Review of trends and targets of complex systems for power system optimization

energies Review Review of Trends and Targets of Complex Systems for Power System Optimization Jan Vysocky * and Stanislav Misak ENET Centre, VSB—Technical University of Ostrava, 708 00 Ostrava, Czechia; stanislav[email protected] *Correspondence: [email protected] Received: 29 January 2020; Accepted: 25 February 2020; Published: 1 March 2020   Abstract: Optimization systems (OSs) allow operators of electrical power systems (PS) to optimally operate PSs and to also create optimal PS development plans. The inclusion of OSs in the PS is a big trend nowadays, and the demand for PS optimization tools and PS-OSs experts is growing. The aim of this review is to define the current dynamics and trends in PS optimization research and to present several papers that clearly and comprehensively describe PS OSs with characteristics corresponding to the identified current main trends in this research area. The current dynamics and trends of the research area were defined on the basis of the results of an analysis of the database of 255 PS-OS-presenting papers published from December 2015 to July 2019. Eleven main characteristics of the current PS OSs were identified. The results of the statistical analyses give four characteristics of PS OSs which are currently the most frequently presented in research papers: OSs for minimizing the price of electricity/OSs reducing PS operation costs, OSs for optimizing the operation of renewable energy sources, OSs for regulating the power consumption during the optimization process, and OSs for regulating the energy storage systems operation during the optimization process. Finally, individual identified characteristics of the current PS OSs are briefly described. In the analysis, all PS OSs presented in the observed time period were analyzed regardless of the part of the PS for which the operation was optimized by the PS OS, the voltage level of the optimized PS part, or the optimization goal of the PS OS. Keywords: optimization methods; energy management; energy storage; microgrids; load control; electric vehicles; optimal power flow 1. Introduction Electrical power system optimization is a popular research topic. Electrical power system experts began to deal with the minimization of the active power loss in electrical power systems (PS), and minimization of the PS operation costs in the 1950s [ 1 , 2 ]. An important milestone in the path to modern power system optimization systems (PS OSs) was defining the Optimal Power Flow (OPF) problem [ 3 , 4 ]. Although the basic solution to the PS optimization problem had already been described, the low computing performance of the computers at that time did not allow for solving extensive complex optimization problems. In the coming decades, the computing performance of computers has been increased. Now, it is possible to solve challenging optimization tasks of large complex PSs, considering many operational constraints of the PS [ 5 ]. Today, PSs are under extensive changes. Conventional thermal power plants, which are easy to control and whose rated powers are in the order of hundreds of MW, are being replaced by a large number of small distributed generation units, whose operational nature is often intermittent (the units using wind or solar energy). Moreover, electrical power consumers require increased quality of the supplied power than before because they use appliances requiring very high power quality [ 6 ]. In addition to large PSs, small isolated microgrids with just a few power sources and consumers are being created today. New devices with Energies 2020,13, 1079; doi:10.3390/en13051079 www.mdpi.com/journal/energies Energies 2020,13, 1079 2 of 22 a special operation character are being connected to PSs, e.g., battery energy storages and electric vehicles. Old analog electricity meters are substituted with new smart electricity meters enabling real-time measurement. All these current changes in PSs, and the effort to minimize operating costs and maximize power system reliability, motivate PS operators to install advanced power-flow control and communication equipment to their PSs, and to use comprehensive PS OSs for optimal control of their PSs. That is why PS researchers are nowadays engaged in creating new PS OSs and improving the abilities of the old ones. To accelerate the development of new PS OSs, these researchers often use applications and function libraries to simulate the operation of a PS in various operation states and to optimize PS parameters. Such applications and function libraries are, e.g., OpenDSS [ 7 ], GridLAB-D [ 8 ], or pandapower [9]. In this paper, we monitor current trends and dynamics of the power system research optimization. The content and form of this review paper have been chosen with respect to the goal of introducing young researchers to current trends of the PS operation optimization research area. It is important to note that the aim of this review paper is to provide a description of the current trends and dynamics of PS OSs, not a description of the current trends and dynamics of PS OSs which are based on the OPF or Unit Commitment optimization problem solution. Many PS OSs which have been analyzed for this review paper are based on these two optimization problems, but there is also many PS OSs which are based on different types of the optimization problem. Specifically, the OPF and Unit Commitment optimization problems are not solved by the PS OSs which do not observe power flows (an example of such a PS OS is an optimization system optimizing power generation of a dam hydroelectric power station, which is limited only by the maximum and minimum value of the water level, output water flow, and generator dynamics and which goal is to maximize profit [10]). The state-of-the-art works, the challenges, and the future trends of OPF research were described in [ 11 , 12 ]. The Unit Commitment research was described in [ 13 ]. In addition, [ 14 – 17 ] also offer some interesting findings. The paper is structured as follows. Section 2describes the process of defining the current dynamics and trends of PS operation optimization research and presents the results of this process. Section 3 describes the characteristics of eleven basic PS-OSs research streams and presents several appropriate representatives of these research streams. Papers presented in individual subsections of Section 3are those that best describe the solution methods of PS operation optimization problems of given research streams and which together form an overall picture of given streams of PS-OSs research. 2. Current Dynamics and Trends of Research In order to analyze the dynamics and trends of PS OSs, 388 papers that present systems optimizing of the operation of a part of the power system, and which were published between December 2015 and July 2019, were collected. The papers for analysis were searched in the Scopus and Web of Science databases. Any paper found here which was suitable for this review was inserted into the review’s paper database. In the scope of the review’s research, any optimization system related to any part of a power system was interesting (regardless of a PS’s voltage level). As such, within the review’s paper database, there are papers presenting PS OSs optimizing of the operation of low-voltage networks systems (e.g., PS OSs optimizing of electric vehicle charging stations) and PS OSs optimizing of the operation of transmission networks systems (e.g., PS OSs optimizing the network power flows by setting the Flexible AC transmission system devices) at the same time. Once all the PS OSs papers relevant for this review had been found and reviewed, all papers in the review’s paper database were analyzed in detail. The aim of this detailed analysis was to determine whether the paper presents a complex PS OS or not. If a paper did not present a complex PS OS, the paper was removed from the review’s database. If a detailed analysis showed that a paper deviates from the review’s research criteria in any way, the paper was also removed from the paper database. Once the detailed analysis was finished, the final paper database was obtained. The final paper database contained 255 papers that meet the review’s research criteria. Energies 2020,13, 1079 3 of 22 Within the detailed analysis of the final paper database, specializations and optimization goals of individual PS OSs were defined. Based on the defined PS OSs’ specializations and goals, 11 groups of PS OSs were identified. These 11 groups (subcategories) are presented in Table 1. A detailed definition of each of these groups is presented in Section 3. Table 1. Topics of 11 identified electrical power system (PS) optimization systems (OSs) groups. Group Mark Name of the Group A PS OSs minimizing the price of electricity/PS OSs reducing PS operation costs B PS OSs optimizing the operation of renewable energy sources C PS OSs regulating the power consumption during the optimization process D PS OSs regulating the energy storage systems operation during the optimization process E PS OSs controlling a special PS action hardware F PS OSs optimizing the microgrid operation G PS OSs regulating the charging/discharging of electric vehicles H PS OSs maximizing the PS operation stability I PS OSs reconfiguring the network topology during the optimization process J PS OSs finding an optimal PS expansion plan K PS OSs using the market clearing during the optimization process After the PS OSs groups identification, individual papers of the paper database were assigned to 11 paper groups (individual paper groups are equivalent to PS OSs groups presented in Table 1) based on the specialization or goal of the PS OS presented in the paper. Two PS OSs papers’ assignment processes were performed. In the first assignment process, a paper was assigned to a relevant paper group based on the main characteristic of the PS OS presented in the paper. In the second assignment process, a paper was assigned to all relevant paper groups based on all identified characteristics of the PS OS presented in the paper. Since in the case of several papers, making a paper assignment decision without a doubt was impossible, the final decision to assign these papers to the most appropriate paper group was burdened with a possible error of subjective decision. However, the number of papers with this unclear assignment decision was low, so any misassignment of these papers would not have a significant impact on the results of the statistical analysis of the review’s paper database. Figure 1shows how many papers have been assigned to each defined paper group in the first assignment process. Figure 2presents the same results using the relative frequency and cumulative relative frequency. Figure 1shows that the highest number of papers has been assigned to group A (PS OSs minimizing the price of electricity/PS OSs reducing PS operation costs, 55 papers). However, it is important to note that group A is intended for the PS OSs papers which cannot be assigned to any of the other ten paper groups in the first assignment process. Since the three largest paper groups contain more than half of all papers of the review’s paper database, the characteristics of these three paper groups (groups A, B, and C) were determined as the most frequently presented in the papers published in recent years. Therefore, these three PS OSs characteristics are considered to be the mainstream of the PS OSs research. The assigned database of papers, which was created in the first assignment process, was later analyzed for the second time, in order to obtain current trends of PS OSs research. For this reason, the frequency of individual PS-OSs characteristics described in the papers of our database was analyzed for individual years of the observed time period. Figure 3presents the frequency of the individual PS-OSs characteristics described in the papers divided by the total number of papers in the observed year. In this figure, the color of graphs of individual PS OSs characteristics is the same as the color of the column of the PS OSs characteristic in Figure 1. This figure shows that the relative frequency of PS OSs group A is gradually increasing, while the relative frequencies of the other PS OSs groups do not change significantly (due to the small extent of the static set for individual observed years, small fluctuations observed in the graphs of individual PS OS characteristics are not considered to be significant). The growth of the relative frequency of Group A can be explained by the idea that current PS-OSs researchers try to differentiate from older papers and standard PS-OSs research streams (i.e., PS Energies 2020,13, 1079 4 of 22 OSs research streams corresponding to paper groups B to K). Authors of recent papers try to create new PS OSs which share the main optimization idea with the older PS OSs (i.e., the minimization of the price of electricity or the reduction of the PS operation costs). However, in these recent PS OSs, unlike in the PS OSs of the standard research streams, there is an unusual secondary optimization target defined or there is an unusual PS part optimized. Regardless, for the whole observed time period, the claim that the PS OSs with usual characteristics (i.e., PS OSs characteristics corresponding to groups B to K) represent the majority of PS OSs presented in the papers of our database is still valid (In 2019, the relative frequency of group A was 35.1%, so, in the last year of the observed time period, the PS OSs with mainstream characteristics represented nearly 2/3 of all PS OSs presented in the papers of our database). Energies2020,13,xFORPEERREVIEW4of22  Figure1.FrequencyofindividualPS‐OSscharacteristicsdescribedinpapersofourpaperdatabase— resultsofthefirstassignmentprocess(papergroupsaremarkedwiththesamelettersasthepaper groupslistedabove,seeTable1.).  Figure2.RelativefrequencyandcumulativerelativefrequencyofPS‐OSscharacteristicsdescribedin papersofourpaperdatabase—resultsofthefirstassignmentprocess(papergroupsaremarkedwith thesamelettersasthepapergroupslistedabove,seeTable1.). Theassigneddatabaseofpapers,whichwascreatedinthefirstassignmentprocess,waslater analyzedforthesecondtime,inordertoobtaincurrenttrendsofPSOSsresearch.Forthisreason, thefrequencyofindividualPS‐OSscharacteristicsdescribedinthepapersofourdatabasewas analyzedforindividualyearsoftheobservedtimeperiod.Figure3presentsthefrequencyofthe individualPS‐OSscharacteristicsdescribedinthepapersdividedbythetotalnumberofpapersin theobservedyear.Inthisfigure,thecolorofgraphsofindividualPSOSscharacteristicsisthesame asthecolorofthecolumnofthePSOSscharacteristicinFigure1.Thisfigureshowsthattherelative frequencyofPSOSsgroupAisgraduallyincreasing,whiletherelativefrequenciesoftheotherPS OSsgroupsdonotchangesignificantly(duetothesmallextentofthestaticsetforindividual observedyears,smallfluctuationsobservedinthegraphsofindividualPSOScharacteristicsarenot consideredtobesignificant).ThegrowthoftherelativefrequencyofGroupAcanbeexplainedby theideathatcurrentPS‐OSsresearcherstrytodifferentiatefromolderpapersandstandardPS‐OSs Figure 1. Frequency of individual PS-OSs characteristics described in papers of our paper database—results of the first assignment process (paper groups are marked with the same letters as the paper groups listed above, see Table 1.). Energies2020,13,xFORPEERREVIEW4of22  Figure1.FrequencyofindividualPS‐OSscharacteristicsdescribedinpapersofourpaperdatabase— resultsofthefirstassignmentprocess(papergroupsaremarkedwiththesamelettersasthepaper groupslistedabove,seeTable1.).  Figure2.RelativefrequencyandcumulativerelativefrequencyofPS‐OSscharacteristicsdescribedin papersofourpaperdatabase—resultsofthefirstassignmentprocess(papergroupsaremarkedwith thesamelettersasthepapergroupslistedabove,seeTable1.). Theassigneddatabaseofpapers,whichwascreatedinthefirstassignmentprocess,waslater analyzedforthesecondtime,inordertoobtaincurrenttrendsofPSOSsresearch.Forthisreason, thefrequencyofindividualPS‐OSscharacteristicsdescribedinthepapersofourdatabasewas analyzedforindividualyearsoftheobservedtimeperiod.Figure3presentsthefrequencyofthe individualPS‐OSscharacteristicsdescribedinthepapersdividedbythetotalnumberofpapersin theobservedyear.Inthisfigure,thecolorofgraphsofindividualPSOSscharacteristicsisthesame asthecolorofthecolumnofthePSOSscharacteristicinFigure1.Thisfigureshowsthattherelative frequencyofPSOSsgroupAisgraduallyincreasing,whiletherelativefrequenciesoftheotherPS OSsgroupsdonotchangesignificantly(duetothesmallextentofthestaticsetforindividual observedyears,smallfluctuationsobservedinthegraphsofindividualPSOScharacteristicsarenot consideredtobesignificant).ThegrowthoftherelativefrequencyofGroupAcanbeexplainedby theideathatcurrentPS‐OSsresearcherstrytodifferentiatefromolderpapersandstandardPS‐OSs Figure 2. Relative frequency and cumulative relative frequency of PS-OSs characteristics described in papers of our paper database—results of the first assignment process (paper groups are marked with the same letters as the paper groups listed above, see Table 1.). Now let’s look at the results of the second assignment process. Figure 4shows how many papers have been assigned to each defined paper group in this assignment process. Figure 5then presents the results of the same process using the relative frequency and cumulative relative frequency. Since the Energies 2020,13, 1079 5 of 22 minimization of PS operation costs is the characteristic shared by the vast majority of PS OSs, group A was not considered as part of the second assignment process. Figure 4shows that the highest number of papers has been assigned to group B (PS OSs optimizing the operation of renewable energy sources, 73 papers). Figure 5shows that the four largest paper groups amount to more than half of all papers in our paper database (specifically, their amount is 55.6%). Energies2020,13,xFORPEERREVIEW5of22 researchstreams(i.e.,PSOSsresearchstreamscorrespondingtopapergroupsBtoK).Authorsof recentpaperstrytocreatenewPSOSswhichsharethemainoptimizationideawiththeolderPSOSs (i.e.,theminimizationofthepriceofelectricityorthereductionofthePSoperationcosts).However, intheserecentPSOSs,unlikeinthePSOSsofthestandardresearchstreams,thereisanunusual secondaryoptimizationtargetdefinedorthereisanunusualPSpartoptimized.Regardless,forthe wholeobservedtimeperiod,theclaimthatthePSOSswithusualcharacteristics(i.e.,PSOSs characteristicscorrespondingtogroupsBtoK)representthemajorityofPSOSspresentedinthe papersofourdatabaseisstillvalid(In2019,therelativefrequencyofgroupAwas35.1%,so,inthe lastyearoftheobservedtimeperiod,thePSOSswithmainstreamcharacteristicsrepresentednearly 2/3ofallPSOSspresentedinthepapersofourdatabase).  Figure3.RelativefrequencyofPS‐OSscharacteristicsdescribedinpapersofourpaperdatabasefor individualyearsoftheobservedtimeperiod—resultsofthefirstassignmentprocess(papergroups aremarkedwiththesamelettersasthepapergroupslistedabove,seeTable1.;thecolorofgraphsof individualPSOSscharacteristicsisthesameasthecolorofthecolumnofthePSOSscharacteristicin Figure1.). Nowlet’slookattheresultsofthesecondassignmentprocess.Figure4showshowmanypapers havebeenassignedtoeachdefinedpapergroupinthisassignmentprocess.Figure5thenpresents theresultsofthesameprocessusingtherelativefrequencyandcumulativerelativefrequency.Since theminimizationofPSoperationcostsisthecharacteristicsharedbythevastmajorityofPSOSs, groupAwasnotconsideredaspartofthesecondassignmentprocess.Figure4showsthatthehighest numberofpapershasbeenassignedtogroupB(PSOSsoptimizingtheoperationofrenewable energysources,73papers).Figure5showsthatthefourlargestpapergroupsamounttomorethan halfofallpapersinourpaperdatabase(specifically,theiramountis55.6%). Figure 3. Relative frequency of PS-OSs characteristics described in papers of our paper database for individual years of the observed time period—results of the first assignment process (paper groups are marked with the same letters as the paper groups listed above, see Table 1.; the color of graphs of individual PS OSs characteristics is the same as the color of the column of the PS OSs characteristic in Figure 1.). Energies2020,13,xFORPEERREVIEW6of22  Figure4.FrequencyofindividualPS‐OSscharacteristicsdescribedinpapersofourpaperdatabase— resultsofthesecondassignmentprocess(papergroupsaremarkedwiththesamelettersasthepaper groupslistedabove,seeTable1.).  Figure5.RelativefrequencyandcumulativerelativefrequencyofPS‐OSscharacteristicsdescribedin papersofourpaperdatabase—resultsofthesecondassignmentprocess(papergroupsaremarked withthesamelettersasthepapergroupslistedabove,seeTable1.). 3.MainResearchStreamsinPowerSystemOperationOptimization ThestatisticalanalysisofpaperspresentedinSection2showedthatthemainstreamofPS‐OSs researcharePSOSswithcharacteristicsofpapergroupsA,B,C,D,E,andF(accordingtothefirst assignmentprocess,thesesixpapergroupscontainmorethan80%ofallpapersofthedatabase).The followingsixsubsectionsofthissectiondetailthesemainstreamPSOSs.Thelesspopulatedpaper groups(G,H,I,J,andK)aredescribedinthelastfivesubsectionsofthissection.Thedescriptionof theselesspopulatedpapergroupsislessdetailedthanthedescriptionofthemostpopulatedpaper groups.OfthetotalnumberofPSOSpapers(255)inthereview’spaperdatabase,111PSOSswere selectedforpresentationinthispapersection.ThepapersthatdescribedthePSOSmost understandablywereselected.  Figure 4. Frequency of individual PS-OSs characteristics described in papers of our paper database—results of the second assignment process (paper groups are marked with the same letters as the paper groups listed above, see Table 1.). Energies 2020,13, 1079 6 of 22 Energies2020,13,xFORPEERREVIEW6of22  Figure4.FrequencyofindividualPS‐OSscharacteristicsdescribedinpapersofourpaperdatabase— resultsofthesecondassignmentprocess(papergroupsaremarkedwiththesamelettersasthepaper groupslistedabove,seeTable1.).  Figure5.RelativefrequencyandcumulativerelativefrequencyofPS‐OSscharacteristicsdescribedin papersofourpaperdatabase—resultsofthesecondassignmentprocess(papergroupsaremarked withthesamelettersasthepapergroupslistedabove,seeTable1.). 3.MainResearchStreamsinPowerSystemOperationOptimization ThestatisticalanalysisofpaperspresentedinSection2showedthatthemainstreamofPS‐OSs researcharePSOSswithcharacteristicsofpapergroupsA,B,C,D,E,andF(accordingtothefirst assignmentprocess,thesesixpapergroupscontainmorethan80%ofallpapersofthedatabase).The followingsixsubsectionsofthissectiondetailthesemainstreamPSOSs.Thelesspopulatedpaper groups(G,H,I,J,andK)aredescribedinthelastfivesubsectionsofthissection.Thedescriptionof theselesspopulatedpapergroupsislessdetailedthanthedescriptionofthemostpopulatedpaper groups.OfthetotalnumberofPSOSpapers(255)inthereview’spaperdatabase,111PSOSswere selectedforpresentationinthispapersection.ThepapersthatdescribedthePSOSmost understandablywereselected.  Figure 5. Relative frequency and cumulative relative frequency of PS-OSs characteristics described in papers of our paper database—results of the second assignment process (paper groups are marked with the same letters as the paper groups listed above, see Table 1.). 3. Main Research Streams in Power System Operation Optimization The statistical analysis of papers presented in Section 2showed that the mainstream of PS-OSs research are PS OSs with characteristics of paper groups A, B, C, D, E, and F (according to the first assignment process, these six paper groups contain more than 80% of all papers of the database). The following six subsections of this section detail these mainstream PS OSs. The less populated paper groups (G, H, I, J, and K) are described in the last five subsections of this section. The description of these less populated paper groups is less detailed than the description of the most populated paper groups. Of the total number of PS OS papers (255) in the review’s paper database, 111 PS OSs were selected for presentation in this paper section. The papers that described the PS OS most understandably were selected. 3.1. PS OSs Minimizing the Price of Electricity/PS OSs Reducing PS Operation Costs This subsection is devoted to OSs that have the goal to minimize the PS operation costs or the electricity price. Many OSs with this goal solve the so-called economic dispatch (ED) problem. For example, OSs in [ 18 – 21 ] solve the real-time ED problem. In [ 21 ], a distributed OS based on a state-based potential game is proposed for the real-time ED problem in smart grids. Under the DC power flow approximation, there is the real-time ED problem with coupled operational constraints formulated as a centralized optimization problem (centralized real-time ED problem). By treating each node in the grid as an agent, centralized real-time ED problem is converted into a state-based potential game by augmenting its objective function in the designed game with a local augmented Lagrange-like function, leading to a distributed algorithm for solving this centralized ED problem. The paper’s authors reveal that the stationary-state Nash equilibrium of the state-based potential game exactly identifies the global optimum of the constrained centralized real-time ED problem. The proposed algorithm is capable of handling both equality and inequality constraints in complicated forms. During optimal solution searching, the OS considers constraints described by the network lines’ capacity limits and the capacity bounds of local generation units and local loads. The paper’s authors tested the OS performance by simulations on the IEEE 9-, 39-, and 118-bus test systems. The results of these tests indicate that the OS can quickly converge to the global optimum even under unreliable communication and plug-and-play operations. The OSs in [ 22 , 23 ] solve the look-ahead ED problem. An OS in [ 24 ] solves the multiple-timescale ED problem using a special stochastic system. Energies 2020,13, 1079 7 of 22 In [ 25 ], an OS based on a comprehensive two-stage robust security-constrained unit commitment approach is presented. The OS minimizes the operation cost of the base case while guaranteeing that the robust solution can be adaptively and securely adjusted in response to continuous load and wind uncertainty intervals, as well as discrete N–K generation and transmission contingency security criteria. The OS is equipped with rigorously formulated corrective capabilities of both non-quick-start and quick-start units. Specifically, unit commitment of quick-start units is adaptively adjusted in the recourse stage for satisfying security constraints under various uncertainties, which introduces mixed-integer recourse to the proposed two-stage robust security-constrained unit commitment model. The proposed model is solved by the combination of the modified Benders decomposition method and the column-and-constraint generation algorithm, which decompose the original problem into a master unit commitment problem for the base case and security-checking subproblems for uncertainties. During optimal solution searching, the OS considers constraints described by the nodal power balance, the maximal active power supplied via the reference bus, the network lines’ capacity limits, the capacity limits of local thermal units, the generation limits of local wind farms, other power generation units’ limits (minimum on/offtime limits, startup/shutdown cost limit, and ramping up/down limit), and many security constraints for handling various uncertainties. The paper’s authors tested the OS performance by simulation on the modified IEEE 118-bus system. The results of this test indicate that the OS’s optimization approach is effective. The paper’s authors also performed robustness performance tests. The results of these tests indicate that a reasonable threshold on the violation of security checking subproblems would guarantee good enough solutions from an engineering point of view, although the modified Benders decomposition does not provide the tightest lower bound and may not guarantee the global optimality. An OS in [ 26 ] minimizes the distribution system (DS) operation costs while coping with high-dimensional uncertainty in a DS with high penetration of RESs. The basic method of PS operation cost reduction is a minimization of active power losses. For example, OSs in [ 27 , 28 ] minimize the power losses through the network reconfiguration. To minimize power losses, an OS in [ 29 ] installs distributed power sources of various types across the DS. An OS in [ 30 ] minimizes the operation costs consisting of several parts. This OS optimally dispatches the active and reactive power of distributed photovoltaic generation (PVG), the switched capacitors, and the voltage regulators in large multi-phase unbalanced DSs to minimize the energy loss, the PVG’s active power curtailment, and the operations of capacitors and voltage regulators, in addition to the elimination of the voltage violations and the reverse power flow. 3.2. PS OSs Optimizing the Operation of Renewable Energy Sources In order to reduce greenhouse gas emissions from power generation, the use of renewable energy sources (RES) is supported world-wide. The price of RES power plants’ technologies has dropped significantly, so the capacity of installed RES power plants is increasing globally. Since the generation of solar and wind power plants is defined by variable weather and not by the PS operator’s needs, the OSs need to be used to achieve the efficient use of RES power plants and their power generation. If the electrical energy produced by a RES power plant cannot be consumed or accumulated nearby the RES power plant site at the time of generation, and this energy cannot be transmitted due to the limited PS transmission capacity, a PS operator curtails the instantaneous power output of the RES power plant. This way, the RES power plant’s total power generation is smaller than it could be, and the RES power plant’s owners are shorter in income. OSs presented in [ 31 – 35 ] help minimize RES power plants curtailment and maximize their total power generation, namely the OSs in [ 32 – 34 ] optimize the photovoltaic power plant operation and the OS in [ 35 ] optimizes the wind power operation. All these OSs are parts of a power management system or an economic dispatch system. The OS in [ 35 ] is based on an unusual approach to implementing a decentralized multi-area dynamic ED problem of a large-scale power system. Usual approaches are based on Lagrangian relaxation, but this OS’s solution method is based on a generalized Benders decomposition framework, a decomposition technique for solving nonlinear programming. Since the OS’s algorithm does not use Energies 2020,13, 1079 8 of 22 the dual relaxation, primal feasible solutions can be obtained after only a few iterations. The generalized Benders decomposition algorithm applied there is modified by introducing a locally optimal cost of each area, which significantly expedites convergence. This approach is applicable for online dispatch of multi-area systems with a hierarchical control structure containing a coordinator (i.e., where each area has a local control center, and these local control centers are coordinated via an upper control center). The OS’s decentralized solution method aims to preserve the decision independence of each area while conducting multi-area dynamic ED but does not aim to compete with the centralized solution methods in computational efficiency. Since the decentralized method presented in this paper is developed for multi-period multi-area dynamic economic dispatch, it can be applied to day-ahead hourly power dispatch or intra-hour look-ahead power dispatch of a multi-area system. During optimal solution searching, the OS considers constraints described by the feasibility-cuts limit, the optimality-cuts limit, and the locally optimal cost of each area subproblem. The paper’s authors tested the OS performance by simulations on a real large-scale power system in China, which is a four-area regional power system with a total wind-power-plant installation capacity of 18 GW. While the electrical energy produced by the first generation of RES power plants was supplied to the PS at a constant subsidized price, in some countries with favorable conditions for the RES power plants operation, there are currently RES power plants under construction which will produce energy to be sold at local energy markets [ 36 ]. To maximize RES power plants owners’ profits on the markets, some OSs use offering strategies. Such OSs are presented in [ 37 – 39 ]. In [ 37 , 38 ], optimal day-ahead offering strategies for wind farms equipped with energy storage systems are presented. The optimization method used in [ 38 ] describes and evaluates an integrated strategy for the day-ahead offering while accounting for the optimal operation of an energy storage system at the balancing stage, where the real-time operation policy for the storage is modeled with linear decision rules. Optimal decision rules and day-ahead offers are obtained jointly. The optimization problem is translated into a stochastic optimization problem where a trade-offis made between the expected profit maximization and the risk-aversion. Subsequently, discretization and linearization methods are employed to eventually obtain the solution of such stochastic optimization problems. This OS neglects the degradation costs of the energy storage system. It uses an assumption of being a price-taker in some European electricity markets. The OS also quantifies the value of the residual energy of the energy storage system. Furthermore, a sensitivity analysis is carried out to analyze the influence of price uncertainty and temporal correlation of wind power generation on profits. During optimal solution searching, the OS considers constraints described by the limit of the residual energy of individual ESSs at each interval, the ESS charging and discharging power limits, the wind power generation curtailment limit, and the limit of wind-farm integration capacity. The paper’s authors tested the OS performance by two case studies which were based on realistic data from the Nord Pool market and wind farms in Denmark. In these case studies, the paper’s authors used 100 scenarios. The results of these case studies indicate that the OS’s strategy is more effective than other existing strategies. An offering strategy in [ 39 ] aggregates a few wind power plants to one virtual power plant. Operation coordination of a RES power plant with a fully-controlled power source is an appropriate way to increase the operational capability of the RES power plant. OSs in [ 40 – 42 ] also use such operation coordination. The OS in [ 40 ] coordinates the operation of hydropower plants with thermal power plants and the OS in [ 41 ] coordinates operation of wind farms and pumped-hydro storage. To maximize the total energy production of a RES power plant, it is also necessary to reduce the power losses of the power plant. To minimize wind farm’s power losses, OSs in [ 43 – 45 ] optimize the design of their internal cable networks. 3.3. PS OSs Regulating the Power Consumption During the Optimization Process Another PS’s part which can be involved in the optimization process is the loads. Individual loads’ power consumption has an intermittent character, similar to the intermittent character of solar or wind power plants’ power generation. However, a load’s power input is primarily defined by the current Energies 2020,13, 1079 9 of 22 needs of the consumer, not by current weather conditions. For some types of electrical appliances, the user may need to run an appliance for a certain time period (e.g., 2 h a day), but the part of the day the appliance is running does not affect the appliance’s utility. Then, PS operators can shift appliances of such types to various parts of day to achieve a power balance throughout the day and minimize the magnitude of the PS’s consumption peak. Power consumers receive financial compensation or pay a lower electricity price for allowing the PS operator to set the operation time of their appliances. To set the operation time of each shiftable load optimally, the load shifts are controlled by OSs. OSs in [ 46 , 47 ] schedule the operation time of loads to reduce PS’s peak power consumption and to flatten the load profile. The OS in [ 47 ] optimally schedules the group of household appliances connected to a microgrid. To achieve optimal scheduling, the OS categorizes the appliances into flexible and non-flexible deferrable loads, according to their electrical components. The OS uses a dynamic scheduling algorithm where users can systematically manage the operation of their electric appliances. The OS algorithm solves two multi-objective optimization problems. The first one targets the activation schedule of non-flexible deferrable loads and the second one deals with the power profiles of flexible deferrable loads. These multi-objective optimization problems are solved by using a fast and elitist multi-objective genetic algorithm (specifically Non-dominated Sorting Genetic Algorithm II). During optimal solution searching, the OS considers constraints described by the limited flexibility of local shiftable loads (individual loads are limited by the total energy demand to complete their task). The paper’s authors tested the OS performance by the simulation of the collaborative system that consists of 40 microgrids registered in the program of the load curve flattening. In this simulation, every registered microgrid includes one flexible deferrable load (e.g., water heater) and a non-flexible deferrable load (e.g., dishwasher). The results of this test indicate that the OS‘s scheduling approach can reach a very flat load curve. OSs in [ 48 – 51 ] optimize PS’s power flows using the residential demand-response service. Specifically, the OS in [ 48 ] controls the power consumption of domestic heat pumps in response to a PS frequency, and the OS in [ 49 ] controls a group of heating, ventilation, and air-conditioning loads. The OS in [ 51 ] combines the centralized and decentralized approach. The OS solves a centralized optimization problem for the independent system operator to minimize the social cost, i.e., the consumers’ discomfort cost and suppliers’ generation cost, subject to the power network operating constraints. The OS’s decentralized energy trading algorithm solves a decentralized optimization problem to maintain the privacy of the consumers and suppliers in the demand response program. This decentralized algorithm searches for the control signals that the independent system operator sends to the local entities. In response, the consumers and suppliers obtain their optimal load and generation levels, respectively. The paper’s authors show that, under some specific control signals from the independent system operator, the decentralized algorithm converges to the unique solution of the OS’s centralized problem. During optimal solution searching, the OS considers constraints described by the limited flexibility of local shiftable loads (individual loads are limited by their demand variation in individual time sloths, and their total energy demand to complete their task) and the minimal and maximal value of active power generated by individual local generators. The paper’s authors tested the OS performance by the simulation on the IEEE 40-bus power system. The results of this test indicate that the OS can decrease both the consumers’ and the generators’ costs and the OS’s algorithm is faster than algorithms based on a centralized approach. Some papers propose an optimum load control and schedule system which controls many loads of various types located in various locations as one large aggregate load. Such systems are presented, for example, in [ 52 – 54 ]. When optimizing PS operation using the load control, technically, the easiest load-control method is to control large compact loads, because this type of control allows changing PS’s total power consumption by hundreds of MW, even when controlling only a small number of loads. This type of load control is used, for example, in OSs [ 55 ], [ 56 ] which control the power demand of large industrial consumers. The OS in [ 56 ] enables cement plants to provide the power regulation or the load following with the support of an onsite energy storage system. OSs in [ 57 , 58 ] then focus on Energies 2020,13, 1079 16 of 22 or the reduction of the PS operation costs). However, in these new PS OSs, unlike in the PS OSs of the standard research streams, an unusual secondary optimization target is defined or there is an unusual PS part optimized. Taking a closer look at the individual PS OSs presented in the papers of our database, we see a growing interest in the impact of uncertainties on the solution to the optimization problem. The impact of uncertainties on the solution of the optimization problem is investigated mainly in PS OSs working with renewable energy sources (especially PS OSs working with wind power plants deal with uncertainties very often [37,38,40,42]). In the future, it would be interesting to analyze new trends in PS OSs’ optimization algorithms. Some recent papers presented new optimization algorithms based on biologically inspired optimization strategies (e.g., optimization algorithms based on the Sine Cosine Algorithm [ 130 ], Particle Swarm Optimization [131], or Flower Pollination Algorithm [132]). Author Contributions: Conceptualization, J.V.; methodology, J.V. and S.M.; validation, S.M.; formal analysis, J.V.; investigation, J.V.; writing—original draft preparation, J.V.; writing—review and editing, S.M.; supervision, S.M. All authors have read and agreed to the published version of the manuscript. Funding: This paper was supported by the following projects: This paper was supported by the following projects: SP2020/129 Students Grant Competition; TACR TN01000007, TK02030039 and TJ02000157, Czech Republic. Conflicts of Interest: The authors declare no conflict of interest. References 1. Kirchmayer, L.K.; McDaniel, G.H. Transmission Losses and Economic Loading of Power Systems. Gen. Electr. Rev. 1951,54, 39–46. 2. Calvert, J.F.; Sze, T.W. A New Approach to Loss Minimization in Electric Power Systems. Trans. Am. Inst. Electr. Eng. Part III Power Appar. Syst. 1957,76, 1439–1446. [CrossRef] 3. Peschon, J.; Piercy, D.; Tinney, W.; Tveit, O.; Cuenod, M. 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