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Optimal design of linear subsynchronous damping controllers for stabilising torsional interactions under all possible operating conditions

2015; Institution of Engineering and Technology; Volume: 9; Issue: 13 Linguagem: Inglês

10.1049/iet-gtd.2014.0824

1751-8695

Huakun Liu, Xiaorong Xie, Liang Wang, Yingduo Han,

Numerical methods for differential equations

IET Generation, Transmission & DistributionVolume 9, Issue 13 p. 1652-1661 Research ArticleFree Access Optimal design of linear subsynchronous damping controllers for stabilising torsional interactions under all possible operating conditions Huakun Liu, Huakun Liu Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorXiaorong Xie, Corresponding Author Xiaorong Xie xiexr@tsinghua.edu.cn Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorLiang Wang, Liang Wang Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorYingduo Han, Yingduo Han Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this author Huakun Liu, Huakun Liu Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorXiaorong Xie, Corresponding Author Xiaorong Xie xiexr@tsinghua.edu.cn Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorLiang Wang, Liang Wang Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this authorYingduo Han, Yingduo Han Department of Electrical Engineering, State Key Laboratory of Power System, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this author First published: 01 October 2015 https://doi.org/10.1049/iet-gtd.2014.0824Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Torsional damping controllers, such as supplementary excitation damping controllers (SEDCs), are widely used to stabilise subsynchronous resonance (SSR) induced by torsional interactions (TIs) between turbo-generators and series-compensated power systems. However, because of the changeable operating situations of a power system, it is a great challenge to design them to guarantee torsional stability under all possible operating conditions. This study proposes a global optimal control-design procedure for tuning SEDCs to accommodate the variation of system conditions. Considering TI is a small-signal stability issue, the non-linear power system is converted into a family of linear parameter varying models and the parameter-tuning task for multiple SEDCs is formulated into a multi-model constrained non-linear optimisation problem. A global optimisation procedure based on genetic algorithm and simulated annealing is designed to efficiently solve this problem and obtain a set of robust or several sets of gain-scheduling SEDCs. The proposed method is applied to a multi-machine series-compensated power system. The results of both eigenvalue analysis and time-domain simulation have fully demonstrated the effectiveness of the optimised SEDCs in stabilising SSR under all possible operating conditions. 1 Introduction Fixed series compensation has been widely applied to enhance transfer capability and transient stability of power systems. However, its interaction with nearby turbo-generator units tends to cause a kind of subsynchronous resonance (SSR) problem, namely, torsional interaction (TI) [1]. To address such an issue, an effective method, namely inductance machine damping unit (IMDU) [2, 3], has been proposed recently. This method eliminates TI by coupling an IMDU to the shaft of a turbo-generator set. It does not need any additional controller. Hence, one of its advantages is that the heavy work in tuning the controller is avoided. Another commonly used method for the same purpose of stabilising TI is to attach torsional damping controllers to the excitation systems of generators. Without installing any additional hardware, they can effectively enhance the modal damping just by upgrading the secondary control systems. Thereby, the delicate tuning of its controllers is necessary to achieve better damping performance. TI is generally recognised as a small-signal stability issue. In other words, its dynamics under a certain operation condition mainly depends on the characteristics of the linearised system model at this particular condition. Therefore, in practice, its stability analysis and control design is conducted based on small-signal (linearised) models derived from the original non-linear system at different operating points. However, the operating condition of a real system changes constantly, which inevitably affects the stability of TI and the performance of the controller. For a practical power system, in consideration of the huge number of operating conditions caused by variation of unit commitment, loading level and grid topology, it is generally a great challenge to design a controller to stabilise SSR under all these situations. To deal with this issue, there are some approaches reported in the previous literature: (i) Designing the controller based on the linearised model derived under a nominal operating point first and then validating its performance at other off-nominal conditions. Since only one linear system should be stabilised, the classical methods, such as the pole-placement technique [4] and linear quadratic regulator (LQR) [5] work effectively. (ii) Gain scheduling. The original non-linear system is linearised at multiple working conditions according to one (several) scheduling variable(s) and thus a family of linear parameter varying (LPV) models can be obtained. For each of the LPV models, a set of control parameters is tuned and then the control system automatically switches (perhaps interpolates) its parameter(s) by observing the scheduling variables. (iii) Robust control, the basic idea of which is to incorporate the constant variation of operating conditions into model uncertainties and then design the controller with explicit consideration of such uncertainties, for instance, the robust static synchronous series compensator (SSSC) supplementary controller in [6] and the thyristor controlled series compensator (TCSC) controller in [7]. (iv) Adaptive control, which usually adopts a certain 'adaptive mechanism' to fit, identify or self-learn the variation of system operating conditions and then automatically adjusts control parameters to achieve satisfactory control performance. Recently, adaptive SSR damping control has been proposed based on the intelligent control approach, such as the multilayer feed-forward artificial neural networks [8], fuzzy logic controller and adaptive neuro-fuzzy inference system [9]. Although there are a lot of publications on robust and adaptive controllers, they have not gained actual applications in the practice of SSR damping control due to their fatal disadvantages; for instance, robust control is mostly conservative and its boundary of uncertainties is hard to define; while adaptive control generally has excessive computation complexity and its convergence cannot be guaranteed. What is more, neither of them is intuitively understandable by field engineers. In contrast, the mentioned methods (i) and (ii) are easy to understand, implement and utilise. So they have been widely applied in practical engineering. However, there is still unresolved issue with them, namely, how to guarantee torsional stability or achieve a globally optimal control performance of the system under all possible operating conditions. In order to address these issues, we here develop an optimal control-design method based on our previously proposed multimodal supplementary excitation damping controller (SEDC) [10, 11]. The major novelty and contribution of our work lies in that: an optimal control-design method is established to tune torsional controllers (SEDCs) for stabilising SSR under all possible operating conditions of the original switched non-linear system based on a family of LPV models derived from a practical power system; and the exponential stability criterion of the controlled system is given theoretically. Then, as a minor contribution of this pour work, we have developed an efficient procedure integrating GASA and gain scheduling to materialise the proposed control-design method. Finally, a practical multimachine series-compensated power system is used as a case study, which fully demonstrates the effectiveness of the proposed method and its resultant controllers. This kind of analysis based on a real system is of great interest for the engineering community, which is another minor contribution of this work. The rest of the paper is organised as follows: Section 2 gives the theoretical background. Section 3 elaborates the problem formulation and our proposed control-design method. In Section 4, case study is conducted to check its feasibility and effectiveness. Section 5 draws some conclusions. 2 Theoretical background 2.1 System modelling A practical power grid is actually a switched non-linear system, which can be generally modelled as following differential-algebraic equations (1)where x, u, y, α are, respectively, state vector, control vector, output vector and parameter vector; gi and hi represent non-linear functions; the finite set I is an index set standing for the collection of discrete operating conditions, and i is the ith operating condition. The original system (1) can be linearised under a certain operating condition, or (x0, u0) (2)where Δ denotes the deviation operation and 2.2 Control configuration With the original non-linear system linearised around different operating conditions, a family of LPV models are obtained, based on which a set of switched linear feedback controller can be synthesised as (3)where L(αk, s) is a transfer function matrix representing the linear control law; αk stands for the different sets of control parameters, which can be switched by the gain-scheduling strategy; the finite set K is an index set standing for the collection of different parameter sets. As shown in Fig. 1, the dashed-line loop represents the artificial controlled system formed by LPV models and linear controllers. With some mathematical manipulations, the closed-loop switched linear systems can be expressed by (4)where , and E represents identity matrix with corresponding dimension. Fig. 1Open in figure viewerPowerPoint Control configuration of the non-linear and linearised systems If such a set of switched linear controllers are applied to the original non-linear system (as denoted with solid lines in Fig. 1), the closed-loop dynamics will actually be determined by (5) 2.3 Stability criterion It is a critical issue that under what condition the controller designed based on the linearised system models (2) is possible to stabilise the original non-linear switched system (1) in a small-signal sense. This has received a sustaining research attention and a comprehensive review could be found in [12], which shows that it is always possible to maintain stability when all the subsystems are stable and switching is slow enough. Actually, it really does not matter if one occasionally has a smaller dwell time between switching, provided this does not occur too frequently. This observation is captured by the concept of 'average dwell-time' in [13]. It was also concluded in [12, 13] by the following corollary: If all the linear subsystems are exponentially stable, then the original switched system remains exponentially stable provided that the average dwell time is sufficiently large. In light of the above-mentioned stability criterion and the fact that in practical systems the operating conditions (including unit commitment, grid topology and loading levels) usually change slowly and not so frequently, it can be concluded that if the controller(s) designed with LPV models can sufficiently stabilise all the switched linearised system, the torsional-interaction type SSR of the original system will be well stabilised under all possible conditions. This is the underlying principle of our control-design method. 3 Control-design problem and its optimal solution 3.1 Formulation of the optimisation problem In this paper, the task of designing the torsional damping controllers, namely SEDCs, is standardised into a constrained non-linear optimisation problem, of which the main points include. 3.1.1 System model The original non-linear switched models of the target power system described by (1), are firstly linearised into multiple LPV models as (2) around all possible operating conditions. Then, as the controllers (3) are applied, multiple closed-loop system models (4) could be obtained. 3.1.2 Control performance Since only torsional modes are concern for this SSR control issue, the control performance is measured by those closed-loop eigenvalues corresponding to torsional modes. 3.1.3 One set of robust control parameters or gain-scheduling strategy In the following work, firstly we will try to find a set of robust parameters to stabilise the system under all operating conditions. If this is not possible or the control performance is dissatisfactory, gain-scheduling strategy will be adopted to meet the control objective. 3.1.4 Constraints Practical constraints on control parameters, for instance, their lower and upper limits and the maximum control energy, should be considered during the control-design process. On the basis of the above mentioned points, the optimisation problem is formulated as follows: first, all possible operating conditions of the target system are normalised as an overall operating scenario set (OOSS), which are then divided into N subsets, denoted as OOSSi (i = 1, …, N), according to the N discrete values (or ranges) of the pre-defined scheduling signal(s). Note: if N = 1, it means no scheduling signal is required and the total operating conditions will be considered at once. Then, for each subset (OOSSi), some typical operating scenarios are selected to constitute a smaller subset called control-design-oriented operating scenario set (COSSi), which is used to tune the control parameters for this particular operating scenario set of OOSSi. During the control-design process, COSSi is dynamically updated to achieve a coordinated controller. On the basis of the COSSi, the control-design task can be finally formulated into a standard constrained non-linear optimisation problem (6)where f is the fitness function to measure the performance of the obtained control parameters over COSS, which is a function of eigenvalues; λij denotes the closed-loop eigenvalue corresponding to torsional modes; Eig{} means the calculation of eigenvalues for the included system; stands for the coefficient matrix of the linear closed-loop model (4), which is derived at any operating condition belonging to COSS; g(α) ≤ 0 are the inequality constraints for the control parameters; subscripts i, j respectively denote the ith torsional mode concerned and the jth operating scenario of COSS. 3.2 Optimal solution based on GASA The control-design problem (6) is a complex non-linear optimisation problem with many LPV models, each of which corresponds to one operating condition of COSSi. Owing to its non-linear characteristic and high dimension of solution space, it is difficult to solve the problem with traditional methods. Although many artificial intelligence algorithms, such as genetic algorithm [14], artificial bee colony algorithm [15] and fuzzy logic [16], were used to solve the small-signal stability optimisation problem, our earlier trial of these methods on our problem reveals that any single one of them can hardly work very well. We find that a suitable algorithm should have the following capabilities simultaneously: (i) global optimisation, (ii) local refinement, (iii) parallel computing and (iv) able to handle implicit functions. So a compound algorithm, namely GASA, which takes advantage of both GA (global searching, evolutionary computation) and SA (local optimisation, stochastic transition), is developed to work out the optimisation problem efficiently and thus tune the multiple control parameters co-ordinately. Since the basic principles of GASA have been well documented in the literature [17–21], only the particular mechanism used for our control-design task is discussed hereafter. 3.2.1 Handling the constrains and re-formulating the optimisation problem The general GA can only be used to handle non-constrained optimisation problem, therefore problem (6) should be re-formulated by using some constraint-handling mechanism. In recent years, many techniques have been proposed to incorporate constraints into evolutionary algorithms (EAs), or specifically GA [22, 23]. In our work, the penalty method is used, so the optimisation problem (6) is re-formulated as follows (7)where Rl is the penalty coefficient used; m is the total number of constraints; , fCOSS() are, respectively, the penalised and unpenalised fitness functions. 3.2.2 Global optimisation procedure based on GASA The flowchart of optimisation procedure is illustrated in Fig. 2, which comprises the following steps: Step 1: All possible operating conditions of the target system, that is, any potential grid topology and unit commitment as well as typical loading levels, are taken into account to generate the OOSS. Step 2: If a scheduling scheme is to be designed, we need select scheduling signal(s) according to system characteristics and/or operating experience. Suppose the scheduling signal has N discrete (groups of) values (or ranges) and then OOSS is divided into N subsets, denoted as OOSSi (i = 1, …, N). A set of optimal control parameters will be designed over each of the subsets, for the first of which, let the number of iterations i: = 1. Step 3: The particular control-design problem defined over OOSSi is solved by the GASA-based optimisation procedure, which is further broken into the following sub-steps: Sub-step 3-1: Evaluate the SSR characteristics of the open-loop system through eigenvalue analysis. Sub-step 3-2: Form initial COSSi by using the operating condition with the worst damping. Sub-step 3-3: Formulate the optimisation problem (7) over COSSi with the above-mentioned method. Sub-step 3-4: Employ GASA to solve the optimisation problem and thus to obtain a set of control parameters over current operating scenario set or COSSi. Sub-step 3-5: Apply the obtained controller to all operating conditions belonging to OOSSi and evaluate the closed-loop SSR characteristics via eigenvalue analysis. Sub-step 3-6: Check if all operating scenarios in current OOSSi are covered. If so, the control parameters are optimised and the next step will be carried out; otherwise, COSSi will be updated by adding the most vulnerable operating condition, which has the largest real part of eigenvalue and thus contributes a lot to the performance index. With the new COSSi, return to sub-step 3-3 and re-design the control parameters. Step 4: Check if I = = N. If so, go ahead to the next step; otherwise, let i: = i + 1 and return to step 3. Step 5: The entire optimisation procedure is completed. The control parameters and the closed-loop SSR characteristics are output. Fig. 2Open in figure viewerPowerPoint Flowchart of the proposed optimal control-design procedure with gain scheduling strategy 4 Case study 4.1 Series-compensated system and its torsional stability control problem The target system for the case study is shown in Fig. 3. It is modified from a practical system in North-China Power Grid. Power Plant A has four steam turbine generators, including two identical 600 MW units (#1 and #2) and other two identical 660 MW units (#3 and #4). They are connected to the main grid via 500 kV transmission networks. To increase transfer capability as well as transient stability, fixed series capacitors are applied to the parallel transmission lines between Power Plant A and Substation B. The compensation degree is 45%. Parameters of the grid are listed in Table 1. Table 1. Parameters of the grid No. Transmission line Impedance, Ω Admittance, S 1 A–B 3.98 + 48.97i −1.35 × 10−3i 2 B–C 2.13 + 26.20i −7.22 × 10−4i 3 C–D 0.65 + 8.06i −2.22 × 10−4i 4 equivalent Z1 0.78 + 18.13i — 5 equivalent Z2 0.98 + 13.17i — Fig. 3Open in figure viewerPowerPoint One-line diagram of the equivalent transmission system The four generators are subcritical air-cooled machines. Each consists of four rotors, that is, a high-and -intermediate-pressure (HIP) turbine rotor, two low-pressure (LPA and LPB) turbine rotors and a generator rotor, thus resulting in three subsynchronous torsional modes, of which the frequencies are listed in Table 2. Moreover the mechanical damping's under three typical loading levels are listed in Table 3. 'No-load (0%)' represents the situation of worst mechanical damping. 'Full-load (100%)' represents the most favourable mechanical damping. '40%' is the minimum load levelling for stable operation of the boiler system of the plant. It represents a median mechanical damping. These three situations have very good coverage of loading levels of a generator. Table 2. Torsional frequencies (in Hz) Modes Mode 1 Mode 2 Mode 3 gen #1, #2 15.33 26.08 30.54 gen #3, #4 15.87 27.52 30.96 Table 3. Mechanical damping (in s−1) Load level, % Mode 1 Mode 2 Mode 3 0 0.017 0.020 0.035 40 0.093 0.1024 0.065 100 0.134 0.155 0.150 By linearising the original series-compensated power system at all possible operating conditions and then applying eigenvalue analysis on the obtained LPV models, we can get an overview of the features of TI, which is summarised as follows: (i) Under all operating conditions, the real parts of eigenvalues corresponding to mode 1 are negative, indicating mode 1 is always stable. (ii) However, eigenvalues for modes 2 and 3 have positive real parts under some system conditions, which means that torsional instability will occur if the system operates under these conditions and that the SSR problem is a multimodal one. (iii) The operating state of the system, including unit commitment, grid topology and loading levels, has great impact on modal damping and stability. For instance, tripping one of the series-compensated lines between A and B will lead to a considerable decrease of the electrical damping of mode 2, thereby resulting in serious SSR risk. (iv) Each torsional mode has a most undesirable (or least damped) condition unique to its own, making it a challenge to design a controller able to accommodate all operating conditions. SEDC is selected as the damping control device to solve the SSR problem of the system. It is a real-time control system that works through the excitation system by modulating the field voltage at the torsional frequencies [10, 11]. Unlike conventional SEDC mentioned in [24, 25] that only has a single control path, we here adopt our previously proposed multimodal SEDC to address the multimodal SSR issue. As shown in Fig. 4, it uses the mechanical speed deviation (Δω1) as the feedback signal and, for each torsional mode concerned, has an independent control channel, which is represented by the following transfer function (8)where the first and second terms are low- and high-pass filters used for extracting subsynchronous (10–40 Hz) signal from the input speed; the third term is a band-pass filter aiming at the ith torsional mode; the fourth and fifth terms represent two band-stop filters to suppress the interference from neighbouring modes; KSi and TSi are, respectively, the control gain and phase-shifting time constant. Fig. 4Open in figure viewerPowerPoint Configuration of the multimodal SEDC The control signals of all channels are summarised and then clipped to form the SEDCs output (uSEDC). This output is next imposed on the output of AVR (uAVR) to form a modulated control signal (uf), which drives the excitation circuit to yield the field voltage (Ef). As a result, the excitation current will have additional components at the concerned torsional frequencies, which, if properly controlled, can generate subsynchronous electromagnetic torque to damp SSR. With all filters well determined based on the measured torsional frequencies, the proper determination of the gains and time constants, that is, KSi, TSi(i = 1, 2,…, M) in (8), is crucial to mitigate the multimodal SSR. For our case, M = 3, which means there are three control paths to be tuned to stabilise three torsional modes. Moreover, since generators #1 and #2 are identical, the parameters of their SEDCs should be the same. This is also the case for generators #3 and #4. Therefore two types of SEDCs (for generators #1/#2 and #3/#4, respectively) should be optimised in a coordinated way. 4.2 Optimisation of SEDCs The control parameters of the two types of SEDCs are optimised following the previously stated procedure, of which some specific manipulations are discussed hereafter. 4.2.1 Operating conditions considered In consideration of all possible system topologies, unit commitment and three typical loading levels of generator, that is, no-load, 40%-load and full-load, the target system has 14 580 operating conditions in total. Two control schemes are to be designed. One is the so-called uni-parameter SEDC, which means that one unique set of control parameters of SEDC is used to stabilise the system at all 14 580 operating conditions. The other is gain-scheduling SEDC. To facilitate its implementation, we select the number of transmission lines between Power Plant A and Substation B as the scheduling signal, which, according to preliminary analyses, does have a great impact on the SSR characteristics. As a result, OOSS is divided into two groups, that is, OOSSi (i = 1, 2 standing for 1 and 2 lines, respectively), each of which has 7290 operating conditions. In addition, a widely-used, but not widely-reported traditional method in practice is applied to design controller, which is so-called 'existing' SEDC. The traditional method works like this: firstly, linearise the original system around the nominal operating point, which is determined according to system analysis and/or engineering experience; secondly, design linear controller based on the linearised model with classic pole-placement method; and thirdly, verify the control performance under off-nominal operating conditions and employ it in practical engineering. In a case study, the nominal operating condition is that all four generators are on line and only one of A–B lines is in service (namely, case 1). 4.2.2 System modelling With SEDC out of service, the original non-linear system is linearised into many an open-loop LPV models as (2), each corresponding to an operating point. Specifically, the state vector Δx and the control vector Δu are (9)where Δδ, Δω, Δψ are, respectively, the electrical angle vector, angular velocity vector and flux linkage vector; Δxe is the state vector of excitation system; Δxn is the state vector of the grid and Δu is the output vector of SEDC. The transfer function (8) of SEDC can also be represented with the following state-space form (10)where Δz is the state vector; α is the control parameters (i.e. the gains KSi and phase-shift time constants TSi); F, G, H are coefficient matrices determined by the control parameters α. As SEDC is put into operation, the closed-loop system model is obtained by substituting (10) into (2) and with some mathematical manipulations, it is expressed as (11)where extended state vector and coefficient matrix 4.2.3 Fitness function For our case study, the fitness function is defined as (12)where σij = Re{λij} is the real part of eigenvalue; OSS represents OOSS or COSS; K is the number of operating conditions belonging to OSS; wi is the weight for different mode; M = 3 is the number of torsional modes concerned. The fitness function (12) approximately stands for the weighted average of the 'swept area' within the time [0, T] by all modes concerned across all operating conditions belonging to OSS. In order to enhance the control performance of the worst-damped mode, extra weight wM+1 is added to the mode with the worst damping. For the case-study system, since mode 1 is always stable while modes 2 and 3 tend to be unstable under some system conditions and mode 2 is the worst damped, we set w1 = 0.15, w2 = 0.35, w3 = 0.25, w4 = 0.25 and T = 20 s. Thus, with the integral operation of dij been worked out, the fitness function (12) can be rewritten as (13) 4.2.4 Constraints on the control parameters Since the power rating of excitation systems is limited, the SEDC output needs to be restricted.