )i d (3.4) and finally for the direct and quadrature axis voltages 0=v q + r a i q e 0 q + x 0 d i d (3.5) 0=v d + r a i d x 0 d i q (3.6) The symbols are detailed in Table Third Order In this model the di erential equations are the following: = b (w ) (3.7) ẇ =(p m p e D(w ))/M (3.8) ė 0 q =( f s (e 0 q) (x d x 0 d )i d + v f )/Tdo 0 (3.9) where the electric power is given by (3.4) and 0=v q + r a i q e 0 q + x 0 d i d (3.0) 0=v d + r a i d x q i q (3.) The symbols are detailed in Table 3.. 5

30 3. Implementing Power System Models in Component to Network Interface The interface with the electrical connector is carried out using the Park s transformation: apple ir i i = apple sin( ) cos( ) cos( ) sin( ) apple id i q (3.2) apple apple vr sin( ) cos( ) = v i cos( ) sin( ) apple vd v q (3.3) where i r and i i correspond to the real part and the imaginary part of the connector current, and v r,v i correspond to the real and imaginary part of the connector voltage. The symbols are detailed in Table Generator Controls Two control blocks for the generators were implemented. An Automatic Voltage Regulator (AVR) and a Turbine Governor (TG). These models are based on the AVR TypeII and TG TypeIII models that can be found in [2]. The symbols are detailed in Table Turbine Governor The turbine governor implemented in can be seen in Fig.3.5. Figure 3.5.: Turbine governor block diagram [2] Fig.3.5 model has three input signals the actual rotor speed (w), the reference rotor speed (w ref ) and the power flow initial mechanical power (p 0 m). The output of the block is the mechanical power that must be applied to the generator (p m ) Automatic Voltage Regulator The automatic voltage regulator is based on the exciter type III from where it has been simplified. It can be seen in figure

31 3.7. Loads Figure 3.6.: Automatic voltage regulator block diagram [2] Fig.3.6 model has three input signals: the actual voltage (v), the reference voltage (v ref ) and the power flow initial field voltage (v 0 f ). The output of the block is the field voltage (v f ) that must be applied to the generator Loads The general equation which all the loads follow is given by S = P + jq = VI =(v r + jv i )(i r ji i ) (3.4) Discerning between real part and imaginary part results in P = v r i r + v i i i (3.5) Q = v i i r v r i i (3.6) In the following subsections di erent types of loads will be presented, the di erence between them will be the way P and Q are defined and calculated. The symbols are detailed in Table Constant P and Q Constant P and Q load is the simplest type of load. During simulation the value of the active power and the reactive power will remain the same Voltage Dependent Load P = constant (3.7) Q = constant (3.8) Voltage dependent loads are loads for which the value of the active power and the reactive power depends on the bus voltage in which they are connected to. P = P 0 (v/v 0 ) p (3.9) Q = Q 0 (v/v 0 ) q (3.20) 7

32 3. Implementing Power System Models in ZIP Load A ZIP load has three components: the constant impedance(z), the constant current(i) and the constant power(p) injections [2]. The model that has been implemented is the following: P = p 0 z(v/v 0 ) 2 + p 0 i (v/v 0 )+p 0 p (3.2) Q = q 0 z(v/v 0 ) 2 + q 0 i (v/v 0 )+q 0 p (3.22) Frequency Dependent Load Based on [2], by di erentiating and filtering the phase angle of the bus the frequency deviation, w, is approximated and the frequency dependency of the loads is introduced as follows: 0=x + ẋ = w/t F (3.23) 2 f n T F ( 0 ) w (3.24) P = P 0 (v/v 0 ) p ( + w) p (3.25) Q = Q 0 (v/v 0 ) q ( + w) q (3.26) Exponential Recovery Load The model implemented is exactly the same proposed in [2]. The values of P and Q recovers from a voltage change according to an exponential curve. For the active power: ẋ p = x p /T p + p s p t (3.27) p = x p /T p + p t (3.28) p s = P 0 (v/v 0 ) s (3.29) p t = P 0 (v/v 0 ) t (3.30) For the reactive power: ẋ q = x q /T q + q s q t (3.3) q = x q /T q + q t (3.32) q s = Q 0 (v/v 0 ) s (3.33) q t = Q 0 (v/v 0 ) t (3.34) 8

33 3.7. Loads Jimma s Load Jimma s Load is very similar to the ZIP load, which was previously explained, however the reactive power depends on the bus voltage time derivate Mixed Load P = p 0 z(v/v 0 ) 2 + p 0 i (v/v 0 )+p 0 p (3.35) Q = q 0 z(v/v 0 ) 2 + q 0 i (v/v 0 )+q 0 p + K v + dv dt (3.36) ẋ =( v/t f x)/t f (3.37) dv dt = x + v/t f (3.38) This load combines the frequency dependency and the bus voltage derivate dependency, as well as, the voltage dependency as follows: ẋ =( v/t fv x)/t fv (3.39) Finally: ẏ = P = K pf Q = K qf dv dt = x + v/t fv (3.40) ( 0 )+y (3.4) 2 f n T ft T ft w = y + 2 f n T ft ( 0 ) (3.42) apple w + P 0 (v/v 0 ) dv + T pv dt apple w + Q 0 (v/v 0 ) dv + T qv dt (3.43) (3.44) Order I Induction Machine The di erential equation is as follows: And the power injections are: = m ( ) 2H m r R v 2 / (r s + r R / ) 2 +(x s + x R ) 2 (3.45) p = r R v 2 / (r s + r R / ) 2 +(x s + x R ) 2 (3.46) q = v2 (x s + x R )v 2 x m (r s + r R / ) 2 +(x s + x R ) 2 (3.47) 9

34 3. Implementing Power System Models in 3.8. Photovoltaic Models First, very simple photovoltaic generators from [2] were implemented (constant PQ, constant PV), but theses models were not detailed enough. So it was decided to implement a more detailed PV model based on previous work [3] Photovoltaic Generators The implemented models were constant PQ and constant PV. The models were designed in in order to perform transient and voltage stability analysis. For this reason, only simplified controllers driving a controllable source were modeled Power Factory Photovoltaic Model This model was provided by Farhan Mahmood and the reference simulation model was developed in Power Factory. The complete documentation can be found in [3]. In comparison with the models from this model includes several additional features such as PV panel s dynamics, which implies the dependency on irradiation and temperature. The model is composed by several sub-blocks: PV panel (composed by several PV cells), DC link, Controller and a Static Generator. The model implemented in is shown and compared with the reference Power Factory model in Fig.3.7 PV Cell The PV cell was modeled as a current source in parallel with a diode and in series with a resistance as shown in Fig.3.8. DC Link The DC link is modeled as a capacitor, where the variation of the DC voltage multiplied by the capacitance provides the current. Controller The controller is aimed to keep the AC voltage and DC voltage at the same reference value. In addition, it calculates the reference direct axis current and the quadrature axis current. 20

35 3.8. Photovoltaic Models Figure 3.7.: Comparison between the model in Power Factory and Figure 3.8.: Model of the PV cell Static Generator This block is an inbuilt model in Power Factory, so the equations which define its behavior were not available for a transparent implementation. The function of this block is to provide a current injection into the network. Given the direct and quadrature axis current references 2

36 3. Implementing Power System Models in the model regulate the actual injected currents. In addition, it includes the interface with the grid (the electrical connector). To sum up, the main two functions of this block are: Interface with the grid. Control the currents. To control the current the following equations were assumed To interface with the grid the following equations were assumed I d =(I dref I d )/T d (3.48) I q =(I qref I q )/T q (3.49) I r =(I d cos( ) I q sin( )) (3.50) Initialization I i =(I d sin( )+I q cos( )) (3.5) In order to initialize the previous equations the following system of algebraic equations was to be solved Software-to-Software Validation V r = V 0 cos( 0 ) (3.52) V i = V 0 sin( 0 ) (3.53) A = V i cos( 0 ) V r sin( 0 ) (3.54) B = V r cos( 0 )+V i sin( 0 ) (3.55) I d =(P 0 B + Q 0 A)/(A 2 + B 2 ) (3.56) I q =(P 0 A Q 0 B)/(A 2 + B 2 ) (3.57) As the models implemented in language are based in other software tools, it has sense that the validation is done by comparing the output results of both softwares when the input system is the same. Software-to-software validation was carried out by designing di erent test scenarios. This procedure required to compare outputs of two systems given the same input (perturbations on the reference signals) and the same system structure (parameters). Ideally, a correct model will provide a one to one match between the corresponding output signals. The following numerical experiments were created. For the synchronous machines: 22

37 3.9. Software-to-Software Validation V ref pulse of at t = [2-2.]. V ref oscillation 0.00sin(0.2) at t=[0-5]. P m pulse of at t = [7-7.]. P m oscillation 0.00sin(0.2) at t = [5-0]. Fault at t=[0-0.] with fault impedance parameters R=20 and X=. Line opening at t = [4-4.]. For the induction machine, solar photovoltaics, TG and the AVR: Fault at t=[3-3.] with fault impedance parameters R=20 and X=. Line opening at t = [8-8.]. For the loads: V ref pulse of at t = [2-2.]. V ref oscillation 0.00sin(0.2) at t=[0-5]. P m pulse of at t = [7-7.]. P m oscillation 0.00sin(0.2) at t = [5-0]. For the detailed solar PV model: Irradiation change from 000 W/m 2 to 500 W/m 2 and back to 000 W/m 2 at t=[ ]. Temperature change from 25 C to 40 C and back to 25 C at t=[ ]. The validation results can be found in Appendix A. In order to provide an example of the results please see Fig.3.9. Discussion: The software-to-software validation proved the capability for power system modeling. Achieving more accurate results than, as it is shown in Fig.3.9. computational power allowed a reduction of the integration step without increasing the computation time, thus providing smoother and more accurate results. To design correct software-to-software experiments the nature of the validated component must be taken into account. 23

38 3. Implementing Power System Models in Generator - 3 v f P P m Q V Figure 3.9.: ZIP-Jimma load software-to-software validation Generator Validation For validating the generator the models used are shown in Fig.3.0. The model and the perturbation are shown together with the corresponding model in. The validation system in Fig.3.0 is composed by four buses. The fault is located in bus number four, which is connected by a power line to bus number two. This structure was chosen in order to avoid to locate the fault directly on the load side, thus being less severe Solar Photovoltaics, TG and AVR For validating the Solar Photovoltaics, Induction Machine, TG and AVR the models used are shown in Fig.3.. The validation system in Fig.3.0 is composed by four buses. The fault is located in bus number four, which is connected by a power line to bus number two. This structure was chosen in order to avoid to locate the fault directly on the load side, thus being less severe Load Validation The models used for validating the load models are shown in Fig.3.2. The validation system in Fig.3.2 is composed by three buses. This test system is simpler because for testing the load models the fault was not required. 24

39 3.9. Software-to-Software Validation GENERATOR VALIDATION SCHEME Model Perturbation Figure 3.0.: and models used to validate the generator models SOLAR & INDUCTION MACHINE VALIDATION SCHEME Model TG & AVR VALIDATION SCHEME Model Figure 3..: and models used to validate the Solar Photovolatics, TG and AVR models 25

40 3. Implementing Power System Models in LOAD VALIDATION SCHEME Model Perturbation Figure 3.2.: and models used to validate the loads models 26

41 3.9. Software-to-Software Validation Table 3..: List of Symbols Symbol Meaning Rotor angle index b Grid angular s speed w Rotor angular speed p m Mechanical power p e Electrical power D Damping coe cient M Mechanical starting time v d Direct axis voltage v q Quadrature axis voltage i d Direct axis current i q Quadrature axis current r a Armature resistance x d Direct axis reactance x 0 d Direct axis transient reactance x q Quadrature axis reactance e q Quadrature axis transient voltage Tdo 0 Transient time constant R Turbine governor drop P Active power Q Reactive power T Time constant K Gain v f Field voltage p Active power voltage exponent q Reactive power voltage exponent p Active power frequency coe cient q Reactive power frequency coe cient T p Active power time constant T q Reactive power time constant s Static active power exponent t Dynamic active power exponent s Static reactive power exponent t Dynamic reactive power exponent Bus voltage angle H m Inertia constant r s Stator resistance x s Stator reactance r R st order cage rotor resistance x R st order cage rotor reactance Slip Torque 27

42

43 4 System Identification and Parameter Estimation 4.. Introduction System identification links mathematical models to real life observations. Particularly, system identification consist in building models of real life dynamic systems from observations of their actual behavior [4]. A system is formed by a set of elements which interact in an integrated structure. There are several kinds of systems: mechanical, electrical, natural, etc. The interest on systems is to study their behavior and to determine how they will interact when subjected to external perturbations. Systems can be excited by input signals and perturbations as depicted in Fig.4.. The difference between input signals and perturbations its how they are originated. Inputs signals are signals which are under the user control, whereas, perturbations are signal which cannot be controlled and appear spontaneously. Finally, the output signals are the response of the system to the stimuli. A car is an example of a system, see Fig.4.2. If we consider a car as the system, the input signals would be the position of the throttle and the breaking pedal. As it can be noticed, those two signals are fully controlled by the driver. The perturbations might be the road inclination, the wind speed, the tra c, etc., and the output would be the speed of the vehicle. Figure 4..: A system with inputs, outputs and perturbations Once the system is observed and its boundaries are clearly defined it is possible to consider models. A model is the representation of a system. Human kind create models of everything they want to study. Mathematical models may consist of a set of equations which relate the output of the system with the inputs and the perturbations. 29

44 4. System Identification and Parameter Estimation A simple mathematical model can consist of a simple transfer function, but nowadays the systems that are being modeled are very complex and the process of creating models is becoming more challenging and important. One key factor for this massive development of models is the increase of computing capacity, which allows us to use these models to perform computations in a very e cient manner. System identification consist in matching the real behavior with the model. For example, estimating the parameters of the model. Figure 4.2.: System identification illustration 4.2. General Optimization Algorithms Notice that in order to perform the system identification and the parameter estimation procedure some general optimization algorithms were used. The function of these general optimization algorithm was to proportionate the best matching between the reality and the model, these algorithm have been used in order to search the parameters which minimize the error between the measured data and the simulated model. The methods explained below are very well-known techniques so only a basic description of them is given. Further information can be found in the cited literature [5], [6], [7], [8] Gradient Descent Method The gradient descent method is a first order optimization method. The objective is to find the local minimum of a a function based on its gradient. The principle is to take steps proportional to the negative of the gradient. This method is simple but slow because it needs to evaluate the gradient at each iteration. Moreover, this method easily gets trapped in local minimuma. To avoid the local minimuma, new methods have been developed. The method works for any number of dimensions. 30

45 4.3. The itesla RaPId Toolbox Particle Swarm Optimization Algorithm, PSO The PSO is a meta-heuristic algorithm for global random optimization [9]. The algorithm makes no prior assumptions and it can search in a large space of candidate solutions. For this reason, finding the optimal solution is not guaranteed. The algorithm works by having a population of candidate solutions (normally called particles). At the beginning these particles are (depending on the algorithm implementation) distributed randomly around the solution space. In each position the particle evaluates its own fitness. In every iteration the particle changes position according some criteria. The criteria to compute the moving speed (direction) depends on two values: The particle s best known position. The system s best position Genetic Algorithms Genetic algorithms are heuristic methods that reproduce the natural process of evolution, which gives rise to their name. The population of candidate solutions is called individuals, where each individual has di erent attributes called chromosomes. These chromosomes can be mutated in order to find the best solution. The algorithm starts with a random generated population (depending on the algorithm implementation), the population in each iteration is called a generation. In every iteration the fitness of each chromosome is evaluated. Then the ones that are more fit are mutated in the next iteration creating a new generation. To carry out the mutation there are two main operators: Mutation: Alters dramatically a gene. Crossover: Exchanges the value of two di erent genes The itesla RaPId Toolbox System identification is an iterative process requiring di erent computational methods. Withing the FP7 funded itesla project, members of SmarTS Lab developed a parameter estimation toolbox for Matlab. In this section a general overview of the toolbox is provided. More details will be available in the users manual. The optimization algorithms available in RaPId are: PSO: Particle Swarm Optimization. A native (manually coded) implementation of the method in RaPId. GAs: Genetic Algorithms. A native (manually coded) implementation of the method in RaPId. 3

46 4. System Identification and Parameter Estimation NAIVE: A naive method. A native (manually coded) implementation of a simple gradient-based method in RaPId. CG: Conjugate Gradient. This method uses the Matlab function Fminun for optimization. NM: Nelder-Mead method. optimization. This method uses the Matlab function Pminsearch for COMBI: Combination. This method allows the combination usage of two methods sequentially, after a determined number of iterations. The idea is to allow the user to utilize a PSO or GA method to find the region of the global minimum, and then a gradient-based method to iterate towards the final solution. PSOext: PSO external. This method uses the Global Optimization Toolbox functions from Matlab. GAext. GA external. This method uses the Global Optimization Toolbox functions from Matlab. Next, the methodology used in order to identify the system will be explained, from creating the model to parameter identification. The main steps in this process are the following:. Collect measurement data (which will be used in order to identify the system). 2. Create the power system model in. 3. Compile an FMU from the model. 4. Create a Simulink model using the FMU block from the FMI Toolbox for Matlab [20]. 5. Start the RaPId Toolbox. 6. Input the settings required by RaPId for the particular case (algorithm, parameters, variables, etc.). 7. Simulate and collect the results. 8. Evaluate the results. Collect measurement data As it was previously explained system identification requires reference data. The reference data will be compared with the simulations one in order to obtain the fitness and modify the parameters. In the ideal case, the data should be measured data, but for the purpose of this thesis the data used as reference was generated from a simulation of the model in Simulink, see Fig.5.5. Observe that, if the data used comes from measurements it will have noise. The noise in the signals makes the identification process harder and, in order to achieve good results, some signal processing should be performed. 32

47 4.3. The itesla RaPId Toolbox Create the power system model in The assumed model to be identified should be created in. This step can be performed according to di erent criteria, including the type of perturbations to be applied to the system and how the user wants to exploit the model using Simulink. Figure 4.3.: model to be converted to FMU In Fig.4.3 two independent systems are simulated at once. The main objective of doing that is to perform the identification process for two types of experiments simultaneously. This allows for a more comprehensive parameter estimation, taking into account all the experiments outputs together, withing the same identification problem. In Fig.4.3 two perturbations are performed to the systems. Using standard library a perturbation signal is injected to the generator. On the top model the perturbation consists in a pulse in the torque of the generator. On the bottom one the perturbation consist in a pulse in the field voltage of the generator. One alternative would be to perform the experiments sequentially by inserting the pertur- 33

48 4. System Identification and Parameter Estimation bations withing the same simulation, but by placing them in two dedicated experiments we avoid any possible interference in between them. Compile an FMU from the model. The itesla RaPId Toolbox uses the models as the mathematical models of real systems via the FMI Toolbox for Matlab. A Functional Mock-up Unit (FMU) is generated by using Dymola which is compliant with the FMI standard (Functional Mock-up interface). The FMI is a tool independent standard for the exchange of dynamic models and for Co-Simulation [2]. The FMI standard provides a common standardized interface for model exchange between di erent software tools for di erent applications [22], realizing this interface through an FMU. When generating a FMU file two types of FMU can be chosen: FMU for Model Exchange: generates lime C-Code or object code containing the dynamic system model in the form of an input/output block. FMU for Co-Simulation: is similar to the FMU for Model Exchange, with the di erence that it includes the solvers together with the models. In order to be able to simulate the models using the solvers provided in Matlab Simulink, the FMU for model exchange was chosen. To generate FMU files the FMI functionalities in Dymola were used. Other programming and simulation environments started to provide this functionality which will be a standard feature in the future [23]. This option in Dymola can be found in the simulation menu as shown in Fig.4.4. Figure 4.4.: Generating an FMU from Dymola 34

49 4.3. The itesla RaPId Toolbox Create a Simulink model using the FMI Toolbox for Matlab The Simulink model needs to utilize an FMU Block for model exchange from the FMI Toolbox for Matlab. The FMU needs to be loaded and configured in this block. After simulation, it redirects the results from the simulation to the toolbox, serving as a link in between Simulink and RaPId. An example of this Simulink model can be found in Fig.5.6. More information can be found in the next chapter. itesla RaPId Toolbox The RaPId toolbox plays a major role as a tool for this thesis to perform parameter identification. It simulates the model contained in the FMU, calculates the fitness of the results against the reference data and, according an optimization algorithm, it modifies the parameters of the FMU model. A simplified diagram is shown in Fig.4.5. More extensive explanation can be found in the documentation of the toolbox, developed withing the FP7 itesla project by KTH SmarTS Lab. Measured Data FMU model Signal Processing Simulate No Fitness criteria satisfied? Algorithm Change Parameters Yes End Figure 4.5.: General diagram of the main internal computation loop used in RaPId In Fig.4.6 the main user interface of RaPId is presented. On the left hand side, the di erent settings and options can be provided and, on the right hand side, the algorithm and results can be instantiated. On the bottom the user must provide output (mandatory) and input (optional) measured data. 35

50 4. System Identification and Parameter Estimation Algorithm Selection Options and Settings Results and Plots Simulink Provide the output measurement data Provide the measurement data Figure 4.6.: Main Graphical User Interface (GUI) of the itesla rapid Toolbox The RaPId Toolbox can be as well used thought a textual interface. 36

51 5 Power System Model Identification 5.. Introduction In this chapter the procedure for power system model parameter identification and physicallybased model aggregation is presented. In addition, the methodology for validating the power system models and the parameters identified will be discussed. The reference measurement data, which will be used in order to identify the systems, was obtained from the Simulink models developed using SimPowerSystems Matlab library. These models have a higher bandwidth and more detailed representation typically used in Electro- Magnetic Transient Programs (EMTP). Thus, the SimPowerSystems Simulink models are considered as the reference. The first step was to define a test system, in this thesis a single machine system has been developed, a one-line diagram is shown in Fig.5.. Figure 5..: Schematic of the testing model structure In the test system, Fig.5., the idea is that the A side remains unchanged for di erent cases. On the contrary, the B side is where the specific component model to be identified is located, thus it will be modified depending on the identification performed. The second step consisted in the implementation of the reference model in Simulink in order to obtain the synthetic reference data. It is important to mention that the library used in Matlab/Simulink for this purpose is SimPowerSystems (SPS). 37

52 5. Power System Model Identification In this thesis three system identification and parameter estimation cases are presented:. Generator parameter estimation: All the component parameters in the test system are known with exception of the generators model parameters. Once the parameters of the generator are identified, they will be used for the load aggregation. 2. Load aggregation: In this case, an unknown load will be placed in the system. The identification will consist in matching the behavior of the load to di erent load models. 3. PV panel parameter estimation. This case is used to illustrate how specific unknown or uncertain parameters of a component model can be identified given su cient knowledge of other model parameters Parameter Estimation Case Generator Generator models in SPS and are not identical. The model has a simplified representation of the generator s dynamics (a 3 rd order model) while the SPS modul uses a very detailed EMTP-type representation. The goal here is to identify the parameters of a simplified dynamic model of a generator that can match the electro-mechanical dynamics of the reference model. To accomplish this goal a known load has been located in the component model side (B) and two perturbations are applied to the system. These perturbation are in fact the experiments designed for this identification case. The first perturbation is a pulse of duration 0.5 seconds of % the nominal torque in the shaft of the generator. The second perturbation is a pulse of duration 0.5 seconds of % the nominal field voltage of the machine. Using the results from these two perturbations, the parameters of the machine are estimated. The most relevant parameters, which rule the electro-mechanical dynamic behavior of the system, depend on the perturbation. Thus, the design of these experiments is relevant for the identification procedure. Perturbations in the torque will primarily excite mechanical dynamics. Hence, for torque perturbations the inertia of the machine is important. Inertia regulates the amplitude and the frequency of the oscillations. The damping coe cient will have similar e ects with higher impact on the amplitude of the oscillations. Fig.5.2 shows simulations results from two systems, the only di erence between both is the inertia of the machine. It can be observed that the machine with lower inertia oscillates faster and with a larger amplitude. Fig.5.3 shows simulation results from two systems, the only di erence between both is the damping coe cient of the machine, it can be observed that when the coe cient is larger the system will be damped more. On the other hand, when tunning the system with respect to field voltage perturbations, the direct axis time constant has an important impact on the output powers. This happens because this constant has a large impact on machine voltage dynamics. This impact can be 38

53 5.2. Parameter Estimation Case Generator.000 Low Inertia Big Inertia.000 W gen Figure 5.2.: Comparison between machines with di erent inertias.000 D=0 D=4.000 W gen Figure 5.3.: Comparison between machines with di erent damping coe cients observed in Fig

54 5. Power System Model Identification 0.02 Low Direct Axis Time Constant High Direct Axis Time Constant Q gen Figure 5.4.: Comparison between machines with di erent direct axis time constant Methodology for Estimating Generator Parameters The generator model used includes 7 parameters and 4 output quantities are measured for each of the experiments (in total 8 outputs must match). This is not a trivial problem and a good approach to formulate the identification problem is needed because the space of possible solutions is immense (7 th dimensional space). Following the general process detailed in Chapter 4, the method used here is stated as follows:. Collect measurement data: this is carried out with Simulink using the test system in Fig.5.5. The data collected was the voltage (magnitude), the rotor speed and, the powers (the machine active power and reactive power). Figure 5.5.: Simulink model of the testing system 2. Create the power system model in, see Fig.4.3 (contains both experiments in order to include them in the identification process simultaneously). The model requires a power flow solution, in this case the results of the power flow were obtained from. 40

55 5.2. Parameter Estimation Case Generator 3. Compile an FMU from the model. This was carried out using Dymola. 4. Create a Simulink model using the FMU block from the FMI Toolbox for Matlab, see Fig.5.6. The only blocks required in order to simulate are: FMUme and Toworkspace. All data inputs and the scopes are included only to monitor the process interactively. FMU block from Modelon Send the simulated data to the toolbox Input data from simulink Scopes to monitor each iteration Figure 5.6.: Simulink model with the FMU of the model 5. Start the RaPId Toolbox. 6. Provide RaPId with settings (algorithm, parameters, variables, etc.). Observation: during this thesis a bug in the FMU files generated by Dymola was found. This bug consists of a problem in the memory handling of the file, for this reason Matlab could not run more than approximately 40 simulations. As a consequence, computationally intensive meta-heuristics algorithms could not be used. In order to proceed with the system identification process an alternative method was developed. It was known that the number of iterations was limited, for that reason it was thought that if a good starting point could be found a gradient descent method could be used. Near an optimum point, a gradient descent method always improves the final solution. In order to find the best start point possible the PSO algorithm was used in a specific way. In RaPId, PSO algorithm is coded in such a way that at first it creates an N -dimensional mesh of equally distributed solutions (N is the number of parameters) and from there it starts iterating for each particle. So, the strategy used was to run the PSO method with a high number of particles (to have a dense populated mesh) and with zero iterations. The result of this experiment was that the best initial particle could be found. 4

56 5. Power System Model Identification In order to have a 7-dimension uniform mesh at least 28 (2 7 ) particles in RaPId were required. Once the initial particle was found, the NM method [24] (a simplex optimization method) was used. The methodology is synthesized in Fig.5.7. Methodology:. Run PSO natively implemented in RaPId with iteration and the maximum number of particles. Why? The PSO coded in RaPId initializes the particles in the n-dimensional space solution in a regular way (not random). N dimensions for N parameters The minimum number of particles is 2 N. The result from the PSO will be the particle with best fitness. 2. This particle is a good starting point for a gradient descent method, a better solution can be found iteratively with this method. The NM method is suggested and executed until the FMI Toolbox crashes, when it crashes last value found can be obtained and the method can be re-runned using these last values as starting point. Note: This methodology must be run several times changing the weights of the fitness function to have different solutions. Figure 5.7.: Methodology used for the identification process Results The numerical results of the generator parameter estimation process are shown in Table 5. Table 5..: Generator parameter estimation results Parameter Value Armature resistance (R a ) Direct axis reactance (X d ) Direct axis transient reactance (Xd 0 ).37 Direct axis transient time constant (Td 0) Quadrature axis reactance (X q ) Inertia coe cient (M) Damping ratio (D) Graphical results are presented in Fig.5.8 and Fig

57 5.2. Parameter Estimation Case Generator Discussion: In Table 5. the values of the identified parameters are shown. The numerical value for all the parameters are inside the normal range for a generator. That means, the values of the parameters are in the standard range and there are no anomalies like negatives parameters. In Fig.5.8 and Fig.5.9 are shown the graphical comparison between the simulations in SPS and. In the graphical comparison the matching is not 00% but the error is acceptable. In Fig.5.8 all the outputs plots match well except for the reactive power, in Fig. 5.9 all the outputs plots match well except for the voltage magnitude. These di erences are due to the di erent nature of the models in SPS (EMTP) and. In order to validate the results of identification, the simulations in both SPS/Simulink and were repeated using perturbations two-times larger than the original experiments used for identification. The results are shown in Fig.5.0 and Fig.5.. V gen P gen Simulink Simulink Q gen Simulink W gen Simulink P m Simulink Figure 5.8.: Comparison between the reference (Simulink) and the identified () model responses for the torque perturbation 43

58 5. Power System Model Identification V gen.00 Simulink P gen Q gen Simulink W gen V f Simulink Simulink Simulink Figure 5.9.: Comparison between the reference (Simulink) and the identified () model responses for the field voltage perturbation V gen P gen Q gen W gen Simulink Simulink Simulink Simulink P m Simulink Figure 5.0.: Validation of the identified generator model parameters for the torque perturbation 44

59 5.3. Load Aggregation V gen P gen Q gen Simulink Simulink W gen V f Simulink Simulink Simulink Figure 5..: Validation of the identified generator model parameters for the field voltage perturbation 5.3. Load Aggregation The load aggregation case will start from the previous test system model. In addition, the load that must be identified will be added in parallel to the existing load. The load to be aggregated represents a feeder in the Scottish Power distribution network. Thus, the load is comprised by the connection of several households loads. The modeled load corresponed to the green feeder in Fig.5.2. The identification process for this case consist in evaluating four di erent load models, which were implemented in as shown in Chapter 3. The RaPId Toolbox is used to obtain numerical results (the mean squared error value) which will help to determine which model is more suitable to provide an accurate aggregate representation of the load. At the end of this identification process there will be 4 results, one for each type of load, and the load model that shows a better fit (least mean squared error) will be the one selected as the aggregate model. To identify the load, two perturbations are used: % torque perturbation at the generator. % field voltage perturbation at the generator. Based on the previous working frame; the methodology used is as follows: 45

60 5. Power System Model Identification Figure 5.2.: Scottish Power distribution grid corresponding to the modeled load [3]. Collect measurement data: this is carried out with Simulink using the test system and adding the Scottish Power distribution network. The load to be identified represents the green feeder in the Scottish Power distribution network [3]. 2. Create the power system model in. The model created is shown in Fig.5.3. Load to be Identified Figure 5.3.: model in the identification process 3. Compile an FMU from the model. 46

61 5.3. Load Aggregation 4. Create a Simulink model using the FMU block from the FMI Toolbox for Matlab. The one used is shown in Fig.5.4. Figure 5.4.: RaPId Simulink model for the load aggregation case 5. Start the RaPId Toolbox. 6. Provide RaPId with settings (algorithm, parameters, variables, etc.): In these cases, to overcome the identification the same strategy described previously was used (PSO+Gradient descent). 47

62 5. Power System Model Identification Load Aggregate Model Identification Results Exponential Recovery Load Model Results The parameters resulting values and the corresponding error obtained from the identification process can be found in Table 5.2: Table 5.2.: Exponential recovery estimated aggregate load model parameters Parameter Value Active power time constant (T p ).398 Reactive power time constant (T q ) Static active power exponent ( s ) Dynamic active power exponent ( t ) Static reactive power exponent ( s ) Dynamic reactive power exponent ( t ) Mean squared error.8627e-007 The corresponding time domain simulation results comparing the reference data (generated using Simulink/SPS) and the simulation results are shown in Fig x 0-3 Simulink 3. x 0-3 Simulink P load 3.09 P load x Simulink.5 x Simulink Q load Q load (a) Torque perturbation (b) Field voltage perturbation Figure 5.5.: Exponential recovery aggregate load model results Discussion: All the parameters numerical values are positive and within an acceptable range. It can be observed that the static parameters are higher than the rest. The mean squared error for this load is low, thus it means that the di erence between the reference data and the aggregate load model is little. This is illustrated in Fig

63 5.3. Load Aggregation Voltage Dependent Aggregate Load Model Results The parameters resulting values and the corresponding error obtained from the identification process can be found in Table 5.3. Table 5.3.: Voltage dependent estimated aggregate load model parameters Parameter Value Active power exponent ( p ) Reactive power exponent ( q ) Mean squared error e-007 The corresponding time domain simulation results comparing the reference data (generated using Simulink/SPS) and the simulation results are shown in Fig x 0-3 Simulink 3. x 0-3 Simulink P load 3.09 P load x Simulink.5 x Simulink Q load Q load (a) Torque perturbation (b) Field voltage perturbation Figure 5.6.: Voltage dependent aggregate load model results Discussion: All the parameters numerical values are positive and within an acceptable range. The active power exponent and the reactive power exponent are above 7, thus the variation of the active power and reactive power is very a ected by the voltage variation on the load. The mean squared error result for this load is low, thus it means that the di erence between the reference data and the aggregate load model is little. This is illustrated in Fig

64 5. Power System Model Identification Frequency Dependent Aggregate Load Model Results The parameters resulting values and the corresponding error obtained from the identification process can be found in Table 5.4. Table 5.4.: Frequency dependent aggregate load model estimated parameters Parameter Value Active power voltage coe cient ( p ) Active power frequency coe cient ( p ) Reactive power voltage coe cient ( q ) Reactive power frequency coe cient ( q ) Filter time constant (T f ) Mean squared error e-007 The corresponding time domain simulation results comparing the reference data (generated using Simulink/SPS) and the simulation results are shown in Fig x 0-3 Simulink 3. x 0-3 Simulink P load 3.09 P load x Simulink.5 x Simulink Q load Q load (a) Torque perturbation (b) Field voltage perturbation Figure 5.7.: Frequency dependent aggregate load model Results Discussion: All the parameters numerical values are positive, except for ( p ), and within an acceptable range. The negative value ( p ) is correct, it means that the load active power is inversely proportional to the frequency of the system. The active power voltage coe cient and the reactive power voltage coe cient are again above 7, thus the variation of the active power and reactive power is very a ected by the voltage variation on the load. The mean squared error result for this load is medium, thus it means that the di erence between the reference data and the aggregate load model is higher than in the previous cases. This is illustrated in Fig

65 5.3. Load Aggregation ZIP aggregate load model Results The parameters resulting values and the corresponding error obtained from the identification process can be found in Table 5.5. Table 5.5.: ZIP aggregate load model estimated parameters Parameter Value Constant Z coe cient for active power (k pz ) Constant I coe cient for active power (k pi ) 0 Constant P coe cient for active power (k pp ) 0 Constant Z coe cient for reactive power (k qz ) Constant I coe cient for reactive power (k qi ) 0 Constant P coe cient for reactive power (k qp ) 0 Mean squared error 5.45e-007 The ZIP load has one main characteristic: k pz + k pi + k pp = (5.) RaPId allows to consider constraints in the form ( min < x < max ). To be able to take into account the equality constraint in (5.), a change of variables was used. Instead of working in the Cartesian coordinate frame, the identification was held in the spherical coordinate frame, setting the constraints as shown in equations (5.2)-(5.4). (These constrains force the solution space to the first /8 of the sphere). 0 <r< (5.2) 0 < < /4 (5.3) 0 < < /4 (5.4) So the final expressions used to represent the constraints are: k pz =(r p sin( p )cos( p )) 2 (5.5) k pi =(r p sin( p )sin( p )) 2 (5.6) k pi =(r p cos( p )) 2 (5.7) This was performed similarly for the reactive power coe cients equality constraints. Discussion: The results show a completely constant impedance behavior of the aggregate load. The mean squared error result for this load are high, thus it means that di erence between the reference data and the aggregate load model is the highest. This is illustrated in Fig.5.6, where it can be observed that for the field voltage perturbation, the system behaves completely di erent to the reference. 5

66 5. Power System Model Identification 3. x 0-3 Simulink 3. x 0-3 Simulink P load 3.09 P load x Simulink.5 x Simulink Q load Q load (a) Torque perturbation (b) Field voltage perturbation Figure 5.8.: ZIP Results Aggregate Load Model Identification Results Analysis The results of the di erent loads are compared in Table 5.6: Table 5.6.: Load Aggregation Error Comparison Type of load Mean squared error Exponential Recovery.8627e-007 Voltage Dependent e-007 Frequency Dependent e-007 ZIP 5.45e-007 Based on the maximum fitness criteria (minimum mean squared error), the aggregate load model that most matches the behavior of the data is the Exponential Recovery Load model. Further experiments can be performed in order to improve the aggregate load model response compared to the reference model. For example, instead of identifying load models independently, the identification could be done using a combination of load models in parallel and identifying which percentage correspond to each type of load Photovoltaic Panel Parameter Estimation Case The last parameter estimation case consist in estimating the parameters of a PV panel when there is an irradiation change and a temperature change. The results will be evaluated and studied using statistical tools and concepts, such as the confidence interval. To identify the PV, two perturbations will be performed: Irradiation change from 000 W/m 2 to 500 W/m 2 and back to 000 W/m 2 at t=[ ]. 52

67 5.4. Photovoltaic Panel Parameter Estimation Case Temperature change from 25 C to 40 C and back to 25 C at t=[ ]. Based on the previous working frame; the methodology used is as follows:. Collect measurement data: this is carried out with Simulink using a reference model developed in Simulink. 2. Create the power system model in. The model created is shown in Fig.5.9. Figure 5.9.: model in the identification process. 3. Compile an FMU from the model. 4. Create a Simulink model using the FMU block from the FMI Toolbox for Matlab, see Fig The only blocks required in order to simulate are: FMUme and Toworkspace. All data inputs and the scopes are included only to monitor the process interactively. Figure 5.20.: RaPId Simulink model for the PV panel parameter identification 5. Start the RaPId Toolbox. 6. Provide RaPId with settings (algorithm, parameters, variables, etc.): In order to proceed with the statistical analysis, the identification process has been performed 50 times 53

68 5. Power System Model Identification using the genetic algorithm with di erent settings each time. Note: the FMU memory issue still appeared, as a consequence, the settings were set to a number of approximately 30 iterations. Photovoltaic Parameters Estimation Results The parameters estimated were: k v : correction factor for the voltage. k i : correction factor for the current. Fig.5.2 shows the results from 50 estimations and the statistical analysis for both variables. It is important to notice that the amplitude of the confidence interval is inversely dependent on the number of samples, that means, the bigger number of samples the narrower the interval is. In this occasion, instead of performing the identification assessment based on the highest fitness criteria (minimum mean squared error), the assessment was performed through statistical analysis. The statistical analysis could be useful in order to evaluate the validity of the parameter estimation results. For example, when the samples have a large variance, this indicates that a change in the parameter will have a negligible influence on the simulation output result. Despite that repeating the estimation process several times o ers the opportunity to obtain more information about the parameters estimated, it also requires more resources and time. It is important to take into account time consumption, a precise identification will require time because more iterations are required. Finally, plotting the results using the mean value of the experiments, it is possible to observe this behaviour Fig

69 5.4. Photovoltaic Panel Parameter Estimation Case Figure 5.2.: Panel parameter identification statistical analysis 55