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Many engineering problems can be expressed as decision-making examples under uncertainty , That is, when the problem parameters are uncertain , The problem of making cost-effective decisions that satisfy certain given constraints . A natural approach is to make a decision that ensures that any possible acceptable value for an unknown parameter is feasible , So take the worst-case paradigm . In some applications , This robust approach produces programs that are easy to handle , You can find the best solution . Such a solution may be very conservative in terms of achievable performance ： The decision may be affected by the extreme value of the parameter , These extreme values are extremely unlikely, but they have a serious impact on the feasible region for seeking the optimal decision .

The second method is to formulate the so-called opportunity constrained decision-making problem . In these questions , It can be tolerated that a set of parameter values with the minimum probability measure in the parameter space can violate the constraints of the problem , Therefore, it is unlikely to achieve . This approach allows for security （ Aimed at the probability of violating the constraint ） And performance （ Cost of decision ） Weigh against . Chance constrained problems are usually nonconvex and difficult to solve , Even if the original problem with known parameter values is convex . However , They can be effectively solved by adopting the so-called scenario method , Random constraints are replaced by deterministic constraints , By sampling the parameter uncertainty . If the number of constraints is large enough , Then the feasibility in the sense of opportunity constraint can be guaranteed with high confidence . contrary , It can also guarantee the cost of the solution obtained by this method .

In this paper , We consider the chance constrained decision-making problem with a specific structure ： One side , We assume that some prior information about the unknown parameters of the decision problem is known , Exist in the form of samples ; On the other hand , We assume that further information about the true values of these parameters can be collected through measurements . We specialize in the scenario approach , So that prior samples and available metrics can be used effectively , To generate feasible regions that meet opportunity constraints . This leads to a two-stage algorithm , It consists of offline pretreatment of samples , Then there is the online part , It needs to be carried out immediately when the measurement is available . The online part is very lightweight in terms of computing time and memory consumption , Therefore, it is suitable for implementation in embedded systems . As an application of choice , We consider the control of micro generators in Distribution Networks .

In the second section , We briefly reviewed the scenario approach , The decision-making problem with limited opportunities is formulated by measurement . In the third section , Shows how to approximate the posterior distribution of unknown parameters , And a fast algorithm for solving chance constrained decision-making problem is analyzed . In section four , It shows the effectiveness of the proposed real-time operation algorithm of distribution network .

Two 、 mathematical model

2.1 Situational approach to opportunity constrained decision making

Consider opportunity constrained decision making ：

among x ∈ Rn It's a decision variable ,f (x) It's convex cost , $w \in \Omega \subseteq \mathbb{R}^{m}$ Are unknown disturbances modeled as random variables , $z \in \mathbb{R}^{l}$ It's a constant term . Let's assume that random variables w The support of is given a σ - Algebra. D also P stay D Defined on the . Last , ∈ (0, 1) Is the expected probability of violating the constraint .

The general chance constrained decision problem is nonconvex , And it is usually difficult to deal with in calculation . Please note that , We assume that linear constraints are affine in random variables . under these circumstances , as long as w The basic distribution of is known , You can get the analysis results , Provide conditions for the opportunity constrained problem to be reformulated as a convex problem . In any other case , Scenario methods are all about transforming random programs into deterministic problems in this form ：

among $\left\{{w}^{(i)}\right\}$ It is a random disturbance N Samples . If N Large enough , Then this mathematical description is equivalent to the above one .

The scene method is obviously not distributed , This means that there is no interference w Make any assumptions about the probability distribution . By the amount that needs to be sampled according to such a distribution $\left\{{w}^{(i)}\right\}$ , About w The distributed information still exists implicitly . This feature of the scene method makes it impossible to obtain a reliable interference first principle model , But applications that can use historical data are very attractive .

2.2 Scenario method with measurement

In some applications , About interference w Online information may be available . for example , Although it may be possible to obtain information about w Prior information of distribution , But some direct measurements may be made when making a decision . We formalize the opportunity constrained decision-making problem with measurement as ：

among y = Hw It is a linear measurement of disturbance , among H It is the rank of the whole bank , $\mathbb{P}[\cdot \mid \cdot]$ The conditional probability . Direct application of scenario method , Such as (3) Medium , The deterministic optimization equation in the following form will be generated ：

among $w_{y}^{(i)}$ Is measured by y = Hw Samples of the determined conditional probability distribution .

The last setting seems to offset the effectiveness of the real-time operation scenario method , Because samples $w_{y}^{(i)}$

Only in measurement y You need to generate . The use of historical samples makes the integration of this new information difficult . Besides , The resulting optimization problem (5) There are still a lot of typical redundancy constraints , This poses a computational challenge to the direct use of scenario methods for fast real-time decision-making . In the next section , We will show how to successfully solve these two problems through the offline preprocessing phase of the sample , Then there is the decision-making step driven by online measurement .

3、 ... and 、 A fast method of opportunity constrained decision making

3.1 Approximate adjustment through affine transformation

3.2 Affine transformation of feasible region

3.3 Two stage decision algorithm

Four 、 Numerical example —— Power distribution network

4.1 Prevent the active power reduction of overvoltage

4.2 Numerical simulation

4.3 Matlab Realization

Only part of the code is shown here , See ： We are shipping your work details

%% ==== Robust optimization ===========

disp(' Robust ')
gmax = (vmax - 1) / max(Rg);
voltageSeries = testCurtailmentStrategy(testGrid, historicalPowerDemands, gmax);
percentilePlot(voltageSeries);
title(sprintf(' Robust optimization —— Power generation ：% 0.3f MW', gmax));
ylabel(' voltage  [p.u.]');

%%  expect 

disp(' expect ')
Ed = mean(historicalPowerDemands, 2);
gmax = min((vmax - 1 + R*Ed)./Rg);
voltageSeries = testCurtailmentStrategy(testGrid, historicalPowerDemands, gmax);
percentilePlot(voltageSeries);
title(sprintf(' Expected optimization  -  Power generation ：%0.3f MW', gmax));
ylabel(' voltage [p.u.]');


%%  Opportunity constraints 

disp(' gaussian ')
dstd = std(R(genBus,:)*historicalPowerDemands);
gmax = min((vmax - 1 + R(genBus,:)*Ed - 1.6449*dstd)./Rg(genBus));
voltageSeries = testCurtailmentStrategy(testGrid, historicalPowerDemands, gmax);
percentilePlot(voltageSeries);
title(sprintf(' Suppose Gauss's chance constrained optimization  -  Electricity generation ：%0.3f MW', gmax));
ylabel(' voltage [p.u.]');