Can a monopoly INCREASE the market surplus compared with a competitive market? Monopolies are often blamed for DWLs(Dead Weight Losses), while competitive markets believed to work without DWLs (assuming zero taxes/subsidies and zero externalities). 

But I think I found an example when ABSENCE of a monopoly (in this case - a regulated natural monopoly) and presence of a competitive market can lead to smaller market surplus (i.e. DWLs). Am I correct? If not - why? 

Let's assume there is a competitive market for a specific medicine. The competition is tough. Economies of scales aren't exploited to their maximum because in this case, only one firm would be able to survive (there is enough demand only for one Gigafactory) and for this reason, the state puts limit on maximum size of factories in order to prevent full exploitation of economies of scale.

So many people can't afford the medicine despite fierce competition. And if you don't buy anything, then obviously you can't have surplus. Not to mention, that in their totality the producers have lower surplus due to higher costs and lower sales.

Now let's suppose that one day the state decided that maybe a monopoly isn't so bad thing if it's properly regulated. So the state let the firms increase their exploitation of economies of scale and soon only one firm was left. Of course, it was regulated in order to prevent abuses. Now the market surplus increased. The surplus of consumers increased because now more people can afford to buy the medicine. The surplus of the producer's also greater than the total surplus of producers of the existed competitive market because the production of the medicine is less costly and it has more buyers.

But if we increased the market surplus compared with the existed competitive market, then what does it mean? Shockingly, it means that said competitive market had DWLs! Even if there were no taxes/subsidies and externalities.

UPDATE 01:

Ok, there is my try to formalize my thought. 

Let's suppose that there is a perfectly competitive market for a particular medicine. There are NO taxes/subsidies or externalities.

The aggregative demand is Qd=1000-P

The aggregative supply is Qs=-2+0.2*P

P = 835 is the equalibrium price, thus we will have Q=165 as equalibrimum quantity.

The total surplus of consumers is equal to (1000 - 835)*165*0.5 = 13612.5

The total surplus of producers is equal to (835 - 10)*165*0.5 = 68062.5

Thus the market surplus is 13612.5 + 68062.5 = 81675

Now let's replace the competitive market with a regulated natural monopoly that can fully exploit economies of scale.

The aggregative demand is the same, it's Qd=1000-P

P=ATC=200 by the law, while the price without dead weight losses is P=183.(3)
It means that Q=800.

The monopolist's MC=-100+5*P

The total surplus of consumers is equal to (1000 - 200)*800*0.5= 320000

The total surplus of the monopoly is equal to 180*800*0.5+800*(200-180)=88000

Thus the market surplus is 320000 + 88000 = 408000.

As you can see, the total market surplus is bigger. Much bigger.

UPDATE 02

I found book "Intermediate Microeconomics" by  John Hey, it seems to support my conclusion. And there is nothing radical. I will quote some places from the chapter 29. 

"One very obvious reason why a single large firm might be more appropriate in some industry is simply that a single large firm might have access to a more efficient technology than lots of small firms. This is a plausible case if centralisation of the industry has consequences in terms of
increased efficiency – contrariwise, if splitting it up into lots of little units causes losses of efficiency and increased bureaucratic costs. ... Even if it behaves like a monopolist – and there is an associated deadweight loss as a consequence – the end product for both consumers and producer might be better."
Under standard assumptions (some of which you state in your question: no externalities, etc.), no. This follows from the First Welfare Theorem.

Perhaps there are departures from standard models that would support something resembling your conclusion, but my guess is that most economists would view any such departure as the absence of “perfect competition”.

It might be a good exercise for you to write this down in a model (even a simple one) to help you figure out what those departures might be.

Regarding the example you put in your edit:

First, your "monopoly" is not a monopoly in the sense that economists understand it. Monopolies have market power, and thus have no supply curve. The firm in your example has a supply curve, and has no market power.

Second, the example you give does not seem to all support your argument. The firms in the two different situations you present seem to have markedly different production technologies (reflected in the differing supply curves). Your argument requires that the firms have the same technology, only that perfect competition prevents the exploitation of "economies of scale" of their (common) technologies. In other words, this result is driven by differences in production technology, and not from the fact that one market is perfectly competitive and the other is monopolistic. (I believe this is what some comments to your question mean when they say that this is a weird comparison.)

Moreover, where exactly is economies of scale in the second situation? How am I supposed to be able to interpret what is going on as the firm being able to take advantage of economies of scale? Bear in mind that in the second situation, you would have the same outcome (with respect to surplus) if there were many firms, as long as the aggregate supply curve were the same. The fact that there is a single firm is not at all important to the result you've found.

Let's illustrate this previous argument specifically in your example. It seems the correct intervention for the state is not to allow the monopoly to continue but to regulate it. Instead, what the state should do is to make the monopolist's production technology available to other firms that want to enter the market to foster competition.

Suppose that the state does this, and another firm enters the market with the same technology. Let's also suppose now that the market is perfectly competitive. Each firm's supply curve must be

$$ q_s = -100 + 5P $$

which makes industry supply

$$ Q_s = -200 + 10P. $$

Given the demand curve $ Q_d = 1000 - P $, the equilibrium price and quantity is

$$ P^* = \frac{1200}{11}; \, Q^* = \frac{9800}{11}. $$

Total surplus is then $$ \frac{1}{2} (1000-20)\cdot \frac{9800}{11}, $$ a $7 \%$ increase in surplus. Surplus should also increase further when more firms enter.